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Which of the following statements is correct about the domains of educational technology? A. Design is the production stage while development is the planning stage. B. Both the design and development are the planning stage, C. Evaluation is synonymous with implementation. D. Utilization is the action phase.
which of the following is not a contribution of technology to the learning process? . A. The quality of learning can be improved B. The delivery of instructions can be more interesting C. The role of the teacher can be changed into a demonstrator. D. The method of teaching and learning becomes more interactive 10. In what way can instructional aids foster learning? A. Reinforce learning B. Entertain students C. Take the place of the teacher D. Holds students in the classroom 11. With the pervasiveness of technologies nowadays, a learner‐centered instruction can be promoted. Which
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Ms. Cruz was hired in a well‐equipped school but she has to start preparing her instructional materials before classes begin. Which of the following is a systematic process in preparing her materials? A. design – utilization – evaluation ‐ development B. design – development – utilization – evaluation C. development – design – utilization – evaluation D. development – utilization – evaluation – design Ms. Briones is planning to integrate technology in her Mathematics class. Which of the following would be her second step? A. set the objectives B. analyze the learners C. utilize the materials with showmanship D. evaluate the performance of the students Which of the following should Ms. Gomez primarily consider in determining her teaching and learning objectives and use of instructional media? A. the learner C. the instructional activity
of the following statements support this approach to teaching? I. It focuses on transformation of facts. II. It supports the use of lecture and drill methods. III. It gives emphasis on collaboration and authentic assessment. IV. Students work on tasks determined and controlled by the teacher. A. I and II only C. II and IV only B. I and III only D. III and IV only 12. Prof. Villamin’s students use cooperative learning, inquiry based and project‐based learning approaches in creating their digital unit plans. What can be developed among the learners through these approaches? A. repetition and active learning B. repetition & information delivery C. information processing and active learning D. construction of knowledge and information exchange
B. the teacher D. the instructional strategy Which is the best reason why teachers state the objectives before using instructional media? A. To be able to practice how to operate the equipment. B. To determine which media to use best. C. To prepare the materials beforehand. D. To secure available materials. Ms. Villegas is thinking of an educational technology that can relay information clearly to her class. Which principle will guide her in the selection of the material? A. interest C. cost effectiveness B. meaningfulness D. communication effectiveness Mrs. Zinampan presented real samples of rocks when she discussed the different forms of rocks. What principle in the selection of instructional material did she apply? A. interest C. cost effective B. B authenticity D. responsiveness Which of the following is a limitation of conventional technologies in teaching and learning? A. They pose problems on storage.. B. They are less abstract and more concrete. C. They are readily available in the environment, around school, and in the home. D. They provide hands‐on learning experiences and emphasize real‐world E. applications
13. Which of these technologies are arranged from the most symbolic to multisensory? A. real objects, print, audio‐visual materials, and visual materials B. visual materials, audio visual materials, print and computers C. visual materials, print, audio‐visual materials and realia D. print, audio, visual materials, and computers 14. Which group of technologies has the highest degree of abstraction? A. book, imaginative literature, programmed instruction B. digital video, film, versatile compact disc C. video, pictures and television D. realia and computer 15. Mrs. Soriano, a Grade V teacher prefers to use textbooks than other instructional materials. What could be her reason for using it? A. Textbooks can be easily duplicated. B. Textbooks quickly become updated. C. Textbooks address the needs of diverse students. D. Textbooks contain most of the materials they need to learn in the course. 16. It is impractical to bring real objects to the classroom so Ms. Simangan constructed a threedimensional visual instead. Which of the following did she construct? A. cartoon C. graphic
St. LouisReviewCenter,Inc. EneelsusBldg. Gen. Luna St., Davao City Tel. no. (082) 224‐2515 or 222‐8732 ED‐TECH POST – TEST
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B. chart D. model 17. If a teacher wants to teach her pupils the skill to organize and integrate related concepts, which of the following is the most appropriate graphic organizer to use? A. timeline C. venn diagram B. fishbone D. semantic webbing 18. Which graphic organizer is used to show how a series of events interact to produce a set of results again and again? A. Series of events chart C. cycle B. Web D. timeline 19. Which instructional aid requires pupils to verbalize? A. graphic C. . model B. diorama D. . video 20. Which of the following is inappropriate in using printed
27. Romalyn is going to discuss about The ADDIE Model to a big class. She is planning to use a technology by which parts of her presentation could be partly hidden to make it more exciting and interesting. What do you think shall she use? A. model C. transparency B. realia D. video 28. Marife wants to make a presentation material wherein more additional transparent sheets with information can be placed over a base transparency. Which one should she make? A. cut‐out C. silhouette B. puppet D. overlay 29. Which one is used with 2D and 3D materials? A. Opaque projector C. digital projector B. overhead projector D. slide projector
visuals such as charts, graphs, and drawings? A. Provide written or verbal cues to highlight important aspects of visuals. B. Allow the students to pass the materials from one person to another. C. Use materials that everyone can see. D. Present the material one at time. 21. Under what category will a globe as an instructional material fall? A. Realia C. solid model B. mock up D. cutaway model 22. Prof. Agustin would like to provide hands‐on experience on the expansion and contraction of matter. Which of the following materials would be the best to use? A. models C. realias B. pictures D. slides 23. Ms. Sarah finds the chalkboard an effective instructional material up to present. However, just like
30. After watching the film, “Muro Ami’, the students of Mrs. Tamaray are expected to show a demonstrative proof of what they have learned. How is the technology used in thissituation? A. entertainment C. instructional B. informational D. entertainment and informational 31. Self made charts and illustrations serve as universal aid for bringing fascinating and exciting experiences in the classroom. To tap the optimum potentials of these materials, which of the following should be avoided? A. Giving due consideration to lettering. B. Presenting materials with accurate facts. C. Giving more importance to austerity over legibility. D. Focusing on the main idea of the lesson presented. 32. Kamyl used overhead transparencies when she presented her assigned topic to class. What type of educational technology are transparencies? A. printed material C. projected material B. graphic material D. non‐projected material
any other materials, it also has its limitations. Which one is it? A. It allows spontaneity, speed and change. B. Absent students cannot keep up with their assignments. C. It is valuable for emphasizing the major points of the lesson. D. It can be used for displaying pictures and important clippings. 24. With which learning style group are manipulatives MOST effective? A. Master style group B. Interpersonal style group C. Understanding style group D. Self‐ expressive style group 25. Which does a pupil use when s/he sings a concept to a familiar tune in order to help himself commit the concept to memory? A. rap C. pop B. jingle D. lullaby 26. Prof. Arcilla would like to use audiocassette tape in teaching a lesson in English. In which activity is audiocassette tape very effective in the teaching‐ learning process? A. in developing listening skills B. in teaching creative writing C. in composing poems D. in building concepts
33. Which instructional material/s is/are MOST fit in contextualized learning? A. TV C. pictures B. Slides D. field trip
34. Ms. Villanueva wants to teach the students the performance of a certain skill such as dancing. Which technology would be the most appropriate and convenient to use? A. film C. television B. video D. printed material 35. Slides are miniature transparencies. They can be created with simple cameras and simple equipment. They display color in a realistic manner. However, they also have some limitations. Which one is it? A. They can be easily updated and revised. B. They can be adapted to group or to individual use. C. They can get out of sequence if handled individually. D. They can be combined with taped narration for greater effectiveness. 36. Mrs. Santos used a film clip in teaching science concepts to her Grade Six class. However, she found out that it was inefficiently used in the classroom. When is a technology considered inefficient? A. When it makes viewing more interesting. B. When it increases the time to master the lesson.
C. When it helps attain the objectives of the lesson. D. When it enhances understanding of new lesson. 37. Prof. Manantan’s lesson in EPP is about “Pagtatanim ng halaman” to her students. How can she make her lesson more interesting and meaningful? A. Have a viewing activity about the lesson. B. Have them read their EPP book. C. Give them a collaborative work. D. Let them listen to a gardener. 38. Prof. Delos Santos would like her students to give more accurate observations about plants in the environment. Which technique would help her attain her objective? A. Bring them to the garden. B. Bring actual plants to class. C. Show colorful pictures to the class.
B. tutorial D. drill and practice 45. 45. Prof. Natividad would like to create a presentation material for his lesson on the types of computer‐ assisted Instruction. Which tool should he use? A. communicative tool C. productivity tool B. Informative tool D. situating tool 46. Prof. De Guzman uses an online learning approach by which content provides links to information at other locations and serves as a focal point for a distance education experience. Which of the following does he use? A. computer‐aided instruction B. web‐based instruction C. self‐paced program D. teleconferencing 47. Mr. Villena searches for related literature by accessing
D. Let the class read books about the topic. 39. Which of the following should be avoided in presenting visuals? A. Show visuals with an element of suspense. B. Shut off the overhead projector when explaining lengthily. C. Present all the materials simultaneously to hold the learners’ interest. D. Erase any writing on the chalkboard or whiteboard when you no longer need it. 40. After listing down the advantages and disadvantages of computers, Mrs. Muñoz decided to purchase a computer for her class. Which do you think is the last consideration in purchasing the equipment? A. Computers can make her more efficient. B. Computers can be a form of entertainment. C. Computers can enhance teaching and learning. D. Computers can be used for interactive presentations.
several databases in the library computer that is connected with other computers that have databases. How is this termed? A. CD ROM search C. mechanical search B. computer search D. online search 48. Which pair of tools provide synchronous communication? A. chatroom and email B. email and bulletin board C. video conferencing and blogs D. instant messaging and chatroom 49. Should Mrs. Reyes allow her pupils to surf the Internet in creating a group newsletter during her English class? Why? A. No, because pupils may just be exchanging messages via email. B. No, because the pupils might open undesirable websites. C. Yes, to allow the pupils to chat with their friends.
41. Marnel prepares his school research works using computer to submit his requirements on time. Does the computer make him productive and efficient? Why? I. Yes, because it can generate its own data. II. Yes, because it can make one’s work easier. III. Yes, because it can perform tasks fast and accurately. A. I and II C. II and III B. I and III D. I, II and III 42. Prof. Aguinaldo would like to integrate technology in writing a friendly letter. How can he do it effectively? A. Let the pupils surf a friendly letter from the Internet. B. Have the pupils write a friendly letter and send it through an email. C. Have the pupils forward a downloaded friendly letter to others via email. D. Let the pupils write a friendly letter using word processing and have it critiqued by E. their peers. 43. Which of the following is known for its strength of giving immediate feedback? A. video C. digital encyclopedia B. story book D. computer‐assisted instruction 44. Which of the following computer‐based instructional material can be used to learn new concepts? A. games C. simulation
D. Yes, as long as it is used effectively. 50. Which of the following should you ask yourself in evaluating the content of an instructional material? A. Do the materials reinforce learning effectively? B. Are the materials of high technical quality? C. Does the content match the curriculum? D. Is it appropriate for the students? 51. Which of the following statements does NOT describe educational technology? i. It includes hardware and software. ii. It refers to the efficiency of teachers in using computers iii. It is the development, application, and evaluation of systems, techniques and aids to improve human learning. A. i only C. Both ii and iii B. ii only D. Both i and iii 52. What should Mr. Asuncion determine first in the selection of media in teaching? A. needs of the students C. technique to be used B. availability of the media D. objectives of the lesson 53. Which is the most important reason why teachers preview materials to be used in class? A. To gain confidence in using them. B. To encourage viewers to be more focused. C. To avoid potential problems that might occur while materials are in use.
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To ensure appropriateness of the materials with the objectives and target audience. 54. After Ms. Raca planned her lesson in English, she found out that the materials at hand do not match her objectives. Which is the best thing that she can do? A. Modify the available materials. B. Teach the lesson the following day. C. Change the objectives to match with the available materials. D. Carry out the lesson as planned and use the materials at hand. 55. Prof. Balagtas used worksheets, manipulatives and models in teaching math to help her students understand the lesson and love the subject. What did she bear in mind when she used these materials? A. appropriateness C. breadth
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verbal symbols, still pictures, visual symbols, models, motion pictures 63. Which is the best way to present instructional materials? ‐>semi‐concrete‐> A. concrete abstract‐>semi‐ abstract B. semi‐concrete‐> concrete ‐> abstract‐> semi‐ abstract C. abstract‐>semi‐abstract‐> semi‐concrete‐> concrete D. concrete ‐>semi‐concrete‐> semi‐abstract ‐> abstract 64. Which of the following technologies provide iconic experiences? A. videos and computer B. books and periodicals
B. balance D. variety 56. Ms. Torres always makes sure that text, animation and color do not confuse students in her presentation materials. Which principle is applied? A. simplicity C. responsiveness B. variety D. cost effectiveness 57. Mrs. Reyes, a librarian, informed the students as well as the teachers that several software are available for classroom instruction and individual learning. Which material is she referring to? A. Computers C. Television set B. CD‐ROM D. VCD and DVD players 58. Susan’s mother tongue is a vernacular. Which of the following materials would be the most efficient and effective material to learn a second language? A. interactive multimedia B. pictures and print materials C. audio compact discs and radio D. printed materials and real objects
C. audio and audio materials D. printed and verbal symbols 65. How can Prof. Ubiña best promote the use of multimedia in teaching Science to her coteachers? A. Sell multimedia at low cost. B. Demonstrate its use to them. C. Explain the literature supporting its use. D. Convince the principal to require the use of technology.
59. Computer can be a good tool for individualized instruction. Which of the following aspects can be a deterrent for its full utilization in the classrooms? A. economic C. social B. physical D. technical 60. With the increasing use of educational technology inside the classroom, what roles are expected of the teacher? A. facilitator C. knowledge giver B. demonstrator D. source of information 61. Which of the following technologies are properly classified? A. computers, compact discs, film, television B. imaginative literature, book, programmed instruction C. versatile compact disc, printed material, diagram, sketches D. digital video, phonograph, compact discs, radio, audio tape 62. Which of the following technologies are arranged from the most concrete to the most abstract? A. motion pictures, verbal symbols, visual symbols, radio, realias B. realias, visual symbols, television, motion pictures, still pictures C. realias, motion pictures, still pictures, visual symbols, verbal symbols
Which of the following is incorrect about the use of real objects and models? A. Familiarize yourself with the object or model. B. Allow passing of a single object around the class. C. Make sure that objects/models are large enough to be seen by the whole class. D. Encourage students’ participation through questioning and having students E. decide the next step. Aaron constructed a three dimensional material to simulate the circulation of blood. Which of the following did he construct? A. A solid model C. mock‐up model B. cutaway model D. cross‐sectional model Which is a two‐dimensional representation of the earth’s geographic and/or political features? A. globe C. mock‐up B. map D. model You asked your students to illustrate what they have understood from what they have read. Which of the following non‐projected visuals are you referring to? A. printed visuals C. models B. graphics D. realias Which software should Dr. Balagtas to manipulate numerical data in the computer? A. Spreadsheet C. word processing B. desktop publishing D. multimedia
66. There are countless things in the environment that you and your students can use to learn from such as trees, globes, pebbles, blocks etc. These real objects and models are really effective if they are utilized properly.
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71. Prof. Silva uses projected visuals such as OHP in presenting her lesson. What could be her main reason in using such an educational technology? A. The materials are readily available. B. Most visuals can be obtained at no cost. C. It is more abstract than any other visuals. D. She can easily prepare her own transparencies in advance. 72. Ms. Pacheco showed a segment of matter in “sine skwela” to her pupils without a follow‐up activity. Thus, the pupils got low in the test. What does this imply? A. TV makes viewing enjoyable. B. TV promotes mastery of the lesson. C. TV induces alienation on the part of the learners. D. TV is effective when learners attain the lesson objectives.
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vocabulary level D. students’ achievement 80. Which statement is true about the opaque projector and overhead projector? A. An opaque projector allows more flexibility than an overhead projector. B. An overhead projector allows more flexibility than an opaque projector. C. Opaque and overhead projectors can instantaneously project 3D visuals well. D. The series of still visuals in an opaque projector are arranged in a fixed pattern but not in an overhead projector. 81. A grade II teacher wanted to show the parts of a seed by using a large, wooden seed visual aid with detachable cotyledons and tiny seed. Under what
73. Which activity is closest to the real thing? A. hear C. watch a demonstration B. view images D. perform in a presentation 74. Your department would like to purchase a computer set as your project. Which of the following advantages of computer will be your last consideration in purchasing it? A. It can enhance the teaching and learning process. B. It can be used for interactive presentation. C. It can be used for research activity D. It can be used for entertainment. 75. Prof. Orencia will have a digitized presentation to pre‐ service teachers. Which of the following will make her presentation appealing and effective? A. Observe maximum use of animations and graphics together. B. Apply as many computer effects per slide as possible. C. Reinforce textual information with graphic
classification does wooden structure fall? A. assembly model C. realia B. cutaway model D. solid model 82. Which term refers to a model which is constructed so as to emphasize a particular part or function? A. audio recording C. mock‐up B. simulation D, realia 83. Which is the best use of computers to students like you? A. They are used for chatting and surfing the net. B. They are used for research and collaboration. C. They are used for playing online games. D. They are used for watching movies. 84. Which statement makes technology ineffective in student learning? A. It develops higher thinking skills. B. It prepares students for the workforce. C. It enhances students’ collaborative skills. D. It decreases achievement in content learning.
organizers. D. Use as many color as possible. 76. Why are computers increasingly becoming pervasive in schools nowadays? A. Schools advocate the use of computers. B. They increase efficiency and productivity. C. Anybody can operate computers without formal training. D. Students have access to computers in school and at home. 77. There are several reasons why teachers are reluctant in using electronic media in the teaching‐learning process. Which is the most common reason? A. The difficulty in integrating them in the curriculum. B. The limited exposure of teachers to new equipment. C. Their incompatibility to diverse needs of the learners. D. The excessive availability of local technology in the community. 78. With the number of senses to be stimulated as criterion, which one should be first in thelist? A. multi sensory aid C. visual aid B. audio‐visual aid D. audio aid 79. Which of the following is considered in terms of technical quality of a material? A. stereotyping C. color and size of text
85. You plan to use instructional materials to a big class‐ size. Which of these will you not use? A. pictures C. 27‐inch television B. projection device D. computer with LCD projector 86. Computers can be classified according to the roles they play namely communicative tool, informative tool, and constructive tool. What is the other role of computes in the options below? A. instructional tool C. utility tool B. situating tool D, application tool 87. Which of the following categories of CAI will you use in your class if your objective is to increase proficiency in a newly learned skill or refresh an existing one? A. tutorial C. simulation B. drill and practice . D. Instructional game 88. Which of the following is an ineffective use of presentation software? A. Darken the room B. Use appropriate pacing C. Read directly from the slides. D. Allow interaction with the learner. 89. Which of the following is NOT an example of communicative tool? A. multimedia encyclopedia B. teleconferencing C. electronic mail D. chat
90. Which is a characteristic of the teaching machines of B. F. Skinner? A. It does not need any feedback. B. It requires teacher’s assistance. C. It is meant for a collaborative work. D. It allows a student to learn at his/her own pace. 91. Why is one‐way delivery of information a misuse of communication tools? A. because the teacher expects the student to study more B. because it requires activities that focus on thinking than responding C. because it enables the users to focus more on higher level cognitive activities D. because this kind of practice lessens interaction capabilities of communication tools 92. Internet consists of thousands of connected computer networks around the world. Which term does NOT refer to Internet? A. A. NET C. “Cyberspace” B. B. Online D. “Information Superhighway” 93. Your class adviser is planning to have an asynchronous communication with your classmates. Which technology tools can she use? A. chat and blog B. chat and instant messaging C. blog and video conferencing D. electronic bulletin board and email 94. In your computer subject, you allow your class to chat as a part of your motivation before discussing them the roles of computer. How is chat used in this context? A. Communicative tool C. Application tool B. Informative tool D. Situating tool 95. Your mother wanted to finish her long dreamed course but she wanted to do it at home during her free time. How could you help your mother in pursuing her dream? A. Encourage her to hire a helper so that she can attend regularly to her class. B. Give up your study so that your mother can attend her classes. C. Enroll her to the school where you enrolled. D. Enroll her in distance education 96. The following statements are true about computer conferencing. Which is an exception? A. It refers to live student interaction with an expert. B. It is also known as discussion forum or bulletin board. C. It also refers to online class discussions, forums or debates D. It permits two or more individuals to engage in asynchronous text‐based dialogue. 97. Which instructional tool application will you introduce to your class if your objective is to help them find and use information resources available in the internet? A. Webquests C. Scavenger Hunt B. Hybrid course D. Distance education 98. Maryjane is looking for an organized instructional program in which the teacher and learners can be physically separated. Which of the following will she choose? A. Distance Education B. Uniform Resource Locator
C. Web Quests D. Computer‐Based Instruction 99. Prof. Ruscoe would like to show Rizal’s museum to the students but due to financial constraint, she couldn’t bring them there. What should she do to make the teachinglearning process more realistic? A. Conduct a virtual tour. B. Use DVD with less resolution. C. Show pictures of the museum to the whole class. D. Go to the museum and relate all observations made. 100. Which of the following should you avoid if you were asked to evaluate the effectiveness of an instructional game after using it in teaching a lesson in high school science? A. Present problems which are relevant to learning B. C. D.
objectives. Allow learners to select different content materials. Provide a cooperative learning atmosphere. Provide a scoring system.
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POST TEST ED‐TECH Which of the following statements is correct about the domains of educational technology?
1. 2. 3 4 5
D B A A B
51. 52 53 54 55
B A D A D
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
D B A C A B D D A D D D C A B C C B C B A C D A C C
56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81
A B A A A B C D A B B C B B A D D D D C B B A C A A
32 33 34 35 36 37 38 39 40 41 42 43 44 45 46
C D B C B C A C B C D D B C B
82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
C B D A B B C A D D B D A D A
12. _____ tainga ang lupa, may pakpak ang balita.
47 48 49 50
D D D B
97 98 99 100
C A A B
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II. Ibinigay ang kahulugan ng salitang may salungguhit.
16. Narinig ko ang alawat ng mga bata sa silid ng mag‐
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FILIPINO POST – TEST
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I. Panuto: Biluganang titikng wastongsagot. 1. Agad na sumigaw ang bata ____ makitang dumating ang kanyang kapatid
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9. Maya‐maya ay sisingaw _____ ang amoy ng Patay. a.
din
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10. Ang dunong ay kailangan ng tao ngunit kailangan _____ niya ang tulong ng Maykapal. a. din b. rin 11. Sino ba ang sumisigaw _____ at nagtatakbuhan ang mga tao. a. doon b. roon
b. Mayroon
para sa kanilang ikabubuti. a. Daw b. raw 14. Ang Pilipinas ay malakas ____ tulad ng Singapore kung karapatan ang Pag‐uusapan. a. din b. rin 15. Unti‐ unti ____ humuhupa ang kanyang galit. a. Ng b. nang
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a. ng b. nang Ang mga mag‐aaral ay nagkasundo _____ sa iminungkahi ng guro. a. din b. rin Ang bawat tao _____ ay kailangang isakatuparan ang kanyang mithiin sa buhay. a. Daw b. raw Ang kirot ay unti‐unti ______ nawawala. a. ng b. nang Ayon kay Jose Rizal, ang mga bata _____ ang siyang pag –asa ng bansa. a. daw b. raw Sa Sabado _____ gabi mawawalan ng kuryente. a. ng b. nang Hindi na nakaramdam ng gutom si Kuya mula _____ siya ay natulog. a. ng b. nang _____ dalang pusa ang Inay nang umuwi. a. May b. Mayroon
May
13. Ang bawat tao sa mundo ___ ay dapat magkaunawaan
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asawang Maria at Jose. a. ingay c. sigaw b. mahinang alingawngaw d. tawanan Ang Itay ay alimbuyaw nang dumating kanina. a. Aburido c. Patakbo b. Masaya d. sumigaw Si Tj ay isang anluwagi nang mapangasawa ni Luisa. a. Guro c. karpintero b. katulong d. pulis Kakarampot ang nakuha kong ulam sa mesa. a. marami c. malalaki b. katiting d. mamhahaba Alumpihit ang Itay habang hinihintay ang Inay. a. Kabang‐kaba c. Di‐mapalagay b. Siyang‐ siya d. Tuwang‐ tuwa Iyon ang kinamihasnan ng babaeng iyon sa bundok kaya di‐makaunawa sa iyo. a. natutuhan c. napag‐aralan b. kinagawian d. nagustuhan Ang alipustahin ang mga dukha ay di kanais‐nais na pag‐uugali. a. layuan c. talikdan b. apihin d. kagalitan Nakita kong pakimod na sumagot ang babae sa dalaga nang mag‐usap sila. a. Paismid c. patawa b. Pangiti d. pasigaw Isang indihente ang tumawag ng aking pansin dahil sa nakakatawang ayos nito. a. Maralita c. mag‐asawa b. Matanda d. paslit Naging Cum Laude si Memi dahil siya ay nagsunog ng kilay gabi‐gabi. a. nagbubunot c. nag‐aahit b. puspos sa pag‐aaral d. nag‐aayuno Parang balat‐sibuyas ang kutis ng babaeng ito. a. namumula sa bilog c. mahaba at payat b. napakaputi at malinis d. pino at malambot Kapit‐tuko sa isat’‐isa habang naglalakad ang
magkasintahang Heart at Echo. a. away nang away b. mahigpit na magkahawak‐kamay c. malayo ang agwat d. patakbo 28. Ang langitngit ng mga bintana ay gumigising nang lubos sa katahimikan ng silid‐aralan. a. Alatiit c. dekorasyon
b.
kulay
d. sira
29. Mataginting na tinanggap ng batang paslit ang pangaral ng guro. a. maingay c. mapayapa b. pasigaw d. paismid 30. Ang paswit ay sa aso, ang Oo ay sa tao. a. Palo c. buto b. Sipol d. sigaw III. Ibinigay ang tamang sa mga tulang bayani 31. Ang tawag sa mga tulang bayani a. dalit c. senakulo b. epiko d. duplo
32. Tinaguriang Joseng Batute ng Pilipinas
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a. Jose Garcia Villa c. Jose Corazon de Jesus b. Francisco Baltazar d. Modesto de Castro Mga sagisag na ginamit ni Rizal a. Piping Dilat c. Pudpod at Plaridel b. Dolores Manapat d. Dimasalang at laong laan Ang “ prinsipe” ng makatang Tagalog a. Modesto de Castro b. Francisco Baltazar c. Fernando bagong Lanta d. Jose Garcia Villa Siya ay tinaguriang “ Ama ng Balarila ng Wikang Pambansa” a. Jose Villa Panganiban c. Severino Reyes b. Lope K. Santos d. Rafael Palma Isang dula noong panahon ng Hapon na isinulat ni Francisco Soc Rodrigo a. Panibugho c. Panday Pira b. Sa pula, Sa Puti d. Luha ng Buwaya Ama ng Katipunan a. Emilio Jacinto c. Apolinario Mabini b. Andres Bonifacio d. Marcelo H. del Pilar Isang uri ng panitikan na nagsasaad ng simulain ng mga bagay o tao sa daigdig a. tula c. alamat b. tibag d. maikling kwento Pinakabantog at pinakamahalagang awit na nasulat ni Francisco Baltazar a. Senakulo c. duplo b. epiko d. Florante at Laura Ang kauna‐unahang Pilipinong manlilimbag a. Marcelo del Pilar c. Jose Maria Panganiban b. Tomas Pinpin d. Emilio Aguinaldo Ang tawag sa ating unang alpabeto a. Alpabetong Romano c. Kartilya b. Alibata d. Romanisasyon Ang taong may “memorya fotograpica” a. Jose Maria Panganiban b. Jose Garcia Villa c. d.
Jose Rizal Corazon de Jesus Jose
43. “Ama ng Dulang Pilipino” a. Julian Balmaceda c. Lope K. Santos b. Severino Reyes d. Emilio Jacinto 44. Siya ay tinaguriang Joseng Sisiw a. Jose Villa Panganiban c. Jose dela Cruz
b.
Pedro Paterno
d. Modesto de Castro
45. Ang kilalang epiko ng mga Muslim
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a. Hudhud c. Hinalawod b. Darangan d. Bantugan Ang kauna‐unahang aklat na nalimbag sa pilipinas a. Pasyon c. Doctrina Christiana b. Barlaan at Josaphat d. Florante at Laura Ama ng Wikang Pambansa a. Emilio Aguinaldo c. Aurelio Tolentino b. Manuel L. Quezon d. Florante at Laura Ang “ Orator ng Pagbabago” a. Graciano Lopez Jaena c. Urbana at Feliza b. Mariano Ponce d. Jose Buhain Isang dulang nagwagi ng kauna‐unahang Gatimpalang Palanca a. Medusa c. Urbana at Feliza b. Tibag d. Hulyo 4, 1946 A.D Ang may‐akda ng “Ang Cadaquilaan ng Dios” a. Emilio Aguinaldo c. Julian Felipe b. Marcelo H. del Pilar d. Lopez Jaena Baston ni Adan Hindi mabilang‐bilang a. buhok c. dahon b. ulan d. palay Ang dalawa’Y tatlo na, Ang maitim ay maputi na Ang bakod ay lagas na a. aso c. matandang tao b. kalabaw d. punong kahoy Ang anak ay nakaupo na Ang ina’y gumagapang pa a. kalabasa c. sanggol b. saging d. aso Kung araw ay bumbong Kung gabi ay dahon a. b.
saging banig
c. ng bahay d. atip paying
55. Isang reyna Nakaupo sa tasa a. kandila c. kasoy b. kapa d. santol 56. Dalawang magkaibigan, unahan nang unahan. a. trak c. bibig b. paa d. mata 57. Dalawang bolang sinulid Umaabot hanggang langit a. bola c. lobo b. mata d. saranggola 58. May ulong walang mukha May katawan, walang sikmura Namamahay nang sadya a. pako c. upo b. palito ng posporo d. talong ako ng alipin 59. Bumili Mataas pa sa akin a. payong b. atip 60. Tubig sa ining‐ining Di mahipan ng hangin a. ilog b. balon
c. sombrero d. bahay
c. ulan d. dagat
61.
76. Ang Kumitang ay isang uri ng awiting bayan. Ito ay may a. b.
eskursiyon iskursiyon
c. exkursion d. excursion
a. b.
scout escout
c. iskawt d. skawt
a. b.
colisiyon kolisiyum
c. koliseum d. coliseum
a. b.
Istadyum Stadium
c. Estadyum d. estadium
a. b.
Matematika Mathematica
c. matimatika d. matemateka
62. 77.
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sa mga sugapa ay mula sa wasak na 66. Karamihan tahanan.
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a. malaki ang sita ng bahay b. maliit lamang ang bahay c. magkahiwalay ang magulang d. walng magulang Matagal na lumagay sa tahimik si Marcia. Ang ibig sabihinay _______. a. matagal na namatay c. hindi na nagpakita b. nag‐asawa na d. nanganak na Alin ang salawikain sa sumusunod: a. Nasa Diyos ang awa Nasa Tao ang gawa b. Di‐maliparang uwak c. May puno walang bunga May dahon walang sanga d. Nag‐bubuhat ng sariling bangko Ang bagong alpabetong Filipino ay may ______ ng letra. a. 20 b. 24 c. 28 d. 30 Ito ay bahagi ng aklat na makikita sa likod. Ito ay talaan ng lahat na mahalagang paksa kasama ang pahina. Ang mga paksa ay nakasulat sa paalpabeto. a. Talatuntunan c. Talahulugan b. Talatinigan d. Talaan ng nilalaman Isang kuwento ng ang gumagapang ay mga hayop na kumikilos at nagsasalita na parang tao. a. Parabola c. kuwento b. pabula d. alamat Ito ay isang uri ng dula na nawawakas a pagkamatay ng pangunahing tauhan. a. komedya c. melodrama b. epiko d. trahedya Isang kuwento hango sa banal na kasulatan na umaakay sa tao sa matuwid na landas ng buhay. Ito ay may aral. a. Anekdota c. parabola b. alamat d. sanaysay
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Isang tagisan ng mga talino 74. katwiran sa pamamaraang patula.sa pamamagitan ng a. balagtasan c. tula b. talumpati d. duplo 75. Si severino Reyes na lalong kilala sa tawag na Lola Basyang ay higit na kilala sa larangan ng: a. dulaan c. pag‐awit b. pagtula d. balagtasan
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karaniwang inaawit sa: a. paghaharana c. paghehele b. pakikidigma d. pamamangka Ang senakulo ay isang panrelihiyon; ito ay naglalayon na a. ipaala ang kapanganakan ni Hesukristo b. ipakita ang pagkakapatiran ng mga Kristiyano at Muslim c. magsalarawan ngmga pinagdaanang buhay at kamatayan ni Hesukristo d. magligtas sa mga kasalanan Sa akda niyang “Guryon”, ipinalintulad ni Idelfonso Santos ang Guryon sa: a. buhay ng tao c. anyo ng pagpapalipad b. tibay ng pisi d. hanging habagat “Unupo si Itim, sinulot ni Pula, heto na si Puti na bubuga‐buga.” Ito ay halimbawa ng isang: a. bugtong c. alamat b. salawikain d. kuwentong bayan “Ang sinuman ay makabubuo ng matibay na lubid kung pagsasamahin ang sinulid.” Ang ibig sabihin ng kasabihang ito ay: a. Mahirap magkaisa ang mga tao. b. Madali ang gumawa ng lubid kung may sinulid c. Kailangan natin ang lubid sa ating mga Gawain. d. Magkakaroon tayo ng lakas kung tayo’y magkakaisa. “Ang taong nagigipit sa patalm kumapit.” Ano ang ibig sabihin ng salawikaing ito: a. Ang kaligtasan ng taong nagigipit ay sa tapang ng dibdib b. Susuungin ng tao kahit ani mang panganib upang malunasan ang kanyang problema c. Malapit sa panganib ang mga taong nagigipit. d. Huwag makiharap sa taong nagigipit sapagkat siya ay siguradong galit Alin sa sumusunod ang hindi tuluyang anyo ng panitikan? a. korido c. kuwentong bayan b. alamat d. maikling kuwento Tukuyin kung anong uri ng panitikan ang “Isang bayabas, pito ang butas.” a. Sawikain c. Salawikain b. Idyoma d. Bugtong “Ano man ang tibay ng piling abaka ay wala ring lakas kapag nag‐iisa.” Isinasasaad ng salawikaing ito ang kahalagaan ng : a. pagkakaisa’t pagtutulungan b. tibay ng dibdib at lakas kahit nag‐iisa c. pagkakaroon ng lakas kahit nag‐iisa d. pagpapalakas ng loob lalo’t nag‐iisa Sa akin lipain doon nagmula Lahat ng pagkain nitong ating bansa Ang lahat ng tao mayaman o dukha Sila’y umaasa sa pawis ko’t gawa. Ano ang ipinahihiwatig ng saknong? a. Lahat ng pagkain ay sa magsasaka nagmumula b. Lahat ng magsasaka ay may lupang sinasaka c. Lahat ng tao’t bagay ay galling sa lupa d. Lahat ng umaasa sa biyayang galling sa magsasaka. Ang kaibigan ko ay isa lamang maralita.
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a. mangmang c. mabait b. maliliit na tao d. mahirap Si Nena ay inaruga ng kanyang lola mula pa noong siya’y maulila. a. pinabayaan c. inalagaan b. pinamigay d. kinuha Palasak na ang desenyong iyan. a. pambihira c. magastos b. pangkaraniwan d. wala sa moda Ang mga salbahe ay kinamuhian niya a. kinakalinga c. kinatatakutan b. kinukumusta d. kinasusuklaman Ang mga kawal na lumabag sa utos ay binigyan ng babala. a. sundalo c. kusinero b. kaibigan d. pulis Nangangamba ka ba na hindi ka niya pagbibigyan? a. nasisiyahan c. nababanas b. natatakot d. naiinis Ang pagpunta sa Saudi Arabia ay di‐gawang biro. a. Madali c. mahirap b. masayang Gawain d. maayos Si Miguel ay sumakabilang buhay na noong Linggo. a. nagpaalam c. nagpunta sa siyudad b. namatay d. nagbayad ng utang Bakit mukhang Biyernes Santo si Marko. a. malungkot c. mukhang masaya b. lumuluha d. tumatawa Nakaririmarim ang nangyaring sakuna sa dagat. a. nakalulungkot c. nakaiinis b. nakatatakot d. nakapangingilabot Ang dayuhang siyang pinakamatalik na kaibigan ni Rizal ay a. Austin Craig c. Otley beyer
Ferdinand Blumentritt d. Don Eulogio Despujl 97. b. Dahil sa tulong at pagmamalasakit ni a. b. c.
Dona Aurora A. Quezon Tandang Sora Luz B. Magsaysay Sa kapakanan ng mga sinalanta ng sakuna, siyay tinaguriang Ina ng Kruz na Pula.
98. Kung ano ang “Urbana at felisa” sa mga tagalog ang a. b.
Lagda c. Bidasari Maragtas d. Hudhud ay siya naman sa mga Bisaya. 99. Sa mga tauhan ng Noli Me Tangere ni Rizal, si a. Basilio c. Capitan Tiago b. Elias d. Simon Ang nagligtas kay Ibarra sa kapahamakan 100. Ang aklat ng mga tinipong tula sa Tagalog ni Lope K. Santos ay pinamagatang a. Damdamin c. Tungkos ng Alaala b.
Puso at Diwa *****
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
B B B B B A B A B A A A B A B B A C B C B B A A B D B A C B A C
51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82
B C A B C B B B C B B C B A A C B A C A B D C A A B C A A D B A
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
D B B B B C D B A A B C B C B A D B
83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
D A D D C B D A B C B A D B A D D B
d. Mga Dahong Ginto T H E E N D *****
WORK HARD, DREAM HARDER
HUMANGROWTHAND DEVELOPMENT POST‐TEST April 2010
Multiple Choices: filipino
1. The process by which certain potentials are inherited
12. Which of the following principles IS NOT considered
from the parents for his development a. Life c. Heredity b. Birth d. Character 2. This theory states that there are 8 basic development stages that the individual has to pass through his life a. Learning Theory b. Psychoanalytic Theory c. Psychosocial Theory d. Cognitive Development 3. Transition age from childhood to adulthood where rapid physical changes and sex maturity occur resulting in changes in ways of feelings, thinking and acting. a. Puberty c. Early adulthood b. Adolescence d. Stage V 4. Modifying an existing scheme after an individual’s interaction with the environment, resulting in the creation of a new scheme. a. Assimilation c. Recognition b. Interaction d. Accommodation 5. Theory stating that a person’s behavior can be motivated by urges towards self satisfaction. a. Psychoanalytic Theory b. Cognitive development theory c. Psychosocial Theory d. Moral development theory 6. The ability of a child to conceptualize the retention and preservation of the same quantity under various changes. a. Recognition c. Assimilation b. Reversibility d. Conservation 7. Refers to the idea that no individual are exactly the same or alike. a. Cognitive theory c. Individual differences b. Exclusivity theory d. Emotional quotient 8. He is known as the Father of Modern I.Q. Test a. Lewis Terman c. Laurence Kohlberg b. Erick Erickson d. Martin Lesley 9. “Intellectual appreciative Experience” is … a. base on the premise that all learning has emotional correlates b. obtained in the field of music, art and literature c. the acquisition and retention o acts and information d. assumes that human activities are based on stimulus and response 10. These statements imply that children at the early learning stage consider parents and teachers as authorities and models. a. Parents and teachers should always coordinate children’s activities b. Parents should enforce strict discipline at home and teachers in school c. Parents and teachers should be the role models at all times d. Parents and teachers should always consult each other with regards the child’s intellectual development 11. Any change in the behavior of an individual a. Learning c. Change b. Response d. Development
under Classical Conditioning by Ivan Pavlov? a. Excitation b. Adhesive Principle c. Stimulus Generalization d. None of the above The reinforcement of a person’s responses by presentation or removal of rewards and punishment. a. Operant conditioning c. Feedback Principle b. Transfer of learning d. Discipline This stimulation of action best explains the behavior of an individual to take what he perceives to be the shortest route to his goals. a. Recognition c. Response b. Assimilation d. Motivation The process by which an individual acquires the social and cultural heritage o the society where he belongs. a. Socialization c. Integration b. Internalization d. Acquisition Philosophy of education’s main function. a. Aid the leaner to build his own personal philosophy b. Definition o goals and setting of directions from which education c. Educations carries on a lifetime cycle d. Provision of academic background prerequisite to learning According to Froebel, kindergarten is also known as “____________? a. children have fun and enjoyment b. Garden where children could grow c. He learning Center for Life d. Where new beginnings begin Which of the following statements is given emphasis by “humanistic education?” a. The great works of man such as the classics should be enjoyed. b. Man should learn the different philosophies of education c. “Build a man who is distinctly civilized, educations and refined” d. Develop man into a thinking individual A teacher who advocates the pragmatic philosophy of education believes that experience should follow learning, thus, she has to? a. require her student mastery of the lessons b. encourage her students to memorize facts c. equip her students with basic skills and abilities d. provide her student with opportunities to apply their skills and abilities How are institutions of learning encouraged to set higher standards over and above the minimum requirement for state recognition? a. Scholastic achievement b. Faculty development c. Academic freedom d. Voluntary accreditation The period of physical, especially sexual, and mental maturation which is characterized by rapid somatic growth is known as a. infancy c. puberty b. early childhood d. adulthood
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a. Darkness c. closed space b. strangers d. height An eye defect characterized by clear vision in one dimension but unfocused vision on the other is called a. myopia c. hyperopia b. astigmatism d. presbyopia Which of the following statements does not apply to adolescents? a. they desire the approval of their peers b. they seek dependence on their parents c. they have a marked sex development d. none of the above As young people mature, society expects them to develop competencies and assume social roles in a conventional manner. a. expectation of parents b. influence of peers groups c. influence of formal education d. cultural demands The founder of the theory of psychology called psychoanalysis was a. Lock c. Freud b. Hume d. leibnitz When the learner reaches a point where no further improvement can be expected, he is in a so‐called a. development crisis c. regression b. learning plateau d. depression Regarding the sexual maturation o boys and girls, teachers should bear in mind that: a. girls mature at a late stage than boys b. girls mature at an earlier stage than boys c. boys and girls mature at the same time d. there are no marked differences in heir time of maturity Rationalization is used by student who a. always give explanation or reason for their failures rather than own their faults b. like to take the blame for their faults c. bribe their elders with promises d. substitute words for deeds Which of the following is true of Abnormal Psychology? a. it studies the cause of personality defects b. it measures the accomplishments of the individual c. it concentrates on the scholastic performance of the individual d. it investigates the educational background of the individual Which of the following is a continuous variable? a. weight c. nationality b. sex d. race Which of the following is true about one’s IQ? a. it remains fairly constant b. it is highly changeable c. it is affected by attitude d. it is never constant Transfer of training easily takes place if the activities involved a. Are different b. Have identical element c. Occur in the same place d. Vary in difficulty
34. When the learner is well‐motivated, he performs his
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task a. with indifference c. with arrogance b. with disinterest d. with enthusiasm A six‐year‐old child who has a mental age of eight years has an IQ of a. 120 b. 130 c. 132 d. 133 The ratio obtained by dividing mental age by chronological age times 100 is called a. derived quotient b. deviation c. intelligence quotient or IQ d. intelligence ratio Which of the following was written by Plato? a. Sic et Non c. The Republic b. The School and Society d. Emile Who among those below asserted that “Education is for complete living” a. Dewey c. Kant b. Spencer d. Froebel The right of an educational institution and its faculty to prescribe the methods/strategies of teaching refers to: a. building style b. choice of curriculum c. academic freedom d. co and extra curricular program The 1987 Constitution provides that religious institution can be given a. with the students’ consent b. with the parent/guardian approval c. with mayor’s permit d. with the school’s support Public schools in the Philippines are the contribution of which colonizer? a. American c. Japanese b. British d. Spanish
42. Hardship allowance is given to a teacher when a. he’s assigned in a depressed area b. he’s given additional teaching load c. he’s in lahar area d. he’s assigned in a hazardous area 43. The ability for quantitative learning of the relations of facts taken from newspaper readings, letter writing and the like is called: a. functional literacy c. Knowledge outcome b. adjustment learning d. Social competence 44. A teacher who gives a uniform assignment to be worked out by all learners in Arithmetic is not observing a characteristic of a good assignment. Which characteristic is overlooked? a. It should be definite b. It should be stimulating c. It should emphasize the essential d. It should provide for individual differences 45. If a student ask a question which the teacher does not have a ready answer, the latter should: a. dismiss the question as irrelevant b. offer a bluff c. admit the fact that he doesn’t know the answer
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ask volunteers to answer the question and do research on it later. The heredity traits acquired by a person in his lifetime; a. are transmissible to his offspring b. reappear in his future grandparent c. Have no influence on the offspring d. Become recessive traits When student are given a chance to settle differences of opinion by discussion, they develop: a. fair play c. irritants b. tolerance d. sociability The school’s responsibility towards teenagers “gang age” is: a. provide the gang all the freedom it needs b. gives classroom activities to give direction to out‐ of‐school youth activities c. supervise gang activities d. set up norms of conduct or the member of the gang In an intelligence test, a 13‐year old girl got a score equivalent to that of a 15‐year old. This means: a. that the girl must be accelerated b. that the girl is 2‐years older mentally c. that the girl has a chronological age of 15 d. that she has a mental age of 13 Which statement is not necessary to achieve the learner’s interest in a learning activity? a. the activity must lead to a practical end b. the activity must be within the ability of the learner c. the activity must fill a need recognized by the learner d. the learner must have the experience that will furnish the background for the activity He is responsible for the theory which recognizes the importance of developing multiple intelligence a. Jean Piaget c. Frederick Freobel b. Howard Gardner d. Sigmund Freud The need to recognize and develop special sensitivity to language, thus helping the learners to use the right word, phrase and/ or graph to grasp new meaning refers to a. visual intelligence c. feelings sensitivity b. linguistic intelligence d. jargon The sensitivity to tone and pitch, allowing one to produce musical scoring is intelligence in? a. musical c. quantitative exercises b. verbal ability d. qualitative analysis One’s ability to do abstract reasoning and manipulate symbols refers to what type of intelligence? a. musical b. personality identification c. mental ability d. mathematical‐logical
d.
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to understand and be able to know the motives of other people’s actions. a. spatial c. logical b. personal d. diametric The type of intelligence which enables a person to understand other person’s feelings, behavior and motivation. a. emotional c. social intelligence b. spatial d. quantitative and qualitative The type of intelligence which characterizes actress, actors, mimes, dancers and people of the Arts? a. bodily‐kinesthetic c. research b. scientific d. emotions An emerging thrust in determining one’s personality, whether pleasant or unwholesome, this type of personality measurement is the wholesomeness of one’s virtues, i.e., values, relationships with other, adjustments to varying situations, behavior an motivations a. emotional quotient (E.Q.) b. intelligence quotient (I.Q.) c. maladjustment personality d. anticipated behavior It is a measurement of personality which is the result by dividing the mental age by the chronological age. a. emotional quotient (E.Q.) b. intelligence quotient (I.Q.) c. multiple Intelligence d. forecasted behavior quotient The teacher must be aware that both heredity and environment represent complex factors, exerting many specific influences on an individual’s growth. Which of the following statements best represents the influence of heredity and environment? a. Heredity counts; environment is less important. b. If the environment is changed, heredity becomes less important. c. The relative influences of heredity and environment can vary widely in an individual’s growth. d. In the long run, both tend to cancel each other’s influences e. None of the above The best possible way to measure the influence of heredity is by: a. keeping the environment constant. b. Ignoring the environment c. Studying only fraternal o normal capability d. Studying only identical twins of normal capability e. Doing none of the above Educators who contributed to the “open education” movement includes: a. b.
55. The ability to perceive how objects are related in order to mentally perceive what is seen, thus creating concrete visual images from memory refers to? a. visual‐spatial intelligence b. musical c. language
logical reasoning
56. The capacity to analyze one’s feelings and thus be able
Neill and piaget c. Bruner and Silberman Kohl and kozol d. All of the above
64. A child’s social skills can be measured by: a. direct observation and parent‐teacher conferences b. c. d.
psychological test adaptive behavior scales A and C above
65. A teacher uses behavioral modification techniques in his classes. Which of the following student behaviors would he find most difficult to change? a. Aggressive tendencies toward classmates b. Poor habits in organizing work materials c. Interrupting a speaker d. Abandoning a project before it is finished
66. Learning‐disabled children most characteristically have:
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a. low IQ b. poor socio‐economic backgrounds c. an average level of intelligence d. minimal brain damage Which of the following is true about educable mentally retarded children? a. Their IQ range between 50 and 70 b. They have short attention spans and experience difficulty in generalizing c. Their reading, writing, and arithmetic skills cannot be improved d. A and B above Which of the following is characteristics of a dyslexic child a. Mirror writing b. listlessness c. Below‐average intelligence d. Hyperactivity Primary reading retardation is presumed to be neurologically based, related to parietal lobe dysfunction? a. Inability to relate sound to letter symbols b. Inadequate auditory information processing c. Left‐right directional confusion d. Speech aphasia Students with secondary reading problems have capacity to read, but are non‐readers because of: a. auditory problems b. congenital defects c. visual‐acuity impairment d. environmental or emotional actors If a teacher accepts Maslow’s theory on the hierarchy of needs, he or she will probably structure objectives to: a. meet both the physiological and intellectual needs of students b. eliminate testing c. eliminate extrinsic motivations d. maintain a certain anxiety level for increased competition The knowledge explosion has led to crowding more and more information into curriculum courses. A likely result is that: a.
the textbook will no longer be the main instructional medium in many classes the child may spend more time in school the teacher may have to rely more on the se of multimedia materials d. all of the above 73. During the learning process the teacher has most control over: b. c.
a. the learners b. the learning environment c. the learning process d. the behavior of the learners 74. Which of the following conditions does NOT contribute to a climate psychologically suited to learning? a. The teacher acts like a “real person.” b. The teacher makes all of the decisions about students’ learning activities. c. The teacher accepts students as they are d. The teacher shows trust in students’ decisions
75. William Glasser advocates the frequent use of classroom meetings, with teacher and students sitting in a small circle. Which one of the following types of discussion would NOT be appropriate in such a setting? a. An educational‐diagnostic conference on the learning weaknesses of individual students. b. An open‐ended meeting for the purpose of exploring and discussing student’s ideas about the curriculum c. A social‐problem‐solving meeting to resolve teacher or student problems elating to the school, the class, or any individual member. A sensitivity‐training meeting for the purpose of helping students ace their school‐related problems and learn how their actions can affect others 76. Which of the following does NOT represent a teacher’s contribution to the emotional environment of the classroom? a. A strident, compelling voice. b. A sustained sense of expectation where student achievement is concerned c. A well‐written lesson plan d. A sense of humor in a tense situation 77. According to Jones, student commitment to accomplishing a learning goal depends on all of the following EXCEPT: a. how interesting the goal is b. how likely it seems that the goal can be accomplished c. what degree of challenge the goal presents d. whether the learner will be able to tell if the goal d.
has been accomplished whether materials are ready assembled for undertaking the goal 78. The teacher who understands the adolescent’s need to conform will: a. use sarcasm as a disciplinary device b. disregard unique responses in discussion and on examinations e.
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83.
establish a learning climate that fosters feelings of security d. lecture students on their weakness o character The best public relations agents for a school are the: a. pupils c. PTA members b. Teachers and pupils d. principals The structured curriculum is in decided contrast to the child‐centered curriculum, which: a. emphasizes fundamental education b. is changeable and is built around student interest and needs c. is oriented to the needs of a democratic society. d. Utilizes the theory of mental discipline According to Bruner, teacher working with young children should a. Push the children to maximum cognitive development as rapidly as possible b. Present all information verbally so the children will listen well c. Present new material from the concrete to the abstract d. Present new information from the abstract to the concrete from the educational viewpoint, intelligence is: a. an abstract concept b. a trait that can be manipulated c. good judgment d. a form of behavior Every taxonomy of educational objectives: a. describes increasingly difficult learning activities b. describes levels of goals for learner development c. suggest evaluation measure for teacher use d. Classifies learning outcomes
84. A mathematics teacher following Gagne’s theory of learning believes that: a. learning can take place under all conditions b. learning is mainly a mater of accurate discrimination c. learning takes place only when the student is in a receptive state d. learning is reinforced chiefly by classical conditioning 85. Under which of the following conditions is a child’s IQ more likely to increase? a. If the emotional climate in the classroom improves b. If the child is given a large “research” project. c. If the child enjoys problem solving and is given ample opportunity for it d. If A and C are true 86. Intelligence is the basis of education. Education is the effective means for national development, hence, a country spends a large portion of its budget for the systematic training of the learner to attain full development Why is education one major concern of every c country? Because a. intelligence has many facets b. intelligence is useful in testing
c.
intelligence is a safe gauge for budgetary allocation d. intelligence test when carefully conducted, can help in determining need for future facilities for national building 87. There are no two individuals who are the same. Individual differences, when early recognize and provided for, enable the teacher to provide different motivations and approaches in guiding the learning process. Each pupil differs physically, mentally, socially and emotionally from other children. Unless the teacher provides for this nature of the learner, no amount of modern approaches in teaching can elicit favorable results. a. The paragraph highlights the need or motivating learning b. Individual differences is an important consideration in guiding the learner c. The above paragraph focuses on teacher‐pupil relationship d. It takes about the nature of the learning process 88. Robert Craig, et al, wrote of the phase of steps in every learning process. These include: 1.) the focusing of attention to the stimulation at hand, 2.) the interplay of the learner and the social factors that surround him, 3.) the acquisition of a new response or behavior he gives to the new learning and 4.) Retention which presupposes that the new learning is acquired. The above paragraph emphasizes a. the learning process b. the steps/phase of how individuals learn c. the manifestations of learning d. why learning is a difficult process 89. Approaches in teaching change from time to time depending on the traditional of sophistication attached to the course being taught. Some mentors believe that the tie tested ways to teaching is effective. Other are easily carried away to use modern approaches in imparting new subject matter. It maybe safe to conclude that once results are realized in teaching, no specific method can be considered the one‐and‐only method to use. When teaching a subject area, it is safe to a. stick to the traditional way b. be modern and most recent c. get results in teaching d. to try any method as they are all theories after all
90. In the early 1980’s programmed teaching became popular in helping teachers to provide for individual differences in learners. The chunks of the subject matter which are divided into units are supposed to help the learner master the lesson, since it is simply to understand the frame of the lessons. No test o mastery of the units are done because the purpose is to provide information on certain subject matter
Would you as a teacher use programmed instruction if you handle a subject on Values Education? a. yes, definitely b. no, not important for the subject matter c. I don’t know d. Why not if the subject matter calls for it 91. The data/subject matter to teach are gathered in different ways, These include historical sources like surveys, systematic observations, experimentation, interviews, etc. to be reliable and valid, the data collected must be organized, properly analyzed and interpreted. From these processes, some conclusion or generalization are done to reveal certain relationships like cause & effect. Data gathering involves: a. tedious and serious study b. easy does it c. data gathered are tested and filed, then verified before being used d. no follow‐up needed 92. Heredity and environment play important roles in the function of human beings. DNA or Deoxyribonucleic Acid is the biological (heredity) band of our genes. Our environment includes the house, school and the community where we live. Whether we become successful or a failure will depend on the interplay of both nature and nurture. If heredity and environment affect the individual, thus, we can conclude that a. both actors play equal roes in one’s life b. one factor, either heredity or environment exerts more influence than the other c. neither factor is important d. nurture and nature are the same 93. The first systematic philosopher to work in the field
94.
95.
96.
97.
of education was a. Socrates c. Plato b. Aristotle d. Rousseau The first state in the world’s history where all human capabilities were allowed to develop freely a. Rome c. Sparta b. Athens d. Germany They are the most practical, pragmatic people who absorbed themselves in the management of their state affairs a. Spartans c. Romans b. Athens d. Chinese Invented the first system of writing in the orient a. Phoenicians c. Greeks b. Chinese d. Romans first to introduce the use of printing press in the Philippines a. Romans c. Greeks b. Chinese d. Japanese
98. conducted the world’s first civil service test a. Greeks c. Chinese b. Romans d. English 99. To develop the capacity of man only for war was the educational aim of the ancient a. Romans c. Athenians b. Spartans d. Chinese
100.
To produce a young man who would be charming in person and graceful in manner, e.g. a beautiful soul in a beautiful body is the educational aim of education of the a. Romans c. Spartans b. Athens d. Italians
*****
T H E E N D *****
WORK HARD, DREAM HARDER
human growthand development
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
c c b d a d c a a c a d c d a b a c d
51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69
c b a d a b c a a b c d d d a c d a d
20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
d c c b b d c b b a a a a b d d c c
70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87
d a d b b a c e c b b c a b c d d b
38 39 40 41 42 43 44 45
b c b a c d d a
88 89 90 91 92 93 94 95
b c b a a c b c
46 47 48 49 50
a b b b a
96 97 98 99 100
b b c b b
LICENSURE EXAMINATION FOR TEACHERS (LET)
Refresher Course ContentArea: MATHEMATICS Focus: ANALYTIC GEOMETRY Preparedby: Daisy de Borja‐Marcelino Competencies: 1. Solve problems involving coordinates of a point, midpoint of a line segment, and distance between two points. 2. Determine the equation of the line relative to given conditions: slope of a line given its graph, or its equation, or any two points on it.
Determine the equation of a non‐vertical line given a point on it and the slope of a line, which is either parallel or perpendicular to it. Solve problems involving a. the midpoint of a line segment, distance between two points, slopes of lines, distance between a point and a line, and segment division. b. a circle, parabola, ellipse, and hyperbola. Determine the equations and graphs of a circle, parabola, ellipse and hyperbola.
3. 4.
5.
I. The Cartesian Plane
Below is a diagram of a Cartesian plane or a rectangular coordinate system, or a coordinate plane.
The two axes separate the plane into four regions called quadrants. Points can lie in one of the four quadrants or on an axis. The points on the x‐axis to the right of the srcin correspond to positive numbers; while to the left of the srcin correspond to negative numbers. The points on the y‐axis above the srcin correspond to positive numbers; while below the ori in corres ond to ne ative numbers. An ordered pair of real numbers, called the coordinates of a point, locates a point in the Cartesian plane. Each ordered pair corresponds to exactly one point in the Cartesian plane.
The following are the points in the figure on the right: A(‐6,3), B(‐2,‐3), C (4,‐2), D(3,4), E(0,5), F(‐3,0).
For numbers 1‐2, use the following condition: Two insects M and T are initially at a point A(‐4, ‐ 7) on a Cartesian plane.
1. If M traveled 7 units to the right and 8 units downward, at what point is it now? Solution: (‐4+7, ‐7‐8) or (‐3,‐15) 2. If T traveled 5 units to the left and 11 units downward, at what point is it now? Solution: (‐4‐5, ‐7‐11) or (‐9, ‐18) II. The Straight Line A. Distance Between Two Points
A. The distance between two points (x1,y1) and (x2,y2) is given by
( x1 x2 ) 2 ( y1 y2 ) 2 .
Example: Given the points A(2,1) and B(5,4). Determine the length AB. Solution: AB = 2 5
2
1 4 2
9
2
3
2
99
18 or
9 2 or 3 2 .
Exercises: For 1‐2, use the following condition: Two insects L and O are initially at a point (‐1,3) on a Cartesian plane. 1. If L traveled 5 units to the left and 4 units upward, at what point is it now?
‐
A) ( 6, 7)
B) (4, 7)
‐ ‐
C) ( 6, 1)
‐
D) (4, 7)
2. If O traveled 6 units to the right and 2 units upward, at what point is it now? A) (7, 5) B) (5,5) C) (‐7, 5) D) (‐5, ‐5)
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3. Two buses leave the same station at 9:00 p.m. One bus travels at the rate of 30 kph and the other travels at 40 kph. If they go on the same direction, how many km apart are the buses at 10:00 p.m.? A) 70 km B) 10 km C) 140 km D) 50 km 4. Two buses leave the same station at 8:00 a.m. One bus travels at the rate of 30 kph and the other travels at 40 kph. If they go on opposite direction, how many km apart are the buses at 9:00 a.m.? A) 70 km B) 10 km C) 140 km D) 50 km 5. Two buses leave the same station at 7:00 a.m. One bus travels north at the rate of 30 kph and the other travels east at 40 kph. How many km apart are the buses at 8:00 a.m.? A) 70 km B) 10 km C) 140 km D) 50 km 6. Which of the following is true about the quadrilateral with vertices A(0,0), B(‐2,1), C(3,4) and D(5,3)? i) AD and BC are equal ii) BD and AC are equal iii) AB and CD are equal A) both i and iii
B) ii only
7. What is the distance between (‐5,‐8) and (10,0)? A) 17 B) 13
C) both ii and iii
D) i, ii, and iii
D) ‐ 0.5
C) 23
B. Slope of a line
a) The slope of the non‐vertical line containing A(x1,y1) and B(x2,y2) is m
y1 y2 x1 x2
or m
y2 y1 x2 x1
.
b) The slope of the line parallel to the x‐axis is 0. c) The slope of the line parallel to the y‐axis is undefined. d) The slope of the line that leans to the right is positive. e) The slope of the line that leans to the left is negative.
C. The Equation of the line In general, a line has an equation of the form ax + by + c = 0 where are not both zero.
a, b, c are real numbers and that a and b
D. Different forms of the Equation of the line General form: ax + by + c = 0. Slope‐intercept form: y = mx + b, where m is the slope and b is the y‐intercept.
Point slope form: y y1
m( x x1 ) where (x1, y1) is any point on the line. y y1 y y1 2 ( x x1 ) where (x1, y1) and (x2, y2) are any two points on the line. x2 x1
Two point form:
Intercept form:
Reminders: A line that leans to the right has positive slope. The steeper the line, the higher the slope is.
x a
y b
1 where a is the x‐intercept and b is the y‐intercept.
p q r The slopes of lines p, q, r are all positive. Of the three slopes, the slope of line p is the lowest, the slope of r is the highest.
A line that leans to the left has negative slope. The steeper the line, the lower the slope is.
t
s
u
The slopes of lines t, s, u are all negative. Of the three slopes, t is the highest, while u has the lowest (because the values are negative.) Exercises St. Louis Review Center-Inc-Davao Tel. no. (082) 224-2515
19
1. What is the slope of 5 x - 4
+ y12
0=
?
B) ‐ 1.25
A)1.25
D) ‐ 0.8
C) 0.8
2. What is the slope of x = ‐ 9? A) 4
B) 1
C) 0
D) undefined
3. What is the slope of y= 12? A) 7
B) 1
C) 0
D) undefined
C) ‐ 0.4
D) ‐ 2.25
4. What is the slope of
x 4
+
y 9
= 1?
A) 0 .4
B) 2.25
E. Parallel and Perpendicular lines Given two non‐vertical lines p and q so that p has slope m1 and q has slope m2. If p and q are parallel, then m1 = m2. If p and q are perpendicular to each other, then m1m2 = ‐ 1. F. Segment division Given segment AB with A(x1,y1) and B(x2,y2).
x1 x2 y1 y2 , ). 2 2 r1 AP r1
The midpoint M of segment AB is M (
If a point P divides AB in the ratio
the formula x
r1 x2 r2 x1 r1 r2
and y
r2
so that
PB
r1 y2 r2 y1 r1 r2
r2
, then the coordinates of P(x,y) can be obtained using
.
G. Distance of a point from a line
The distance of a point A(x1,y1) from the line Ax + By + C = 0 is given by
d
Ax1 By1 C A
2
B2
.
Exercises 1. Write an equation in standard form for the line passing through (–2,3) and (3,4). a. 5x – y = ‐ 13 b. x – 5y = 19 c. x – y = ‐5
d. x – 5y = –17
2. Write an equation in slope intercept form for the line with a slope of 3 and a y‐intercept of 28. a. y = –3x + 28 b. y = 0.5x + 28 c. y = 3x + 28 d. y = 3x + 21 3. Write the equation in standard form for a line with slope of 3 and a y‐intercept of 7. a. 3x – y = –7 b. 3x + y = 7 c. 3x + y = 7 d. –3x + y = –7 4. Which of the following best describes the graphs of 2x – 3y = 9 and 6x – 9y = 18? a. Parallel b. Perpendicular c. Coinciding d. Intersecting 5. Write the standard equation of the line parallel to the graph of x – 2y – 6 = 0 and passing through (0,1). a. x + 2y = –2 b. 2x – y = –2 c. x – 2y = –2 d. 2x + y = –2 6. Write the equation of the line perpendicular to the graph of x = 3 and passing through (4, –1). a. x – 4 = 0 b. y + 1 = 0 c. x + 1 = 0 d. y – 4 = 0 7. For what value of d will the graph of 6x + dy = 6 be perpendicular to the graph 2x – 6y = 12? a. 0.5 b. 2 c. 4 d. 5 III. Conic Section A conic section or simply conic, is defined as the graph of a second‐degree equation in x and y.
In terms of locus of points, a conic is defined as the path of a point, which moves so that its distance from a fixed point is in constant ratio to its distance from a fixed line. The fixed point is called the focus of the conic, the fixed line is called the directrix of the conic, and the constant ratio is called the eccentricity, usually denoted by e. If e < 1, the conic is an ellipse. (Note that a circle has e=0.) If e = 1, the conic is a parabola. If e > 1, the conic is hyperbola. A. The Circle
1.
A circle is the set of all points on a plane that are equidistant from a fixed point on the plane. The fixed point is called the center, and the distance from the center to any point of the circle is called the radius.
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2.
Equation of a circle a) b)
3.
2
2
general form: x + y + Dx + Ey + F = 0 2 2 2 center‐radius form: (x – h) + (y – k) = r where the center is at (h,k) and the radius is equal to r.
Line tangent to a circle A line tangent to a circle touches the circle at exactly one point called the point of tangency. The tangent line is perpendicular to the radius of the circle, at the point of tangency.
Exercises For items 1‐2, use the illustration on the right.
y 2
1. 5
1. Which of the following does NOT lie on the circle? 1
a. (3,‐1) c. (2,‐1)
b. (3,0) d. (3,‐2)
0. 5
x
0 - 3.5
-3
- 2.5
-2
- 1.5
-1
- 0.5
0
0.5
2. What is the equation of the graph?
x 3 1 2 c. ( y 1) x 3 1 a.
y2
1
1.5
2
2.5
3
3.5
-0. 5
b.
( y 1) 2 x 3 1
d.
( y 1)
2
-1
x 3 1
-1. 5
-2
B. The Parabola
1.
Definition. A parabola is the set of all points on a plane that are equidistant from a fixed point and a fixed line of the plane. The fixed point is called the focus and the fixed line is the directrix.
2.
Equation and Graph of a Parabola 2
a)
The equation of a parabola with vertex at the srcin and focus at (a,0) is the right if a > 0 and opens to the left if a < 0.
b)
The equation of a parabola with vertex at the srcin and focus at (0,a) is upward if a > 0 and opens downward if a < 0.
c)
The equation of a parabola with vertex at (h , k) and focus at (h + a, k) is The parabola opens to the right if a > 0 and opens to the left if a < 0.
(y – k) = 4a(x – h).
d)
The equation of a parabola with vertex at (h , k) and focus at (h, k + a) is
(x – h) = 4a(y – k).
e)
The parabola opens upward if a > 0 and opens downward if a < 0.
f)
Standard form:
(y – k) = 4a(x – h) or (x – h) = 4a(y – k)
g)
General form:
y + Dx + Ey + F = 0, or x + Dx + Ey + F = 0
2
2
y = 4ax. The parabola opens to
2
x = 4ay. The parabola opens
2
2
2
2
3. Parts of a Parabola a)
The vertex is the point, midway between the focus and the directrix.
b)
The axis of the parabola is the line containing the focus and perpendicular to the directrix. The parabola is symmetric with respect to its axis.
c)
The latus rectum is the chord drawn through the focus and parallel to the directrix (and therefore perpendicular to the axis) of the parabola.
d)
In the parabola y =4ax, the length of latus rectum is 4a, and the endpoints of the latus rectum are (a, ‐2a) and (a, 2a). 2
In the figure at the right, the vertex of the parabola is the srcin, the focus is F(a,o), the directrix is the line containing LL ' , the axis is the x‐axis, the latus rectum is the line containing CC ' .
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The graph of x
2
16
2
y.
3
The graph of (y‐2) = 8 (x‐3).
C. Ellipse
1.
An ellipse is the set of all points P on a plane such that the sum of the distances of P from two fixed points F’ and F on the plane is constant. Each fixed point is called focus (plural: foci).
2. Equation of an Ellipse
a) If the center is at the srcin, the vertices are at ( a, 0), the foci are at ( c,0), the endpoints of the minor axis are at (0, b) and b
2
a2 c2 ,
then the equation is
x2 a
2
y2
1.
b2
b) If the center is at the srcin, the vertices are at (0, a), the foci are at (0, c), the endpoints of the minor axis are at ( b, 0) and b
2
a 2 c 2 , then the equation is
x2 2
y2
1.
2
b a c) If the center is at (h, k), the distance between the vertices is 2a, the principal axis is horizontal and
b
2
a 2 c 2 , then the equation is
( x h) 2 a
2
( y k )2 b2
1.
d) If the center is at (h, k), the distance between the vertices is 2a, the principal axis is vertical and 2
b2
4.
2
a c
2
, then the equation is ( y 2k )
a
2
( x b 2h) 1 .
Parts of an Ellipse For the terms described below, refer to the ellipse shown with center at O, vertices at V’(‐a,0) and V(a,0), foci at F’(‐c,0) and F(c,0), endpoints of the minor axis at B’(0,‐b) and B(0,b), endpoints of one latus rectum at G’ (‐c,
b
2
) and G(‐c,
a
2
H’ (c,
a) b)
c)
b
a
b
V’ -
2
a
B 0b
2
(c, ba )
F’ -
O
2
(c, ba )
Fc0 Va0 x 2
(c, ba )
2 (c, ba )
) and the other at
B’ 0 -
2
) and G(c, b ).
a
The center of an ellipse is the midpoint of the segment joining the two foci. It is the intersection of the axes of the ellipse. In the figure above, point O is the center. The principal axis of the ellipse is the line containing the foci and intersecting the ellipse at its vertices. The major axis is a segment of the principal axis whose endpoints are the vertices of the ellipse. In the figure,
V 'V is the major axis and has length of 2a units. The minor axis is the perpendicular bisector of the major axis and whose endpoints are both on the ellipse. In the figure, B ' B is the minor axis and has length 2b units.
d)
The latus rectum is the chord through a focus and perpendicular to the major axis. G ' G and H ' H are the latus rectum, each with a length of
2b 2 a
.
y
y
(2,6) (8,5)
(0, 3) (-4,9) 5
(-5, 0)
(-6,4)
(4,9) 5
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O (-4,-9) 5
(2,1)
(12,1)
(-4, 0)
x (0, -3)
(4,-9) 5
x
O (8,3) (2,-4)
22
The graph of
x2
25
y2
9
1.
The graph of
( x 2) 2 100
( y 1) 2 25
1.
4. Kinds of Ellipses a) b)
Horizontal ellipse. An ellipse is horizontal if its principal axis is horizontal. The graphs above are both horizontal ellipses. Vertical ellipse. An ellipse is vertical if its principal axis is vertical.
D. The Hyperbola
1. 2.
A hyperbola is the set of points on a plane such that the difference of the distances of each point on the set from two fixed points on the plane is constant. Each of the fixed points is called focus. Equation of a hyperbola a) If the center is at the srcin, the vertices are at ( a, 0), the foci are at ( c,0), the endpoints of the minor axis are at (0, b) and b
2
c2 a2 ,
then the equation is
x2 a
2
y2 b2
1.
b) If the center is at the srcin, the vertices are at (0, a), the foci are at (0, c), the endpoints of the minor axis are at ( b, 0) and b
2
c 2 a 2 , then the equation is
y2 a
2
x2 b
2
1.
c) If the center is at (h, k), the distance between the vertices is 2a, the principal axis is horizontal and b 2 2
, then the equation is ( x 2h )
a
( y 2k ) 1 .
b 2 d) If the center is at (h, k), the distance between the vertices is 2a, the principal axis is vertical and b
(y k)
then the equation is
2.
a2
2
c2 a2
2
( x h) b2
2
c2 a2 ,
1
Parts of a hyperbola For the terms described below, refer to the hyperbola shown which has its center at O, vertices at V’(‐a,0) and V(a,0), foci at F’(‐c,0) and F(c,0) and endpoints of one latus rectum at G’ (‐c,
H’ (c,
b
2
a
) and H(c,
b
b2 a
) and
G(‐c,
b2 a
) and the other at
2
a
).
a) The hyperbola consists of two separate parts called branches.
b) The two fixed points are called foci. In the figure, the foci are at ( c,0).
c) The line containing the two foci is called the principal axis. In the figure, the principal axis is the x‐axis. d) The vertices of a hyperbola are the points of intersection of the hyperbola and the principal axis. In the figure, the vertices are at ( a,0).
e) The segment whose endpoints are the vertices is called the transverse axis. In the figure V 'V is the transverse axis.
f) The line segment with endpoints (0,b) and (0,‐b) where b
2
c 2 a 2 is
called the conjugate axis, and is a
perpendicular bisector of the transverse axis. g) The intersection of the two axes is the center of the hyperbola .
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h) The chord through a focus and perpendicular to the transverse axis is called a latus rectum. In the figure, G ' G is a latus rectum whose endpoints are
3.
G’ (‐c,
b
2
a
) and G(‐c,
b
2
a
) and has a length of
2b
2
.
a
The Asymptotes of a Hyperbola
Shown in the figure on the right is a hyperbola with two lines as extended diagonals of the rectangle shown.
These two diagonal lines are said to be the asymptotes of the curve, and are helpful in sketching the graph of a hyperbola. The equations of the asymptotes associated with
equations of the asymptotes associated with
y2 a2
x2 b2
x2 a
2
1 are y
y2
b b 1 are y x and y x . Similarly, the b2 a a a a x and y x . b b
69
(-9,6)
F’(-6 0)
(-3 0)
(3 0)
(9,6)
F(0,6)
3y
x
0
3y
x
0
(0,3)
F(6 0)
x
O (0,-3) (6 -9) F’(0-6 )
The graph of
x2
9
y2
27
1.
The graph of
y2
9
x2
27
1.
PRACTICE EXERCISES Directions: Choose the best answer from the choices given and write the corresponding letter of your choice.
For items 1‐5, use the illustration on the right.
3 M
1. Which of the following are the coordinates of A? a. (1,2) b. (2,1) c. (‐3,3) d. (2,‐3)
x
2. What is the distance between points M and T? a. 61 units b. 6 sq. units c. 51 units d. 8 units 3. Which of the following points has the coordinates (‐3,‐1) a. M b. A c. T d. H 4. Which of the following is the area of the triangle formed with vertices M, A and H? a. 5 sq. units b. 10 sq. units c. 5 units d. 10 units 5. Which of the following is the equation of the line containing points A and T? a. y= 2 b. x=2 c. y+2x=3 d. y‐2x+3=0 6. Suppose that an isosceles trapezoid is placed on the Cartesian plane as shown On the right, which of the following should be the coordinates of vertex V? a. (a,b) b. (b+a, 0) c. (b‐a,b)d. (b+a,b) 7. The points (‐11,3), (3,8) and (‐8,‐2) are vertices of what triangle? a. Isosceles b. Scalene c. Equilateral
d. Right
8. What is the area of the triangle in #7? a. 40.5 sq units b. 41.8 sq units
d. 46.8 sq units
c. 42 sq units
9. Which of the following sets of points lie on a straight line? a. (2,3), (‐4,7), (5,8) b. (‐2,1), (3,2), (6,3) c. (‐1,‐4), (2,5), (7,‐2)
‐3
‐2
‐1
H
A
‐10 1 ‐23
2
3 T
y O(0,b) x
0
D(a,0) E(b,0)
d. (4,1), (5,‐2), (6,‐5)
10. If the point (9,2) divides the segment of the line from P1(6,8) to P2(x2,y2) in the ratio r = St. Louis Review Center-Inc-Davao Tel. no. (082) 224-2515
2 1
3 7
, give the coordinates of P2. 24
a. (16,–12) b. (–16, 15) c. (14,15) d. (12,–12) 11. Give the fourth vertex, at the third quadrant, of the parallelogram whose three vertices are (‐1,‐5), (2,1) and (1,5). a. (‐3,‐2) c. (‐3,‐4) c. (‐4,‐1) d. (‐2,‐1) 12. The line segment joining A(‐2,‐1) and B(3,3) is extended to C. If BC = 3AB, give the coordinates of C. a. (17,12) b. (15,17) c. (18,15) d. (12,18) 0 13. The line L2 makes an angle of 60 with the L1. If the slope of L1 is 1, give the slope of L2. 0.5 0.5 0.5 0.5 a. (3 + 2 ) b. (2 + 2 ) c. –(2 + 3 ) d. –(3 + 3 ) 0 14. The angle from the line through (‐4,5) and (3,m) to the line (‐2,4) and (9,1) is 135 . Give the value of m. a.7 b. 8 c. 9 d. 10 15. Which equation represents a line perpendicular to the graph of 2x + y = 2? a. y = ‐ 0.5x – 2 b. y = –2x + 2 c. y = 2x – 2 d. y = 0.5x + 2 16. Which of the following is the y – intercept of the graph 2x – 2y + 8 = 0? a. ‐ 4 b. ‐ 2 c. 2 d. 4 17. Which of the following may be a graph of x – y = a where a is a positive real number? a. b. c. d. y y
y
y
x
x
x
x
18. Write an equation in standard form for a line with a slope of –1 passing through (2,1). a. x + y = –3 b. –x + y = 3 c. x + y = 3 d. x – y = –3 b. For items 19‐22, use the illustration on the right. y 19. Which of the following are the coordinates of A? A a. (1,1) b. (1,‐1) c. (‐1,1) d. (‐1,‐1) 20. What is the distance between points A and H?
T
2
1. 5
1
0. 5
61 units
a.
b. 6 sq. units
x
0 - 3.5
-3
- 2.5
c. 51 units d. 8 units 21. Which of the following points has the coordinates (‐2,‐2)? a. M b. A c. T d. H
-2
- 1.5
-1
- 0.5
0
0.5
1
1.5
2
2.5
3
3.5
-0. 5
-1
H
M
-1. 5
-2
22. Which of the following is the equation of the given graph? a.
2 .
y x
2
b.
2 .
y x
2
c.
2 .
y x
2
23. Which of the following is the equation of the line containing points M and T? a. y= 2 b. x=2 c. y‐2x‐2=0 24. What is the shortest distance of a. 1 unit
x2
a. (0,‐4)
y2
12
x2
4
26. What are the x‐intercepts of a. none
4
2
d. y+2x+2=0
y2
9 b. 2
c. 3 units
d. 8 units
c. (0,4)
d. (4,0)
c. 3
d.
1?
b. (‐4,0)
x2
2.
y x
8 y from x 3 ?
b. 2 units
25. Which of the following is a focus of
d.
1?
27. Which of the following is a graph of a hyperbola? a.
4 b.
y
y
15
15 10
10 5
5
x
0
- 25
- 20
- 15
- 10
-5
0
- 30
x
0 - 30
5
10
15
20
25
- 25
- 20
- 15
- 10
-5
0
5
30
10
15
20
25
30
-5
-5 -10
-10 -15
-15
c.
d.
y
y
2 2
1. 5 1. 5
1 1
0. 5 0. 5
x
0 - 3.5
-3
-2 .5
-2
-1.5
-1
-0.5
0
0.5
x
0
1
1.5
2
2.5
3
3.5
-0. 5
-1
-1. 5
-2
-3 .5
-3
-2 .5
-2
-1.5
-1
-0 .5
0
0.5
1
1.5
2
2.5
3
3.5
- 0. 5
-1
- 1. 5
-2
28. Which of the following is an equation of an ellipse that has 10 as length of the major axis and has foci which are 4 units away from the center?
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25
a.
y2
25
x2
9
1
b.
y2
9
x2
16
1
c.
y2
5
x2
3
1
y2
d.
16
x2
25
1
For items 29‐31, consider the graph on the right. 29. Which of the following is the equation of the graph?
25x 2 2500 2 b. 100x 25 y 2500 2 2 c. 100 y 25x 2500 2 2 d. 100x 25 y 2500 a. 100 y
y
10
2
2
- 20
- 10
0
30. What are the x‐intercepts of the graph? a. none b. 2 c. 5
d.
x
0 - 30
c. hyperbola
30
d. Parabola
32. Which of the following is the center of the graph shown on the right? a. (0,0) b. (0,10) c. (10,0) c. (0,‐10) 33. Which of the following is a focus of the graph shown on the right? a. (0,0) b. (0,10) c. (0,5) c. (0,‐10)
20
- 10
10
31. What kind of figure is shown on the graph? a. circle b. ellipse
10
y
10
x
0 - 30
- 20
- 10
0
10
20
30
- 10
34. What is the area of the shaded region? a. 4 units c. 16 units
b. 4 square units d. 16 square units
LICENSURE EXAMINATION FOR TEACHERS (LET)
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Refresher Course Content Area: MATHEMATICS Focus: ADVANCED ALGEBRA Prepared by: Daisy de Borja‐Marcelino
Competencies: Show mastery of the basic terms, concepts and equations in Advanced Algebra involving radicals, rational exponents and functions. Solve, evaluate and manipulate symbolic and numerical problems in above areas by applying fundamental principles and processes.
KEY IDEAS Some Helpful Tips in Answering the LET
1 2 3 4 5 6 7 8
Read the question/s or the items carefully and understand what they say. Determine what is/are wanted or what is/are asked for. Find out what is/are given and which data are needed to solve the problem. Reason out what processes (operations) to apply and the order in which they are to be applied. Summarize the problem by means of an open number sentence. Compute carefully. Check each step in the computations. Decide the reasonableness of the result. Check the result by seeing to it that the result satisfies all the conditions of the problem.
RATIONAL EXPONENTS n
th
If a is a real number and n is any positive integer, the symbol a denotes the n power of a. The real number a is called the base and n is called the exponent. In symbols,
a
a a a ... a .
n
n factors
Examples:
a)
4 33 3 3 4 3 or 81. 1
b)
1 1 1 1
. 2 2 2 2 2 2 math math e) math
c)
3
2m
2m 2m .3 2 . d) 1.3 1.3 1
.
2m m
e)
2m .
2 m 2 .
Note that any base raised to the power of 1, is just the base. Moreover, any base raised to the power of 0 is 1, 0 while 0 is indeterminate. Examples: a) 9
0
2
1. 3
e) h op e
0
1
b) 4
1.
f)
4.
1
0
c) (‐m) = ‐ m.
38.230 1 .
2
3
g) j o y
5 1
2
3
d) z = 1. 5
j o y .
h)
1 4
0
m a
1
a.
1
4
Laws of Exponents If a and b are real numbers and m and n are positive real numbers, then the following are true.
a
n m
a mn .
aman
a
a0 If m If n
If
n m
a
Examples: a)
and and
2 3
3
2
d) 3
2
mn
a
, then a n
am
a 0 , then
0 , then
4
m
46.
an an
a
n
a
n m
23 2 2
b)
5m 0
e)
A. 5
20 m 9 n 3 s 14
10 s 6 n8 m 10
0
.
n
b . n
.
If
b 0 , then
35 3
2
232 25 32.
a an . b bn
2
2
2
c) (3 x 4 ) = 3 x 4 .
352 33 27.
Exercises 5 1. In the expression 8m , 5 is called the ________. A. base B. coefficient 2. Evaluate
ab a
n
a0 1 .
1.
250 m 2
n
.
1
a nm
f).
C. constant
32 35
325 33
1 33
1 27
D. exponent
0
. 12 4 ‐2
B. 23m n s
C. m
2
20
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D. undefined
27
.
1
1
1
( )4 = 3a + 4b 4 . Which of the following is his misconception?
3a 4b 4
3. Anthony wrote
A. It is possible to factor out the exponent. B. It is possible to distribute exponents over a sum. C. The exponents should be multiplied with the base. D. The coefficient inside the parentheses should be added. 4. Is
5r 4 3 3s
2
equal to
25r 6 9s 5
? Why or why not?
A. Yes, for the exponents inside the parenthesis and the numerical coefficients should be added to 2. B. No, for the exponents inside the parenthesis and the numerical coefficients should be subtracted from 2. C. Yes, for the exponents inside the parenthesis should be added and the numerical coefficients should be raised to 2. D. No, for the exponent inside the parenthesis should be multiplied by 2 and the numerical coefficients should be raised to 2.
x12
5. Explain why
4
is not equal to x ?
x3
A. Because the exponents should be added. B. Because the exponents should be multiplied. C. Because the numerator should be divided by the denominator. D. Because the exponent of the numerator should be subtracted by the exponent of the denominator. 5 3 2 4 2 6. Which of the following is the product of x y z and x y z ? 10 12 2 10 12 2 7 7 3 7 7 3 A. 2x 2y 2z B. x y z C. 2x 2y 2z D. x y z 7. Give the product of A. 5a
4
B. 5a
4
5a and a 3 2a 5 11a.
10a 5 55a 2 10a 6 55a 2
8. Marlon claims that
10a 6 55a 5 D. 5a 10 a 55a C. 5a
3
3
8 23 25 is equal to 2 . Is he correct? Why or why not?
A. Yes, for the exponents inside the parenthesis and the numerical coefficients should be added to 5. B. No, for the exponents inside the parenthesis should be multiplied by 2 and the numerical coefficients should be raised to 5. C. No, for the exponents should be multiplied. D. Yes, for the exponents should be added. The mentioned laws of exponents also hold when m and n are positive rational numbers. Examples: The following are true if there is no zero denominator. a)
12 12 12 12 1 3 3 3 3 3 .
b)
2 5 7 23 53 m m m 3 3 m 3 .
2
c)
5
2 1
53
5
1
1
5
3 3
3
.
p7
d)
3
53
5 3
p7
2
7
p7 .
p7 2
e)
e 3 2
e
2 2
3 3
e 0 1 .
f)
e 3 2
3 4
2
314 x 24 y 34 34 x8 y12 .
5y
2
art 5 art 5 g) 2 xy xy 5
3x y
.
h)
x 4 x
11 y 4
1
11 y 5 y
x
4
1
x
4
6y
1 3y
x
4
.
2
Negative Exponents ‐n
If a is a nonzero real number and n is any rational number, then a =
1 an
.
To simplify algebraic expressions with negative exponents is just to express the given expression into an equivalent quantity where the exponents become positive. Examples: Simplify the following such that they only have positive exponents.
a)
3
Solution: a)
3
1
2
1 2
1 1
3
b)
7
b)
7
.
3
1
4
c)
3 4
2
1 3
7
.
2m 4
c) 2m
1 4
1
1
.
2m 4
4
Examples: Simplify the following such that they only have positive exponents. Assume nonzero bases and no denominator is zero.
a)
16 2 4
4
b)
m m
4 3
c)
a a
2
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d)
t t
3 3
28
4
16 2 m 4 at 3
e)
4
3 2 3
16 2
4 =
m g
f)
4 m a t
1
3
g)
1 3
g h
5
3
2
h) x 2
2
Solution: a)
4
4
4
a
c)
a
2
2
= a
2
4
1 2
4 4 4
1.
4
a1 2 a 3
d)
16 2 m 4 at 3
e)
m 4 g 3
m 4
1 3 =
f)
1
g h
g)
1
5
3
2
=
b)
g3 g
1
( 5 )(
1 2
1 h
)
( 3)(
4 m a t
1
3
1 2
3 2 3 =
4
)
m
4
t 3 t 3
= m
2
m 43 m 1
1 m
.
= 1.
4 4 m 3 a 3t 3
4 3
16 m t
g 31h 3 m
4 3
4
4 m t
4
4 3
g 4 h3
a3
.
m
.
m4
5
3
2
m 3
4 4 m 3 aa 2t 3
g 31 h3 = 5
m 4
2
2
3
.
2 2
h) x 2
=
1
x 2
1
Note that
x 2
x 2
2
or
is equal to
1
x 2 1
x 2
or
1
x 2 x 2
1 x2 4x 4 but
Whenever the exponent of a base is in rational form
.
1
a
where b
b
1
is not equal to
x2 4 x 4
x2
1 4x
1
.
4
0, the expression can always be expressed in
radical form. RADICALS 1 n
If n is a positive integer and a is a real number for which a n is defined, then the expression
a is called a
1
radical, and
n
a = an .
The symbol
is a radical sign, the number a is the radicand and n is the index of the radical
Examples: a) In the expression
3
n
a.
5 , the number 3 is called the index and 5 is the radicand. 1
4
b) The expression 27 4 can be written as 27 . When a radical notation has no index, it is understood that n=2 or we are going to extract the square roots of the radicand. Examples: a)
2
49
1
49.
b)
32
2 3 3.
Simplified radicals An expression with radicals is simplified when all of the following conditions are satisfied. Exponents of the radicand and index of the radical have no common factor except 1. The radicand has no factor raised to a power greater than or equal to the index. All indicated operations have been performed (if possible). No denominator contains a radical. The radicand has no fractions. m
Radical notation of a n m
If a is a real number, m is an integer and
n
a is a real number, then a n =
a
m
n
=
n
am .
Examples: Write each exponential expression using radical notation. 3
1
a) a 3
b) m
c)
52
c)
52 =
3
1
Solution: a) a 3 =
3
4
3
a.
b) m
4
3
= (4
m ) 3 .
3
2
53 =
5 2
or
53 .
Examples: Write each radical expression using exponential notation and simplify. a)
22
b)
3
27
c)
3
x6
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1
1
22 = 22 .
Solution: a)
1
6
27 = (27) 3 = 33 3 3 .
3
2
b)
3
x6 = x 3 = x 2 .
c)
a = a = a and if n is an odd positive integer then n
n
If n is an even positive integer, then
n
n
a =a.
Examples: Simplify each of the following and give all the roots.
625
a)
b)
4
81
169 x 6
c)
d)
27 x 6 y 9
3
Solution:
252 25 .
a)
625
c)
169 x 6 = 132 x 3
4
81 =
4
34 = 3.
= 13x . 3 x y =3x y . 2
3
2 3
3
27 x 6 y 9 = 3
3
d)
b)
3 3
2 3
For all real numbers a and b, and positive integers m and n for which the indicated roots are real numbers, the following are true.
( a)( b) n
n
a
n
b
= n ab .
n
a
n
b
mn
a
mn a .
where b is not equal to zero.
Examples: The following are true.
3 5 15.
a)
4
b)
81x
24 81x 8 81x .
c)
12 x 6
x 3 12
.
Examples: Simplify each of the following. 3
a)
8
b)
125
Solution:
a)
3
8 125
b)
4
3
8
3
53
32 y 5 243
=
4
3
(2)
3
5
16 2 y 4 y 4
81 3
4
243
2 5
4
32 y 5
2
. 5
24 2 y 4 y 4
4
3
3
2 y 4
2y
4
3 3
=
= 2 y 54 y . 3 3
2y 4 2y
3
4
Operations on Radical Expressions Addition and Subtraction Radicals with the same radicand and the same index are called like radicals. Like radicals are added or subtracted by using the distributive property of real numbers. Moreover, only like radicals can be combined. Examples: a) Give the sum of
3
7 , 3 3 7 , 2 3 7 and 4 3 7 .
15 +3 15 ‐ 15 .
b) Evaluate 10
c) If the lengths of the sides of a triangle is Solution: a)
3
24 cm, 2 6 cm and 4cm, give its perimeter.
7 +3 3 7 +2 3 7 + 4 3 7 =10 3 7 .
b) 10 15 +3 15 ‐ 15 = 12 15 . c) Given a triangle, its perimeter is determined by adding the lengths of its sides. Hence,
4 6 cm + 2 6 cm + 4cm
24 cm + 2 6 + 4cm = =2
6 cm + 2 6 cm+ 4cm = 4 6 cm + 4cm.
Multiplication of Radicals Note that
n
ab =
n
a
n
b allows multiplication of radicals with the same index.
Examples: Give the product of the following in simplest form. Take only the positive roots. a)
6 8 9 10
Solution: a)
b)
23
m 2 (3 m
3
3
m2 )
c)
(3 7 + 8 ) (3 7 ‐ 8 )
5 54 4 5 216 5 . 6 8 9 10 = 6 9 8 10 54 80 54 16
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b)
2 3
m [3 m 2
33
2
m ]=2
3
2
m m
2 3 =
c)
(3 7 + 8 ) (3 7 ‐ 8 ) = 9 7 2
3
m m 2
2
23
3
m ‐(6) 3 m4
2m 6m 3 m .
82
= 9 7
8
=63‐8=55.
Exercises 1. In the expression A. base
34 5 , 5 is called the ________. B. index
C. radicand
D. root
2
2. Write
3 5 as a radical expression. A.
3. Write
5
2
35
B.
5
32
C.
3
3
D.
25 3
3
5 y as an exponential expression.
5 y
3 5
A.
B.
15 cm
B. 8
3m
55 y 3
D.
3 3 5
5 y
15 cm
C. 10
15 cm
D. 12
15 cm
C. 16
3m
D. 20
3m
3 m. Give its perimeter.
5. The side of a square measures 4 A. 4
C.
15 m, 5 15 m and 15 m. What is the perimeter of the triangle?
4. The sides of a triangle measure 3 A. 5
1 3 5
5 y
B. 8
3m
FUNCTIONS A relation is a set of ordered pairs (x, y) such that for every first element x, there corresponds at least one y. The set of all first elements is called the domain of the relation, whereas the set of second elements is the codomain of the relation. A function is a relation such that for every first element x of the ordered pair (x, y), there corresponds a unique
second element . The set of all first elements is called the domain of the function, whereas the set of second elements is the range of the yfunction. Tests for a Function There are some tests that can determine whether a relation is a function or not. We have a function if no two pairs in the set consisting of ordered pairs have the same first components. Examples: a) The relation {(Mr. Cruz, Mark), (Mr. Cruz, Mary), (Mr. Gonzales, Art), (Mrs. Tan, Alice), (Miss Peralta, Niko)} is not a function because more than one ordered pair have the same first component‐ Mr. Cruz. b) The relation {(‐1,0), (0, 1), (1, 2), (2, 3), (3, 4)} has no ordered pair that has the same first component. Thus, it is a function. Moreover, its domain is the set {‐1,0,1,2,3} and its range is {0,1,2,3,4}. c) Consider the relation {(‐3, ‐9), (‐2, ‐4), (‐1, ‐1), (0, 0),(1, ‐1), (2, ‐4), (3, ‐9)}. This relation is considered a function because it has no ordered pair that has the same first component. A relation may be described by a set of ordered pairs. A function is described by a set of ordered pairs with no two pairs having the same first components. Example: The table below shows the relation of the distance traveled by a car for a given length of time. d = rt Number of Hours 1 2 3 4 5 6 7 8 9 10 Distance
60
120
180
240
300
360
420
480
540
600
It can be observed that the distance traveled depends upon the number of hours or time. We say that distance is a function of time. The relation can be expressed as d = rt or d = 60t in this particular example. Example: The area of a square is a function of the length of its side. 2 A=s Side 1 2 3 4 Area 1 4 9 16
5 25
If a relation described by an equation or defined by a rule, a functional relationship exists if a change in the independent variable x causes a change in the dependent variable y. Another method of showing the relationship between the elements of two sets is by means of an arrow diagram.
Example: An arrow diagram for the relation y = 8x, where x is in set of whole numbers from 0 to 3, is shown on the right.
0
0
1
8
2
16
3
24
x St. Louis Review Center-Inc-Davao Tel. no. (082) 224-2515
31
2
0 2
Example: An arrow diagram for the relation {(x, y) │ y = x + 2} is shown on the right where x = ‐ 3, ‐ 2,‐1,0,1,2,3 and y = 2,3,6,11.
‐1
3
1
‐2
6
2
‐3
11
3 A relation described by an arrow diagram is a function if: 1 2
there exists a one‐to‐one correspondence between the elements of the two sets there exists a many‐to‐one correspondence between the elements of the two sets.
Another method of identifying a function from a mere relation is through its graph. A graph of a relation is a function if a vertical line is drawn through the graph will intersect the graph in no more than one point. The most fundamental way to graph a function is to plot points. Once the behavior of the graph of the function becomes familiar, graphing becomes easier. Note that in graphing functions, we include all possible real numbers in the domain. We may start by creating a table of values in order to find out the behavior of the function. It is very important to choose different numbers to get a clear picture of the graph. That is, it is helpful to generate as many points possible. Example: Graph f
1 x, y y 3x . 2
Solution: We note that f
b= X f(x)
1 x, y y 3x can be expressed in terms of slope and m = 3 and intercept 2
of points, we consider first a table of values. 0.5 . By plotting ‐2 ‐1 0123
‐6.5
‐3.5
2.5
‐0.5
5.5
8.5
y 8
6
4
2
x
0 - 14 -
12
- 10
-8
-6
-4
-2
0
2
4
6
8
10
12
14
-2
-4
-6
-8
Note that the when a vertical line is drawn through the graph, it will intersect the graph in no more than one point. Hence, the given relation is a function. Example: If we have f(x) = x, this function is called an identity function defined by f = {(x, y) │ y = x}. Let us graph the identity function by assigning to x the values 0, 1, 2. Hence, the corresponding values of y are 0, 1 and 2 respectively. Moreover, the line is determined by the following points. X f(x)
‐2 ‐2
‐1 ‐1
0123 0123 y 8
6
We now have the graph on the right.
4
2
x
0 - 14 -
12 -
10 -
8
-6
-4
-2
0
2
4
6
8
10
12
14
-2
-4
-6
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If f is defined by f = {(x, y) │ y = b} where the range of the function f consists of one number, then f is a constant function whose graph is a straight line parallel to the x‐axis. Example:
Graph f(x) = 4.
Solution: We now have first a table of values.
‐2 4
x f(x)
‐1 4
0123 4444
y
5
4
3
2
1
x
0 -7 -
6 -
5 -
4 -
3 -
2 -
1
0
1
2
3
4
5
6
7
-1
-2
-3
Non‐example: Graph
y
2
x.
Solution: We now have first a table of values. x y
0 0
1 1
1 ‐1
2
2
2
3
‐ 2
3
4
3
4 2
‐ 3
2
y 4
3
2
1
x
0 -3 -
2 -
1
0
1
2
3
4
5
6
7
8
9 1
0 1
1 1
2
-1
-2
-3
-4
Observe that the graph of the said set of points shows that if any vertical line drawn through the graph intersects the graph at more than one point. Hence, it is not a function. The graph of a relation shows a function if any vertical line drawn through the graph intersects the graph at no more than one point. Non‐examples: The graphs below do not define functions. y 2. 5
y
4
2 3
1. 5 2 1 1 0. 5 x
0 -8 -
7 -
6 -
5 -
4 -
3 -
2 -
1
0
1
2
3
4
5
6
7
-1
x
0 -4.5
-4
-3.5
-3
- 2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
-0. 5 -2 -1 -3 -1. 5 -4
PRACTICE EXERCISES Directions: Choose the best answer from the choices given. Write the corresponding letter of your choice. 3
1. In the expression 9m , 3 is called the ________. A. base B. coefficient
2. Evaluate
C. constant
D. exponent
17 73 7 7 ?
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B. 7
A. 7
12 32 m n
3. Which of the following is equal to
A. 0
2
C.
7
2
4
7
D.
77
2
? C. mn
B. 1
3
2
D. m n
3
3
4. Which of the following is equal to A.
4
23
B.
3
24 ?
24 3
5. Give the index of the expression A. 1 6. In the expression A. Index
24
C.
34m .
B. 2
C. 3
D. 34 m
888tm , 888tm is called the __________. B. Radicand
A. 0
C. Radical sign
5
7. Which of the following is equal to
5
D. Exponent
?
B. 1
8. The length of a rectangle is
5
C.
D. 2
5
3 3 m and its width is 2 2 m. Which of the following is the area of the rectangle?
2
A. 10 m
B.
5 6 m2
C.
9 6 m2
D.
9. Rosalinda was asked to get the perimeter of a rectangle whose width is 2 Which of the following should be her answer?
7 10 mm
A.
23
D.
10 mm
B. 14
C. 14
6 6 m2
5 mm and whose length is 5 5 mm.
5 mm
D. 10
5 mm
10. Is the sum of 2 , 3 , and 7 equal to 42 ? Why or why not? A. No, because the indices should be multiplied.
B. No, because the terms should not be combined for these are not like radicals. C. Yes, because the radicands should be added and the indices should be copied. D. Yes, because the terms have no coefficient and the radicands should be added. 11. Write as an exponential expression:
3x
2 3
A.
B.
7m
1 2 3
C.
7 m, 3 7 m and
2
3 3x 2
D.
3x 3
28 m. Give the perimeter of the triangle.
7m
B. 7
13. The side of a square measures 5
3x 2 .
3x
12. The sides of a triangle measure 2 A. 5
3
7m
C. 9
7m
D. 35
22 cm. Give its perimeter.
A. 10 22 cm B.15 22 cm C. 20 22 cm D. 20 11 cm 24k 2k 14. A secretary can type 3 words in 3 minutes. How many words can she type in a minute? A. 3
22k
B. 3
26k
C.
1 3
23 k
D.
1 3
23
8x7 y 6 15. Simplify
3
2 x y x 2
A.
.
27 2
3
2 x y 2
3
B. ‐
2 3
x
3
C.
8x3 y 3 x
16. Which of the following is the index of the expression A. 1
B. 2
17. In the expression A. Index
3
27 3
4
2 x y 2
D.
3
x
3
3 5
h o p e ? C. 3
D. 4
8m , 8m is called the __________. B. Radicand
C. Radical sign
D. Exponent
4 2 18. Which of the following is equal to 3 5 3 5 ? 6
A.
310
6
B. 6
6
10
C.
35
C.
35
8
D.
3 25
D.
35
4
19. Which of the following is equal to
35 4
?
35 4
A. 0
B. 1
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34
20. Evaluate
23 13 x y
3
. 5
A. 0
B. 1
C.
x2 y
D. x 3 y
21. Which of the following sets of ordered pairs describes a function? A. {(‐1, 4), (2, 8), (2, 20), (9, 36), (‐3, ‐ 12)} B. {(24, 6), ( 20, 5), (16, 4), (12, 3)} C. {(1, 2), (‐1, 3), (1, 4), (2, 3), (2, 4)} D. {(1, ‐ 1), (‐1, 1), (4, ‐ 2), (4, 2), (9, 3), (9, ‐ 3)} 22. Which of the following sets of ordered pairs DOES NOT describe a function? A. (5,(‐20), (9,(0, 36), B. {( {(‐‐1, 3, ‐4), 2),(2, (‐2,8), ‐ 1), 1, 0), 1)}(5, ‐ 12)} C. {(24, 6), ( 20, 5), (16, 4), (12, 3)} D. {(1, 1), (1, 2), (1, 4), (2, 3), (2, 4)} 23. Which of the following sets of ordered pairs describes a function? A. {(0.5, 4), (1, 8), (1.5, 20), (2, 36), (‐1, ‐ 12)} B. {(4, 6), ( 4, 5), (16, 4), (3, 3)} C. {(‐1, 2), (‐1, 3), (1, 4), (2, 3), (2, 4)} D. {(0.25, ‐ 1), (0.5, 1), (0.75, ‐ 2), (1, 2), (9, 3), (9, ‐ 3)} 24. Which of the following sets of ordered pairs DOES NOT describe a function? A. {(1.5, 1), (2, 1), (2.5, 1), (3, 1), (3.5, 1)} B. {(‐3, ‐ 2), (‐2, ‐ 1), (‐1, 0), (0, 1)} C. {(24, 6), ( 20, 5), (16, 4), (12, 3)} D. {(9, 1), (9, 2), (9, 4), (9, 3), (9, 4)} 25. Which of the following describes a function? A. x y
1 ‐2
1 ‐4
2 ‐6
4466 ‐8
1 ‐2
3 4
5 ‐6
7 8
1 2
2 4
3 6
1234 8
10
12
1 0.5
2 1
3 1.5
4512 2
2.5
3
‐10
‐12
‐14
B. x y
9
11
‐10
13
‐14
12
C. x y
14
D. x y
3.5
26. Which of the following describes a function? A. x y
‐1 2
‐2 4
‐3 6
‐4 8
10
‐5 12
‐6 14
‐1 2
1 4
‐2 6
2334 8
10
12
14
1 ‐2
‐2 ‐4
3 ‐6
‐1 8
10
1 ‐0.5
2 ‐1
3 ‐1.5
4512 ‐2
7
B. x y C. x y
‐2
‐3 12
4 14
D. x y
‐2.5
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‐3
‐3.5
35
27. Which of the following arrows or mapping diagrams DOES NOT specify a function? A.
1 2 3 4
B.
6 7 8 9
‐2 ‐1
9 4 0
0
C.
‐2
D.
4
2 ‐3 3 ‐4 4
9
3
2
6
3
9
16
4
28. Which of the following arrows or mapping diagrams specifies a function? A.
B.
2 4 6
2
9
C.
1 4 5
D.
4
1
1
9
4
C.16
7
8 4 2
29. Which of the following arrows or mapping diagrams DOES NOT specify a function? A. B.
6 7 8 9
6 7 8 9
C.
‐1
9
0 1
4 0
D.
‐2 2 ‐3 3 ‐4 4
3
‐4
2
9
‐9
3
12
‐
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36
30. Which of the following graphs is a function? A.
C. y
y
2 2. 5 1. 5 2 1 1. 5 0. 5 1 x
0 -3 .5
-3
-2 .5
-2
-1 .5
-1
-0 .5
0
0.5
1
1.5
2
2.5
3
3.5 0. 5
-0. 5 x
0 -1
-3 .5
-3
-2 .5
-2
-1 .5
-1
-0 .5
0
0.5
1
1.5
2
2.5
3
3.5
-0. 5 -1. 5
-1 -2
-1. 5
B.
D.
y
3
y
2. 5
2
2
1. 5 1 1
0. 5
-3
-2 .5
-2
-1 .5
-1
-0 .5
0
x
0 -3
-2
-1
0
1
2
3
x
0 -3 .5
0.5
1
1.5
2
2.5
3
3.5
-1
-0. 5
-1
-2
-1. 5 -3
31. Which is NOT a graph of a function? A. B.
-6 -5
100 80 60 40 20 0 -4 -3 -2 -1 -20 0 -40 -60 -80 -100 -120
C.
D.
30
4
25
5
25 20
0 -4
15 1234
30
2
20 -3
-2
-1
-2
0
1
2
3
4
15
10
6
10 -4
5 -6 -5
-4 -3 -2
0 -1 0 -5
5
-6 123456
-6 -5
-8
-10
0 -1 0 -5
-4 -3 -2
-10
1
2
3
2
2.5
4
5
6
-10
32. Which of the following is NOT a graph of a function? A.
C.
y
.
2
y
2
1. 5
1. 5
1
1 0. 5
x
0 -3 .5
0. 5
-3
-2 .5
-2
- 1.5
-1
-0 .5
0
0.5
1
1.5
3
3.5
-0. 5
-1
- 0. 5
0
-1
x
0 - 1. 5
0. 5
1
1. 5 -1. 5
-0. 5
-2
B.
D. 1
y 3
y
0. 5 2
x
0 - 1. 5
-1
- 0. 5
0
0. 5
1
1
1. 5
-0. 5
x
0 -3
-2
-1
0
1
2
3
-1 -1
-1. 5 -2
-2 -3
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33. Which of the following graphs represents a function? A.
3
C.
y
3
y
2 2
1 1
x
0 -3
-2
-1
0
1
2
x
0
3 -3
-2
-1
0
1
2
3
2
3
-1 -1
-2 -2
-3 -3
y
C. 3
D. y
5
2 4
1 3
x
0 -3
-2
-1
0
1
2
3
2
-1 1
-2 x
0 -3
-2
-1
0
1
-3
34. Which of the graphs below best represents the following scenario? Chok was at home when he decided to visit his friend. After a few hours, he was already at his friend’s house which is a few kilometers away from his home. He stayed there for a few hours and went back home.
A.
C.
Distance
Distance
Time
Time B.
D.
Distance
Distance
Time
Time
35. Which of the graphs below best represents the following scenario? Avel is running at a steady rate and then comes to a hill, which causes him to run at a slower rate. Once he reaches the top of the hill, he runs down the hill very fast. Upon reaching the bottom of the hill, he resumes his srcinal pace.
A.
B.
Speed
Speed
Time Time
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C.
D.
Speed
Speed
Time
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Time
39
Content Area: MATHEMATICS Focus: Probability and Statistics Prepared by: Daisy de Borja‐Marcelino
LET Competencies: Counting Techniques Experiment: any activity that can be done repeatedly (e.g. tossing a coin, rolling a die). Sample space: the set of all possible outcomes in an experiment. Example: In rolling a die, the sample space is S = {1, 2, 3, 4, 5, 6}. Sample point: an element of the sample space. Example: In rolling a die, there are 6 sample points. Counting Sample Points 1.
Fundamental Principle of Counting (FPC)
If a choice consists of k steps, of which the first can be performed in n1 ways, for each of these the second can be performed in n2 ways, for each of these the third can be performed in n3 ways…, and for each of these the kth can be made in nk ways, then the whole choice can be made in n1n2n3…nk ways. Example: In how many ways can two dice fall? Ans. : 6 ∙ 6 = 36 ways 2. Permutation Permutation is an arrangement of objects wherein the order is important. a. Linear Permutation If n objects are to be arranged r objects at a time, then the number of distinct arrangements is given by nPr
! ! , . ! = ! 10 9 8 720
=
Example: In how many ways can the first, second and third winners may be chosen with 10 contestants? 10P3 b. Circular Permutation If n objects are to be arranged in a circular manner, then the number of distinct arrangements is (n ‐ 1)!
Example: In how many ways can 6 people be arranged around a circular table? Answer: (6 ‐ 1)! = 5! c.
Permutation with Repetitions
The number of permutations of n things of which n1 are one of a kind, n2 second of a kind, …, nk of a kth kind is
n! n1!n2 !...nk ! Example: How many different permutations are there in the word WAGAYWAY if all letters are to be taken? Answer: 3.
8! 2 !3 ! 2 !
Combination
Combination is the arrangement of objects regardless of order. In other words, the order of arranging the objects is not important. If n objects are to be arranged r at a time, the number of distinct combinations is given by: nC r
=
! !! , .
Example: In how many ways can a committee of 3 be chosen from 7 persons? Answer:
7!
3! 4 ! Probability Probability: the likelihood of occurrence of an event. If E is any event, then the probability of an event denoted by P(E) has a value between 0 and 1, inclusive. In symbols, 0 ≤ P (E) ≤ 1 If P(E) = 1, then E is sure to happen. If P(E) = 0, then E is impossible to happen. Moreover, if the probability that E will not happen is P(E’), then P (E) + P (E’) = 1. Theoretical Probability
Theoretically, the probability of an event E, denoted by P(E), is defined as P(E) =
n( E ) n( S )
where n(E) = number of favorable outcomes St. Louis Review Center-Inc-Davao Tel. no. (082) 224-2515
40
n(S) = number of possible outcomes Exercises 1. A coin is flipped once. a. How many possible outcomes are there? b. What are these outcomes? c. How many of these outcomes consist of a face facing up? d. What is the probability that a tail faces up? 2. A spinner can land on any of the numbers 1 through 12 with equal likelihood. What is the probability that the spinner lands on a/an a. odd number? b. number greater than 12? c. prime number? e. number divisible by 3 and 4?
d. number divisible by 3 or 4? f. number between 3 and 9?
3. A box contains cards printed with the letters of the word “PHILIPPINES,” so that “P” is one 3 cards, “H” is on one card, and so on. A card is drawn from the box. a. How many possible outcomes are there? b. What are these outcomes? c. Are these outcomes equally likely? d. What is the probability that the card drawn is i. a “P” ii. a “H” ii. a “I” iv. not a “P” STATISTICS Statistics is the branch of mathematics used to summarize quantities of data and help investigators draw sound conclusions. Its two main branches are descriptive statistics and inferential statistics. A sample is a specified set of measurements or data, which is drawn from a much larger body of measurements or data called the population. Kinds of Sampling 1. Random sampling techniques are used to ensure that every member of the population has an equal chance of being included in the sample. A random sample is said to be representative of the entire population. The two methods of random sampling are lottery method and the use of the table of random numbers. 2. Systematic sampling is a technique which selects every nth element of the population for the sample, with the starting point determined at random from the first n elements. 3. Stratified random sampling is a technique of selecting simple random samples from mutually exclusive groupings or strata of the population. Graphical Representations of Data Graphs are used in mathematics to show relationships between sets of numbers. Graphs are useful in the field of statistics because they can show the relationships in a set of data. 1. Histogram ‐ a graphical picture of a frequency distribution consisting of a series of vertical columns or rectangles, each drawn with a base equal to the class interval and a height corresponding to the class frequency. The bars of a histogram are joined together, that is, there are no spaces between bars. 2. Bar Chart‐ uses rectangles or bars to represent discrete classes of data. The length of each bar corresponds to the frequency or percentage of the given class or category. The categories are in turn placed in either horizontal 3. Frequency Polygon‐ a special type of line graph, where each class frequency is plotted directly above the midpoint or class mark of its class interval and lines are then drawn to connect the points.
‐
4. Pie Chart an effective way of presenting categorized (qualitative) distributions, where a circle is divided into sectors ‐ pie‐shaped pieces ‐ which are proportional in size to the corresponding frequencies or percentages. 5. Pictogram‐ known as picture graph where picture symbols are used to represent values. MEASURES OF CENTRAL TENDENCY
A measure of central tendency is a single, central value that summarizes a set of numerical data. It describes a set of data by locating the middle region of the set. Measures Definition How to find Advantages Disadvantages of Central Tendency Mean The sum of the data Ungrouped data: A single, unique Not appropriate for divided by the number of value that is skewed distribution data representative of all as it is affected by = the scores extreme scores or outliers Grouped Data: Stable from group to group =
Median
The middle number of the set when the data are
̅
∑
̅
∑
Ungrouped data:
Mayfurther be used in computations More stable from group to group than
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Not necessarily representative of all 41
arranged in numerical order
The middle for the
Grouped Data:
Mdn = L + Mode
The number that occurs most frequently in the data
the mode
scores
Appropriate for skewed distribution
Unstable from group to group
Cannot be used in further analyses
Ungrouped data: The most frequent score
Easy to obtain
Not necessarily representative of all scores
Grouped Data: The class mark of the Cannot be used in class interval with the further analyses highest frequency Example: The scores for five students on a quiz are 40, 20, 30, 25, and 15. To find the mean score for this group of students, we first find the sum of the scores: 40+20+30+25+15=130 We then divide the sum by 5, the number of scores:
130 5
26 .
Example: Determine the median for the set of scores 82, 81, 80, 87, 20 Solution: We arrange the scores in Example 1 in order fro lowest to highest: 20, 80, 81, 82, 87 The middle number in this arrangement is 81 and it is the median. Exercises 1.
The mean score on a set of 10 scores is 71. What is the sum of the 10 test scores?
2.
The mean score on a set of 13 score is 77. What is the sum of the 13 test scores?
3.
The mean score on four of a set of five scores is 75. The fifth score is 90. What is the sum of the five scores? What is the mean of the five scores?
4.
Two sets of data are given. The first set of data has 10 scores with a mean of 70, and the second set of data has 20 scores with a mean of 80. what is the mean for both sets of data combined?
MEASURES OF VARIABILITY
A measure of variation or variability describes how large the differences between the individual scores. The common measures of variability are range and standard deviation. Measures of Variation
Definition
How to find
Advantages
Disadvantages
Range
The difference between the highest score and the lowest score
Ungrouped data: R = HS ‐ LS
Easy to compute
Unstable
Gives a unique value
Not representative of the set of data
Grouped Data: Easy to understand
Standard deviation
The square root of the variance of the set of data
R = Upper Limit of the Highest Class Interval ‐ Lower Limit of the Lowest Class Interval Ungrouped data:
Not used in further computations
Most stable
∑ ̅
Gives a unique value
Grouped Data:
Most representative
∑ ̅ S=
Used in further computations
S=
Affected by extreme scores More
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difficult
compute understand
to and
42
Exercises 1. On a quiz, the following scores were made in a class of 10 students: 72, 83, 86, 97, 90, 70, 65, 71, 80, 86. For this set of scores, give the a. mean b. median c. mode d. range e. midrange f. standard deviation 2. Jao, Dhei, Nelfe, Dada and Ched are all in the same statistics class. Their scores for the first two exams in the class are listed in the accompanying table. The first exam had a mean of 84 and a standard deviation of 6, whereas the second exam had a mean of 78 and a standard deviation of 4. __________________________________ Exam 1 Exam 2 __________________________________ Jao 84 78 Dhei 90 74 Nelfe 66 78 Dada 78 70 Ched 84 78 __________________________________ a. b. c. d. e.
Who improved on the second exam? Who improved the most on the second exam? Who did not improve on the second exam? Considering both exams, who did the poorest? Who performed the same on both exams?
PRACTICE EXERCISES 1. In a political science survey, voters are classified into six income categories and five education categories. In how many
different ways can a voter be classified? A. 11 B. 25 C. 30 D. 45 2. The number of permutations of the word “probabilities” is A. 13 B. 2!3! C. 13! D. 13!/(2!3!) 3. The number of ways seven visitors can be seated on a round table is A. 6! B. 7! C. 6!7! D. 6!/2 4. In a certain town, 40% of the people have brown hair, 25% have brown eyes, and 15% have both brown hair and brown eyes. A person is selected at random from the town. If he has brown eyes, what is the probability that he does not have brown hair? A. 3/8 B. 2/5 C. ½ D. none of these 5. A die is tossed. If the number is odd, what is the probability that it is prime? A.
2
B.
3
3
C.
4
2
D.
5
1 3
6. An accounting professor can assign grades of A, B, C, D, or F to student’s examinations. In how many ways can the professor assign grades to three different student examinations? A. 120 ways B.125 ways C. 25 ways D. 50 ways 7. A bag contains 15 red beads, 30 white beads, 20 blue beads, and 7 black beads. If one of the beads as drawn at random, what is the probability that it will be white or blue? A.
25 36
B.
15 42
C.
13
D.
40
17 52
8. A whole number is chosen at random from the whole numbers from 1 to 50. What is the probability that it is an even square? A. 0.07 B. 0.06 C. 0.02 D. 0.09 9. A pair of dice is tossed. If the numbers appearing are different, find the probability that the sum is even.
A.
1 2
B.
2 3
C.
2 5
D. none of these
10. If repetitions are not permitted, how many 3 digits number less than 400 can be formed from the digits 2, 3, 5, 6, 7 and 9? A.30 B.126 C.40 D. 162 11. In how many different ways can a true‐false test be answered if there are five items, assuming that the student give an answer for each item. 5
2
A.5! B.2! C. 5 D. 2 12. There are five musical numbers in a program. The number of ways these numbers can be presented is 5
A. 5 B.5! 13. The numbers of permutations of the word “probabilities” is
A.13
B. 13!
D. 5! / 3!2!
C. 2
C.2! 3!
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D. 13! / 2!3!
43
14. The number of ways seven students can be seated on a round table is A. 6! B.6! 7! C.7! D. 6! /2 15. How many different signals, each consisting of 6 flags hung in a vertical line, can be formed from 4 identical red flags and 2 identical blue flags? B. 6 4 !
A. 6!
C.4! 2!
D. 6! 4!2!
C.1
D. 1 52
16. The probability of getting a black heart is B. 1 4
A.0
17. In a certain town, 40% of the people have brown hair, 25% have brown eyes, and 15% have both brown hair and brown eyes. A person is selected at random from the town. I he has brown eyes, what is the probability that he does not have brown hair? A. 3 8
B. 1 2
C. 2 5
D. None of these
For items 18‐20: The scores of 10 students in a Math quiz are as follows: Student Score
A 18
B 32
C 10
DE F 15
27
GH 18
I 23
28
29
J 30
18. What is the median score? A. 18 B. 20 C. 23 D. 25 19. What is the mode of the scores? A. 18 B. 20 C. 23 D. 25 20. What is the range of scores? A. 18 B. 22 C. 23 D. 25 21. Jojo aims to get an average of 90 in 5 unit tests. The results of the first four unit tests are as follows: 89, 92, 86, and 91. What should his grade in the fifth unit be for his average to be at least 90? A. 91 B. 92 C. 93 D. 94
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LICENSURE EXAMINATION FOR TEACHERS (LET)
Refresher Course Content Area: MATHEMATICS Focus: Calculus Prepared by: Daisy de Borja‐Marcelino LET Competencies:
Solve for the roots of a given quadratic equation Solve problems on quadratic equations
Determine an equation given a set of roots which are imaginary/complex numbers Perform operations involving exponential and logarithmic functions Solve for the solution set of a given inequality n Determine the rth term of the expansion (a + b) Solve problems involving variations Determine the number of positive and negative roots of a given polynomial
Equations An equation that contains at least one variable is called an open sentence. Equations b & c above are examples of open sentences. In equation b, only ‐1 makes the sentence true or satisfies the equation. However, more than one number might satisfy an equation. For example, +2 and ‐ 2 satisfy the equation x 4 0 . Any number that satisfies an equation is called a solution or root to the equation. The set of numbers from which you can select replacements for the variable is called the replacement set. The set of all solutions to an equation is called the solution set to the equation. To solve an equation means to find all of its solutions. 2
QUADRATIC EQUATION 2 An equation of the form ax + bx + c = 0 where a 0, a ,b, and c are constants, is a quadratic equation. ROOTS OF QUADRATIC EQUATIONS To solve a quadratic equation means to find the value of x (unknown) that will satisfy the given equation. The values of x that will make the equation true are called the roots or solution of the quadratic equation. Methods of Finding the Roots of a Quadratic Equation 1. Factoring (Use this method if the quadratic equation is factorable) 2
Example: Determine the roots of x ‐ 8x ‐ 15 = 0 Solution: Factoring the left side of the equation, (x ‐ 5) (x – 3) = 0 Equating each factor to zero, (x ‐ 5) = 0 (x – 3) = 0 x=5 x=3 2. Quadratic Formula The quadratic formula is
x
b b 2 4ac 2a
BINOMIAL FORMULA n
To obtain the terms of the binomial expansion (a + b) , we use the binomial formula:
n
(a + b) = a
n
na n 1 b
n ( n 1)a n 2 b 2 2!
n ( n 1)( n 2)a n 3 b 3 3!
... nab n 1 b n
n
THE rth TERM OF THE EXPANSION (a + b)
rth term =
n ( n 1)(n 2)...(n r 2)a n r 1 b r 1 ( r 1)!
EXPONENTIAL FUNCTIONS
The exponent i al f unct i on f wi t h base b i s denot ed by f(x) = bx, wher e 1, and x i s any r eal number . b >0 , b St. Louis Review Center-Inc-Davao Tel. no. (082) 224-2515
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x
Properties of f(x) = b
f has the set of real numbers as its domain.
f has the set of positive real numbers as its range.
f has a graph with a y‐intercept of (0,1).
f is a one‐to‐one function.
f has a graph asymptotic to the x‐axis.
f is an increasing function if b>1 and f is a decreasing function if 0
An exponential function has a constant base and a variable exponent. The Natural Exponential Function
For all real numbers x, the function defined by
f ( x) e x is called the natural exponential function. Note that e
is an irrational number and its accurate value to eight places is 2.71828183. EXPONENTIAL EQUATIONS An equation where the unknown quantity appears in an exponent is called an exponential equation.
SOLVING EXPONENTIAL EQUATIONS To solve an exponential equation is to find the value of the unknown quantity in the given equation. LOGARITHMIC FUNCTIONS
For x > 0, onl y i f ay = x.
1, we have y = log a (x) i f and
b > 0, and b
Note: 1. If the base of the logarithm is not indicated it is understood that the base is 10. 2. If the base of the logarithm is the number e, then it is called a natural logarithm and it is written as f(x) = ln x. Properties of
f x logb x
f has the set of positive real numbers as its domain. f has the set of real numbers as its range. f has a graph with a x‐intercept of (1,0). f is a one‐to‐one function. f has a graph asymptotic to the y‐axis. f is an increasing function if b>1 and f is a decreasing function if 0
Remarks
Logarithmic functions are the inverse of exponential functions. We can use the rules of exponents with logarithms. The two most common logarithms are called common logarithms and natural logarithms. Common logarithms have a base of 10, and natural logarithms have a base of e. Equation in exponential form can be rewritten in logarithmic form, and vice versa.
The exponent i al f or m of
is
The l ogar i t hmi c f or m of
is
.
.
2
Example1: The exponential equation 7 = 49 may be written in terms of a logarithmic equation as log7 (49) = 2.
‐3
Example 2: The exponential equation 9
1 = ‐3 729
=
1 93
or
1
may be written in terms of a logarithmic equation as log9
729
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Basic Properties of Logarithms 0
Property 1 : loga (1) = 0 because a = 1. 0
Example 1: In the equation 22 = 1, the base is 22 and the exponent is 0. Remember that a logarithm is an exponent, and the corresponding logarithmic equation is log22 (1) = 0, where the 0 is the exponent.
Example 2: In the equation
2 5
0
= 1, the base is
2 5
and the exponent is 0. Remember that a logarithm is an exponent,
and the corresponding logarithmic equation is log 2 1 = 0. 5 1
Property 2: loga (a) = 1 because a = a 1
Example 3: In the equation 7 = 7, the base is 7, the exponent is 1, and the answer is 7. Since a logarithm is an exponent, and the corresponding logarithmic equation is log7 7 = 1 1
Example 4: Use the exponential equation m = m to write a logarithmic equation. If the base m is greater than 0, then logm (m) = 1. x
x
x
Property 3: loga (a) = x because a = a 2
2
2
Example 5: Since 9 =9 , we may write the logarithmic equation with base 9 as log9 9 = 2. 2
2
2
Example 6: Since you know that 11 =11 , we may write the logarithmic equation with base 11 as log1111 = 2. INEQUALITIES Any relation expressed using the symbols <, >, > or < is called an inequality. An absolute inequality is an inequality which is always true. A conditional inequality is one which is true only for certain values of the variable involved. 1. 4 > 3 is an absolute inequality 2. x > 3 is a conditional inequality PROPERTIES OF INEQUALITIES Let a, b, c, & d be real numbers. The following hold. 1. Trichotomy Property a > b or a < b or a = b 2. a > b if a ‐ b > 0 a < b if a – b < 0 3. a. If a> 0 and b> 0, then a + b> 0 and ab>0. b. If a < 0 and b < 0, then a+b< 0 and ab> 0 4. Transitivity If a < b and b < c then a < c. 5. Addition Property If a < b and c < d, then a + c < b+ d 6. Multiplication Property If a < b and c > 0, then ac < bc If a < b and c < 0, then ac > bc SOLVING INEQUALITIES To solve an inequality means to find the value of the unknown that will make the inequality true. POLYNOMIAL FUNCTION The function defined by the equation n
n‐1
f(x) = a0x +a1x
n‐ 2
+ a2x
2
+ . . .+ an‐2x + an‐1x + an
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where n is a nonnegative integer and a0, a1,
. . ., an
are constants, a0,
0 is a polynomial function in x of degree n.
The zeros or roots of f(x) are the numbers that will make f(x) = 0.
The Number of Positive and Negative Roots of a Polynomial Function If f(x) is a polynomial function with real coefficients, then the following are true. The number of positive real zeros of f(x) is either equal to the number of variations in sign in f(x), or to that number
diminished by a positive even integer.
The number of negative real zeros of f(x) is either equal to the number of variations in sign in f(‐x), or to that number
diminished by a positive even integer.
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LICENSURE EXAMINATION FOR TEACHERS (LET)
Refresher Course Content Area: MATHEMATICS Focus: ARITHMETIC, NUMBER THEORY AND BUSINESS MATH Prepared by: Daisy de Borja‐Marcelino
Competencies: 1. Simplifying expressions involving series of operations 2. Solve problems involving a. GFC and LCMF b. prime and composite c. divisibility d. inverse and partitive proportions e. compound interest f. congruence g. linear Diophantine Equation 3. Apply Euler’s function and theorems, or Fermat’s theorem in solving problem.
THE NUMBER SYSTEM
Example: Some examples of imaginary numbers are:
25 , 49 ,
negative numbers , 3i, ‐ 7i.
Example: Simplify: 2 (3 + 2i) – 5 (4 – 6i) Solution: 2 (3 + 2i) – 5 (4 – 6i) = ( 6 + 4i) – (20 – 30i) = 6 + 4i – 20 + 30i = ‐ 16 + 34i. Rational numbers are numbers which can be expressed as quotient of two integers, or can be expressed as fractions in simplest forms. Examples are 8, ‐ 3, 3.45, and
2 3
.
Irrational numbers are numbers which cannot be expressed as fractions in simplest forms. Examples are and
3 3
3 , 4 7 , , e
.
Set of Natural/Counting numbers: {1, 2, 3, 4, … }. This set contains the numbers that we use in counting; also called natural numbers. Set of Whole Numbers: { 0 , 1, 2, 3, …}. This set is the union of the number zero and the set of counting numbers. Set of Integers: { … , ‐ 3, ‐2, ‐ 1, 0, 1, 2, 3, …}. This set is the union of the set of counting numbers, their negatives, and zero
II. THE COUNTING NUMBERS A. Divisibility. An integer is divisible by a certain divisor (also an integer) if it can be divided exactly by that divisor. That is, the remainder is zero after the division process is completed. To illustrate, the integer 12 is divisible by 1, 2, 3, 4, 6, and 12. St. Louis Review Center-Inc-Davao Tel. no. (082) 224-2515
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To determine whether the integer is divisible by a certain integer or not, you may use the following divisibility rules. An integer is divisible by a) 2 if it ends with 0, 2, 4, 6, or 8. (Examples: 134 or 12 or 12,330 or 4) b) 3 if the sum of the digits is divisible by 3. (Examples: 132 or 18 or 12,330 or 45) c) 4 if the last two digits form a number which is divisible by 4. (Examples: 13,412 or 12,332) d) 5 if it ends with 0 or 5. (Examples: 135 or 10 or 12,330 or 495) e) 6 if it ends with 0, 2, 4, 6, 8 and the sum of the digits is divisible by 3.(Examples: 134 or 12) f) 7 if the difference obtained after subtracting twice the last digit from the number formed by the remaining digits is divisible by 7. (Examples: 14 or 364) g) 8 if the last three digits form a number which is divisible by 8. (Examples: 24160 or 5328) h) 9 if the sum of the digits is divisible by 9. (Examples: 9, 432 or 18,504 or 270) i) j) k) l)
10 if it ends with 0. (Examples: 120 or 7, 890 or 1, 230) 11 if the difference between the sum of the digits on the even powers of 10 and the sum of the digits on the odd powers of 10 is divisible by 11. (Examples: 2123 or 2816 or 94369 or 36465) 12 if it is both divisible by 3 and 4. (Examples: 413,412 or 112,332) 15 if it is both divisible by 3 and 5. (Examples: 150 or 350)
Remarks: Divisibility rules for two or more relatively prime numbers (GCF is 1) may be combined to serve as a divisibility rule for their product. Example: The rules for 3, 4, and 5 may be combined to serve as the rule for their product which is 60 since 3, 4, and 5 are relatively prime. Exercises: Put a check mark on the space provided for, if the integer on the first column divides exactly the integer on the top row.
456
36,720
800,112
456
2 3
10 11
4 5 6 7 8 9
12 14 24 32 45 77
36,720
800,112
Even numbers are whole numbers which can be divided exactly by two whole numbers. Odd numbers are whole numbers which cannot be divided exactly by two whole numbers. 3
Example: If n is odd, which of the following is true? 2 I. n is odd II. n is odd A) II only B) I and II only
2
III. n + 1 is odd
C) I only D) I and III only
Example: If x is an odd integer and y is an even integer, which of the following is an odd integer? 2
A. 2x‐y 2 B. x +
y-
1
D.
x-
C. 1
3
x + y
B. Factors and Multiples. In the number sentence 2 x 3 = 6, the numbers 2 and 3 are called factors, while 6 is their product. Or we say, 2 and 3 are divisors of 6. Moreover, we say that 6 is a multiple of 2 and 3. 2
Example: How many factors does 4 have? A) 2 B) 4 2 Answer: (C). The factors of 4 = 16 are {1, 2, 4, 8, 16}. Example: What are the multiples of 6? Answer: The multiples of 12 are {12, 24, 36, 48, …}
a. b. c. d. e.
c) 5
D) 16
Exercises Fill in the blanks with either 7 or 42. ______ is a factor of ______. ______ is divisible by ______. ______ is a divisor of ______. ______ is a multiple of ______. ______ divides _______.
C. Prime and Composite Numbers
Prime numbers are counting numbers that have exactly two factors in the set of counting numbers: 1 and itself. Composite numbers are counting numbers that have more than two factors in the set of counting numbers. St. Louis Review Center-Inc-Davao Tel. no. (082) 224-2515
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The numbers 0 and 1 are special numbers. They are neither prime nor composite. Example: What is the sum of prime numbers less than 15? A) 4 B) 5 C) 6 D) 14 Answer: The number 2,3,5,7, 11 and 13 are prime number less than 15. Hence, the answer is C.
D. Prime Factorization. This is a process of expressing a number as product of prime factors.
Example: Solution:
Express 24 as product of prime factors. 3 3 24= 2 x 2 x 2 x 3 = 2 x 3 or 3 x 2 .
Fundamental Theorem of Arithmetic Every composite whole numbers can be expressed as the product of primes in exactly one way (the order of the factors is disregarded). E. The Greatest Common Factor (GCF)
The GCF of two or more numbers is the largest possible divisor of the given numbers. Example: Determine the GCF of 12 and 42. Solution: 24 = 2 x 2 x 3 42 = 2 x 3 x 7 GCF: 2x3=6 Example: What is the greatest integer that can divide the numbers 18, 24 and 36? Solution: 18 = 3 x 3 x 2 24 = 3 x 2 x 2 x 2 36 = 3 x 3 x 2 x 2 GCF: 3 x 2 = 6 F. Least Common Multiple (LCM). The LCM of two or more numbers is the smallest possible number that can be divided by the given numbers. Example: Give the LCM of 20 and 30. 2 Solution: 20 = 2 x 2 x 5 = 2 x 5 30 = 2 x 3 x 5 2 LCM: 2 x 3 x 5 = 60.
Example: What is the smallest integer that can be divided by the numbers 24, 36 and 54? 3 Solution: 24 = 2 x 2 x 2 x 3 = 2 x 3 2 2 36 = 2 x 2 x 3 x 3 = 2 x 3 3 54 = 2 x 3 x 3 x 3 = 2 x 3 3 3 LCM: 2 x 3 = 216 G. Relatively Prime. Two numbers are relatively prime if their GCF is 1. The numbers themselves may not be prime. The numbers 12 and 49 are relatively prime. Example:
Which of the following pairs are relatively prime to each other? A)15 and 36 B) 23 and 51 C) 231 and 27 D) 121 and 330
III. INTEGERS
Consecutive integers are two or more integers, written in sequence, in which each integer after the first is 1 more than the preceding integer. Examples:
1,2,3,4,5, 6 3, 4, 5, 6, 7, 8 –
–
–
–
4, 3, 2, 1, 0, 1, 2, 3
x, x+1, x+2, x+3, x+4, x+5
The absolute value of a number x, denoted by x , is the undirected distance between x and 0 on the number line.
–
5
–
4
–
3
–
2
–
1
0
1
2
3
4
5
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It is also defined as
x if x ≥ 0
x =
– x if x < 0
Examples: Evaluate each of the following. a)
│2│ = 2
c) │0│ = 0
b)
│– 7│ = 7
d) – │ –15│ = –15
A. Multiplication. The product of two integers with like signs is a positive while the product of two integers with unlike signs is negative.
Example: Example:
(‐4) x 7 = (‐28) (‐8) x (‐5) = 40
or (‐4) (7) = (‐28) or (‐4) 7 = (‐28) or (‐8) (‐5) = 40 or (‐8) (‐5) = 40
B. Division. The quotient of two integers with like signs is a positive while the quotient of two integers with unlike signs is negative.
Example:
(‐72) (‐8) = 9
Example:
(‐123) 3 = ‐ 41
C. Addition. The sum of two integers with like signs is the sum of their absolute values with the common sign prefixed before it.
The sum of two integers with unlike signs is the difference of their absolute values with the sign of the integer with the larger absolute value prefixed before the difference. Example: (‐3) + (‐23) = (‐26) Example: (‐34) + 12 = (‐22) D. Subtraction. Express subtraction statements as addition statements and follow the procedure in addition. (That is, change the sign of the subtrahend to its opposite, and proceed to addition.) Example: (‐12) – (‐3) = (‐12) + 3 = ‐ 9
Exercise: What number should a) be added to (‐12) to yield 26? b) be subtracted from (‐2) to yield 5? c) be multiplied by (‐4) to yield (‐36)? d) be divided by (‐2) to yield 30? E. P‐E‐MDAS. P‐E‐MDAS stands for “Parenthesis‐Exponent‐Multiplication Division Addition Subtraction. When two or more operations are involved in a single expression, operations are performed in the order of P‐E‐ MDAS. That is, we perform first the operation inside the parenthesis (or any grouping symbol), then followed by determining the power of the number which is raised to a given exponent, then followed by multiplication/division, and lastly by the addition/subtraction. Should there be multiplication and division only, perform the operation from left to right. Should there be addition and subtraction only, perform the operation from left to right. Example:
Simplify 20 + 100 ( 5 – 63 3 + 12)
Solution:
20 + 100 ( 5 – 63 3 + 12) = 20 + 100 ( 5 – 63 9 + 12) = 20 + 100 ( 5 – 7+ 12) = 20 + 100 ( (–2) + 12) = 20 + 100 10 = 20 + 10 = 30.
2
2
1. Two bells ring at 5 P.M. For the rest of the day, one bell rings every half hour whereas the other rings every 45 minutes. When is the first time, on that same day, that both bells ring at the same time again? a. 6:30 P.M. b. 8:30 P.M. c. 8:45 P.M. 2. Which is true? a. The set of prime factors of 6 is {1,2,3}
d. 9:00 P.M.
c. All prime numbers are odd numbers.
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b. The product of irrational and rational is irrational.
d. 3.14 is a rational number.
3. Which of the two‐digit numbers below when inserted in the blank will make 38__09 divisible by 3? a. 98 b. 84 c. 34 d. 60 4. Which of the following number is divisible by 45? a. 300,000,000,450 b.600,000,000,045
c. 100,200,600,090
d. 400,450,000,000
th
th
5. On its anniversary, a certain store offers a free sandwich for every 4 customer and a free softdrink for every 6 customer. After 75 customers, how many had received both free sandwich and softdrink? a. 30 b. 18 c. 12 d. 6
IV. FRACTIONS Kinds of Fractions
As to relation between the numerator and the denominator a. Proper – the numerator is less than the denominator b. Improper – the numerator is equal to or greater than the denominator As to relation of the denominators of two or more fractions
1
3
,
a.
Similar – the denominators are equal. Examples:
b.
Dissimilar – the denominators are not equal. Examples:
5
5
4
,
5
3 4
,
5 8
,
5 6
Other classes a.
Equivalent – fractions having the same value. Examples:
3 4
b.
6
,
8
,
12
.
14
Mixed – composed of a whole number and a proper fraction . Examples:
3 5 2 , 5 . 4 8
Rules involving Zero a. Zero numerator and non‐zero denominator – the value is zero b. Zero denominator – no value, undefined c. Zero value – the numerator is zero A. Multiplication of Fractions. Multiply numerator by numerator and denominator by denominator to get the numerator and denominator respectively of the product
Example:
12 25
5
60
6
150
2
or
5
.
B. Division of Fractions. Multiply the supposed dividend by the reciprocal of the supposed divisor.
Example:
36 25
6 10
36
25
Exercises: Evaluate the following.
a)
10 6
360 12 =
150
or
5
2
2 5
.
2 1 32 4 5 c)
2
b)
1 3
1
7 18
2 19 5 3 21 20 5 1 d) 3 1 15 6
D. Changing Dissimilar Fractions to Similar Fractions. Determine the LCM of the denominators. Then with the said LCM as the denominator, express each fraction to its equivalent.
Example:
Solution:
1
Express
1 3
4 4
3 =
,
4 12
3 4 ,
,
5 6
to similar fractions.
3 3
4 3
=
9 12
and
5 2
=
10
6 2 12 4 9 10 Therefore, the similar fractions are , and . 12 12 12
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E. Addition of fractions. Convert the fractions to similar fractions. Then add the numerators to obtain the numerator of the sum and copy the denominator.
1
3
Evaluate
Solution:
The LCM is 12, so convert the addends to similar fractions with 24 as the denominator.
1 3
3
+
4
+
+
3
5 6
4
=
+
5
Example:
4 12
.
6
9 12
10
+
12
23
or
12
1
11 12
.
F. Subtraction of Fractions. Convert the fractions to similar fractions. Then subtract the numerators to obtain the numerator of the difference and copy the denominator.
7
Example: What number should be subtracted from
1
to obtain
12
2
?
Solution: Let the desired number be x. Then, the equation is given by
7 12
‐x=
1 2
x=
7 12
‐
1 2
=
7 12
6
1
=
12
12
.
G. Fraction as Part of a Whole
Example: What is
3 4
of 28?
Solution: Let the desired number be m. Then, the equation is given by M =
3 4
28
3
28
4
1
84
21 .
4
Example: What part of 24 is 4? Solution: Let the desired number be q. Then, the equation is given by q 24 = 4
q=
4 24
or
1 6
.
H. Simplifying Fractions A fraction is in simplest form if the numerator and the denominator are relatively prime (their GCF is 1). Thus, to simplify fractions, multiply by the fraction whose numerator and denominator are the reciprocal of the GCF of the numerator and the denominator of the given fraction.
Example: The simplest form of
12 18
is
2 3
because
12 18
26 3 6
2 3
.
I. Ordering Fractions
Two fractions are equivalent if their cross products are equal. Otherwise, that fraction the numerator of which was used to get the greater of the two cross products is the larger fraction. Exercises 1. A 100‐m wire is cut into two parts so that one part is ¼ of the other. How long is the shorter piece of wire? a. 120m b. 80m c. 25m d. 20m 2. Luis left ½ pan of a cake on the table. Dada ate ¾ of it. What fraction of cake was left? a. 1/8 b. 3/8 c. ¼ d. ½ 3. If
and
are equivalent fractions, what is the value of n?
a. 13 4. Mr. dela Cruz owned
b. 14 of a business. He sold
c. 20
d. 21
of his share in the business at a cost of P1M. What is the total cost
of the business? a. P 6M b. P7M c. P 8M 5. Arrange the fractions 5/8, 4/5, 3/4 in increasing order. a. 5/8, 4/5, 3/4 c. 3/4, 4/5, 5/8 b. 4/5, 3/4, 5/8 d. 4/5, 5/8, 3/4 St. Louis Review Center-Inc-Davao Tel. no. (082) 224-2515
d. P 9M
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6. Which of these fractions has the largest value? a. 3/5 b. 11/16 c. 7/10 d. 5/8 7. Mark spent his monthly salary as follows: 3/5 for food and allowances, 1/3 for his child’s education and house rental. If his monthly salary is P15, 000, how much would he left at the end of the month? a. P 1,000 b. P2,000 c. P5,000 d. P 14,000 8. Chedy and Dada run for President for their organization. Chedy got 1/3 of the votes. If Dada got 300 votes, how many students voted for Chedy? a. 900 b. 200 c. 150 d. 100
V. DECIMAL NUMBERS
A. The Decimal Numbers and the place value chart
The place value chart d n a s u o h T d re d n u H
d n a s u o h T n e T
100 000
10 000
s d n a s u o h T
s d e r d n u H
s n e T
s h t n e T
s e n O
1 000 100 10 1
s h t d e r d n u H
0.1
0.01
s h t d n a s u o h T
s th d n a s u o h T n e T
0.001 0.0001
s d n a s u o h T d e r d n u H
0.00001
The number 0.8 is read as “eight tenths” and .214 as “two hundred fourteen thousandths”. The number 0.8 is equal to .800. The number 0.8 is greater than 0.214. Exercise:a) Arrange the following decimal numbers in ascending order: 0.5, 0.343, 0.142, 0.5254 b) In 2.3456, what digit is in the thousandths place? B. Addition and Subtraction of Decimal Numbers. Addition of decimals is done by writing them in a column so that their decimal points are aligned. Thus aligned, digits with the same place values would be in the same column, and the addends (or the minuend and the subtrahend) are added (or subtracted) as if they were whole numbers. C. Multiplication of Decimal Numbers. To multiply decimals, multiply the numbers as if they were whole and so place the decimal point in the result as to have as many decimal places in it as there are in the factors combined. D. Division of Decimal Numbers.
To divide a i. decimal by a whole number, do as in dividing whole numbers but writing the decimal point directly above that of the dividend. ii. number by a decimal, multiply both dividend and divisor by that power of ten such that the divisor becomes the least whole number, and then proceed as in (i) above.
VI. CONVERSION A. Fraction to Decimal. Divide the numerator by the denominator.
Exercises: Convert the following to decimal: a) 3/5 = ______ b) 5/6 = ______
c)
7/8 = ______
B. Decimal to fraction
a) Terminating – multiply the number by a fraction (equal to one) whose numerator and denominator is a multiple of 10 such that the numerator of the product is a whole number.
Example:
Convert 0.15 to fraction.
Solution:
0.15
100 100
=
15 100
3 20
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b) Repeating decimal number Example:
Convert 0 .5 to fraction
Solution:
Let n = 0 .5 =
0.5555555.
5.5555555.
10 n =
‐ n = 0.5555555. ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ 9n= 5
5
n= Hence, 0 .5 is equal to
9
5. 9
Exercises 1. Jeepney fares are computed as follows: P7.50 for the four kilometers plus P0.50 for every additional kilometer thereof. How much should Au pay for a ride that covers 10 kilometers? a. P8.00 b. P9.50 c. P10.00 d. P10.50 2. Which of the following is 0.3 of ? a. b. c. d. 3. Which of the following is between 3 and 4?
5
a.
b. ‐ 3.5
c.
d.
π
15 2
4. Evaluate 14.8 + 3.95 + .003. a. 5433 b. 753 c. 446 d. 18.753 5. Carmen bought 4 kilograms of rice at P31.45 per kilo and 6 kilograms of salt at P22.35 per kilo. If she gave a P1000 bill to the cashier, how much change did she get? a. P8.00 P9.50 c. P120.10 d. P740.10 6. Each capsule of a certainb.commercial vitamins contains 0.6 mg of calcium. In how many pieces of capsules can 22.2 mg of calcium be distributed? a. 8 b. 37 c. 50 d. 105 7. Which of the following is equal to 2.4545454545… ? a.
5
b. 2
11 1
5
c.
11
44
1 is equal to _____________. 25 a. 0.0051 b. 0.006 c. 0.51 9. Which of these numbers is greater than ¼? 2 a. .04 b. (1/2) c. 1/8 8. The expression
d. 2
45
44 45
1
100
+
1000
+
d. 0.051 d. 1/0.04
VII. PERCENT Per Cent – literally meaning “per hundred”, it is one way of writing fractions in which the denominator which is required to be 100 is written as “%”, and read as “per cent”.
Since 1 = 100% hence
3 4
75 1 100 4
= 75%
A. CONVERSION Percent to Decimal Number. Divide the number by 100%. Note that 100% = 1.
Example: Convert the following to decimal: a) 35% c) 8.37% b)
Solution:
6
1 4
%
d)
1 4
%
a) (35%) 100% = 0.35 b) (6 ¼ %) 100% (6.25%) 100% = .0625 c) (8.37) 100% = .0837
d)
(1/4%)
100%
(0.25%)
100% = .0025
Decimal Numbers to Percent. Multiply the decimal number by 100%. Note that 100% = 1. Example:
Convert the following to percent. St. Louis Review Center-Inc-Davao Tel. no. (082) 224-2515
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a) Solution:
0.25
b) 0.143
c) 2.03
a) 0.25 x 100% = 25% c) 2.03 x 100% = 203%
d) .005
b) 0.143 x 100% = 14.3% d) .005 x 100% = 0.5%
Exercises: Fill in the blanks so that the entries in each row are equal. Fraction Decimal Percent A 4/7 B 160% C 0.95 D 6/11 E ½% B. Percentage. Percentage is a percent of a given number. The given number is called the base. The percent is called the rate.
Formula: Percentage (P) = Base (B) x Rate (R) and
Example: Solution:
What is 50% of 140? N = 0.50 x 140 = 70.00 = 70
Example:
36 is 10% of what number?
Solution:
36 = 0.10
Example:
22 is what percent of 88?
Solution:
22 = P
P
22 88
N
N
36 0.10
360
88 0.25 100% 25%
Example: Mr. Ballaran receives a 10% increase in his salary. With the increase, he now receives P22, 000. How much is his monthly salary before the increase? Solution: present salary = previous salary + increase 22, 000 = previous salary + (10% of previous salary) 22,000= previous salary (1 + .10) 22 000 1.1 = previous salary 20,000 = previous salary
C. Discount
The discussion on DISCOUNT is very similar with the discussion on PERCENTAGE. Original Price/ Marked Price/ List price ‐ as the Base Rate of Discount ‐ as the Rate Discount Selling Price
as the Percentage ‐‐ Original Price minus Discount
Example:
A skirt with an srcinal price of P250 is being sold at 40% discount. Find its selling price.
Solution:
S.P.
Example: price? Solution:
An item has a selling price of P 210.00. If the selling price is 70% of the srcinal price, what is its srcinal
Example:
A shirt is being sold at P 199.95. If its srcinal price is P 430, find the rate of discount.
Solution:
Discount ==O.P. 430– –S.P. 199.95 = 230.05
= Original price ‐ Discount = 250 – (0.40 x 250) = 250 ‐ 100 = 150
Selling price is 70% of the srcinal price 210 = 0.70 O.P. O.P. = 210 0.70 = 300. Therefore, the srcinal price is P300.
Rate of Discount =
230.05 430
100% 53.5%
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D. Simple Interest Interest (I) is the amount paid for the use of money or the money earned for depositing the money. Principal (P) is the money that is borrowed or deposited. Time (t) is the number of days/months/years for which the money is being borrowed/deposited and interest is calculated.
I = Prt,
P=
t=
r=
Example: Give the interest of P10,000 for three at 5.5% per year. Solution: I=Prt I = (10 000) (.055) (3) I = P1 100.
simple years
Example:
Determine the amount of the principal if the interest at 10% per annum after 8 months is P3,600.
Solution:
I=Prt
P
I rt
=
3600 (0.10)(8 / 12)
= 45,000
Compound Interest ( Final Amount = P[ 1 + r ] n )
Compound interest is different from simple interest because after the first interest calculation, the interest is added to the principal, so interest is earned on previous interest in addition to the principal. Compound Interest rates may be given as annual (1 time a year), semiannual (2 times a year), quarterly (4 times a year), monthly (12 times a year), and daily (365 times a year). Example: If P500is invested at 8% compounded semiannually, what will the final amount be after three years? Final Amount
= P[ 1 + r ]n = 500[ 1 + (8% / 2)]3 * 2 = 500[ 1 + 0.04 ]6 = 500[1.27] = 635
Exercises 1. John bought a jacket for Php 850.00. If he was given a discount of 15%, what was the srcinal price? a. P8,500.00 b. P1,000.00 c. P900.00 d.P765.00 2. In a basket, there are 15 santol, 12 balimbing, and 3 durian. What percent of the fruits are durian? a. 10% b. 12.5% c. 12% d. 15% 3. A certain mobile phone model was sold for P4,000 in 2000. Two years later, the same mobile phone model sold for P2,800. What was the percent decrease of the price? a. 15% b. 30% c. 20% d. 35% 4. If is 50% of a larger figure, which of the following is the larger figure? a. b. c. d. 5. A senior class of 50 girls and 70 boys sponsored a dance. If 40% of the girls and 50 % of the boys attended the dance, approximately what percent attended? a 44 b. 46 c. 42 d. 40 6. Which of the following is equal to a. 2.5
1 2 %? 2
b. 0.25
c.
5 2
d. 0.025
7. Sarah’s earning P 9,200 a month will receive a 15% increase next month. How much will her new salary be? a. P 10,500 b. P 10,530 c. P 10,580 d. P 10,560 8. How much is 37% of 80% of 24? a. 7.1 b. 1.92 c. 19.2 d. 71 9. According to the latest survey, 60% of the cancer patients were smokers. If there were 180 smoking cancer patients, how many cancer patients are there in all? a. 90 b. 108 c. 240 d. 300 10. Which of the following is 70% of 50? a. 7 b. 17.5 c. 35 d. 71 11. Twenty four is 12% of what number? a. 40 b. 150 c. 200 d. 400 12. Thirty six is what percent of 90? a. 32.4% b. 40% c. 45% d. 76% 13. In a mathematics test of 40 items, Mavic got 90%. How many items did Mavic get? a. 7 b. 28 c. 36 d. 360 14. Mr. Mabini receives a 10% increase in his salary. With the increase, he now receives P13,200. How much is his monthly salary before the increase? a. P12 000 b. P 13, 500 c. 14, 100 d. P14, 520
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15. According to the latest survey, 60% of the cancer patients were smokers. If there were 180 smoking cancer patients, how many cancer patients are there in all? a. 70 b. 150 c. 300 d. 360
VIII. RATIO AND PROPORTION
A ratio is a comparison of two or more quantities.
A proportion is a number sentence stating the equivalence of two ratios.
Note that in ratio, we are comparing quantities of the same units and that the ratio is expressed in terms of integers. Examples:
a) The ratio of 12 days to 3 weeks is 12:21 or 4:7. b) The ratio of 3 meters to 180 cm is 300:180 or 5:3. c) The ratio of 2 hours to 25 minutes is 120:25 or 24:5. d) The ratio of 1 ½ to 4 ½ is 1:3.
A. Direct Proportion. As one quantity increases, the other increases also.
Example: Solution:
Find the value of x if 15:20 = 14 : x. Equate the product of the means and the product of the extremes. Then solve for x. Thus, (15) (n) = (20) (14)
x Example: Solution:
20 14 15
= 18.6 or
18
2 3
.
A car travels at an average rate of 260 km in 5 hours. How far can it go in 8 hours, if traveling at the same rate? 260 : 5 = x : 8 (5) x = (260) (8)
x
260 8 5
= 416.
Example: If the ratio of teachers to students in a school is 1 to 18 and there are 360 students, how many teachers are there? Solution: Let x be the number of teachers,
1 18
x
360
or 1 : 18 = x : 360
(18)x = (360)1 x = 360/18 = 20 teachers B. Inverse Proportion. As one quantity increases, the other decreases.
Example: victims?
If the food is sufficient to feed 10 flood victims in 15 days, how many days would it last for 8 flood
Solution:
Equate the product of the terms in the first condition to the product of the terms in the second condition. Thus, we have: (10 victims)(15 days) = (x) (8 victims)
x
10 15 8
= 18.75 days
C. Partitive Proportion. One quantity is being partitioned into different proportions.
Example:
A wood 120 m long is cut in the ratio 2:3:5. Determine the measure of each part.
2 Solution:
·
120
10 1 3
·
120
10 1
=
24
1
=
36
1
= 24 m
= 36 m
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5
·
120
10 1
=
60
1
= 60 m
Example: A wire is cut into three equal parts. The resulting segments are then cut into 4, 6, and 8 equal parts respectively. If each of the resulting segments has an integer length, what is the minimum length of the wire? A)
24
B) 48
C) 72
D) 96
Solution: Each third of the wire is cut into 4,6 and 8 parts respectively, and all the resulting segments have integer lengths. This means that each third of the wire has a length that is evenly divisible by 4, 6, and 8. The smallest positive integer that is divisible by 4, 6, and 8 is 24, so each third of the wire has a minimum length of 24. So, the minimum length of the whole wire is three times 24, or 72.
Exercises 1. A 300 m ribbon is cut into four pieces in the ratio 1:2:3:4. Give the length of the shortest piece.2. If there are 18 boys and 45 girls in the gym, what is the ratio of the girls to the boys? a. 2:5 b. 2:3 c. 5:2 d. 3:7 2. What one number can replace x in 2: x = x: 32? a. 2 b. 6 c. 4 d. 8 3. If 5 men can do a job in 12 days, how long will it take 10 men to complete this task, assuming that they work at the same rate? a. 20 days b. 6 days c. 2 days d. 0.06 day 4. If 3 kg of oranges cost as much as 5 kg of chicos, how many kg of oranges would cost as much as 60 kg of chicos? A. 100 B. 36 C. 7.5 D. 4 5. If 2/5 mm in a map represents 120 km, how many km will be represented by 2 mm? A. 600 km B. 300 km C. 96 km D. 24 km 6. In a Mathematics Club, the ratio of boys to girls is 3:5. If there are 240 members, how many are girls? A. 90 B. 144 C. 150 D. 450
7. A photographer wishes to enlarge a picture 18 cm long and 12 cm wide so that it will be 36 cm wide. How long will the enlarged picture be? A. 54 cm B. 72 cm C. 24 cm D. 6 cm 8. If 8 secretaries can type 800 pages in 5 hours, how long would it take for 12 secretaries to type 800 pages at the same rate? A. 7 1/2 hours B. 3 1/3 hours C. 10 hours D. 2 1/2 hours THE THEORY OF CONGRUENCES
If a and b are integers, m a positive integer and m (a – b), we say that “a is congruent to b modulo m”. In symbols, we write this as a b (mod m). CONGRUENCE was invented by Karl Friedrich Gauss at the beginning of the th 19 century and is a convenient statement about divisibility.
The following are equivalent and may be used interchangeably. a b (mod m). m (a – b) or (a – b) is divisible by m. a = b + mk, k Z. Theorem: If a and b are integers and m a positive integer then a remainder upon division by m. Let m be a positive integer. A collection of m integers
b (mod m) if and only if a and b leave the same
a1 , a2 ,..., an is called a complete residue system modulo m if
every integer b modulo m is congruent to exactly one of the elements in the collection.
Properties of Congruence
Congruence is an equivalence relation in the set of integers; that is, congruence is reflexive, symmetric and transitive with respect to integers. In the following, let a, b, c, and d be integers and m a positive integer.
If a b (mod m) then
a+c b+c (mod m). ac bc (mod m). r r a b (mod m) where r is a positive integer.
If a b (mod m) and c d (mod m), then a + c (b + d) (mod m).
ac (bd) (mod m).
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If ac bc (mod m), then a b mod
m d
where d = (c, m).
The following are some applications of congruence. a) Finding the units digit (or hundreds digits) of a very large number written in exponential form; and b) Finding the remainder when a very large number is divided by another number. Two of the most prolific mathematicians in Number Theory are Pierre de Fermat and Leonhard Euler. FERMAT’S AND EULER’S THEOREMS
Theorem 5. (Fermat’s Little Theorem) Let p be a prime number and a Z . If p does not divide a, then p–1 a 1 (mod p) . Theorem 6. (Fermat’s Second Theorem). Let p be a prime number and a Z . If p and a are relatively prime, then p a a (mod p) . DEFINITION OF
( m)
Let m be a positive integer greater than 1. The number of positive integers less than and relatively prime to m is the value of Euler’s totient or function at m and is denoted by ( m ) . Remarks: If p is prime, then ( p ) = p – 1. Theorem 7. Euler’s Theorem: If n is a positive integer and the greatest common divisor of a and n is 1, then
a
n
1mod n .
LINEAR DIOPHANTINE EQUATIONS An equation in one or more unknowns having integral solutions is called a Diophantine equation, in honor of
Diophantus of Alexandria. Theorem 8. Given two integers a and b where (a , b) = d. The linear Diophantine equation ax + by = c has an integral solution if and only if dc. Theorem 9. If the equation ax + by = c has a solution x = x0 , y = y0, form
x x0
y
y0
b d a d
t ,
t
Z
t ,
t
Z.
then any other solutions can be expressed in the
and
Example: To determine the integral solution of 24x + 138y = 18, we note that since (24,138) = 6 and 618. Then we know that it has solution. We now have the following.
138==15(18) (24)++618 24 18 = 6 (3). Observe that, 6
= 24 – 1(18) = 24 – [138 ‐ 5 (24)] = (‐1)(138) + 6(24)
Moreover, 18 = 3(6) = 3[(‐1)(138) + 6(24)] =(‐3)(138) +(18)(24) Thus,
y0 = ‐ 3
and
x0 = 18
Hence, the solution of the equation is of the form y = ‐ 3+23t and x = 18– 4t where t is an integer. There are problems which can be solved using linear Diophantine equations as working equations. The following steps may be used in solving word problems which involve linear Diophantine equations in two unknowns/variables: Step 1. Represent the unknown values using any two variables. Step 2. Form the equation using the condition given in the problem. St. Louis Review Center-Inc-Davao Tel. no. (082) 224-2515
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Step 3. Solve the resulting linear Diophantine equation. Step 4. Determine the solution/s to the problem using the results in step 3.
Theorem: (Wilson’s Theorem) If p is a prime, then (p‐1)!
1mod p .
Exercises 1. Which of the following is true? A. 34
3mod 5
B.
54
5mod 25
C. 308
0mod 11
D. 3 4mod 15
2. Mavic argues that 15 9mod 6 . Is she correct? Why? A. Yes, because 6 divides 15 ‐ 9. C. No, because 6 does not divide 15 + 9. D. No, because 15
B. Yes, because 6 divides 15 + 9.
3. Which of the following is congruent to 11 modulo 13? A. ‐ 7 B. ‐ 5 4. Which of the following must be the value of n if
9 61 .
C. ‐ 2
D. 4
n6 1 0mod 7 ?
A. x is divisible by 7 B. x is prime
C. x is relatively prime with 7 D. x is any integer greater than 7 105
5. What is the remainder when 5 is divided by 3? A. 1 B. 2 100
6. What is the units digit of 3 A. 1
C. 3
D. 5
B. 3
C. 5
D. 9
B.4
C.6
D. 11
?
7. What is 12 ? A. 1
SET A Exercises. Choose the letter of the best answer.
1. What is the sum of the first four prime numbers? a. 11 b. 26 c. 17 d. 28 2. Which of the following is NOT true about the sum of two consecutive positive odd integers? a. it is even b. it is only divisible by 12 c. it is divisible by 4 d. it is always divisible by 1 3. In a sequence of starts and stops, an elevator travels from the first floor to the fourth floor and then to the second floor. From there, the elevator travels to the third floor and then to the first floor. If the floors are 3 meters apart, how far has the elevator traveled? a. 21 m b. 24 m c. 28 m d. 32 m 4. An orange light blinks every 4 seconds. A blue one blinks every 5 seconds while a red one blinks every 6 seconds. How many times will they blink together in two hours? a. once b. 2 times c. 10 times d. 60 times 5. If one prime factor of 84 is 3, what are the other prime factors?
a. 2 and 3
b. 2 and 7
c. 3 and 5
d. 4 and 7
6. A television show reports the following temperature for 5 cities: Beijing London Chicago Philippines Moscow 0 0 0 0 0 ‐6 C ‐9 C 2 C 0 C 30 C Which city is the coldest? a. Beijing b. Chicago c. London d. Moscow 7. If the sum of a certain number and 7 is divided by 4, the quotient is 3. What is the number? a. 5 b. 12 c. 15 d. 18 8. Which of the following numbers has the largest value? a. –22 b. –10 c. –75 d.3 9. Which of the following numbers has the least value? a. –22 b. –10 c. –75 d.3 10. What is the difference in the elevation between the top of a mountain 51 meters above sea level and a location 28 meters below sea level. a. 23 m b. 33 m c. 79 m d. 89 m 11. A pack of P50‐bills is numbered from RV628 to RV663. What is the total value of the pack of bills, in pesos? a. 35 b. 36 c. 1750 d. 1800 2 2 3 12. Simplify: [ 5 81 3 – 5 3 + 2] (4 – 2 ) a. 15 ¼ b. 4 c. ‐ 6 d. – 15 1/4 13. If each container contains b. 68 kg
5 2 kg of flour, how many kg of flour are there in 12 container? 3 b. 70kg
c. 72 kg
14. Eighteen is 2/3 of what number? St. Louis Review Center-Inc-Davao Tel. no. (082) 224-2515
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62
a. 6 b. 12 c. 6 d. 27 15. What part of an hour has passed from 2:48 am to 3:20 am? a. 7/8 b. 1/3 c. 8/15 d. 8/25 16. Clarita spent one‐sixth of her money in one store. In the next store, she spent three times as much as she spent in the first store. If she had 80 pesos left, how much money did she have from the start? a. 240 pesos b. 252 pesos c. 300 pesos d. 360 pesos 17. Philip has obtained the following grades: 1.4, 1.7, 1.8 and 2.5. What must be his fifth grade so that his average is 1.7? a. 2.1 b. 1.9 c. 1.5 d. 1.1 18. Out of the 20 numbers, 6 were 2.5’s, 4 were 3.25’s and the rest were 2.2’s. Give the arithmetic mean of the numbers. a. 2.5 b. 2.65 c. 10 d. 22 19. Ron bought X number of notebooks at P23.00 each, Y pad papers at 18.45 each, and Z ballpens at P8.25 each. If he gave an amount of P1000 to the cashier, how much change did he receive? a. P 434.25 c. 1000 – [(23.00)(X) + (18.45) (Y) + (8.25) (Z)] b. P 334.25 d. none of these 20. A bag has a selling price of P60.00. If the selling price is 75% of the srcinal price, what is its srcinal price? a. P80 b. P120 c. P200 d. P280 21. Mr. de Borja, a store owner, advertises a polo‐shirt srcinally sold for P200 for P170 only. What rate of discount is he giving? a. P 30 b. P15 c. 30% d. 15% 22. Ja bought an article for P400 and sold it for P500. What rate of profit did she enjoy in that deal? a. P100 b. 100% c. 25 % d. 20% 23. The price of an item is increased by 70% and then offered at 40% discount. What happened to the srcinal price? a. There is an increase of 30%. c. There is an increase of 2 %. b. There is an increase of 28%. d. There is a decrease of 32%. 24. How much should Allan invest so that his money earns P2,250 deposited at 6% for 9 months? a. P 50,000 b. 37,500 c. P 135 d. P 101.25 25. Dan sells a real estate. He receives a monthly salary of P10,000 plus a commission of 1/5 % of his net sales for that month. Find his gross pay for a month during which his net sale is one million pesos. a. P 2,000 b. P 12,000 c. P 200,000 d. P 210,000 26. There are 20 million Filipinos who are qualified voters. If 25% of the population are qualified voters, how many are not qualified voters? a. 80 million b. 60 million c. 15 million d. 5 million 27. Three cavans of rice for a family of six members last for 5 weeks. At this rate, how many weeks will 4 cavans of rice last a family of 8 members? A. 4 B. 5 C. 5 1/3 D. 6 28. If the assembly, ratio of boys to girls is 1:4. What percent of the assembly are the boys? A. 10% B. 20% C. 25 % D. 80% 30
29. What is the remainder when 11 is divided by 31? A. 1 B. 2
C. 10
D. 101
2061
30. Which of the following is the remainder when 5 is divided by 7? A. 1 B. 2 C. 3 D. 6 31. If y is the remainder when 47 is divided by 6, what is the remainder when 19 is divided by y? A. 1 B. 2 C. 4 D. 5 32. Which of the following is a value of z such that the congruence A. 9 B. 22 33. What is the remainder when 18! + 2 is divided by 19? A. 0 B. 1
3z 1mod 13 is NOT true?
C. 30
D. 48
C.3
D. 97
34. A certain the number of sixes and nines added to there give aafter sumthe of switch? 126. If the numbers of sixes and nines are interchanged, new sum is 114. How manyare of each were A. Ten sixes and 6 nines B. Four sixes and Twelve nines C. Seven sixes and nine nines D. Six sixes and Ten nines
SET B Exercises. Choose the letter of the best answer. 1. How many prime numbers are less than 37? a. 9 b. 10 c. 11 d. 12 2. In a series of card games. Marlon starts out with P200 and wins a total of P450. If he later loses P350, wins P60 and loses P150, how much cash does Marlon have? a. 0 b. P150 c. P210 d. P300 3. Your score in a game is ‐ 6. How many points must you earn to get a score of 10? a. ‐6 b 15 c. 16 D. 22
4. Arrange the fractions 5/12, 3/7, 2/5 in decreasing order. a. 2/5, 5/12, 3/7 c. 3/7, 5/12, 2/5
b. 5/12, 2/5, 3/7
d. 2/5, 3/7, 5/ 12
5. Edwin, Doms and Lon weigh 45 kg. If Edwin and Doms weigh 11
1 5
kg and
13
3 4
kg, respectively, what is the weight of
Lon in kilograms? St. Louis Review Center-Inc-Davao Tel. no. (082) 224-2515
63
a.
20
1
b.
20
24
19
c.
20
6. Which of the following should be multiplied to a. 6
4
3 4
A. 2
1 3
d.
69
19 20
c. 6
d. 75
1mod m ?
B. 3
3
9 20
so that the product is 57?
b. 12
7. Which of the following is a value of m if 15
8. Alex works on his assignment
42
C. 23
D. 32
hours a day, what part of the day does he spend doing his assignment?
a. 1/8 b. 1/7 c. 5/36 9. What value of p will satisfy the equation 0.2 (2p + 1470) = p?
d.
15/28
a. 294 b. 490 c. 560 d. 1470 10. A blouse srcinally priced at P600 is being sold at a discount of 30%. How much would you pay if you buy that blouse? a. P 30 b. P180 c. P 420 d. P 570 11. A pair of slippers with a selling price of P120 is sold at 40% discount. What is its srcinal price? a. P 48 b. P72 c. P 200 d. P 300 12. An item is offered at 20% discount. Later, it is offered at 30% discount. If the new selling price is P112, what is the first srcinal price? a. P162 b. P200 c. P224 d. P1866.67 13. To have a 25% profit, the vendor should sell the item at P80.00. How much is his profit? a. P20.00 b. P60.00 c. P16.00 d. P64.00 14. Minda deposited P50,000 in a bank that pays a simple annual interest of 14
1 5
%. How much
money will she have in the bank after five years. a. P 85,500 b. P35,500 c. P36,250 d. P86,250 15. The ratio of cows and carabaos in the field is 4:9. If there are 468 cows and carabaos in the field, how many are carabaos? A. 52 B. 117 C. 144 D. 324 16. In the class, the ratio of boys to girls is 6:5. If there are 90 girls, how many persons are in the class? A. 75 B. 108 C. 165 D. 198 17. Edwin painting a wall at 9:00 a.m. and was able to finish painting 3/5 of it at 10:30 a.m. Continuing at this rate, at what time will he finish? A. 10:45 a.m. B. 11:30 a.m. C. 11:45 a.m. D. 12:15 a.m. 200
18. What is the remainder when 4 is divided by 15? A. ‐ 1 B. 1 C. 2 3x 19. If x is any positive integer, then 2 + 1 is _______ divisible by 8. A. Always B. Never C. Sometimes
D. 4 D. Equivalently
20. What is 29 ? A. 0
B. 1
C. 2
21. Which of the following is the remainder when A. 0 B. 2
D. 28
2 50 1 is divided by 7? C. 4
D. 5
22. Which of the following is equivalent to the pair of congruence x A. x C. x
1mod 12 3mod 12
B. x D. x
1mod 4 and x 2mod 3 ?
2mod 12 5mod 12
23. Which of the following is NOT true if a is a positive integer?
A. a divides a
5
C. a
B. a divides a 1 . 5
5
amod10
5
D. a and a have the same units digit
24. Which of the following gives a remainder of 2 when divided by 5 and a remainder of 12 when divided by 13? A. 22
B.38
C. 77
D. 92
25. What is the remainder upon dividing the sum 2!4!6!... 2008!2010! by 5? A. 1 B. 2 C. 3 D. 4 109 26. What is the least residue if 17 is a multiple of 6? A. 1 B. 2 C. 3 D. 5 27. Which of the following has an integral solution?
13mod 77 5mod 36
A. 21 x
B. 3 x
C. 3 x
D. 3 x
28. How many integral solutions does 18 x A. 2
B. 3
4mod 81 4mod 5
21mod 24 have? C. 6
D. 10
29. Which of the following has a solution if the variables are positive integers? A. 5x + 30y = 18 B. 8x + 10y = 15 C. 22x + 4y = 28 D. 101x 37 y 3819 St. Louis Review Center-Inc-Davao Tel. no. (082) 224-2515
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30. When 16! is divided by 17, the remainder is ______. A. 0 B. 1 C. 17 D. 18 31. A John’s transcript shows x number of 3‐unit courses and y number of 5‐unit courses for a total of 64 units. Which of the following may appear in the transcript? A. 2 x’s and 18 y’s C. 11 x’s and 3 y’s
B. 13 x’s and 5 y’s D. 9 x’s and 8 y’s
32. Which of the following is a value of x if
x
2
1mod8 ?
A. 2 B. 5 C. 6 D. 10 33. When students in a certain college are grouped by 2’s, 3’s, 4’s, 5’s or 6’s at a time, there remain, 1,2,3,4, or 5 students respectively. When the students are grouped by 7’s, no is student left. What is the smallest possible number of students in the school? A. 227
B.1,534
C. 1,379
D. 2,778
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LICENSURE EXAMINATION FOR TEACHERS (LET)
Refresher Course Content Area: MATHEMATICS Focus: Plane and Solid Geometry Prepared by: Daisy de Borja‐Marcelino Basic Ideas
The undefined terms, point, line, and plane are geometric ideas and they are visually represented by a tiny dot, a thin wire, and a smooth flat surface, respectively. Points are labeled by means of capital letters, lines by naming any two of its points, and planes by naming at least three of its points. The subsets of a line are ray, segment, and the line itself. Space is the set of all points.
A ..
B
.
A.
Line
AB
B
A .
.
Ray
AB
B.
Line segment
AB
Some postulates on points, lines and planes: The Distance Postulate. To every pair of different points there corresponds a unique positive number. The Ruler Postulate. The point of a line can be placed in correspondence with the real numbers in such a way that (1) to every point of the line there corresponds exactly one real number; (2) to every real number there corresponds exactly one point of the line; and (3) the distance between any two points is the absolute value of the difference of the corresponding numbers. The Ruler Placement Postulate. Given two points P and Q of a line, the coordinate system can be chosen in such a way that the coordinate of P is zero and the coordinate of Q is positive. The Line Postulate. For every two points there is exactly one line that contains both points. The Plane Postulate. Any three points lie in at least one plane, and any three noncollinear points lie in exactly one plane.
The Plane Separation Postulate. Given a line and a plane containing it. The points of the plane that do not lie on the line form two sets such that (1) each of the sets is convex, and (2) if P is in one of the sets and Q is in the other, then the segment PQ intersects the line. The Space Separation Postulate. The points of space that do not lie in a given plane form two sets, such that (1) each of the sets is convex, and (2) if P is in one of the sets and Q is in the other, then the segment
PQ intersects the plane.
The following are also true. (a) Every plane contains at least three noncollinear points. (b) Space contains at least four noncoplanar points. (c) If two points of a line lie in a plane, then the line lies in the same plane. (d) If two different planes intersect then, their intersection is a line.
Some theorems on points, lines and planes:
If A,B, and C are three different points of the same line, then exactly one of them is between the other two.
The Point‐Plotting Theorem. Let
AB be a ray, and let X be a positive number. Then there is exactly one point P of AB
such that AP x . Every segment has exactly one mid‐point. If two different lines intersect, their intersection contains only one point. If a line intersects a plane not containing it, then the intersection contains only one point. Given a line and a point not on the line, there is exactly one plane containing both. Given two intersecting lines, there is exactly one plane containing both. The First Minimum Theorem. The shortest segment joining a point to a line is the perpendicular segment. In a given plane, through a given point of a given line, there is one and only one line perpendicular to the given line. The Perpendicular Bisector Theorem. The perpendicular bisector of a segment, in a plane, is the set of all points of the plane that are equidistant from the end points of the segment. St. Louis Review Center-Inc-Davao Tel. no. (082) 224-2515
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Through a given external point there is at least one line perpendicular to a given line. Through a given external point there is at most one line perpendicular to a given line. If M is between A and C on a line L, then M and A are on the same side of any other line that contains C. If B and C are equidistant from P and Q, then every point between B and C is equidistant from p and Q. If a line is perpendicular to each of two intersecting lines at their point of intersection, then it is perpendicular to the plane that contains them. Through a given point of a given line there passes a plane perpendicular to the given line. If a line and a plane are perpendicular, then the plane contains every line perpendicular to the given line at its point of intersection with the given plan. Through a given point of a given line there is only one plane perpendicular to the line. The Perpendicular Bisecting Plane Theorem. The perpendicular bisecting plane of a segment is the set of all points equidistant from the end points of the segment. Two lines perpendicular to the same plane are coplanar. Through a given point there passes one and only one plane perpendicular to a given line. Through a given point there passes one and only one line perpendicular to a given plane. The Second Minimum Theorem. The shortest segment to a plane from an external point is the perpendicular segment. Two parallel lines lie in exactly one plane. In a plane two lines are parallel if they are both perpendicular to the same line. Let L be a line and let P be a point on L. Then there is at least one line through P, parallel to L.
Angle. It is a union of two non‐collinear rays that have a common endpoint. o
o
An acute angle measures more than 0 but less than 90 . o A right angle measures exactly 90 . o o An obtuse angle measures more than 90 but less than 180 . o Two angles are complementary if the sum of their measures is 90 . o Two angles are supplementary if the sum of their measures is 180 . Two angles are adjacent if they have a common vertex, common side but no common interior points. Two angles form a linear pair if they are both adjacent and supplementary.
Some postulates on angles:
The Angle Measurement Postulate. To every angle there corresponds a real number between 0 and 180.
The Angle Construction Postulate. Let
AB be a ray on the edge of the half‐plane H. for every number r between 0
AP , with P in H, such that mPAB r . The Angle Addition Postulate. If D is in the interior of BAC , then mBAC
and 180 there is exactly one ray
mBAD mDAC .
The Supplement Postulate. If two angles form a linear pair, then they are supplementary.
The transversal line is a line intersecting two or more coplanar lines at different points
1 3 5 7
2 4
6
Interior angles:
3, 4, 5, 6
Exterior angles:
1, 2, 7, 8
Corresponding angles: 1 and 5, 2 and 6
3 and 7, 4 and 8
8
Alternate interior angles:
3 and 6, 4 and 5
Alternate exterior angles: 1 and 8, 2 and Same‐side interior angles:
7
3 and 5, 4 and 6
Some theorems on angles and transversal lines:
Given two lines cut by transversal. If a pair of alternate interior angles are congruent, then the lines are parallel. Given two lines cut by transversal. If a pair of corresponding angles are congruent, then a pair of alternate interior angles are congruent. Given two lines cut by transversal. If a pair of corresponding angles are congruent, then the lines are parallel. Given two lines cut by transversal. If a pair of interior angles on the same side of the transversal are supplementary, the lines are parallel. If two parallel lines cut by transversal, each pair of corresponding angles are congruent. In a plane, if a line intersects one of two parallel lines in only one point, then it intersects the other. Every right angle has measure 90, and every angle with measure 90 is a right angle. If two angles are complementary, then both are acute. Any two right angles are congruent. If two angles are both congruent and supplementary, then each is a right angle. The Supplement Theorem. Supplements of congruent angles are congruent. The Complement Theorem. Complements of congruent angles are congruent. St. Louis Review Center-Inc-Davao Tel. no. (082) 224-2515
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The Vertical Angle Theorem. Vertical angles are congruent. If two intersecting lines form one right angle, then they form four right angles. The Angle Bisector Theorem. Every angle has exactly one bisector. The Isosceles Triangle Theorem. If two sides of a triangle are congruent, then the angles opposite these sides are congruent. If two angles of a triangle are congruent, then the sides opposite them are congruent.
If M is between B and C, and A is any point not on BC , then M is in the interior of BAC . The Exterior Angle Theorem. An exterior angle of a triangle is greater than each of its remote interior angles.
Polygon. It is a closed plane figure with three or more sides, consists of segments (sides) that meet only at their endpoints (vertices) such that every side contains only two vertices, and every vertex is on exactly two sides. Name of Polygon Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon
Number of Exact Sides 3 4 5 6 7 8
Name of Polygon nonagon decagon undecagon dodecagon pentadecagon
Number of Exact sides 9 10 11 12 15
Notes: a) In a triangle, the sum of the lengths of two sides is always greater than the length of the third side. 2 2 2 b) In a right triangle with legs x, y and hypotenuse z: x + y = z (Pythagorean triple) o o c) The sum of the interior angles of an n‐gon is (n‐2) 180 . The sum of the interior angles of a triangle is 180 o while the sum of the interior angles of a quadrilateral is 360 .
A polygon is equilateral if the measures of all its sides are equal.
A polygon is equiangular if the measures of all its angles are equal.
A polygon is regular if all its sides and angles are correspondingly congruent.
Some postulates on polygons The SAS Postulate. Every SAS correspondence is a congruence. The ASA Postulate. Every ASA correspondence is a congruence. The SSS Postulate. Every SSS correspondence is a congruence. The Parallel Postulate. Through a given external point there is only one parallel to a given line. The Area Postulate. To every polygonal region there corresponds a unique positive real number. The Congruence Postulate. If two triangles are congruent, then the triangular regions determined by them have the same area. The Area Addition Postulate. If two polygonal regions intersect only in edges and vertices (or do not intersect at all), then the area of their union is the sum of their areas. The Unit Postulate. The area of a square region is the square of the length of its edge. The Unit Postulate. The volume of a rectangular parallelepiped is the product of the altitude and the area of the base. The Cavalieri’s Principle. Given two solids and a plane. Suppose that every plane parallel to the given plane, intersecting one of the two solids, also intersects the other, and gives cross sections with the same area. Then the two solids have the same volume. The SAA Theorem. Every SAA correspondence is a congruence. The Hypotenuse‐Leg Theorem. Given a correspondence between two right triangles, if the hypotenuse and one leg of one of the triangles are congruent to the corresponding parts of the second triangle, then the correspondence is a congruence. If two sides of a triangle are not congruent, then the angles opposite them are not congruent, and the larger angle is opposite the longer side. If two angles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the larger angle. The Triangle Inequality. The sum of the length of any two sides of a triangle is greater than the length of the third side. The Hinge Theorem. If two sides of one triangle are congruent, respectively, to two sides of a second triangle, and the included angle of the first triangle is larger than the included angle of the second, then the third side of the first triangle is larger than the included angle of the second. The Converse Hinge Theorem. If two sides of one triangle are congruent respectively to two sides of a second triangle, and the third side of the first triangle is longer than the third side o the second, then the included angle of the first triangle is larger than the included angle of the second. For every triangle, the sum of the measures of the angles is 180.
Each diagonal separates a parallelogram into two congruent triangles. In a parallelogram, any two opposite sides are congruent. In a parallelogram, any two opposite angles are congruent. In a parallelogram, any two consecutive angles are supplementary. The diagonals of a parallelogram bisect each other. St. Louis Review Center-Inc-Davao Tel. no. (082) 224-2515
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Exercises 1. Which of the following has a definite length? A) ray B) line 2. A plane is determined by ______________. i. a line and a point. iii. any three points. A) ii only B) iv only 3. In angle LON, what is the vertex? A) L B) O
C) angle
D) line segment
ii. two intersecting lines. iv. a line and a point not on it. C) ii and iii
D) ii and iv
C) N
D) cannot be determined
4. These shapes are arranged in a pattern.
Which of these shapes are arranged in the same pattern? A) B) C) D) 5. Which of these triangles cannot be isosceles? A) acute B) right
C) obtuse
D) scalene
6. Two angles that are complementary _____________________. i. each acute. ii. congruent. iii.are have equal measures. iv.are have measures that add up to 90 . A) ii only B) iv only C) ii and iii D) i and iv 7. Refer to the figure on the right. Given: m 1 = 57° and m 3 = 80°. What is m 5? A) 43° B) 63° C) 137° D) 180° 0
8. A quadrilateral MUST be a parallelogram if it has ________________________. A) one pair of adjacent sides equal B) one pair of parallel sides C) two pairs of parallel sides C) two adjacent angles equal 9. NCTE is a parallelogram. If m N = 67°, then m T = _____. A) 113° B) 90° C) 67° D) 23° 10. Which two properties of the rhombus do not hold true for other parallelograms? A) Diagonals are perpendicular and congruent. B) Diagonals are congruent and bisect each other. C) Diagonals are perpendicular and bisect each other. D) Diagonals are perpendicular and each bisects the angles of the parallelogram. 11. The supplement of an angle is three times its measure. What is the measure of its supplement? A) 22.5° B) 45° C) 67.5° D) 135°
12. Refer to the figure on the right. If lines r and s are parallel, which of the following pairs of angles are congruent?
A)
4 and 5
B)
4 and 3
C)
1 and 8
D)
3 and 5
13. The angles of a triangle are in the ratio 2: 3: 5. What is the largest angle? A) 36° B) 54° C) 90° D) 99° 14. An angle measures 65.5°. What is the measure of its supplement? A) 24.5° B) 25. 5° C) 114.5° D) 124.5° 15. Which refers to a pair of lines that intersect and form 4 right angles? A) parallel lines B) perpendicular lines C) intersecting lines D) skew lines 16. Which of the following quadrilaterals best describes a square? A) Its diagonals are perpendicular to each other. B) It is an equiangular rhombus. C) Its diagonals are congruent. D) It has four right angles. 17. If ABC is an isosceles triangle with a right angle at B, then A) is the hypotenuse. B) AC = BC C) angle BCA measures 45°. D) AB = ½ AC 18. What is the measure of each interior angle of a regular pentagon? A) 108° B) 140° C) 180° D) 540° 19. What is the sum of the measures of the interior angles of a regular heptagon? m A) 1260° B) 1080° C) 900° D) 112.5°
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t h 69
20. In the adjoining figure, if m = 63° and h = 134°, then a must be equal to _____. A) 46° B) 71° C) 109° D) 117° 21. Which of the following statements is true about parallel lines? A) They form a right angle. B) They do not intersect at all. C) They are skew. D) None of these. 22. In the adjoining figure, the quadrilateral is a parallelogram. What is the value of x? A) 150 B) 75 C) 60 D) 50 23. Which of the following can be lengths of the sides of a triangle? A) {3, 4, 9} B) {5, 5, 10} C) {‐12, 6, 8}
D) {3, 18, 20}
24. The measures of the exterior angles of a triangle are in the ratio 2:3:4. What is the measure of the widest angle? A) 40° B) 80° C) 160° D) 170° G. Perimeter.The distance around a polygon is called a perimeter. To obtain this, we just add the length of all the sides of the polygon. In the case of equilateral polygons, we just multiply the length of one side to the number of sides. Example: Solution:
Determine the perimeter of a triangle with sides 7, 12, 18. P = 7 + 12 + 18 = 37 units.
Exercises:
Complete the table below.
Regular Polygon a) square b) c) nonagon d) e) heptagon
Length of a Side 13 cm 7 cm
Perimeter 56 cm 54 cm 45 cm 28 cm
9 cm
H. Areasof Trianglesand Quadrilaterals
a)
triangle = (1/2) (base) (height)
b)
rectangle = (length) (width) or (base) height)
c)
square = (side)(side) or (1/2)(d1)(d2)
d)
parallelogram = (base) height)
e)
rhombus = (1/2)(d1)(d2)
f)
trapezoid = (1/2) (upper base + lower base)(height)
Example: Solution:
Give the area of a square whose perimeter is 100 m? 2 Since the perimeter of the square is 100 cm, then each side measures 25 m. Hence, its area is (25 m) = 2 625 m .
Example:
Determine the area of a triangle whose base is 10 mm and its altitude is thrice its base.
Solution:
A=
Example: Solution:
Give the area of a rectangle whose width is 5 cm and its length exceeds 4 times the width by 3. A = lw = [4w + 3] (w) = [(4)(5) + 3] (5) = (23) (5) = 115 sq. cm.
Example:
Determine the area of the trapezoid whose bases are 6 cm and 10 m, while the altitude is 7 m.
Solution:
A = (1/2) (b1 + b2) (height) = (1/2) (6 + 10) (7) = 56 m
1
2
bh
1
2
(10) [(3) (10)]
150 mm2 .
2
I. Circle.This is a set of points in a plane, equidistant from a fixed point. The fixed point is called the center, and the fixed distance is the length of the radius.
Circumference is the distance around the circle. Formula: 2 r 2 Area is the interior region bounded by the circle. Formula: r Terms related to a circle:
Center – the fixed point Radius – segment (or distance) drawn from the center to any point on the circle Chord – segment whose endpoints are any two points on the circle St. Louis Review Center-Inc-Davao Tel. no. (082) 224-2515
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Diameter – a chord that passes through the center (the longest chord) Secant – a line/ray/segment that cuts the circle in two points Tangent – a line that touches the circle in only one point Note: The diameter is twice as long as the radius. Exercises: In the figure on the right, identify the following: center, radius, chord, diameter, secant, tangent.
Example: Solution:
Determine the circumference of a circle with radius 10 mm. C = 2 r = 2 (3.1416) (10 cm) = 62.832 mm
Example: Solution:
Give the area of the circle if its circumference is 18 m. Since the circumference is 18 cm, then its radius is 9 m. Hence, 2 2 2 A = r = (3.1416) (9m) = 254.4696 or 254.47 m
Exercises 1. Complete the table below. Radius Diameter 1. 2 cm 2. 6 dm 3. 4. 5. 9 m
Circumference (in )
Area (in )
50 m 36 km
2
2. A diameter of a circle is also a ________. A) radius B) secant C) chord 3. Which of the following sets of numbers can be sides of a right triangle?
D) tangent
A. 4, 5, 6 B. 6 , 6 2 C. 3, 6 3 D. 2, 2, 3 , 3 , 2 4. The length and width of a rectangle are (3x – 2) and (2x + 1). What is its perimeter? 2 2 A. 5x‐1 B. 10 x ‐ 2 C. 6x –x ‐2 D. 6x –x +2 5. The area of a square is 32x. Which of the following could be the value of x? A. 2 B. 6 C. 3 D. 4 6. If the area of one circle is twice of another circle, what is the ratio of the area in percent of the smaller to larger circle? A. 70% B. 25% C. 75% D. 50% J. Volume of Solids. It describes how much space a three dimensional figure occupies. 3
a)
cube
= (side)
b)
rectangular prism
= (length)(width)(height) or (area of the base)(height)
c)
pyramid
= (1/3) (area of the base) (height)
d)
sphere
= (4/3)( )(radius)
e)
cylinder
= (radius) (height)
f)
cone
= (1/3) (radius)2 (height)
3
2
Example: Solution:
What is the volume of a cube whose edge is 4 mm? 3 V = (side) 3 = (4 mm) 3 = 64 mm
Example:
The length of a rectangular box is 20 cm. Its width exceeds 1/4 of the length by 5 cm, while the height is 7 cm less than 1/2 of the length. What is its volume?
Solution:
Volume = (length)(width)(height) = (20 cm) = (20
Example:
1 20cm 5cm 4
1 20cm 7cm 2 3
cm) (10 cm) (3 cm) = 600 cm 2
The area of the base of a pyramid is 48 cm while the height is 6 cm. What is its volume?
Solution:
V = (1/3) (area of the base) (height) 2 = (1/3) (48 cm ) (6 cm ) = 96 cu. cm.
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Example: Solution:
Determine the volume (in ) of a sphere whose diameter is 12 m. 3 V = (4/3)( )(radius) and since diameter is 12 cm, then the radius is 6 m. 3 3 = (4/3)( )(6 cm) = 288 m
Example:
A cone has a base diameter of 32 cm and its height is 3 cm less than 3/4 of the radius. What is its volume?
Solution:
V = (1/3) (radius) (height) 2
=
3 1 (16 cm)2 16cm 3cm 3 4
= 768 cm
3
K. Lateral and Total Surface Areas of Solids.
Solids a) cube b) rectangular prism c) pyramid d) sphere e) cylinder f) cone
Lateral area 2 4s (perimeter of the base)x height 1/2 (perimeter of the base) slant height None (Circumference of the base) (slant height) 1/2 (Circumference of the base) (slant height)
Total surface area 2 6s 2B + ph B + 1/2 p (slant height) 2 4r 2 2r + 2r (slant height) r2 + r (slant height)
Exercise 1. Determine the volume, lateral and surface area of the following: (All units are in cm.) Express your answers in , if possible.
edge length = 10cm,
base edge = 10 m width = 3cm, height = 4cm
radius = 5
3
2
slant height = 13m altitude = 12m
radius = 5 height = 12
Volume (in cm )
=
radius = 5 slant height = 5 altitude = 4
2
Lateral Area (in cm )
2
Total Surface Area (in cm )
a) cube b) rectangular prism c) square pyramid d) sphere e) cylinder f) cone PRACTICE EXERCISES
1.
What will happen to the area of a circle if its radius is doubled? a. The area will be doubled. c. The area will be quadrupled. b. The area will remain the same. d. The area will be reduced to half.
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mm
2.
What is the reason for your answer in item number 1? a. The area of the circle varies directly to the radius. b. The area of the circle varies inversely to the radius. c. The area of the circle varies directly to the square of its radius. d. The area of the circle varies inversely to the square of its radius. 3. If a side of square measures 5mm, which of the following is its perimeter? 2 2 a. 20mm b. 25mm c. 20 mm d. 25mm 2 4. If a square has an area of 144m , which of the following is the measure of its side? a. 12mm b. 12cm c. 1,200 cm d. 1.2m 5. If the base of a triangle measures 20 centimeters and its height measures 1 inch, which of the following is its area? a. 20 cm b. 25.4 cm c. 20 cm2 d. 25.4 cm2 A string 1mm long is used to form a shape that would give a maximum area. What shape is it? a. square b. rectangle c. circle d. parallelogram 7. If the length of the side of a square is reduced to half, what will happen to its perimeter? a. It will be tripled. c. It will be doubled. b. It will remain the same. d. It will be reduced to half. 8. The area of a new circle is nine times its srcinal area. What made this happen? a. The radius was tripled. c. The value of π was squared. b. The radius was multiplied by 9. d. The radius was reduced to half. 9. A regular polygon is inscribed in a circle. Suppose we increase continuously the number of its sides, what do you think will happen? a. The regular polygon with infinitely many sides will never exist. b. It is impossible for a regular polygon with infinitely many sides to be inscribed in a circle. c. The perimeter of the regular polygon will be greater than the circumference of the circle. d. The perimeter of the regular polygon will be approximately equal to the circumference of the circle. 10. Which of the following is the most reasonable weight of an adult woman? a. 12 lbs b. 120 lbs c. 1,200 lbs d. 12,000 lbs 11. Which of the following is the nearest approximation of 10 kilometers? 6.
a. 5 miles b. 10miles c. 16 miles d. 20 miles 12. Consider a rectangle with a fixed perimeter. Assume that we allow continuous variations of either its width of length. Which of the following is NOT possible? a. The area of the new rectangle will be greater than the srcinal area. b. The area of the new rectangle will be smaller than the srcinal area. c. The area of the srcinal rectangle will be equal to the area of all new rectangles that can be formed. d. The area of the srcinal rectangle will be equal to the area of some new rectangles that can be formed. 13. What is the reason for your answer in item number 12? a. Rectangles possibly have the same perimeter but of different area. b. Rectangles with the same perimeter always have the same area. c. If rectangles have larger perimeter, then their area is wide. d. If rectangles have small perimeter, then their area is small. 14. A largest circle is to be made out of a square having sides of 20 cm. Approximate the materials that will be wasted. a. 400.00 sq. cm b. 314.00 sq. cm c. 86.00 sq. cm d. 22.80 sq. cm 15. A circle with radius 12 cm and a rectangle with width 16 cm have equal areas. Which of the following is the approximation of the length of the rectangle? a. 4.19 cm b. 9.00 cm c. 4.71 cm d. 28.26 cm 16. 2. One hectare is equivalent to 10,000 square meters. How many hectares are in a rectangular field which is 750 m wide and 800 m long? A. 120 B. 56 C. 60 D. 40 17. An aquarium is 40 cm high. Its length is twice its height, while its width is half its height. How much water is needed to fill in the aquarium in cu. cm.? A. 64,000 B. 32,000 C. 16,000 D. 8,000 18. What is the volume of a cube if one face has a perimeter of 36 cm? A. 729 cu. cm. B. 216 cu. cm. C. 46,656 cu. cm D. 5,832 cu. cm 19. How much water can be filled with a cone whose diameter is 24 cm and whose height exceeds 2/3 of the radius by 2 cm? A. 480 cu. cm. B. 540 cu. cm. C. 660 cu. cm. D. 900 cu. cm.
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LICENSURE EXAMINATION FOR TEACHERS (LET)
Refresher Course Content Area: MATHEMATICS Focus: Algebra Prepared by: Daisy de Borja‐Marcelino
LET Competencies: Solve for the roots of a given quadratic equation Solve problems on quadratic equations Determine an equation given a set of roots which are imaginary/complex numbers Perform operations involving exponential and logarithmic functions Solve for the solution set of a given inequality n Determine the rth term of the expansion (a + b) Solve problems involving variations Determine the number of positive and negative roots of a given polynomial
Equations An equation that contains at least one variable is called an open sentence. Equations b & c above are examples of open sentences. In equation b, only ‐1 makes the sentence true or satisfies the equation. However, more than one number might satisfy an equation. For example, +2 and ‐ 2 satisfy the equation x 4 0 . Any number that satisfies an equation is called a solution or root to the equation. The set of numbers from which you can select replacements for the variable is called the replacement set. The set of all solutions to an equation is called the solution set to the equation. To 2
solve an equation means to find all of its solutions. QUADRATIC EQUATION 2 An equation of the form ax + bx + c = 0 where a 0, a ,b, and c are constants, is a quadratic equation. ROOTS OF QUADRATIC EQUATIONS To solve a quadratic equation means to find the value of x (unknown) that will satisfy the given equation. The values of x that will make the equation true are called the roots or solution of the quadratic equation. Methods of Finding the Roots of a Quadratic Equation 1. Factoring (Use this method if the quadratic equation is factorable) 2
Example: Determine the roots of x ‐ 8x ‐ 15 = 0 Solution: Factoring the left side of the equation, (x ‐ 5) (x – 3) = 0 Equating each factor to zero, (x ‐ 5) = 0 (x – 3) = 0 x=5 x=3 2. Quadratic Formula The quadratic formula is
x
b b 2 4ac 2a
BINOMIAL FORMULA n
To obtain the terms of the binomial expansion (a + b) , we use the binomial formula:
n
(a + b) = a
n
na n 1 b
n ( n 1)a n 2 b 2 2!
n ( n 1)( n 2)a n 3 b 3 3!
... nab n 1 b n
n
THE rth TERM OF THE EXPANSION (a + b)
rth term =
n ( n 1)(n 2)...(n r 2)a n r 1 b r 1 ( r 1)!
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EXPONENTIAL FUNCTIONS
The exponent i al f unct i on f wi t h base b i s denot ed by f(x) = bx, wher e b >0 , b 1, and x i s any r eal number . x
Properties of f(x) = b
f has the set of real numbers as its domain.
f has the set of positive real numbers as its range.
f has a graph with a y‐intercept of (0,1).
f is a one‐to‐one function.
f has a graph asymptotic to the x‐axis.
f is an increasing function if b>1 and f is a decreasing function if 0
An exponential function has a constant base and a variable exponent.
The Natural Exponential Function
For all real numbers x, the function defined by
x f ( x) e is called the natural exponential function. Note that e
is an irrational number and its accurate value to eight places is 2.71828183.
EXPONENTIAL EQUATIONS An equation where the unknown quantity appears in an exponent is called an exponential equation. SOLVING EXPONENTIAL EQUATIONS To solve an exponential equation is to find the value of the unknown quantity in the given equation. LOGARITHMIC FUNCTIONS
For x > 0, onl y i f ay = x.
b > 0, and b
1, we have y = log a (x) i f and
Note: 1. If the base of the logarithm is not indicated it is understood that the base is 10. 2. If the base of the logarithm is the number e, then it is called a natural logarithm and it is written as f(x) = ln x. Properties of
f x logb x
f has the set of positive real numbers as its domain. f has the set of real numbers as its range. f has a graph with a x‐intercept of (1,0). f is a one‐to‐one function. f has a graph asymptotic to the y‐axis. f is an increasing function if b>1 and f is a decreasing function if 0
Remarks
Logarithmic functions are the inverse of exponential functions. We can use the rules of exponents with logarithms. The two most common logarithms are called common logarithms and natural logarithms. Common logarithms have a base of 10, and natural logarithms have a base of e. Equation in exponential form can be rewritten in logarithmic form, and vice versa.
The exponent i al f or m of
is
The l ogar i t hmi c f or m of
is
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2
Example1: The exponential equation 7 = 49 may be written in terms of a logarithmic equation as log7 (49) = 2.
‐3
Example 2: The exponential equation 9
=
1 93
or
1 729
may be written in terms of a logarithmic equation as log9
1 = ‐3 729 Basic Properties of Logarithms 0
Property 1 : loga (1) = 0 because a = 1. 0
Example 1: In the equation 22 = 1, the base is 22 and the exponent is 0. Remember that a logarithm is an exponent, and the corresponding logarithmic equation is log22 (1) = 0, where the 0 is the exponent.
Example 2: In the equation
2 5
0
= 1, the base is
2 5
and the exponent is 0. Remember that a logarithm is an exponent,
and the corresponding logarithmic equation is log 2 1 = 0. 5 1
Property 2: loga (a) = 1 because a = a 1
Example 3: In the equation 7 = 7, the base is 7, the exponent is 1, and the answer is 7. Since a logarithm is an exponent, and the corresponding logarithmic equation is log7 7 = 1 1
Example 4: Use the exponential equation m = m to write a logarithmic equation. If the base m is greater than 0, then logm (m) = 1. x
x
x
Property 3: loga (a) = x because a = a 2
2
2
Example 5: Since 9 =9 , we may write the logarithmic equation with base 9 as log9 9 = 2. 2
2
2
Example 6: Since you know that 11 =11 , we may write the logarithmic equation with base 11 as log1111 = 2. INEQUALITIES Any relation expressed using the symbols <, >, > or < is called an inequality. An absolute inequality is an inequality which is always true. A conditional inequality is one which is true only for certain values of the variable involved. 1. 4 > 3 is an absolute inequality 2. x > 3 is a conditional inequality
PROPERTIES OF INEQUALITIES Let a, b, c, & d be real numbers. The following hold. 1. Trichotomy Property a > b or a < b or a = b 2. a > b if a ‐ b > 0 a < b if a – b < 0 3. a. If a> 0 and b> 0, then a + b> 0 and ab>0. b. If a < 0 and b < 0, then a+b< 0 and ab> 0 4. Transitivity If a < b and b < c then a < c. 5. Addition Property If a < b and c < d, then a + c < b+ d 6. Multiplication Property St. Louis Review Center-Inc-Davao Tel. no. (082) 224-2515
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If a < b and c > 0, then ac < bc If a < b and c < 0, then ac > bc SOLVING INEQUALITIES To solve an inequality means to find the value of the unknown that will make the inequality true. POLYNOMIAL FUNCTION The function defined by the equation n
n‐1
f(x) = a0x +a1x
n‐ 2
+ a2x
2
+ . . .+ an‐2x + an‐1x + an
where n is a nonnegative integer and a0, a1,
. . ., an
are constants, a0,
0 is a polynomial function in x of degree n.
The zeros or roots of f(x) are the numbers that will make f(x) = 0.
The Number of Positive and Negative Roots of a Polynomial Function If f(x) is a polynomial function with real coefficients, then the following are true. The number of positive real zeros of f(x) is either equal to the number of variations in sign in f(x), or to that number
diminished by a positive even integer.
The number of negative real zeros of f(x) is either equal to the number of variations in sign in f(‐x), or to that number
diminished by a positive even integer. LICENSURE EXAMINATION FOR TEACHERS (LET)
Refresher Course Majorship: MATHEMATICS Prepared by: Daisy de Borja‐Marcelino FOCUS: Mathematical Investigation and Problem Solving LET COMPETENCIES:
1. Cite differences between problem solving and mathematical investigations. 2. State patterns observed as conjectures. 3. Solve non‐routine problems. CONTENT UPDATE I. Problem Solving
Problem solving is defined as a set of actions to be done to perform the task or to solve the problem. It is a process of applying acquired knowledge to a new or unfamiliar situation. I. Polya’s Problem‐Solving Principles 1. Understand the problem
Understanding the given problem is a very important principle in order to solve it correctly. The following questions may be helpful in the analysis of a given problem: Do you understand all the words used in stating the problem? What are you asked to find or show? Can you restate the problem in your own words? Could you work out some numerical examples that would help make the problem clear? Could you think of a picture or diagram that might help you understand the problem? Is there enough information to enable you to find a solution? Is there extraneous information? What do you really need to know to find a solution? Is there yet another way to state the problem? What does key word really mean? 2. Devise a plan
The following strategies may be of great help as you learn the art of problem solving. ● guess and check ● make a table orderly list ● ● make draw aan diagram ● look for a pa ern
variable ● ● use worka backward ● eliminate possibilies
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3. Carry out the plan
To carry out the plan you devised earlier be careful and be patient to make it work. If it doesn’t work after several trials, then discard it and try a new strategy. 4. Look back
Looking back is an important step in developing problem‐solving skills. Once you have solved the problem make it a habit to go over your solution and polish it. II. Mathematical Investigation
A mathematical investigation of a problem or of an open‐ended situation is a sustained exploration of the problem or situation. Stages in a Mathematical Investigation 1. Getting Started ‐ Attaining familiarity with the situation to be investigated. ‐ Producing instances, maybe starting from the simplest or whatever is interesting. ‐ Deciding on what is worth pursuing. 2. Exploring Systematically ‐ Systematic listing/ drawing; ‐ Organizing relationships in tables or graphs; and ‐ Looking for a pattern or relationship. 3. Making Conjecture ‐ Making general statements about patterns or relationships observed in the cases considered. A conjecture is a generalization obtained inductively, which has not been validated or proven true. 4. Testing/ Verifying Conjectures ‐ Checking consistency of conjectures using existing cases; ‐ Predicting results for untried cases for which data are available. 5. Explaining/ Justifying Conjectures ‐ Explaining why the conjectures made will work for new or all cases 6. Reorganizing ‐ Simplifying/ generalizing the approach ‐ Seeing the connection among the conjectures 7. Elaborating ‐ Extending the investigation by considering other aspects of the investigation 8. Summarizing ‐ Involves an account or summary, written or oral, of what has been obtained in stages 2 – 7, with some reference on the experiences in stage 1.
MAJORSHIP: MODERN GEOMETRY LET COMPETENCIES:
1. 2. 3. 4. 5.
Give characteristics of non‐Euclidean geometry which are not found in Plane Euclidean Geometry. Define and illustrate concepts in linear algebra Apply properties of matrices in performing matrix operations Evaluate determinants Perform modular clock arithmetic
MODERN GEOMETRY Non‐Euclidean geometry Non‐Euclidean Geometry is any geometry that is different from Euclidean geometry. The two most common non‐ Euclidean geometries are elliptic geometry and hyperbolic geometry. A. Hyperbolic Geometry
Hyperbolic geometry is known as saddle geometry or Lobachevskian geometry. It differs in many ways from Euclidean geometry, often leading to quite counter‐intuitive results. Some of the remarkable consequences of this geometry's unique fifth postulate include: Moreover, in this field, more than one distinct line through a particular point will not intersect another given line.. St. Louis Review Center-Inc-Davao Tel. no. (082) 224-2515
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1. The sum of the three interior angles of a triangle is strictly less than 180°. Moreover, the angle sums of two distinct triangles are not necessarily the same. 2. Two triangles with the same interior angles have the same area. B. Elliptic Geometry
In Elliptic geometry there are no lines that will not intersect, Some theorems in Elliptic Geometry o
1.
The angle sum of any triangle is more than 180 .
2.
Given two lines perpendicular to line CG. By the parallel postulate for elliptic geometry, these two lines meet at a point A. Then every line through A is perpendicular to line CG.
C. Projective Geometry
Projective geometry is the most general and least restrictive in the hierarchy of fundamental geometries. It is an intrinsically non‐metric geometry, whose facts are independent of any metric structure. Under the projective transformations, the incidence structure and the cross‐ratio are preserved. In particular, it formalizes one of the central principles of perspective art: that parallel lines meet at a point called an ideal point. Consequently, the five initial axioms in Euclidean Geometry resulted to the following axioms. 1. 2. 3. 4. 5.
Any two distinct points determine one and only one line. Any two distinct coplanar lines intersect in one and only one point. Any line not in a given plane intersects the plane in one and only one point. Any two distinct planes intersect in one and only one line. Any three noncollinear points, also any line and a point not on the line, determine one and only one plane.
MATRICES AND MATRIX OPERATIONS Definition
A matrix is defined as a rectangular array of elements. The entries, also called elements, may be real, complex or functions. If the arrangement has m rows and n columns, then the matrix is of order m x n (read as m by n). A matrix is enclosed by a pair of parameters such as ( ) or [ ]. It is denoted by a capital letter.
a11 a 21 A am1
a12
a 22
aij
am 2
a1n
amn a2 n
A = [ aij ] TYPES OF MATRICES 1. The ROW MATRIX: This matrix has only one row. Example: [1 6 4 3] This is a 1 X 4 row matrix 2. The COLUMN MATRIX: This matrix has only one column.
Example:
10 2 7 8
This is 4 x 1 column matrix
3. The RECTANGULAR MATRIX: This has two or more rows with two or more columns.
Example
2 3 5 2 0 8 1 4
This is a 2 X 4 matrix, because it contains two rows and four
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8 11 22 0 7 6 95 10 0 45 49 68
This is a 3 X 4 matrix
4. The SQUARE MATRIX: This is a special case of a Rectangular Matrix; here the number of rows is equal to the number of columns.
Example:
A =
a d g
b
c
e
f
i
h
B=
1 5 8 8
3
2
4
1
3
1
1 1 1 5 3 1
Here A and B are square matrices of order 3 and 4 respectively 5. The DIAGONAL MATRIX: This is a square matrix where all its non‐diagonal elements are 0 Example:
(a)
8 0 0 (b) 0 1 0 0 0 8
92 0 0 1
(c)
5 0 0 0
0
0
0
6
0
0
0 3 0 0 0 10
These are diagonal matrices of order 2, 3 and 4 respectively. 6. The SCALAR MATRIX: This is a diagonal matrix where all the elements on its leading diagonal to bottom right are of equal value. Example:
(a)
2 0 0 2
(b)
3 0 0 0 3 0 0 0 3
(c)
6 0 0 0
0
0
0
6 0
0 6
0 0
0
0
6
7. The IDENTITY MATRIX: This is a scalar matrix where the elements on its leading diagonal (the diagonal running from top left to bottom right) are 1 and the rest are of value 0 Example:
1 0 0 0
0 0
0
1
0
0 1 0 0 0 1 0
This is an identity matrix I4 of order 4.
PROPERTIES OF MATRIX ADDITION Theorem Let A, B, C, and D be matrices of the same size, m x n.
1. A + B = B + A
(Commutativity)
2. A + (B + C) = (A + B ) + C
(Associativity)
3. There is a unique m x n matrix O such that A+ 0 = A for any m x n matrix A. The matrix O is called the m x n additive identity or zero matrix.
4. To each m x n matrix A, there is a unique m x n matrix D such that A+D=0 We write D as (‐ A), so that A + D = 0 can be written as A + (‐A) = 0. The matrix (‐A) is called the additive inverse or negative of A.
PROPERTIES OF MATRIX MULTIPLICATION
Theorem Let A, B, and C be matrices of the appropriate sizes.
1. A(BC) = (AB)C 2. A(B + C) = AB + AC
(Associativity) (Right Distributivity)
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3. (A + B)C = AC + BC
(Left Distributivity)
PROPERTIES OF MULTIPLICATION BY A SCALAR Theorem Let A and B be matrices of the appropriate sizes, and let r and s be scalars.
1. r(sA) = (rs)A (Associativity) 2. (r + s)A = rA + sA (Distributivity I) 3. r(A + B) = rA + rB (Distributivity II) 4. A(rB) = r(AB) = (rA)B
Definition
Let A = [ai j ] be an n x n matrix. We define the determinant of A (written det(A) or A
det A =
A
=
(
)
a
1 j1
a
2
j2
... a
n
) by
jn
where the summation ranges over all permutations j1 j2 …jn of the set S = {1,2,…n}.The sign is taken as + or – according to whether the permutation j1 j2…jn is even or odd.
Second‐order Determinant
If A is the square matrix of order two
either det A or a1 a2
a1 a 2
b1 ,is defined by a1 a2
b2
b1
then the determinant of A, denoted by
b2
b1 = a b – a b 1 2 2 1 b2
Example:
Compute the determinant:
PROPERTIES OF DETERMINANT T
1. The determinant of a matrix and its transpose are equal, that is, det(A ) = det(A). 2. If two rows (columns) of A are equal, then det (A) = 0 3. If a row (column) of A consists entirely of zeros, then det (A) = 0
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4.
The determinant of a diagonal matrix is the product of the entries on its main diagonal.
5. If matrix A = [ai j] is upper (lower) triangular, then det (A) =a11a22…ann; that is The determinant of a triangular matrix is the product of the elements on the main diagonal. 6. If a multiple of one row of A is added to another row to produce a matrix B, then det B = det A.
7. If two rows of A are interchanged to produce B, then det B = ‐ det A.
8. If one row of A is multiplied by k to produce B, then det B = k det A.
9. The determinant of a product of two matrices is the product of their determinants; That is, det(A B) = det(A) det(B)
MODULARCLOCKARTITHMETIC
Clock (or modular) arithmetic is arithmetic you do on a clock instead of a number line. On a 12‐hour clock, there are only twelve numbers in the whole number system. However, every number has lots of different names. For example, the number before 1 is 0, so 12 = 0 on a 12‐hour clock
In clock arithmetic, you can add, subtract, and multiply, you can divide by some numbers. Additionand Subtraction
Addition and subtraction work the same as on number line. For example, to add 9 and 7, start at 0, count 9 along the line, and then count 7 more. You are at 16.If you count on a 12‐hour clock, you will be at 4. To add negative numbers, use the minus (‐) sign to change direction. To subtract on a clock, first find standard (positive) names for the two numbers, count clockwise for the first one, and count counter clockwise for the second.
In ordinary arithmetic, the additive inverse of 4 is ‐4. In mod 12 arithmetic, the additive inverse of 4 is 8. In either system, the sum of a number and its additive inverse is zero. Rows corresponding to additive inverses are opposites of one another, save for the number 0.
Addition Mod 12 ( 12‐ hour clock)
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What branch of philosophy is closely related to education because it is concerned with human knowledge in general and the criteria of truth?
13 The educational theory, in response to progressivism, which concerns with the revival of effort in the direction of teaching the fundamental tools of learning as the most indispensable type of education a. Essentialism c. Philosophical analysis b. Perennialism d. Social traditionalism 14 This philosophy believes that the schools should srcinate policies and progress which would bring about reform of the social order a. reconstructionism c. Existentialism b. Progressivism d.
a. Metaphysics c. Epistemology b. Logic d. Ethics This is the philosophical study of the universe and everything in it a. Cosmology c. Metaphysics b. Ethics d. Axiology What philosophy posits that a social teacher is a revered person, central to the educative process and therefore must be excellent mentally and morally in personally conduct and conviction? a. Realism c. Essentialism b. Idealism d. Progressivism What would the aim of a realist education likely be? a. Contribute to the development of mind and self b. Provide students with the knowledge he will need to survive in the natural world c. Give students insights into their traditions and
Perennialism 15 This is the logical conclusion if we are to apply the philosophy of martin Buber to education a. Education will be characterize by imposition and rules b. Education will be characterize by mutual unfolding of persons c. Education will be characterize by pretensions d. Education will be characterize by self motivation 16 Man as an embodied spirit means a. Man is a body and soul b. Man is a rational animal c. Man is a totally, unique core or center, source wellspring of initiative and meaning d. Man is subjective 17 Man as an embodied sprit implies that education should
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POST TEST SOCIAL SCIENCES L.E.T APRIL 2010
1
2
3
4
adapting students to society a. Develop the intellect / mind Meet the need of a growing child rather than b. Develop man’s individuality emphasize the subject matter c. Develop essential physical and intellectual skills 5 This philosophy emphasizes the teaching of d. Develop the total man controversial issues to prepare students for a 18 human beings are living in a human world implies progressive structuring of the social order that a. Progressivism c. Reconstructionism a. Education must help man give meaning to his b. Social traditionalism d. Social Experimentalism world of others 6 The philosophy whose criterion of truth is “ seeing b. Education must develop skills to survive in this is believing” world a. Essentialism c. Progressivism c. Education must be relevant to this world b. Native realism d. Existentialism d. Education must be in the context of societal 7 This is the systematic consistent explanation of all needs the facts of experience. Its technical term is reason 19 This philosophy contends that the intellectual which is considered as the best criterion of truth faculty of man is higher that the physical faculties a. Pragmatism c. Correspondence a. Idealism c. Existentialism b. Consistency d. Coherence b. Realism d. Logical Positivism 8 This philosophy contends that truth is universal an 20 The philosophical study of the essence and nature unchanging and therefore a good education is also of man universal and constant a. Rational psychology a. Perennialism c. Existentialism b. Epistemology b. Progressivism d. Essentialism c. Philosophy of human person 9 Which philosophy has for its curriculum a body d. Social Philosophy intellectual subject matter that are conceptual on 21 The branch of philosophy that studies the morality subject, essential for the realization of mental and of human acts moral development? a. Aesthetics c. Ethics a. Realism c. Existentialism b. Cosmology d. Theodicy b. Idealism d. Pragmatism 22 This is the philosophical study of human knowledge 10 A branch of Philosophy that studies the morality of and the criteria of truth human action a. Axiology c. Logic a. Epistemology c. Ethics b. Epistemology d. Metaphysics b. Metaphysics d. Cosmology 23 This contemporary philosophical theme in 11 The area of Philosophy that specifically deals with education contends that education must involve in the problem of human values its processes the total human being‐his subjectivity, a. criteriology c. Theodicy the rational, the material and the emotional aspect b. Epistemology d. Axiology for man is a unity and not a duality 12 The aim of this philosophy is to reduce statements a. Genuine dialogue about education to empirical terms b. Man as a being in the world a. Reconstructionism c.Philosophical c. Man as an embodied spirit analysis d. The socials and the inter human being b. Existentialism d. Social traditionalism St. Louis Review Center-Inc-Davao Tel. no. (082) 224-2515 83 d.
24 This philosophy proclaims the spiritual nature and the universe. It holds that the good, the true and the beautiful are permanently part of the structure of an unchanging universe a. Existentialism c. Materialism b. Idealism d. Realism 25 This philosophy posits the know ability of the world and everything in its as they are in themselves and their existence independent of human mind a. Existentialism c. Materialism b. Idealism d. Realism 26 “The central aim of education should be to develop the power of thought” this would likely be the aim
36 It is a systematic consistent explanation of all the facts of experience. Its technical name is reason. This is believed to be the ultimate criterion of truth a. Coherence c. Naïve Realism b. Consistency d. Pragmatism 37 This educational philosophy encourages accumulation of knowledge and thinking and must apply criteria for moral evaluation. Suggested methods are questioning and discussion, lecture and the project, whether done singly or in group a. Existentialism c. Materialism b. Idealism d. Pragmatism 38 This theory views education as a recurring process
of what educational practitioner? a. Existentialism c. Progressivist b. Perennialist d. Reconstructivist 27 This educational theory is based on the philosophy grounded on experience and the interaction of the person with his environment. It posits that education must use past experiences to direct future experiences a. Existentialism c. Realism b. Idealism d. progressivism 28 This philosophy is a way of viewing and thinking about life in the world so that priority is given into individualism and subjectivity. It believes that human beings are the creator of their own experiences a. Existentialism c. Realism b. Idealism d. Reconstruction 29 This emerging social values in education stresses
based on eternal truths; thus, the school’s curriculum should emphasize the recurrent themes of human life a. Perennialism c. Progressivism b. Pragmatism d. Reconstructionism 39 The philosophical study of human mind a. Axiology c. Metaphysics b. Epistemology d. Rational psychology 40 The philosophical study of being, its nature and essence a. Axiology c. Metaphysics b. Epistemology d. Rational Psychology 41 A sociologist would be most interested in studying a. Artifacts of the fast civilization b. Group behavior c. Psychological problems d. Economic institution
that man exist through the other and for others for he becomes actuated through relations a. Existential Dialogue c. Socialization b. Personalization d. Vigilance 30 To what particular concept must education be based according to contemporary philosophy? a. Existential dialogue b. Social Change c. Supremacy of the human person d. Synergy 31 This emerging social value in education stresses the importance if the “we‐experience” ad the converging of the worlds a. Authentic being c. Personalization b. Existential dialogue d. Synergy 32 The essentialist would likely have this as the ultimate aim of education a. To fit man to perform, justly, skillfully and
42 Interrelated social structures that fit together to form an integrated whole a. Structural functionalism b. Conflict theory c. Symbolic Interactionism d. Disruptive Functionalism 43 Development of self and adaptation of individual to society a. Structural functionalism b. Conflict theory c. Symbolic Interactionism d. Disruptive Functionism 44 Which is NOT a method in sociological inquiry a. Experimental study c. Intuitive study b. Longitudinal Study d. Ex‐post Facto study 45 Which of the following give some difficulties to a sociological researcher?
magnanimous in all times in peace and war To develop the power of thought. To search for an disseminate the truth c. To give all pupils insight into their traditions d. To meet the needs of the growing child 33 The philosophical belief that reality is precisely what as it appear to be and adheres to the belief that “seeing is believing” a. Coherence c. Naïve Realism b. Consistency d. Pragmatism 34 It is belief that when an idea agrees with its object, it is proof of its truth. However, it is definition of truth not a criteria a. Coherence c. Naïve Realism b. Consistency d. Correspondence 35 The belief that the ultimate criterion of truth is if an idea works then it is true a. Coherence c. Naïve Realism b. Consistency d. Pragmatism
a. Complexity of phenomena b. Constant change c. Unpredictability of behavior d. All of the above A cluster of behavior patterns related to the general culture of a society and yet distinguishable from it a. Sub‐culture c. Norm b. Culture shock d. mores Which group would most likely have a counter culture? a. Musician c. Singers b. Drugs users d. Dancers The complex whole which includes knowledge, beliefs, arts moral, laws, customs and other capabilities and habits acquired by man as a member of the society a. Norm c. Culture b. Folkways d. Mores When one consider the culture of another group as inferior to his own, this connotes a. Ethnocentrism c. Assimilation b. Cultural relativism d. Acculturation
b.
46
47
48
49
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50 The rightness or wrongness of what one does depend on where he is doing it a. Cultural integration c. Cultural relativism b. Cultural lag d. Cultural traits 51 The culture that includes undesirable behavior of the member of society. Formally condemned but widely practiced a. Real culture c. Sub‐culture b. Ideal culture d. Counter culture 52 The process by which a member learns the norms of the group a. Accommodation c. Compartmentalization b. Aggregation d. Socialization
64 Which of the following statements is NOT true about Filipino families today? a. Male authority has declined b. Division of labor has changed c. Separation has become more common d. Has cease to be socialization unit in society 65 Which of the following is NOT an ethical issue that must be confronted when doing social research? a. Are the subjects being paid enough? b. What degree of risk, pain or harm is involved? c. To what extent are the subjects being deceived? d. Will there be disclosure of confidential or
53 A role or status assigned according to heredity traits without regard to individual preference, ability or performance a. Achieved c. Attached b. Ascribed d. Applied 54 An achieved position in a society a. Heir to the throne c. Movie actor b. Presidential son d. First lady 55 Refers to the difficulty people have in meeting their role obligations a. Role conflict c. Role strain b. Role set d. Role ambiguity 56 When a supervisor is tasked to report unsatisfactory performance of her co‐workers who are also her friends. She may experience a. Role play c. Role strain b. Role conflict d. Role ambiguity 57 Social control imposed on social deviants to
personally harmful information? 66 In most instances of participant observation the researchers: a. Hide their true identity b. Do not hide their true identity c. Pay their informants for information d. Acts as therapist to the subject 67 A research technique in which the investigators enter to the activities of the group at the same time they study the groups behavior is a. Participants observation b. A semi‐structured (open‐ended) Interview c. A structured interview d. A data discussion 68 Participant observation: a. A research tool used to follow‐up on expected findings b. A research interview in which the investigators
maintain a. Social stigma c. Social order b. Social strata d. Social mobility 58 Which of the following is NOT true? a. The deviant behavior of one generation may become the norm of the next b. All forms of deviant are bad c. Deviant behavior is one way of adapting a culture to a social change d. Some deviation are due to the failure of the socialization process to integrate the cultural norms into individual’s personality 59 A teenager in a “good” neighborhood of stable families and conventional people who rejects middle class norms and become a delinquent is an example of a. Individual deviation c. Counter deviation b. Group deviation d. Relative deviation
ash a list of questions, but is free to vary them or make up new ones that become important during the course of the interview c. A research technique in which the investigators enter into activities of the group at the same time they study the group’s behavior d. A research interview determined entirely in advance and followed rigidly 69 A research interview determined entirely in advance and followed rigidly is: a. Participants observation b. A semi structured (open‐ended) interview c. A structured interview d. A data discussion 70 A research interview in which the investigators ask a list of questions, but is free to vary them or make up new ones that become important during the course interview, that is
60 A case of compulsive drug addict is an example if a a. Primary deviant c. Tertiary deviant b. Secondary deviant d. All of the above 61 Which is NOT a characteristic of a social group? a. Physical collection of people b. Shares a common purpose and conscious of each other c. Share some common characteristics d. Member interact with one another 62 Which is NOT a characteristic of the relationship among members of a primary group? a. Personal c. Segmental b. Informal d. Sentimental 63 Which is an example of members if a primary group? a. Peer group c. School administration b. Family d. Neighbors
a. Participant observation b. A semi structured (open‐ended) interview c. Structured interview d. A data discussion 71 A conversation between two or more individuals in which one party attempts to gain information from others(s) by asking of questions is known as; a. A survey c. Participant observation b. An interview d. Hypothesis testing 72 Sociological and anthropological be misused because it: a. Is hard to understand b. Is often used selectively by groups or individuals to support certain motives c. Is often ambiguous d. Has few applicants 73 A survey by their nature usually a. Are cross‐sectional b. Are longitudinal c. Involve secondary methodology d. Involve participants observation
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74 A cross‐sectional study is a. A research technique in which investigators enter into activities of the group at the same time they study the groups behavior b. Research examines a population at a given point in a time c. Research which examines a population, or portion thereof, is questioned in order to reveal specific facts about itself d. Research that investigates a population at several intervals over relatively a long period of time 75 Research that investigates a population at several intervals over a relatively long period of time is called a. A cross‐sectional study c. Secondary Analysis b. A residual analysis d. Longitudinal research 76 Survey are used when: a. Other measure do not produce statistical significance b. The findings of the study must be repeatable c. It is desirable to discover he contribution and interrelations of certain variables among large number of people d. Procedures having cross‐cultural applications needed 77 A research method is which a population, or portion thereof, is questioned in order to reveal specific facts itself is known as:
85 If apes can be taught to use language, it denies the views that: a. Only human can think b. Only human have culture c. Apes do not have history to pass on d. Apes are stupid 86 Which of the following statements about culture is not true? a. Every social group must have a culture on its own in order to function b. Every individual participates in a number of different culture c. Meeting the social expectation of several cultures is often source of tree Families do not have their own culture but instead reflect the culture of the larger society Promised to “make this country great again.” Ruled for about twenty years a. Diosdado Macapagal c. Ramon Magsaysay b. Ferdinand Marcos d. Carlos Garcia The form of government established by Aguinaldo was change from dictatorial to: a. Democratic c. Republic b. Sociologist d. Revolutionary Which provides he Tagalog shall be the official language of the republic? a. Constitution of 1935 c. Biyak na Bato b. Makabula d. Malolos Which of the following were the greatest naval battles in history during Japanese occupation of the Philippines? d. 87
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a. A survey c. Experiments b. Participant observation d. Residual analysis 78 Which of the following is NOT one of the three main methods of research used by sociologists? a. A survey c. Experiments b. Participants observation d. Residual analysis 79 A statistical technique to make all significant groups in as society represented in a sample proportion to their numbers in the large society is known as: a. Random sampling b. A stratified random sample c. Sampling d. Randomized response 80 The goal in designing a research project should be: a. To prove a point b. To provide a collection for all necessary and sufficient data to test the stated hypothesis c. To selectively isolate the variables in order to
a. Battle of Leyte gulf c. Claveri b. Surigao Strait d. Dela Torre 91 The most cultured of the reformist, wrote a socio‐ historical novel based on facts he gathered in the Philippines a. Marcelo H. Del Pilar c. Mariano Ponce b. Jose Rizal d. Jose Ma. Paganiban 92 The editor of La Solaridad and a great orator, he initiated the reform movement a. Marcelo H. Del Pilar c. Emilio Jacinto b. Jose Rizal d. Graciano Lopez‐ Jaena 93 The generation of the spirits was common among ancient Filipinos. Memories of their dead relatives were kept alive through craved idol. This practice was called a. Cult of the dead c. Nature Worship
prove the hypotheses are either correct or incorrect To prove hypothesis false. If they cannot proven false, they must be true An independent variable is a. A testable statements about the relationship between two or more empirical variables b. Anything that can be change c. A variable that changes for reason that have nothing to do with another variable d. A variable that changes in response to changes in another variable Which of the following is not a sign? a. A clenched first c. A flag b. A knock on the door d. Yawning The most important symbols are a. Action c. Actions b. Words d. Behavior The principal means through which culture is transmitted from generation to generation is a. Actions c. Language b. Diffusion d. Behaviors
b. Mass of the dead d. Divination 94 The practice of having surnames was started during Spanish regime through the order of a. Gov. Gen Jaudenes c. Gov. Gen Claveria b. Gov. Gen Dela Torre d. Gov. Gen Polavejia 95 As promised by Gen. Mc Arthur‐ he would return to liberate the Filipinos after he could train troops, in Australia. The Leyte landing signified the liberation of the Philippines, was well as the return of the commonwealth government. Who was the commonwealth president who came with Mc Arthur? a. Romulo c. Osmeňa b. Roxas d. Quezon 96 Men and women of pre‐colonial times had weakness for personal adornment. A side from jewels, tattooing of the body was common. In the Visayas tattooed men were called a. Umalohokan c. Abi b. Pintados d. Plista
d. 81
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97 Memories of brutality which was so in human during the Japanese regime were always associated with a particular Japanese group called a. Kalibapi c. Kamikaze b. Kempetai d. Samurai 98 Membership in the Katipunan come in grades: first (katipon’s); second (kawal); and third (bayani). Each grade designed as well as password so they could maintain their secrecy of the movement. The password for the kawal grade was? a. Rizal c. Kawal b. Gomburza d. Sundalo 99 Pre‐colonial trade in the Philippines was
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d a c d a b b b a
prosperous. Business transactions made use of: a. Money system c. Credit system b. Barter system d. Banking System 100 The use of magic charms was very common among pre‐Spanish Filipinos. One of these is the anting‐ anting or agimat which is meant to: a. Insure a man against a weapon of every kind b. Made a man lovable to all ladies c. Made a man invisible d. Made him walk in storm or sea without getting wet ***** T H E E N D ***** WORK HARD, DREAM HARDER Social science 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
C A B B C B D A A C D C A A B C D A A C C B C B d b d a c c d a c c d a
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a d b c c b c b a a a c c d a a a c c b b b c b d c a d b b c c d d b d
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101 What branch of philosophy is closely related to education because it is concerned with human knowledge in general and the criteria of truth? a. Metaphysics c. Epistemology b. Logic d. Ethics 102 This is the philosophical study of the universe and everything in it a. Cosmology c. Metaphysics b. Ethics d. Axiology
114 This philosophy believes that the schools should srcinate policies and progress which would bring about reform of the social order a. reconstructionism c. Existentialism b. Progressivism d. Perennialism 115 This is the logical conclusion if we are to apply the philosophy of martin Buber to education a. Education will be characterize by imposition and rules b. Education will be characterize by mutual unfolding of persons c. Education will be characterize by pretensions d. Education will be characterize by self
103 What philosophy posits that a social teacher is a revered person, central to the educative process and therefore must be excellent mentally and morally in personally conduct and conviction? a. Realism c. Essentialism b. Idealism d. Progressivism 104 What would the aim of a realist education likely be? a. Contribute to the development of mind and self b. Provide students with the knowledge he will need to survive in the natural world c. Give students insights into their traditions and adapting students to society d. Meet the need of a growing child rather than emphasize the subject matter 105 This philosophy emphasizes the teaching of controversial issues to prepare students for a progressive structuring of the social order
motivation 116 Man as an embodied spirit means a. Man is a body and soul b. Man is a rational animal c. Man is a totally, unique core or center, source wellspring of initiative and meaning d. Man is subjective 117 Man as an embodied sprit implies that education should a. Develop the intellect / mind b. Develop man’s individuality c. Develop essential physical and intellectual skills d. Develop the total man 118 human beings are living in a human world implies that a. Education must help man give meaning to his world of others b. Education must develop skills to survive in this
a. Progressivism c. Reconstructionism b. Social traditionalism d. Social Experimentalism 106 The philosophy whose criterion of truth is “ seeing is believing” a. Essentialism c. Progressivism b. Native realism d. Existentialism 107 This is the systematic consistent explanation of all the facts of experience. Its technical term is reason which is considered as the best criterion of truth a. Pragmatism c. Correspondence b. Consistency d. Coherence 108 This philosophy contends that truth is universal an unchanging and therefore a good education is also universal and constant a. Perennialism c. Existentialism b. Progressivism d. Essentialism 109 Which philosophy has for its curriculum a body intellectual subject matter that are conceptual on subject, essential for the realization of mental and moral development? a. Realism c. Existentialism b. Idealism d. Pragmatism 110 A branch of Philosophy that studies the morality of human action a. Epistemology c. Ethics b. Metaphysics d. Cosmology 111 The area of Philosophy that specifically deals with the problem of human values a. criteriology c. Theodicy b. Epistemology d. Axiology 112 The aim of this philosophy is to reduce statements about education to empirical terms a. Reconstructionism c. Philosophical analysis b. Existentialism d. Social traditionalism 113 The educational theory, in response to progressivism, which concerns with the revival of effort in the direction of teaching the fundamental tools of learning as the most indispensable type of education a. Essentialism c. Philosophical analysis b. Perennialism d. Social traditionalism
world Education must be relevant to this world Education must be in the context of societal needs 119 This philosophy contends that the intellectual faculty of man is higher that the physical faculties a. Idealism c. Existentialism b. Realism d. Logical Positivism 120 The philosophical study of the essence and nature of man a. Rational psychology b. Epistemology c. Philosophy of human person d. Social Philosophy 121 The branch of philosophy that studies the morality of human acts a. Aesthetics c. Ethics b. Cosmology d. Theodicy
POST TEST SOCIAL SCIENCES L.E.T APRIL 2010
c. d.
122 This is the philosophical study of human knowledge and the criteria of truth a. Axiology c. Logic b. Epistemology d. Metaphysics 123 This contemporary philosophical theme in education contends that education must involve in its processes the total human being‐his subjectivity, the rational, the material and the emotional aspect for man is a unity and not a duality a. Genuine dialogue b. Man as a being in the world c. Man as an embodied spirit d. The socials and the inter human being 124 This philosophy proclaims the spiritual nature and the universe. It holds that the good, the true and the beautiful are permanently part of the structure of an unchanging universe a. Existentialism c. Materialism b. Idealism d. Realism
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125 This philosophy posits the know ability of the world and everything in its as they are in themselves and their existence independent of human mind a. Existentialism c. Materialism b. Idealism d. Realism 126 “The central aim of education should be to develop the power of thought” this would likely be the aim of what educational practitioner? a. Existentialism c. Progressivist b. Perennialist d. Reconstructivist 127 This educational theory is based on the philosophy grounded on experience and the interaction of the person with his environment. It posits that
137 This educational philosophy encourages accumulation of knowledge and thinking and must apply criteria for moral evaluation. Suggested methods are questioning and discussion, lecture and the project, whether done singly or in group a. Existentialism c. Materialism b. Idealism d. Pragmatism 138 This theory views education as a recurring process based on eternal truths; thus, the school’s curriculum should emphasize the recurrent themes of human life a. Perennialism c. Progressivism b. Pragmatism d. Reconstructionism
education must use past experiences to direct future experiences a. Existentialism c. Realism b. Idealism d. progressivism 128 This philosophy is a way of viewing and thinking about life in the world so that priority is given into individualism and subjectivity. It believes that human beings are the creator of their own experiences a. Existentialism c. Realism b. Idealism d. Reconstruction 129 This emerging social values in education stresses that man exist through the other and for others for he becomes actuated through relations a. Existential Dialogue c. Socialization b. Personalization d. Vigilance 130 To what particular concept must education be based according to contemporary philosophy?
139 The philosophical study of human mind a. Axiology c. Metaphysics b. Epistemology d. Rational psychology 140 The philosophical study of being, its nature and essence a. Axiology c. Metaphysics b. Epistemology d. Rational Psychology 141 A sociologist would be most interested in studying a. Artifacts of the fast civilization b. Group behavior c. Psychological problems d. Economic institution 142 Interrelated social structures that fit together to form an integrated whole a. Structural functionalism b. Conflict theory c. Symbolic Interactionism
a. Existential dialogue b. Social Change c. Supremacy of the human person d. Synergy 131 This emerging social value in education stresses the importance if the “we‐experience” ad the converging of the worlds a. Authentic being c. Personalization b. Existential dialogue d. Synergy 132 The essentialist would likely have this as the ultimate aim of education a. To fit man to perform, justly, skillfully and magnanimous in all times in peace and war b. To develop the power of thought. To search for an disseminate the truth c. To give all pupils insight into their traditions d. To meet the needs of the growing child 133 The philosophical belief that reality is precisely
d. Disruptive Functionalism 143 Development of self and adaptation of individual to society a. Structural functionalism b. Conflict theory c. Symbolic Interactionism d. Disruptive Functionism 144 Which is NOT a method in sociological inquiry a. Experimental study c. Intuitive study b. Longitudinal Study d. Ex‐post Facto study 145 Which of the following give some difficulties to a sociological researcher? a. Complexity of phenomena b. Constant change c. Unpredictability of behavior d. All of the above 146 A cluster of behavior patterns related to the general
what as it appear to be and adheres to the belief that “seeing is believing” a. Coherence c. Naïve Realism b. Consistency d. Pragmatism 134 It is belief that when an idea agrees with its object, it is proof of its truth. However, it is definition of truth not a criteria a. Coherence c. Naïve Realism b. Consistency d. Correspondence 135 The belief that the ultimate criterion of truth is if an idea works then it is true a. Coherence c. Naïve Realism b. Consistency d. Pragmatism 136 It is a systematic consistent explanation of all the facts of experience. Its technical name is reason. This is believed to be the ultimate criterion of truth a. Coherence c. Naïve Realism b. Consistency d. Pragmatism
culture of a society and yet distinguishable from it a. Sub‐culture c. Norm b. Culture shock d. mores 147 Which group would most likely have a counter culture? a. Musician c. Singers b. Drugs users d. Dancers 148 The complex whole which includes knowledge, beliefs, arts moral, laws, customs and other capabilities and habits acquired by man as a member of the society a. Norm c. Culture b. Folkways d. Mores 149 When one consider the culture of another group as inferior to his own, this connotes a. Ethnocentrism c. Assimilation b. Cultural relativism d. Acculturation 150 The rightness or wrongness of what one does depend on where he is doing it a. Cultural integration c. Cultural relativism b. Cultural lag d. Cultural traits
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151 The culture that includes undesirable behavior of the member of society. Formally condemned but widely practiced a. Real culture c. Sub‐culture b. Ideal culture d. Counter culture 152 The process by which a member learns the norms of the group a. Accommodation c. Compartmentalization b. Aggregation d. Socialization 153 A role or status assigned according to heredity traits without regard to individual preference, ability or performance a. Achieved c. Attached
165 Which of the following is NOT an ethical issue that must be confronted when doing social research? a. Are the subjects being paid enough? b. What degree of risk, pain or harm is involved? c. To what extent are the subjects being deceived? d. Will there be disclosure of confidential or personally harmful information? 166 In most instances of participant observation the researchers: a. Hide their true identity b. Do not hide their true identity c. Pay their informants for information
b. Ascribed d. Applied 154 An achieved position in a society a. Heir to the throne c. Movie actor b. Presidential son d. First lady 155 Refers to the difficulty people have in meeting their role obligations a. Role conflict c. Role strain b. Role set d. Role ambiguity 156 When a supervisor is tasked to report unsatisfactory performance of her co‐workers who are also her friends. She may experience a. Role play c. Role strain b. Role conflict d. Role ambiguity 157 Social control imposed on social deviants to maintain a. Social stigma c. Social order b. Social strata d. Social mobility 158 Which of the following is NOT true?
d. Acts as therapist to the subject 167 A research technique in which the investigators enter to the activities of the group at the same time they study the groups behavior is a. Participants observation b. A semi‐structured (open‐ended) Interview c. A structured interview d. A data discussion 168 Participant observation: a. A research tool used to follow‐up on expected findings b. A research interview in which the investigators ash a list of questions, but is free to vary them or make up new ones that become important during the course of the interview c. A research technique in which the investigators enter into activities of the group at the same time they study the group’s behavior
a.
The deviant behavior of one generation may become the norm of the next b. All forms of deviant are bad c. Deviant behavior is one way of adapting a culture to a social change d. Some deviation are due to the failure of the socialization process to integrate the cultural norms into individual’s personality 159 A teenager in a “good” neighborhood of stable families and conventional people who rejects middle class norms and become a delinquent is an example of a. Individual deviation c. Counter deviation b. Group deviation d. Relative deviation 160 A case of compulsive drug addict is an example if a a. Primary deviant c. Tertiary deviant b. Secondary deviant d. All of the above 161 Which is NOT a characteristic of a social group?
A research interview determined entirely in advance and followed rigidly 169 A research interview determined entirely in advance and followed rigidly is: a. Participants observation b. A semi structured (open‐ended) interview c. A structured interview d. A data discussion 170 A research interview in which the investigators ask a list of questions, but is free to vary them or make up new ones that become important during the course interview, that is a. Participant observation b. A semi structured (open‐ended) interview c. Structured interview d. A data discussion 171 A conversation between two or more individuals in which one party attempts to gain information from
a. b.
others(s) by asking of questions is known as; a. A survey c. Participant observation b. An interview d. Hypothesis testing 172 Sociological and anthropological be misused because it: a. Is hard to understand b. Is often used selectively by groups or individuals to support certain motives c. Is often ambiguous d. Has few applicants 173 A survey by their nature usually a. Are cross‐sectional b. Are longitudinal c. Involve secondary methodology d. Involve participants observation
Physical collection of people Shares a common purpose and conscious of each other c. Share some common characteristics d. Member interact with one another 162 Which is NOT a characteristic of the relationship among members of a primary group? a. Personal c. Segmental b. Informal d. Sentimental 163 Which is an example of members if a primary group? a. Peer group c. School administration b. Family d. Neighbors 164 Which of the following statements is NOT true about Filipino families today? a. Male authority has declined b. Division of labor has changed c. Separation has become more common d. Has cease to be socialization unit in society
d.
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174 A cross‐sectional study is a. A research technique in which investigators enter into activities of the group at the same time they study the groups behavior b. Research examines a population at a given point in a time c. Research which examines a population, or portion thereof, is questioned in order to reveal specific facts about itself d. Research that investigates a population at several intervals over relatively a long period of time 175 Research that investigates a population at several
185 If apes can be taught to use language, it denies the views that: a. Only human can think b. Only human have culture c. Apes do not have history to pass on d. Apes are stupid 186 Which of the following statements about culture is not true? a. Every social group must have a culture on its own in order to function b. Every individual participates in a number of different culture c. Meeting the social expectation of several
intervals over a relatively long period of time is called a. A cross‐sectional study c. Secondary Analysis b. A residual analysis d. Longitudinal research 176 Survey are used when: a. Other measure do not produce statistical significance b. The findings of the study must be repeatable c. It is desirable to discover he contribution and interrelations of certain variables among large number of people d. Procedures having cross‐cultural applications needed 177 A research method is which a population, or portion thereof, is questioned in order to reveal specific facts itself is known as:
cultures is often source of tree Families do not have their own culture but instead reflect the culture of the larger society 187 Promised to “make this country great again.” Ruled for about twenty years a. Diosdado Macapagal c. Ramon Magsaysay b. Ferdinand Marcos d. Carlos Garcia 188 The form of government established by Aguinaldo was change from dictatorial to: a. Democratic c. Republic b. Sociologist d. Revolutionary 189 Which provides he Tagalog shall be the official language of the republic? a. Constitution of 1935 c. Biyak na Bato b. Makabula d. Malolos 190 Which of the following were the greatest naval battles in history during Japanese occupation of the Philippines?
a. A survey c. Experiments b. Participant observation d. Residual analysis 178 Which of the following is NOT one of the three main methods of research used by sociologists? a. A survey c. Experiments b. Participants observation d. Residual analysis 179 A statistical technique to make all significant groups in as society represented in a sample proportion to their numbers in the large society is known as: a. Random sampling b. A stratified random sample c. Sampling d. Randomized response 180 The goal in designing a research project should be: a. To prove a point b. To provide a collection for all necessary and sufficient data to test the stated hypothesis c. To selectively isolate the variables in order to
a. Battle of Leyte gulf c. Claveri b. Surigao Strait d. Dela Torre 191 The most cultured of the reformist, wrote a socio‐ historical novel based on facts he gathered in the Philippines a. Marcelo H. Del Pilar c. Mariano Ponce b. Jose Rizal d. Jose Ma. Paganiban 192 The editor of La Solaridad and a great orator, he initiated the reform movement a. Marcelo H. Del Pilar c. Emilio Jacinto b. Jose Rizal d. Graciano Lopez‐ Jaena 193 The generation of the spirits was common among ancient Filipinos. Memories of their dead relatives were kept alive through craved idol. This practice was called a. Cult of the dead c. Nature Worship
prove the hypotheses are either correct or incorrect To prove hypothesis false. If they cannot proven false, they must be true 181 An independent variable is a. A testable statements about the relationship between two or more empirical variables b. Anything that can be change c. A variable that changes for reason that have nothing to do with another variable d. A variable that changes in response to changes in another variable 182 Which of the following is not a sign? a. A clenched first c. A flag b. A knock on the door d. Yawning 183 The most important symbols are a. Action c. Actions b. Words d. Behavior 184 The principal means through which culture is transmitted from generation to generation is a. Actions c. Language b. Diffusion d. Behaviors
b. Mass of the dead d. Divination 194 The practice of having surnames was started during Spanish regime through the order of a. Gov. Gen Jaudenes c. Gov. Gen Claveria b. Gov. Gen Dela Torre d. Gov. Gen Polavejia 195 As promised by Gen. Mc Arthur‐ he would return to liberate the Filipinos after he could train troops, in Australia. The Leyte landing signified the liberation of the Philippines, was well as the return of the commonwealth government. Who was the commonwealth president who came with Mc Arthur? a. Romulo c. Osmeňa b. Roxas d. Quezon 196 Men and women of pre‐colonial times had weakness for personal adornment. A side from jewels, tattooing of the body was common. In the Visayas tattooed men were called a. Umalohokan c. Abi b. Pintados d. Plista
d.
d.
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197 Memories of brutality which was so in human during the Japanese regime were always associated with a particular Japanese group called a. Kalibapi c. Kamikaze b. Kempetai d. Samurai 198 Membership in the Katipunan come in grades: first (katipon’s); second (kawal); and third (bayani). Each grade designed as well as password so they could maintain their secrecy of the movement. The password for the kawal grade was? a. Rizal c. Kawal b. Gomburza d. Sundalo 199 Pre‐colonial trade in the Philippines was
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a c c d c b c a c
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d a c d a b b b a
prosperous. Business transactions made use of: a. Money system c. Credit system b. Barter system d. Banking System 200 The use of magic charms was very common among pre‐Spanish Filipinos. One of these is the anting‐ anting or agimat which is meant to: a. Insure a man against a weapon of every kind b. Made a man lovable to all ladies c. Made a man invisible d. Made him walk in storm or sea without getting wet ***** T H E E N D *** * * WORK HARD, DREAM HARDER Social science 1 c 51 2 a 52 3 b 53 4 b 54 5 c 55 6 b 56 7 d 57 8 a 58 9 a 59 10 c 60 11 d 61 12 c 62 13 a 63 14 a 64 15 b 65 16 c 66 17 d 67 18 a 68 19 a 69 20 c 70 21 c 71 22 b 72 23 c 73 24 b 74 25 d 75 26 b 76 27 d 77 28 a 78 29 c 79 30 c 80 31 d 81 32 a 82 33 c 83 34 c 84 35 d 85 36 a 86 37 38 39 40 41
b a d c b
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a d b c c b c b a a a c c d a a a c c b b b c b d c a d b b c c d d b d b d a a b
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It has reference to what teachers do in planning, implementing and evaluating instruction. a. Teaching c. Teaching strategies b. Curriculum d. Instruction The orderly process directing learners to develop their skills and habits so that they will be assisted in acquiring knowledge and attitudes. a. Instructional Media b. Instructional Method c. Teaching Techniques
14. It is “control by enforcing obedience or orderly conduct or training that corrects and strengthens? a. Management c. Techniques b. Discipline d. Strategies 15. When students are asked to respond to incomplete statements or questions that are presented in oral/ written form a. Open‐ended Statement b. Close‐procedure c. PAC Strategy d. Structured Activity 16. These are all the experience which children have under the direction of a school
d. Instructional System Facial Expression, writing on the board, and oral expression of the teacher is an example of a. Teaching Behavior b. Technical Skills of teachers c. Instruction d. Instructional System Learning to draw, drive a car, play tennis, cook and type a poem often taught in is an example of a. Cognitive Learning c. Verbal Learning b. Motor Skill Learning d. Social Learning Responding to telephone calls, writing one’s name, reading a book orally is an example a. Cognitive Learning c. Verbal Learning b. Motor skill learning d. Serial Learning A process wherein the pupil’s attention and interest are aroused and directed to a definite purpose.
a. Curriculum c. Learning b. Instruction d. Socialization 17. The subjects mater, not the child is important in this type of curriculum a. Correlated curriculum b. Subject‐centered curriculum c. Experience curriculum d. Fused curriculum 18. The child‐instead of the subject‐matter is important in this kind of curriculum a. Correlated curriculum b. Core curriculum c. Experience curriculum d. Fused curriculum 19. It is a unified curriculum where subject matters from different subject field are treated unitary of the same curriculum a. Core curriculum
PRINCIPLES AND STRATEGIES OF TEACHING POST‐TEST SEPTEMBER 2010
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a. Learning c. Method b. Motivation d. Principle 7. Contains a statement of results to be accomplished and specific means by which these results are to be attained under direction and guidance. a. Method c. Technique b. Lesson Plan d. Principle 8. Could be the means of developing good study habits and independence in work as well as preparing the pupils for the job to be done a. Review c. Assignment b. Drill d. Recitation 9. The act of repeating from memory the reciting of a lesson and often described as a session lesson hearing a. Review c. Assignment b. Recitation d. Drill 10. A teaching procedure dealing with first‐hand experiences pertaining to material obtained from experimentation a. Demonstration Method b. Laboratory Method c. Discovery Method d. Deductive Method 11. Starts with generalization and principles or from general to particular a. Inductive Method c. Classical Method b. Deductive Method d. Problem Method 12. Students enact situations that arise in daily living, where values may be clarified, insights are developed and decision‐making is practiced a. Simulation Game c. Demonstration b. Role Playing d. Inquiry Process 13. Encouraging students to search for and see relationships that are not obvious; also it stretches the intellect of students a. Open ended Questions b. Recall‐ Questions c. Explanatory Questions d. Descriptive Question
b. Integrated Curriculum c. Broad field curriculum d. Fused curriculum 20. Teacher’s initiative, imagination, puppet shows, play, reading and animated cartoons can be examples of enriching the curriculum under these resources a. Specializing Resources b. Creative Resources c. Human Resources d. Reading Resources 21. A curriculum considered basics for all students, that all must get them a. Broad field curriculum b. Core curriculum c. Integrated Curriculum d. Experience Curriculum 22. The whole body of experience utilized by the
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school to attain the aims of education a. Psychology c. Socialization b. Curriculum d. Methods Formal education starts when the child a. begins to talk b. reaches the age of six years old c. first enters school d. begins to be inquisitive That aspect of curriculum that has to do with the preservation of the best in our culture, customs and traditions has been borrowed from a. Sociology c. Psychology b. Sociometry d. Ethics The curriculum must take into consideration the a. aim of education b. learning process c. motives and incentives d. instincts The curriculum is a. all‐embracing c. all power b. encompassing d. selective
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27. In the traditional school, the focus of attention was on the a. child c. method b. subject matter d. book 28. Curriculum objects are formulated in the light of our a. past history b. educational policy and philosophy c. experience as a nation d. needs in school 29. Which of the following questions encourages reflective thinking? a. What are the parts of a complete flower? b. c. d.
39. Teaching method which proceeds from the details of a lesson towards the generalization is called: a. Inductive c. problem‐solving b. deductive d. debate 40. A teaching method which proceeds from a generalization, principle or rule is: a. inductive c. project b. deductive d. process 41. The recent approach in teaching Social Studies is called a. discovery c. process b. conceptual d. formal‐education 42. A method of teaching which aptly applies to
What do we use to observe matter? In what ways can help his community Why are machine‐made goods cheaper than those made by hands? 30. The success of the pupils in formulating generalization greatly depends on: a. the interest of the pupils b. the devices used c. the subjects matter d. the teacher’s skillful questioning 31. In the inquiry method, the initiation phase calls for the teacher to set the stage for: a. finding solutions to problem b. raising of problems c. gathering data d. formulating generalization 32. Method is dependent upon: a. classroom techniques
lessons needing experiments is called: a. problem‐solving c. observation b. laboratory d. demonstration 43. What type of lesson is presented wherein the learner meets the learning experience through understanding, analysis, and generalizations of facts presented? a. review c. developmental b. drill d. deductive 44. What lesson is presented when the teacher takes up the previous learning experiences of the learners in a recognized pattern of presentation? a. Drill b. developmental c. review d. discussion procedure 45. A lesson which aims to focalize skills to make them fixed to the point of mastery is
b. teacher’s expectation c. theoretical assumptions d. available textbooks 33. To lead the students to the desired behavior, method must be implemented through: a. selected technique b. the curriculum c. the discussion of the teacher d. careful observation 34. Which is not true regarding the project method? a. Many worthwhile projects are impossible because of the materials needed b. The project method should be used occasionally but not regularly c. The pupil or the class should carry the chief responsibility of planning the project d. The project method is adaptable to all units in the curriculum
a. problem‐type c. review b. drill d. experimental The law of exercise is aptly applied in a a. review lesson c. drill lesson b. assignment d. check‐up A type of review which presents the sum‐total of all activities previously presented a. integrated c. daily b. cumulative d. drill What recent technique o teaching calls for acting out of a situation where the participants aim to uncover a problem of great importance to the class? a. panel c. role‐playing b. debate‐form d. lecture‐form What technique of in‐service training for teachers involves the identification and solution of common problems by them, thru live‐in sessions,
35. The laboratory method is also called: a. the research methods b. the deductive method c. the development method d. the problem method 36. In the unit method, actual learning takes place in: a. orienting the pupils b. collecting, discovering and recording data c. summarizing the unit d. organizing the unit or study 37. A statement of objectives, learning experience and the means of attaining results of teaching is called a. procedure c. outcomes b. lesson plan d. strategy 38. Teaching aids which the teacher uses to make learning meaningful, productive and interesting is known as: a. device b. technique c. method d. learning continuum
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conferences, and speeches of consultants? a. buzz session c. seminar b. workshop d. professional meeting 50. The non‐verbal symbols used to maximize learning are referred to as a. Instructional devices b. Classrooms techniques c. Field trips d. Educational media 51. Graphic material which are eye‐catching and which use slogans and topics presented in bold letterings and strong colors to serve as reminders of standards and / or important events are called a. poster c. projector b. film strips d. objects 52. What contemporary aid to teaching utilizes carefully‐planned materials where each step of learning requires repetition and practice until such step is thoroughly learned? a. programmed instruction b. Keypunching c. Educational Television d. Educational hardware
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53. The Stimulus‐Response theory of learning which involves the association between a conditioned stimulus and a response thru the repeated presentation of the stimulus was advocated by whom? a. Edward Thorndike c. Burrhus Skinner b. Ivan Pavlov d. Wolfgang Kohler 54. What plan of promoting pupils is committed to encouraging the learners to progress from grade to grade without needless repetition a. non‐graded scheme b. individualized c. heterogeneous grouping
64. What teaching method helps the learners draw generalization from a discipline with the end in view of applying the same similar situations in the future? a. discovery approach b. process approach c. conceptual approach d. problem‐solving approach 65. Which subjects is in the elementary and secondary school levels mostly concerned with the study of societal problems and issues which are significant to the learners as member of society? a. Modern Mathematics
d. acceleration Differentiated assignments, tutorial and remedial work to would‐be‐failures are not considered in the individualized Instruction Scheme a. Yes c. Maybe b. No d. Sometimes 56. A part of a daily lesson which serves as a carry‐ over for the next day of what has been presented is the a. review b. drill c. assignment or agreement d. lesson proper 57. A good learning environment is one a. free from distraction c. disturbing noise b. aver decorated d. dilapidated 58. The proper handling of the physical condition ad instructional materials in the classroom to effect
b. Social Studies c. Filipino d. Character Education 66. Which of these are considered with two essential dimension of science teaching? a. observing and inferring b. seeing and observing c. reading and researching d. knowledge and performance 67. Which of these is not a process in science teaching? a. Measurement b. Communication skill c. Controlling variables d. None of these 68. Of the process involve in the modern approach to science instruction, which one utilizes the most number of scientific processes.
learning refers to a. teaching method b. Classroom management c. Discipline grouping d. Guidance‐oriented 59. What refers to the process o directing immediate personal desires, interests or wishes for the purpose of achieving an effective action? a. discipline c. supervision b. teaching d. management 60. What characteristics an effective type of discipline? a. vital, sympathetic, humane b. formal and strict c. inhibited d. imposed 61. Which of these is not a quality of a good teacher? a. mastery o the subject matter
a. prediction c. inference b. experimentation d. hypothesis 69. Give the main difference of these two objectives: “ to teach the importance of proper nutrition for good health “ “ to give the importance of proper nutrition for good health” a. The first objective is general while the second is specific. b. The first objective is hard to do while the second is easy c. The first objective needs a longer time while the second doesn’t d. The first objective is teacher behavior while the second pupil behavior. 70. Which of the objectives below show overt behavior? a. To appreciate the value of democracy.
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b. broad background of liberal education c. aims to enrich himself thru teaching d. understand the nature of the learners 62. Which of these is a good personal qualification of a teacher? a. resourceful, creative and intelligent b. rich, capricious and luxurious c. complaining, demanding and scornful d. materialistic 63. Which of these is included among the professional ethics o school teachers? a. professional jealousy b. integrity c. engaging in business pre‐judicial to his teaching duties d. gossip mongering
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To understand the importance of a constitution c. To recite he preamble of the constitution d. To show love to one’s country 71. The basis by which content is outlined and institutional procedures are developed is the: a. lesson plan c. objectives b. basic text d. instructional materials 72. An objective MUST specify: a. What the learner must do or say. b. What the teacher must do or say c. What projects are to be accomplished d. What the learner must understand 73. “Given ten photographs of biological cells, the pupils will be able to identify six of them as plant or animal cells.” The underlined phrase is a : a. terminal behavior b. standard or acceptable performance c. d.
condition for learning an accomplishment to be realized.
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74. “ To make statement” as an objective in an English Lesson that is: a. specific c. correct b. vague d. none of the above 75. What is the most fitting condition of learning for this behavior: “to conclude that plants need sunlight in order to live”? a. with the must of materials b. given a set of pictures c. after reading the book d. realistic 76. Which of the following is not a criterion of a well‐ formulated objective?
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89. Which of the following statements is correct? a. Method is probably more important in college than in the elementary b. Method is more important in the elementary than in high school or college c. Method is more important in college than in high school d. Method is less important than a lesson plan 90. What encourages the child to think, rationalize and make proper decisions? a. drill b. Appreciation lesson c. Memorization
a. attainable c. interesting b. observable d. realistic Which task below is not in the psychomotor domain? a. imitation c. manipulation b. evaluation d. articulation The growth of attitudes or values is in the: a. cognitive domain b. psychomotor domain c. affective domain d. behavioral domain The domains of behavior do not come in isolation. This statements is : a. True c. False b. Acceptable d. Partly true “Will a person do it freely without any type of coercion?” This is: a. a cognitive question
d. Problem‐ oriented strategies 91. The following except one are the factors that determine the choice of a method. Which is the exception? a. nature o the learners b. school equipment and facilities c. educational background of the teacher d. Subject matter 92. How well a teacher tells a story depends on: a. Techniques c. the method used b. the plot d. classroom 93. Which of the following statements is correct? a. Method is synonymous with technique b. A device is a teaching method c. Method can be standardized d. There is no single best method 94. When a teacher reviews a lesson, she is utilizing the law of:
b. an affective question c. a psychomotor question d. a behavioral question “ To develop appreciation of poetry” is a : a. general aim c. nature aim b. specific aim d. serious aim Which aim below does not belong to the group? a. To enumerate the uses of common garden tools b. To express opinion politely c. To explain the significance of the story d. To identify the parts of a flower. Which objective below is not realistic? a. To respect places of worship b. To sing the national anthem correctly c. To give the importance of cleanliness d. To cite ways to show love one’s country Which objective below is not specific?
a. Readiness c. effect b. exercise d. multiple response 95. In which situation is the law of readiness best applied? a. The teacher gives the aims of the lessons to be taken up b. The teacher announces he subject matter at the start of the period c. The teacher waits or the children to be ready before teaching her lesson d. The teacher presents a song, related to the lesson 96. Which of the glowing is not an am in the inductive method a. To delay judgments until truth is given b. To enable pupils discover important truths for themselves. c. To help student/pupil to carry out an
a. To describe some of farming procedures b. To define terms comprehensively c. To pay tax promptly d. To know the life cycle of a moth. What phrase below is a standard of performance? a. Solve the problem correctly within 10 minutes b. Identify and sketch the curve c. With the use of a ruler d. After several examples A visible activity shows : a. overt behavior b. covert behavior c. confident behavior d. artificial behavior Which infinite below is not behavioral? a. to describe c. to compare b. to select d. To believe Which objective below needs improvement a. To prepare a seed box b. To develop skill in embroidery c. To plan a noon meal d. To make an apron
investigation by themselves independent of the teacher d. To make relationship of ideas clear to pupils 97. In the inductive method, what does the child do during the comparison and “abstraction” step? a. Recalls information and directs himself to the activities to be accomplished b. Perceives the common element present in the cases given c. Applies the principles learned to other problems or exercises d. Draw conclusion in his own words 98. The deductive method uses the following steps: a. statement of the problem, generalization, inference, verification b. statement of the problem, inference , generalization, verification c. inference, statement of the problem, generalization, verification d. inference, statement of the problem, verification, generalization
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99. In reality, the type of study method is: a. an inductive procedure b. a deductive procedure c. a traditional method d. a question and answer method 100. The Herbartian formal steps corresponds to the steps of: a. the inductive method b. teaching an appropriate lesson c. the deductive method d. the project method
“A thing becomes wasted when you fail to take advantage of it Principle and strategies
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PROFESSIONAL EDUCATION Pre‐board EXAM April 2010
MULTIPLE CHOICE 1. You are very much interested in a quality professional development program for teachers. What characteristic should you look for? a. Prescribe by top educational teachers b. Dependent on the availability of funds c. Required for renewal of professional license d. Responsive to identified teacher’s needs. 2. To ensure high standards of teachers’ personal and professional development, which of the following measures must be implemented? I. A school head plans the professional development of his/her teachers. II. Every teacher formulates his/her own professional development plan III. The implementation of what is leaned in a training must be monitored. a. I only II and III b. I and III d. II only 3. As a community leader, which of the following should a teacher NOT do? a. Support effort of the community to improve their status in life. b. Make herself aloof to ensure that her decisions will not be influenced bu the community politics. c. Solicit donation from philanthropists in the community. d. Play an active part in the activities of the community. 4. In a highly pluralistic society, what type of learning environment is the responsibility of the teacher? I. Safe II. Gender‐biased III. Secure a. I and II c. II only b. I, II and III d. I and III 5. A teacher is said to be “trustee of the cultural and educational heritage of the nation and is under obligation to transmit to learners such heritage”. Which practice makes the teacher fulfill such obligation? a. Use interactive teaching strategies. b. Use the latest educational technology. c. Observe continuing professional education d. As a class, study the life of Filipino heroes. 6. Which actions show that political factors affect schools as agents of change? I. The strengthening of the teaching of English in Philippines school. II. The introduction of mandated subjects such as Rizal in the curriculum III. The practice of mainstreaming IV. The turnover of day care centers for DSWD to DepEd for supervision. a. I and III c. II and III b. I and II d. II and IV 7. For more efficient and effective management of school as agents of change, one proposal is for the DepEd to cluster remote stand‐alone schools under one lead school head. Which factor has the strongest influence on this proposal? a. Psychological c. Geographical b. Historical d. Social
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What does the acronym EFA imply for schools? a. The acceptance of exclusive schools for boys and for girls. b. The stress on the superiority of formal education over that of alternative learning system. c. Practice of inclusive education d. The concentration on formal education system The wide acceptance of “bottom up” management style has influenced schools to practice which management practice? a. Exclusion of politicians from the pool of guest speakers during graduation exercises. b. Prescription of what ought to be done from the Center Office. c. Involvement of students, parents, teachers, and community in school planning d. Allowing schools to do what they think is best Large class size in congested cities is a common problem in our public schools. Which measure/s have schools taken to offset the effects of large class? I. The deployment of more teachers II. The implementation of 1:1 pupil textbook ratio III. The conduct of morning and afternoon sessions a. I, II and III c. III only b. I and II d. II only The failure of independent study with most Filipino students may be attributed to students’ a. unpreparedness fro schooling b. ambivalence c. high degree of independence d. high degree of independence on authority The schooling incidents in school campuses abroad have made school to rethink the curriculum. Which is believed to counteract such incidents and so is being introduced in schools? I. Inclusion of socio‐emotional teaching II. The emphasis on the concept of competition against self and not against others III. Focus on academic achievement and productivity a. I and III c. I and II b. II and III d. I, II and III Widespread abuse of Mother Earth prompted schools to teach sustainable development. which one does this prove about schools? a. The curricula of schools are centered on Mother Earth. b. Schools can easily integrate sustained development in their curriculum. c. Sustained development cannot be effectively taught in the classroom. d. Environment factors influence the school as an agent of change. A father tells his daughter “You are a woman. You are meant for the home and so for you, going to school is not necessary.” Is the father CORRECT? a. It depends on the place when the daughter and the father live. b. Yes, women are meant to be a mother only. c. No. today women can take on the jobs of men. d. No, there is gender equality in education.
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15. Is there a legal basis for increasing the teacher’s starting salary to P18,000 an months? a. No, it is a gift to teachers from Congress b. Yes, R.A 7836 c. No, it is simply an act of benevolence from President GMA d. Yes, the Phil. Constitution 16. As provided for the Educational Act of 1982, how are the institutions of learning encourage to set higher standards of equality over the minimum standards required for state recognition? a. Granting of Special Permit b. Academic freedom c. Continuing Professional Education d. Voluntary accreditation 17. Despite of opposition from some school official, DepEd has continuously enforced the “no collection of fees” policy during enrolment period in public schools. In this policy in accordance with EFA goals? a. No, it violates the mandate of equality education b. Yes, it somewhat eliminates gender disparities c. Yes, it supports equitable access to basic education d. No. it does not support parent of adult education 18. “Specialization is knowing more and more about less and less”. Hence, it is better to be a generalist, claims Teacher F. Which Philosophy does Teacher F. subscribe to? a. Existentialism c. Essentialism b. Perennialism d. Progressivism 19. Mencius believed that all people are born good. This thought on the innate goodness of people makes it easier to our pupils. a. teach c. like b. Respect d. motivate 20. The specialization requires of every professional teacher for him/her to be competent is in line with which pillar of learning? a. Learning to know b. Learning to be c. Learning to live together d. Learning to do 21. Which pillar of learning is aimed at the wholistic development of man and his complete fulfillment? a. Learning to be b. Learning to know
24. Which violate(s) the principle of respect? I. Teacher A tells her students that what Teacher B taught is wrong. II. To retaliate, Teacher B advises students not to enroll in Teacher A’s class. III. Teacher C secretly gives way to a special favor (e.g. add 2 points to grade) requested by student A who is vying for honors. a. II and III c. I and II b. I, II and III d. I and III 25. Which is/are in accordance with the principle of pedagogical competence? I. Communication of objectives of the course to students II. Awareness of alternative instruction strategies III. Selection of appropriate methods of instruction a. I and III c. III only b. I, II and III d. II and III 26. With a death threat over his head, Teacher D is directed to pass an undeserving student. Which will a utilitarianist do? a. Pass the student, why suffer the threat? b. Don’t pass him; live by your principle of justice. You will get reward, if not in this life, in the next. c. Pass the student. That will be of use to the student, his parents and you. d. Don’t pass him. You surely will not like someone to give you a death threat in other to pass. 27. Teacher A knows of the illegal activities of a neighbor but keeps quiet in order not to be involved in any investigation. Which foundational principle of morality does Teacher A fail to apply? a. The end does not justify the means b. The end justifies the means c. Always do what is right d. Between two evils, do the lesser evil 28. To earn units for promotion, a teacher pays her fee but does not attend class at all. Does this constitute professional growth? a. Not immediately but yes after promotion b. It depends on the school she is enrolled in c. No, it is simply earning MA units for promotion d. Yes, just enrolling in an MA program is already professional growth 29. If a teacher asks more higher‐order questions, he has to ask more questions.
c. Learning to live together d. Learning to do 22. Material development at the expense of human development points to the need to do more in school. a. “Learning to do” b. “Learning to know” c. “Learning to live together” d. “Learning to be” 23. A student complains to you about his failing grade. When you recomputed you found out that you committed an error in this grade computation. Your decision is not accept the erroneous computation before the student and so leave the failing grade as is for fear that you may lose credibility. Is this morally right? a. No, the reason for not accepting the error before the students in flimsy.
c. convergent a. fact b. close d. concept 30. Misdemeanor has a “ripple effect”. This implies that as a classroom manager, a teacher a. reinforces positive behavior b. responds to misbehavior promptly c. is consistent in her classroom management practice d. count 1 to 10 before she deals with a misbehaving student 31. Based on Edgar Dale’s “Cone of Experience”, which activity is farthest from the real thing? a. Watching demo c. Video disc b. Attending exhibit d. Viewing images 32. The students of Teacher Y scan an electronic encyclopedia, view a film on subject, or look at related topics at the touch of a button right there in the classroom. Which device/s des teacher Y’s class
b. No, the end does not justify the means c. Yes, the end justifies the means d. Yes, as a teacher you must maintain your credibility
have? a. Teaching machines b. CD c. Video disc d. Videotaped lesson
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33. Which is an INAPPROPIATE way to manage off‐task behavior? a. Redirect a child’s attention to task and check his progress to make sure he is continuing work b. Make eye contact to communicate what you wish to communicate c. Move closer to the child to make him feel your presence d. Stop your class activity to correct a child who is no longer on task 34. To be an effective classroom manager, a teacher must be friendly but must at the same time be . c. analytical a. confident b. businesslike d. buddy‐buddy 35. Which software is needed when one wants to perform automatic calculations on numerical data? a. Database b. Spreadsheet Program c. Microsoft Word d. Microsoft Powerpoint 36. Which of the following questions must be considered in evaluating teacher‐made materials? a. In the material new? b. Does the material simulate individualism? c. Is the material expensive? 37. Kounin claims that “with‐it‐ness” is one of the characteristics of an effective classroom manager. What is one sign of “with‐it‐ness”? a. Giving attention to students who are having difficulty with school work b. Seeing only a portion of the class but intensively c. Knowing where instructional materials are kept d. Aware of what’s happening in all parts of the classroom 38. Which of these is one of the ways by which the internet enables people to browse documents connected by hypertext links? a. URL b. Browser c. Welcome page d. World Wide Web 39. Which characteristics must be primary considered as a choice of instructional aids? a. Stimulate and maintain students interests b. Suited to the lesson objectives c. Updated and relevant to Filipino setting
43. The task of setting up routine activities for effective classroom management is a task that a teacher should undertake a. as soon as the students have adjusted to their schedules b. on the very first day of school c. every day at the start of the session d. every homeroom day 44. Teacher S uses the low‐profile classroom control technique most of the time. what does this imply? a. She is reactive in her disciplinary orientation b. She manages pupils personalities c. She reacts severely to a misbehaving student d. She stops misbehaving without disrupting lesson flow 45. When teacher tries to elicit clarification on a student response or solicits additional information, which of these should be use? a. Directing c. Structuring b. Probing d. Cross examining 46. Which priority criterion should guide a teacher in the choice of instructional devices? a. Novelty c. Attractiveness b. Cost d. Appropriateness 47. Which learning activity is MOST appropriate if a teacher’s focus is attitudinal change? a. Fieldtrip c. Role play b. Exhibit d. Game 48. Teacher H strives to draw participation of every student into her classroom discussion. Which of these student needs is she trying to address? The need to . a. shoe one’s oral abilities to the rest of the class b. feel significant and be part of a group c. get everything and be part of a group d. be creative 49. Instead of teacher giving this comment a student response. “You’re on the wrong track!”, what should be teacher do? a. Change the question to an easier one b. Redirect the question by calling another student to recite c. Probe to redirect the response into a more productive area d. Pause, ask the question, lecture, then ask the question again 50. If curriculum is designed following the traditional approach, which feature(s) apply(ies)?
d. New and skillfully made 40. You can exhibit referent power on the first day of school by a. telling them the importance of good grades b. giving your students a sense of belongingness and acceptance c. making them feel you know what you are taking about d. reminding your students your authority over them again and again 41. I would like to use a model to emphasize particular part. Which of these would be MOST appropriate? a. Regalia c. Stimulation b. Audio recording d. Mock up 42. What must a teacher do to ensure orderly transitions between activities? a. Allow time for the students to socialize in between activities
The aims of the curriculum professionals and experts are set by Interested groups (teachers, students, communities) are assumed to agree with the aims of the curriculum III. Consensus building in not necessary a. III only c. I, II b. I, II, III d. I, III 51. I want my student to develop the ability to look at a problem from various perspectives. Which approach will be MOST fit? a. Behaviorist approach b. Computer‐based Education c. Modular approach d. Cognitive approach
I.
II.
b. Have the materials ready at the start of the activity
c. Assign fewer exercise to fill the allotted time d. Wait for students who lag behind St. Louis Review Center-Inc-Davao Tel. no. (082) 224-2515
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52. One’s approach to teaching is influenced by Howard Gardner’s MI Theory. What is he/she challenged to do? I. To come up with 9 different ways of approaching lesson to cater to the 9 multiple intelligence II. To develop all student’s skill in all nine intelligences III. To provide worthwhile activities that acknowledge individual difference in children a. I, II and III c. II only b. II, III d. III only 53. If my approach to my lesson is behaviorist, what features will dominate my lesson? I. Copying notes III. Lecturing II. Reasoning IV. Demonstration a. III, IV c. I, II, III, IV b. I, III, IV d. II, III, IV 54. You practice inclusive education. Which of these applies to you? I. You accept every student as full and valued member of the class and school community II. Your special attention is on learners with specific learning or social needs III. Your address the needs of the class as a whole within the context of the learners with specific learning or social needs a. II only c. I only b. I and II d. I and III 55. School curriculum reflects the world’s economic and political integration and industrialization. What does this point in curriculum development? a. The trend towards the classical approach to curriculum development b. The trend toward the globalization and localization c. The trend toward participatory curriculum development d. The shift in the paradigm of curriculum development from a process‐oriented to a product‐oriented one 56. You choose cooperative learning as a teaching approach. What thought is impressed on your students? a. Interaction is a must, but not necessarily face to‐face interaction b. Student’s success depends on the success of the group c. Student’s individuality evaluate how effectively their group worked. d. The accountability for learning is on the group not on the individual 57. What principle is violated by overusing the chalkboard, as though it is the only education technology available? a. Isolated use c. Variety b. Flexibility d. Uniformity 58. Which statement applies a CORRECTLY to Edgar Dale’s ”Cone of Experience” a. The farther you are from the base, the more direct the learning experience becomes. b. The farther you are from the bottom, the more direct the learning experience becomes. c. The closer you are from the base, the more indirect the learning experience becomes d. The closer you are from the base, the more direct the learning experience becomes
59. “When more senses are stimulated, teaching and learning become more effective.” What is an application of this principle? a. Appeal to student’s sense of imagination b. Use multisensory aids c. Make your students touch the instructional material d. Use audiovisual aids because the eyes and the eras are the most important senses in learning. 60. Which is a classroom application of the theory of “operant conditioning”? a. Help student see the connectedness of facts, concepts, and principles b. Create a classroom atmosphere that elicits relaxation c. Reinforce a good behavior to increase the likelihood that the learner will repeat the response d. Make students learn by operating manipulatives Read the following teacher‐student situation 61. TEACHER: Why is the process called photosynthesis? STUDENT: I don’t know Which questioning technique should be the teacher use? a. Clarification c. Prompting b. Multiple response d. Concept review 62. Here is the test item. “From the data presented in the table, from generalizations that are supported by the data”. Under what type of question does this item fall? a. Convergent c. Application b. Evaluative d. Divergent 63. I want to teach concepts, patterns and abstractions. Which method will be MOST appropriate? a. Discovery c. Direct instruction b. Indirect instruction d. Problem solving 64. Teacher A teaches English as a Second Language. She uses vocabulary cards, fill‐in‐the‐blanks sentences, dialogues, dictation and writing exercises in teaching a lesson about grocery shopping. Based on this information, which of the following is a valid conclusion? a. The teacher wants to make her teaching easier by having les talk b. The teacher emphasizing reading and writing skills c. The teacher is teaching in a variety of ways because not all students learn in the same manner d. The teacher is applying Bloom’s hierarchy of cognitive learning 65. Teacher A an experienced teacher, does daily review of past lessons in order to a. introduce a new lesson b. reflect on how he presented the previous lessons c. provide his pupils with a sense of continuity d. determine who among his pupils are studying 66. I combined several subject areas in order to focus on a single concept for interdisciplinary teaching. Which strategy did I use? a. Reading‐writing activity introduction b. c. Thematic Unit method d. Problem‐centered learning
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67. To teach the democratic process to the pupils, Biag Elementary School decided that the election of class officers shall be patterned after local elections. There are qualifications set for candidates, limited period for campaign and rules for posting campaign materials, etc. Which of the following did the school use? a. Symposium c. Pole playing b. Simulation d. Philips 66 68. Which are effective methods in teaching student critical reading skills? I. Interpret editorial II. Read and interpret three different movie reviews Read a position paper and deduce underlying assumptions of the position papers a. II and III c. I and II b. I and III d. I, II and III 69. Here is a test item “The improvement of basic education should be the top priority of the Philippine government. Defend or refute this position”. Under what type of question does this test item fall? a. Low‐level c. Analysis b. Evaluative d. Convergent 70. When I teach, I often engage in brainstorming. Which do I avoid? a. Break down barriers b. Increase creativity c. Generate many ideas d. Selectively involves pupils 71. Teacher S teaches a lesson in which students must recognize that ¼ is the same 0.25. They use this relationship to determine that 0.15 and 0.20 are slightly less than ¼. Which of the following concept/s is/are being taught? a. Numeration skills b. Place value of decimals c. Numeration skills of decimals and relationships between fractions and decimals d. Relationship between fraction and decimals 72. What is the best way to develop math concept? a. Solving problems using multiple approaches b. Solving problems by looking for correct answer c. Learning math as applied to situations, such as being a tool of science d. solving problems by applying learned formulas 73. After the reading of a selection in the class, which of these activities can enhance students creativity. I. Reader’s theater II. Reading aloud III. Silent reading a. I and II c. I only b. II only d. III only 74. Teacher C, a Reading teacher, advised he class to “read between the lines”. What does she want her pupils to do? a. Make an educated guess b. Determine what is meant by what is stated c. Apply the information read d. Describe the characters in the story 75. To nurture students’ creativity, which activity should a teacher AVOID? a. Ask “hat if…” questions b. Ask divergent thinking questions III.
c. Emphasize the need to give right answers d. Be open to “out‐of‐this‐world” ideas
76. Teacher R wants to develop his student’s creativity. Which type of questions will be MOST appropriate? a. Synthesis questions b. Fact questions c. “What if….” questions d. Analysis questions 77. In my attempt to develop creative thinking skills, I want to test fluency of ideas. Which activity for my students will be MOST appropriate? a. Solve this math problem b. List animals covered with hair in 1 minute c. Solve this puzzle d. Compare pictures 1 and 2. Where are the differences? 78. You want your students to answer the questions at the end of a reading lesson. “What did I learn did?”,”What still puzzle me?”, “What did I enjoy, hate accomplish in the class today”?,”How did I learn from the lesson?”.Which of the following are you asking them to do? a. Work on an assignment b. Make journal entry c. Work on a drill d. Apply what they learned 79. After reading an essay. Teacher B wants to help sharpen her students’ ability to interpret. Which of these activities will be most appropriate? a. Drawing conclusions b. Making inferences c. Getting the main idea
d. Listing facts separately from opinion Read the following then answer the question 80. A man and his son are driving in a car. The car crashes into a tree, killing the father and seriously injury his son. At the hospital, the boy needs to have surgery. Looking at the boy, the doctor says (telling the truth),”I cannot operate on him. He is my son. How can this be? ASWER: The doctor is the boy’s mother. The above brain twister helps develop critical reading skills. Which activity was used? a. Comparing b. Classifying c. Inferring meaning d. Looking for cause and effect 81. Research says that mastery experiences increase confidence and willingness to try similar or more challenging tasks as reading. What does this imply for children’s reading performance? a. Children who have not mastered the basic skills are more likely to be motivated to read in order to gain mastery over basic skills. b. Children who have mastered basic skills are more likely to be less motivated to read because they get fed up with too much reading. c. Children who have a high sense of self‐ confidence are not necessarily those who can read d. Children who have gained mastery over basic skills are more motivated to read 82. The value that students put on reading is critical to their success. In what way/s can teachers inculcate his value for reading? I. Sharing the excitement of read‐aloud II. Showing their passion for reading III.
Being rewarded to demonstrate the value of reading c. I, II and III d. II only
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83. Bruner’s theory on intellectual development moves from enactive to iconic and symbolic stages. Applying Bruner’s theory. How would you teach? a. Be interactive in approach b. Begin with the abstract c. Begin with the concrete d. do direct instruction 84. A person who has painful experiences at the dentist’s office may become fearful at the mere sight of the dentist’s office building. Which theory can explain this? a. Generalization b. Operant Conditioning c. Attribution theory d. Classical conditioning 85. Which is/are the basic assumption/s of behaviorists? I. The mind of newborn child is a blank state. II. all behaviors are determined by environmental events III. The child has a certain degree of freedom not to allow himself to be shaped by his environment. a. III only c. II only b. I and II d. I and II 86. If a student is encourage to develop himself to the fullest and must satisfy his hierarchy of needs, the highest needs to satisfy according to Maslow is . a. psychological need c. belongingness b. self‐actualization d. safety needs 87. In a Social studies class. Teacher I presents a morally ambiguous situation and asks student what they would do. On whose theory is Teacher I’s technique based? a. Bandura c. Kohberg b. Piaget d. Bruner 88. Teacher F is convinced that whatever a student performs a desired behavior, provide reinforcement and soon the student learns to perform the behavior on his own. On which principle is Teacher F’s conviction based? a. Environmentalism c. Cognitivism b. Behaviorism d. Constructivism 89. Bandura’s social learning theory, states that children often imitate those who I. have substantial influence over their lives II. belong their peer group III. belong to other races IV. are successful and seem admired a. IV only c. I and II b. I and IV d. II and IV 90. According to Erikson, what years are critical for the development of self‐confidence? a. High school years b. College years c. Preschool years d. Elementary school years 91. Which of the following does NOT describe the development of children aged 11 to 13? a. They exhibit increased objectivity in thinking b. They shift from impulsivity to adaptive ability c. Sex difference in IQ become more evident d. They show abstract thinking and judgment
92. Teacher H begins a lesson on tumbling, demonstrating front and back somersaults in slow motion and physically guiding his students through the correct movements. As his students become more skillful, he stands back from the man and gives verbal feedback about how to improve. With Vygotsky’s theory in mind, what did Teacher H do? a. Apprenticeship b. Guided participation c. Peer interaction d. Scaffolding 93. What does Gagne’s hierarchy theory propose for effective instruction? a. Be concerned with the socio‐emotional climate in the classroom b. Teach beginning with the concrete c. Sequence instruction d. Reward good behavior 94. William Glasser’s control theory states that behavior in inspired by what satisfies a person’s want at any given time. What then must a teacher do to motivate students to learn? a. Make teaching‐leaning interactive b. Avoid giving assignments c. Organize a curriculum in a spiral manner d. Make schoolwork relevant to students’ basic human needs. 95. Soc exhibit fear response to freely roaming dogs but does not show fear when a dog is on a leash or confined to a pen. Which conditioning process is illustrated? a. extinction c. acquisition b. generalization d. discrimination 96. Based on Freud’s theory, which operate/s when a student strikes a classmates at the height of anger? a. Ego c. Id and Ego interact b. Id d. Superego 97. Bernadette enjoyed the roller coaster when he and her family went to Enchanted Kingdom. The mere sight of a roller coaster gets her excited. Which theory explains Bernadette’s behavior? a. Operant conditioning b. Social learning theory c. Attribution theory d. Pavlovian conditioning 98. According to Frued, with which should one be concerned if he/she has to develop in the students a correct sense of right and wrong? I. III. Id a. I Super and II ‐ego II. Ego c. I b. II d. III 99. When small children call animals “dog”, what process is illustrated on Piaget’s cognitive development theory? a. reversion c. accommodation b. assimilation d. conservation 100.Researchers found that when a child is engaged in a learning experience a number of areas of the brain are simultaneously activated. Which of the following is/are implication/s of this research finding? I. Make use of field trips, guest speakers II. Do multicultural units of study III. Stick to the “left brain and right brain” approach a. I and III c. I and II
b. I only
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101.Which appropriate teaching practice flows from this research finding on the brain: “The brain’s emotional center is tied into its ability to learn”. a. Establish the discipline of being judgmental in attitude b. Come up with highly competitive games where winners will feel happy c. Tell the students to participate in class activities or else won’t receive plus points in class recitation d. Create a learning environment that encourages students to explore their feeling and ideas freely 102.Research on Piagetian tasks indicates that thinking becomes more logical and abstract as children reach the formal operations stage. What is an educational implication of this finding? a. Engage children in analogical reasoning as early as preschool to train them for higher order thinking skills (HOTS) b. Learners who are not capable of logical reasoning from ages 8 to 11 lag behind in their cognitive development c. Let children be children d. Expect hypothetical reasoning for learners between 12 to 15 years of age 103.Research says: “People tend to attribute their successes to internal causes and their failures to external causes.”Based on this finding, what should be taught to students for them to be genuinely motivated to succeed? a. Tell them the research finding when applied will make them genuinely motivated b. Convince them that genuine motivation is the only factor that matters for a person to succeed c. Make them realize that failure is a part of life d. Make them realize that both success and failure are more a function of internal causes. 104.Which characterize/s a learning environment that promotes fairness among learners of various cultures, family background and gender? I. Inclusive II. Exclusive III. Gender‐sensitive a. I only c. I and III b. III only d. II and III 105.Which of the following steps should be completed first in planning an achievement test? a. Define the instructional objective b. Set up a table of specialization c. Select the types of test items to use d. Decide on the length of the test 106.The computed r fro scores in Math and Science in 0.92. What does this mean? a. Math score is positive related to Science score b. The higher the Math score, the lower the Science score c. Math score is not in any way related to Science score d. Science score is slightly related to math score 107.Which types of test is most appropriate if Teacher Y wants to measure student’s ability to organize thoughts and ideas? a. Short answer type of test b. Extended response essay alternative response c. d. Modified Limited response essay
108.With assessment of affective learning in mind, which does NOT belong to the group? a. Cloze test c. Reflective writing b. Moral dilemma d. Diary entry 109.I want to test student’s synthesizing skills. Which has the highest diagnostic value? a. multiple choice test c. Essay test b. Performance test d. Completion test 110.
is an example of a leafy vegetable.
I. II. III. IV. A. B. C. D.
Why is this test item poor? The test item does not pose a problem to the examinee There are variety of possible correct answer to this item the language used in the question is not precise The blank is near the beginning of a sentence I and III II and IV I and IV I and II
111.
What follows is a multiple choice type of test. Some test items . a. are too difficult b. are objective c. are poorly constructed d. have multiple defensible answers What makes the multiple choice type of test poor? a. The options are not grammatically connected to the stem b. The stem fails to present a problem c. There are grammatical clues d. The options are not parallel 112.If a teacher wants to measure her students’ ability to discriminate, which of these is an appropriate type of test item as implied by the direction? a. “Outline the Chapter on The Cell”. b. “Summarize the lesson yesterday”. c. “Group the following items according to shape”. d. “State a set of principle that can explain the following events”. 113.A test item has a difficult index of 0.89 and a discrimination index of 0.44. What should the teacher do? a. Reject the item b. Retain the item c. Make it a bonus item d. Make it a bonus item and reject it 114.Which form of assessment is consistent with the saying “The proof of the pudding is in the eating”. a. Contrived c. Traditional b. Authentic d. Indirect 115.
Who is best admired for outstanding contribution to world peace? a. Kissinger c. Kennedy b. Clinton d. Mother Teresa
a. b. c. d.
What is WRONG with this item? Item is overly specific Content is trivial Test item is option‐based There is a cue to the right answer
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116.Student’s score were as follows: 82, 83, 84, 86, 88, 84, 83, 85. The score 86 is the. a. mode c. median b. average d. mean 117.Which text form would you choose if you want to have a valid and reliable test based on the table below? Test Form Validity Index Reliability Index A .47 .68 B .87 .57 C .20 .86 D .40 .41 E .63 .07
a. A only b. B only
c. A and D d. B and E 118.A mathematicians test was given to all Grade V pupils to determine the contestants for the Math Quiz Bee. Which statistical measure should be used to identify the top 15? a. Mean percentage score b. Quartile Deviation c. Percentile Rank d. Percentage Score 119.Nellie’s score is within x±1 SD. To which of the following groups does she belong? a. Below average b. Average c. Needs Improvement d. Above average 120.Use the inbox below to answer the question that follows: Percentage Grades for Final Examination
40 100
a. b. c. d.
70
80
90
Which of the following statement is TRUE about the plot of grades above? The median is a score of 80 and the range is 60. The median is a score of 70 and the range is 60. The median is a score of 80 and the range is 20. The median is a score of 70 and the range is 20.
121.Which can be said of Arielle who obtained a score of 75 out of 100 items in a Grammar objective test? a. She performed better than 25% of her classmates b. She answered 75 items in the test correctly c. Her rating is 75 d. She answered 75% of the test items correctly 122.The criterion of success in Teacher D’s objective is that “the pupils must be able to spell 90% of the words correctly”. Ana and 24 others in the class spelled only 40 out of 50 words correctly while the rest scored 45 and above. This means that Teacher D a. attained her lesson objective b. did not attain her lesson objective because of the pupils’ lack of attention c. failed to attain her lesson objective as far as the 25 pupils are concerned
123.If the scores of your test follow a negatively skewed score distribution, what should you do? Find out ? a. why your items were easy b. why most of the scores are high c. why most of the scores are low d. why some pupils scored high 124.Principal A is talking about “grading on the curve” in a faculty meeting. What does this expression refer to? a. A student mark compares his achievement to his effort. b. A student’s grade or mark depends on how his achievement compares with the achievement of other students in a class. c. A student’s grade determines whether or not a student attains a defined standard of achievement d. A student mark tells how closely he is achieving to his potential. 125.Which tests determine whether students accept responsibility for their own behavior or pass on responsibility for their own behavior to other people? a. Thematic tests b. Sentence‐completion tests c. Stylistic test d. Locus‐of‐control tests 126.Which of the one weakness of self‐supporting personality checklists? a. Many personality measures have built‐in lie scales b. They lack stability c. They may not get true information because individuals can hide or disguise feelings d. They have poor internal consistency 127.Which of these can measure awareness of values? a. Sociogram b. Moral dilemmas c. Projective techniques d. Rating scales 128.Marking on a normative basis means that a. the normal distribution curve should be followed b. some should fall c. some get high marks d. the grading is based on a present criteria 129.Which process enhances the comparability of grades? a. Using a table specifications b. Determining the level of difficulty of the tests c. Giving more HOTS (higher order thinking skills) d. Constructing departmentalized examinations for each subject area. SITUATIONAL Situation 1‐ In a faculty meeting, the principle told his teacher: “We need to improve our school performance in the National Achievement Test. What should we do? The teacher gave varied answers as follows: 1. Let’s give incentives and rewards to students
who get a rating of 85% 2.
Let’s teach them to accept complete
3.
Let’s make the school environment
4.
Let’s make use of the experimental methods
responsibility for their performance
d. attained her lesson objective because of her effective spelling drill
conducive for learning of teaching
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130.Which response/s come/s from a behaviorist? a. #2 and #4 c. #3 and #4 b. #1 and #2 d. #1 and #3 131.On which educational philosophy is response #1 anchored? a. Existentialism c. Progressivism b. Essentialism d. Bahaviorism 132.If you learned toward a progressivist philosophy, with which response would you agree? a. #2 b. #3 c. #4 d. #1
Situation 2- One principle in the utilization of technology of the classroom is appropriateness of material or activity. 133.Teacher C wants his students to master the concept of social justice. Which series of activities will be most effective? a. Pretest‐teaching‐posttest b. Pretest‐teaching‐posttest‐re‐teaching for unlearned concepts‐posttest c. Review‐pretest‐teaching‐posttest d. Teaching‐posttest 134.Teacher A likes to show how the launching of spaceships takes place. Which of the following materials available is most fit? a. Model c. Replica b. Mock‐up d. Realia 135.Teacher B likes to concretize the abstract concepts of an atom. She came up with a concrete presentation of the atom by using wire and plastic balls. How would you classify Teacher B’s visual aids? a. Chart c. Model b. Replica d. Realia
Situation 3- After reading and paraphrasing Robert frost’s “Stopping by the Wood on a snowy Evening”. Mr. Sales asked the class to share any insight derived from the poem . 136.The class was asked to share their insights about the poem. The ability to come up with a n insight stems from the ability to a. analyze the parts of a whole b. evaluate the worthiness of a thing c. relate and organize things and ideas d. comprehend the subject that is being studied 137.To ask the class any insight derived from the poem is based on the theory of a. realism c. conditioning b. behaviorism d. constructivism 138.On which assumption about the learner is Mr. Marquez’s act of asking the class to share their insight based? a. Learners are like empty receptacles waiting to be filled up b. Learners are meant to interact with one another c. Learners have multiple intelligence and varied learning styles d. Learners are producers of knowledge not only passive recipients of information
Situation 4- Principal E wants her teachers to apply constructivism in teaching
139.On which assumption/s is the principal’s action anchored? I. Students learn by personally constructing meaning of what is taught. II. Students are construct and reconstruct meaning based on experiences III. Students derive meaning from the meaning that the teacher gives a. II only c. I, II, and III b. I and II d. I only 140.Which materials will her teachers LEAST prefers? a. Controversial issues b. Open‐ended topics c. Unquestionable laws d. Problem or cases 141.Which concept/s of the learner will Principal E NOT accept? I. “Empty vesse!” II. “Tabula rasa” III. Candle to be lighted a. III only c. II only b. I only d. I and II
Situation 5- Study the matching type of test then answer the 3 questions that follow: Column A Column B 1. equilateral triangle A. With 3 equal sides 2. right triangle B. With 5 equal sides 3. octagon C. Has 90- degree angle 4. pentagon 5. heptagon 6. poly
D. Means many E. with 7 sides F. with 8 sides
142.How can you make the items homogeneous? a. Increase the number of items in Column B b. All items should be on polygons c. Remove the word triangle in items #1 and #2 in column A d. The word “gon” must be included in column B 143.What is the main defect of this matching test? a. the matching type is an imperfect type b. the items are NOT homogeneous c. the items quite easy d. an obvious pattern is followed in the answering 144.Which should be done to improve the matching type of test? a. Capitalize the items in Column A b. Items in Column A and B should be exchanged c. Drop #6 item in Column A d. The item in Column A should be increased
Situation 6- Below the template for Scoring Rubric. 5-Demonstrate complete understanding of the problem. All requirements of task are included in response 4-Demonstrate considerable understanding of the problem. All requirements of task are included 3- Demonstrate partial understanding of the problem. Most requirements of task are included 2- Demonstrate little understanding of the problem. Many requirements of task are missing 1- Demonstrate no understanding of the problem 0-No response/task not attempted
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145.Which of these is/are essential in constructing a scoring rubric? I. Description of criteria to serve as standard II. Clear descriptions of performance at each level III. Levels of achievement (mastery) IV. Rating scheme a. I, II, III c. I, II, III, IV b. I, II d. I only 146.Which statement is TRUE of the rubric? a. It is developmental b. It is analytical c. It is both holistic and developmental d. It is holistic 147.Which is TRUE of the scoring rubric? I. It describes criteria of levels of achievement II. It has a rating scheme III. It limit itself to 4 levels of achievement a. I and II b. I and III c. II and III d. I, II and III
Situation 7- Study the table on item analysis for non-attractiveness and nonplausibility of distracters based on the results of a try-out test in Science. The letter marked with a asterisk is the correct answer. Item No. 1 Upper 27% Lower 27%
A 10 6
B 4 5
C 1 2
D 1 2
E 0 0
148.The table shows that the test item analyzed . a. has a positive discrimination index b. has a negative discrimination index c. is extremely easy d. is extremely difficult 149.Based on the table, which is the most effective distracter? a. Option D b. Option A c. Option C d. Option B 150.Based on the table, which group got more correct answer? Upper group a. It cannot be determined b. c. Lower group d. Data are not sufficient to give an answer
When you get right down to the root of the meaning of the word "succeed," you find that it simply means to follow through.
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GENERAL SCIENCE POST – TEST Multiple Choices:
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These are living things that use sunlight, chlorophyll, water and carbon dioxide to produce food. a. Autotrophs c. Heterotrops b. Consumers d. Food chain It is describe as the lifeline of the body. IT is the body’s “pickup” and delivery system.” a. Blood c. circulatory system b. nervous system d. heart It carries the oxygen‐rich blood to the head, arms, chest and down to the waist and the legs. a. heart c. aorta b. ventricles d. arteries They are the transmitters of message from the different parts of the body to the brain and vice versa. a. spinal cord c. brain b. neurons or nerve cells d. arteries Which part of the brain controls the following activities: breathing, blood pressure, heart rate, alertness. a. brainstem c. cerebrum b. hypothalamus d. spinal cord Refers to a sequence of organism in a community that constitutes a feeding chain. a. photosynthesis c. consumers
15. What is a eat haze? a. A reflection caused by pollutants in the air b. A distorted image resulting from the bending o sun’s light rays by changes in air temperature c. A movement o warm air over a vast expanse of land d. Caused by extremely high temperature common in dessert areas 16. What sort of rock formation do the world’s greatest mountain ranges consist of? a. Magma c. Fold eruptions b. Chalk deposit d. Slip formation 17. What is the fore that wears down mountains?
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a. Earthquake c. Volcanic eruptions b. Erosion d. Deforestation How are volcanic island formed? a. Collision of two oceanic plates b. Cooling of lava by seawater c. Volcanic eruptions d. Accumulation of corals When the Theory of plate Tectonics was generally accepted? a. 1900’s c. 1950’s b. 1930’s d. 1980’s The weathering away of rocks by water, wind and ice. a. Denudation c. Volcanic rock b. Erosion d. Metamorphic rock How do hormones work? a. By releasing adrenaline b. By controlling cell chemistry
b. ecosystem d. food chain c. By regulating water loss Is a group of interacting plants, animals and human d. By controlling blood pressure in a particular area. 22. Name the male and female sex hormones a. ecological community c. living organism a. Sperm cell & Ovum c. Chromosomes b. environment d. food b. Testosterone d. Red & white blood chain cells Excessive presence of carbon dioxide in the air, 23. The unit of measurement of energy in a given an trapping heat near the earth’s surface causing a rise mount of food in temperature in the environment. a. Pound c. Olfactory system a. El Nino c. Deforestation b. Kilo d. calorie b. “Greenhouse Effect” d. Weather 24. Nitrogen compounds known as the building blocks disturbance of proteins The Earth’s shield against sun’s harmful radiation. 25. The growth of roots towards water is an example a. Atmosphere c. Ozone layer of? b. Air d. Forest a. Chemotropism c. Hydrotropism The use of product containing ______ is b. Geotropism d. Phototropism discouraged because they contribute to the 26. DNA means depletion of_____. a. Data nurturing analysis c. Deoxyribonucleic a. Chlorofluorocarbon‐solar radiation acid b. Gas‐ozone layer b. Deoxytribonucleic acid d. Deotrixyl nucleic c. Ozone layer air acid d. Chlorofluorocarbon‐ozone layer 27. What are the three products of oxygen when it has What causes high and low tides? been burned? a. Earth’s rotation on its axis a. Water, carbon dioxide and air b. Moon’s gravitational pull b. Energy, water and carbon dioxide c. Sun’s solar energy c. Energy, carbon and oxide d. Earth’s gravitational pull d. Energy, air and water How is coral a tool formed? 28. In flowering plants, fertilization happens in the? a. Volcanic eruption a. Pollen tube c. Ovules b. Corals growing around a volcanic island b. Stamen d. Pollen grain c. Underwater bedrock formations 29. The development of egg without fertilization d. Earthquake a. Mitosis c. Spermatogenesis What is a long shore drift? b. Parthenogenesis d. Mitochondria a. Movement o sand and shingles along the coast 30. Which of the following is a source of energy needed b. Sand bars for photosynthesis? c. Accumulation o sad at the river mouth a. Water c. Light d. Island formed by volcanic eruptions b. Soil d. Fertilizer How does an occlusion form? 31. Chemistry is primarily concerned with the a. Cold air moving up from the ground composition and changes of? b. Cold front pushing warm air up of the ground a. Nature c. Man c. Unbalance electrical reaction in the air b. Matter d. Earth d. Cold and warm air mixing in the atmosphere St. Louis Review Center-Inc-Davao Tel. no. (082) 224-2515 108
32. A scientific theory is a. A hypothesis not yet subjected to experimental test b. An idea that correctly predict the result c. An imagination d. A guess 33. Which of the following units of measure is equivalent to cubic centimeter? a. Milligram c. Millimeter b. Milliliter d. Centiliter 34. Which of the following is NOT a compound? a. acetic acid c. magnesium b. alcohol d. Zinc Oxide
45. When a force is applied to a body, several effects are possible. Which one of the following effect CAN”T occur? a. the body rotates b. the body changes direction c. the body increase its mass d. the body changes shape 46. It is the reluctance of the object to change either its’ state of rest or uniform motion in a straight line a. Force c. Inertia b. Friction d. Motion 47. This law states that energy cannot be created nor destroyed but only changes from one form to
35. The easier the atom to receive electrons is measured by its? a. Elecrtonegativity c. Number of shells b. Atomic radius d. Valence electrons 36. The willingness o an atom to receive electron is measured by its? a. Electronegativity c. Atomic size b. Ionization potential d. Electron affinity 37. A molecule is said to be polar or dipole if? a. Its positive and negative charges are at different places b. It possesses polar bonds c. Its’ polar bond have unsymmetrical charge distribution d. All of the above 38. Which of the following is NOT a physical property of water? a. freezing point at 0 degree C
another a. Energy law b. Kinetic Theory of Matter c. Law of Conservation Energy d. None of the above 48. This law states that matter is made up of a large number of molecules which are in continuous motion a. Boyles’s Law c. Law of Conservation Energy b. Kinetic Theory d. None of the above 49. The lowest possible temperature that a substance can reach a. Freezing point c. Steam point b. Absolute Zero d. Threshold 50. It is the amount of heat required to raise the temperature of kg. of a substance by degree C a. Calorie c. Specific heat capacity b. Watt d. Joule
b. boiling point at 100 degrees C c. its’ heat of fusion at 80 cal/g. d. its’ density at 4 degrees at 1 lb/cu. Ft 39. Which of the following statements is true? a. Molecular weight does not influence boiling and melting point of a substance b. Boiling and melting point tend to increase with molecular weight c. Boiling and melting point tend to decrease with molecular weight d. None o the above 40. A Mole is the amount of substance or a mass of a substance that contains? 23 23 a. 6.02 x 10 particles c. 6.02 x 10 particles 23 23 b. 60.2 x 10 particles d. 60.2x 10 particles 41. It is the measure of the amount of matter in an
51. The relationship of give‐and‐take of living organism in the biosphere is a balance of nature called________. a. universal relationship b. symbiotic relationship c. spontaneous relationship d. abiogenetic relationship 52. Process of removing excess odor in water. a. sedimentation c. distillation b. chlorination d. aeration 53. Which of the following statement is CORRECT? a. As altitude increases, atmospheric pressures corresponding b. Throughout the available space, gas tends to contract c. Equal chances are always given to all in life d. Shadow is formed when a colored object is projected against the wall
object a. Weight c. Volume b. Mass d. Quantity 42. It is the distance traveled by the body per unit time and tell how fast or slow the body moves a. Velocity c. Acceleration b. Speed d. None of the above 43. The rate of change of the distance traveled per unit time in a stated direction a. Velocity c. Acceleration b. Speed d. None of the above 44. This law states that the force acting upon an object is equal to the product o the mass and acceleration of the object nd a. Newton’s 2 law of motion rd b. Newton’s 3 law of motion st c. Newton’s 1 law of motion d. None of the amount
54. The earth rotates on its axis from west to east. This causes the sun to _______? a. appear with a fiery orange color b. cause the appearance of solar eclipse c. rise room the east and sets in the west d. emit solar radiation 55. One of these planets has the greatest gravitational pull. Which one is it? a. Mars b. Earth c. Mercury d. Jupiter 56. It is the law which explains why one can pull a piece without topping a glass in a quick motion. a. energy in motion c. law of inertia b. gravity d. force 57. Which of the following is NOT a source of energy? a. water c. geothermal heat b. nuclear d. inertia at rest 58. Which instrument will one use to convert mechanical energy to electrical energy? a. rotor b. generator c. motor d. circuit
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59. Sun’s energy is generated by? a. nuclear fission b. sun enacting with gravity c. nuclear fusion reaction d. reaction with magnetic field 60. Application of energy is called _______? a. work c. kinetics b. inertia d. simulation 61. The falling of any form of water from the air to the earth’s surface a. Condensation c. Water vapor b. Precipitation d. Rainwater 62. The part of the atmosphere that filters the
76. It is a segment of DNA molecule which controls the appearance of a given trait a. Chromosomes c. Gametes b. Genes d. Zygotes 77. Group of similar cells performing similar functions together a. Organs b. System c. Nucleus d. Tissue 78. The diffusion of water through a semi permeable membrane a. osmosis c. Transfusion b. permeability d. Capillary 79. It shows the complex food relationship of organism
ultraviolet rays of the sun a. stratosphere c. Ozone layer b. Troposphere d. Ionosphere The point in the earth’s orbit nearest to the sun a. Solstice c. Aphelion b. Eclipse d. Perihelion A property of minerals which gives off rays of light when exposed to ultraviolet light. a. Luminescence c. Radiation b. Phosphorescence d. Fluorescence Which process involves chemical weathering? a. Carbonation c. Hydration b. Oxidation d. All of the above How long does it take for the earth to complete one rotation a. 365 days c. 24 hours b. 30 days d. 12 hours What is the principal function of gravity in the
in a given area and the cyclic flow of food through organisms a. Food chain c. Food pyramid b. Food web d. Biological cycle 80. Which of the following does NOT occur to both respiration and fermentation? a. energy is released b. sugar is broken down c. carbon dioxide is produced d. alcohol is formed 81. Energy removal is best illustrated in a. boiling of liquid substances b. changing water to ice c. changing water to stem d. none of the above 82. Refers to the maximum amount of solute expressed in grams that can be dissolved in 100 grams of water at a specific temperature
universe? a. Provision for energy b. Keeps the stars and other heavenly bodies in orbit c. Causes movement in space d. Part of universal design 68. What does the word “monsoon” mean? a. Moon will soon come c. Seasons b. Rains d. Wet weather 69. Its’ discovery enable geologist to date rocks accurately a. layering c. Radioactivity b. Evolutionary staging d. Carbon‐ dating 70. It occurs when the earth is between the sun and the moon, with the earth’s shadow cast over the moon.
a. Solubility c. Molarity b. Stability d. Molality Compounds with the same molecular formula but with different structural formulas a. Cellulose c. Polymers b. Isomers d. Monomers The most penetrating type of radiation given of by radioactive elements a. Alpha particle c. Gamma particle b. Beta particle d. None of the above The basic unit for expressing the masses o individual atoms a. Atomic number c. Nucleus b. Atomic mass unit d. Atomic weight A substance that speeds up a chemical reaction without itself undergoing a chemical change a. catalyst c. Electrolytes b. Enhancer d. Ionizer
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a. Total eclipse c. Eclipse b. Lunar eclipse d. Partial Eclipse 71. Male and female reproductive part of a flower a. Pollen grains and ovules c. Pollen grains and pistil b. Stamen and pistil d. Stamen and ovules 72. In the human body, the cell that most nearly resembles a one‐celled animals a. Red Blood cell c. Nerve cell b. White Blood cell d. Antibodies 73. The main energy source of a plant‐eating animals a. Glucose b. Starch c. Cellulose d. Glycogen 74. These are cellular secretions which help regulate the breakdown and buildup of various substance in the body a. Enzymes c. Plasma b. Amino Acids d. Hormones 75. It is the energy source of the cell which it uses for growing, reproducing and other activities a. Adenosine Triphosphate c. Chloroplast b. Amino Acids d. Sunlight
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87. The temperature at which the vapor pressure of the liquid is equal to the pressure of the surroundings atmosphere a. Melting point c. Boiling Point b. Critical point d. None of the above 88. the warming of the earth’s surface due to an increase in atmospheric carbon dioxide a. “Greenhouse effect” c. Atmospheric pressure b. Ozone d. El Nino phenomena 89. Describe the force of gravity on an object a. mass c. capacity b. weight d. pressure 90. When gaseous molecules are compressed, they tend to? a. increase in volume c. repel each other b. decrease in volume d.attract and liquefy 91. It is the union of two light nucleus to form a heavier nucleus, resulting in a mass defect and release of energy a. Radiation c. Nuclear Fusion b. Nuclear Fission d. Radioactivity
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92. Which of the following statements is a characteristic of an electronic spectrum? a. They all travel at the same speed in fire space b. The exhibit diffraction and interference phenomena c. They follow the laws of refraction and reflection d. All of the above 93. The term “RADAR” is derived from the phrase? a. “Radiation Detection and Ranging “ b. “Radiation Diffusion and Ranging” c. “Radio Diffraction and Resolution” d. “Radiation Diffraction and Resolution” 94. A material whose ability to conduct electricity lies
science and health
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between those of conductors and insulators a. Integrated Circuits c. Semiconductors b. Silicon Chips d. Insulators 95. “LASER” is derived from the phrase? a. Light Amplification by Stimulated Emission of Radiation b. Light Application by Simulated Emission of Radiation c. Light Amplification by Simulated Ejection of Radiation d. None of the above 96. What is the color of a transparent substance? a. The color of the light it absorbs b. The color of light it reflects c. The color of light it transmit d. The color of light it refracts 97. What is a rotating electromagnetic called? a. Motor b. Rotor c. Phasor d. Sensor
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98. What happens with the centripetal force when sped is doubled? a. remains the same c. triples b. force is increase 4x d. force is doubled 99. What is an electrochemical cell in which the reacting materials can be renewed by the use o reverse current a. Storage cell c. Fuel cell b. Primary cells d. Chemical cell 100. What will make an object move in a circular path? a. Central force c. Frictional force b. Gravitational d. Electromagnetic force
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