New Edition
Countdown Level Six Maths
Teachi T eaching ng Guide
Shazia Asad
Contents Introduction Introduction .................... ............................... ...................... ..................... ..................... ...................... ..................... ..................... ............... iv Format of the guide ........................................................................................v Lesson Planning Planning ............................. ........................................ ...................... ..................... ..................... ...................... .................... ......... vi Chapter 1: Elementary Elementar y Set Concepts ........................... ............. ............................ ........................... ..............2 .2 Chapter 2: Types of Sets ............................ .............. ........................... ........................... ........................... ......................2 .........2 Chapter 3: Natural Numbers and Whole Numbers ........................... .............. .................4 ....4 Chapter 4: Whole Numbers and the Number Ray ........................... ............. ....................7 ......7 Chapter 5: Factors and Multiples........................... ............. ............................ ........................... .....................9 ........9 Chapter Chapter 6: Factorization Factorization ..................... ................................ ...................... ..................... ..................... .................... ......... 11 Chapter Chapter 7: Least Common Multiples Multiples ..................... ............................... ..................... ..................... .......... 12 Chapter Chapter 8: Integers Integers ...................... ................................ ..................... ...................... ...................... ..................... ................. ....... 14 Chapter Chapter 9: Operations Operations on Integers Integers ..................... ................................ ...................... ...................... ............... 16 Chapter Chapter 10: Square Roots of Whole Whole Numbers Numbers..................... ................................ .................. ....... 18 Chapter Chapter 11: Ratio and Proportion Proportion ..................... ................................ ..................... ..................... ................. ...... 19 Chapter Chapter 12: Day-to-Day Day-to-Day Arithmetic .................... ............................... ...................... ..................... ............. ... 22 Chapter Chapter 13: Profit Profit and Loss ...................... ................................ ..................... ...................... ...................... ................ ..... 24 Chapter Chapter 14: Simple Simple Interest Interest ...................... ................................ ..................... ...................... ..................... ................ ...... 25 Chapter Chapter 15: Introduction Introduction to Algebra Algebra.................... ............................... ...................... ..................... ............. ... 26 Chapter Chapter 16: Operations Operations Using Using Algebra Algebra ..................... ................................ ...................... ................... ........ 27 Chapter Chapter 17: Linear Equations Equations.................... ............................... ...................... ..................... ..................... ............... .... 29 Chapter Chapter 18: Basic Concepts of Geometry ...................... ................................ ..................... .............. ... 30 Chapter Chapter 19: Construction of Line Segments ..................... ................................ ..................... .......... 32 Chapter Chapter 20: Angles .................... ............................... ...................... ..................... ..................... ...................... ..................... .......... 33 Chapter Chapter 21: Triangles ..................... ................................ ...................... ..................... ..................... ...................... ................ ..... 35 Chapter Chapter 22: Circles ...................... ................................. ..................... ..................... ...................... ...................... ................... ........ 37 Chapter Chapter 23: Mensuratio Mensuration: n: Area ..................... ............................... ..................... ...................... ..................... ............ 38 Chapter 24: Mensuration: Volume .......................... ............ ............................ ........................... .................. ..... 40 Chapter Chapter 25: Graphs Graphs .................... ............................... ...................... ..................... ..................... ...................... ..................... .......... 42 ii
Introduction Professional development improves a teacher’s depth, knowledge, and instructional decision-making. Judgement and leadership skills are two of the many facets of a professionally trained teacher. In order to be an effective teacher, one must be actively engaged in content learning. It is closely connected to delivering lessons in the t he classroom, and learning as well while doing so. Many brilliant ideas can be picked up from the students themselves and the lesson plan modified accordingly. Mathematics Mathematics should become part of ongoing classroom routines, routines, outdoor play, and activities involving day-to-day life. Teachers also benefit from working with colleagues who can question, challenge, support, and provide a network of resources for each other. is teaching guide has been developed keeping in mind the needs of a teacher while using the textbook. It is a reference which pre-empts the teacher’s queries that might emerge during the course of using this series. Use it as a ‘guide by your side’ and not as a ‘sage on the stage’. e guide follows the format given on the following page. I hope the teachers will find the guide useful and enjoy their teaching even more. Shazia Asad
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Format of the guide IMPROVISE / SUPPLEMENT / METHODOLOGY / ACTIVITY
Teaching techniques and activities mentioned in the manual are to be utilized by implementing, improvising or supplementing. KEYWORDS / TERMINOLOGIES
Usage of mathematical language and alternative terms EQUATIONS / RULES / LAWS
Quick recall of numbers, facts, rules, and formulae ASSOCIATION
Ability to adapt the aforementioned facts to mathematical usage CONCEPT LIST
Flow charts, pictorial representations, and steps of mathematical procedure BACKTRACK QUESTIONS
An adaptable approach to calculations, investigating various techniques, and methodologies FREQUENTLY MADE MISTAKES
To be able to pre-empt the pitfalls the students might encounter in the course of a particular topic SUGGESTED TIME LIMIT
Suggested duration of classes will be mentioned but it is entirely up to the teacher to evolve her own time considering the level of her students.
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Lesson Planning Before we even begin to write on lesson planning, it is important to talk about teaching and the art of teaching. A. Furl
irst understand by relating to day-to-day routine and then learn. It is vital vita l for the teachers teac hers to realize reali ze the need nee d to incorporate incor porate meaningful meaning ful teaching by relating to daily routine. Another R is re-teaching and revising, which is covered under the supplementary/continuity category. Effective teaching stems from engaging every student in the classroom. is is only possible if you have a comprehensive lesson plan. How you plan your work and then work your plan is the building block of teaching. ere are three integral facets to lesson planning: curriculum, instruction, and evaluation. F
1. CURRICULUM
A curriculum must meet the needs of the students and objectives of the school. It must not be over ambitious, or haphazardly planned. is is one of the major shortfalls when planning to teach maths according to the curriculum. 2. INSTRUCTIONS The instruction methods used are, for example, verbal explanation,
material-aided explanation, teach-by-asking philosophy. e methodology adopted by teachers is a reflection of their skills. I will not mention experience as even the most experienced teachers could be adopting flat and short-sighted approaches; approaches; the same s ame can be said s aid for beginner teachers. e best teacher is the one who works out the best plan for the class customized to the needs of the students. is will only be successful if the teacher is proactive and learning and re-learning the content the whole year round and reinvents the teaching methodology on a regular basis. 3. EVALUATION
is is the tool that tells how effective the teachers have been in their attempt to conduct the topic. e evaluation programme is not just a test of the student but also indicates how well the lesson has been taught. v
B. Long-term Lesson Plan
e long-term lesson plan encompasses the entire term. Generally, schools have their coordinators plan out the core syllabus and the unit studies. Core syllabus comprises of the topics to be taught during the term. Two things which are very important while planning this are the time frame and whether the students have prior knowledge of the topic. An experienced coordinator would know the depth of the topic and the ability of the students to grasp it in the given time frame. Allotting the exact number of lessons for a topic is essential, as more time spent on a relatively easy topic could affect the time needed for a difficult topic when it is taken up. C. Suggested Unit Study Format
Week
Dates
Month
Number of Days
Remarks
D. Short-term Lesson Planning
is is where the course teacher comes in. e word lesson comes from the Latin word ‘lectio’ which means action of reading. e action of conducting a topic takes plave in the classroom, what and how the students will be taught. Following is a suggested format for planning a topic. It should be noted that each school and the concerned teacher may have their own way of doing things. is should be respected but at the same time teaching can be developed by improvising from other sources. Study the following outline for this to take effect. 1. TOPIC
is could just be the title of the topic e.g. volume of 3–D shapes.
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2. OVERVIEW
e initial concern while planning for a topic should be ‘how much the students already know about the topic’. If it is an introductory topic, then recall a preceding topic that may lead to the topic being taught. For the topic given above, the teacher could write the following: e students have prior knowledge of the properties of 3-D shapes e.g. cube, cuboid, cylinder. ey can identify the dimensions of each shape viz. their length, breadth, height, and radius. 3. OBJECTIVES
is highlights the aims of the topic. Example:
• • • •
to calculate the volume of a cube and cuboid to calculate the surface area of a cube and cuboid to calculate the volume of a cylinder to calculate the surface area of a cylinder
4. TIME FRAME
Ascertaining the correct time frame would make or break a lesson plan. Generally, class dynamics vary from year to year so flexibility is important. e teachers should draw their own parameters but can adjust the time frame depending on the receptivity of the class to the topic at hand. e model plan could say • 5 classes for teaching the volume of a cube and cuboid, • 6 to 7 classes for teaching the surface area of a cube and cuboid, • 5 classes for teaching the volume and surface area of a cylinder. Note that introduction takes longer but when the sessions continue, the teacher may find that the students learn faster and the time frame initially set may actually be reduced. For example, for the topic of cylinders one should know in advance that it will take less time. 5. METHODOLOGY
is means how you would demonstrate, discuss, and explain a topic. • e introduction to calculate the volume of a cube and cuboid should be done using real-life objects that the students should be encouraged to bring. vii
•
e students must be encouraged to feel the surfaces and the space within the objects and also measure the dimensions of the shapes. • e formula can then be introduced on the board with the students noting it in their exercise books. e same procedure may be followed for the topic of cylinders. 6. RESOURCES USED
• • • • •
Every day objects and models of cubes, cuboids, and cylinders. Exercises A, B, and C. Worksheets made from source X. Assignment / project on making diagrams of the shapes. Test worksheets made from source X.
7. CONTINUITY
• Alternate sums will be done from Exercises A, B, and C for class work. • e remaining alternate sums to be done for homework. It may be noted that class work should comprise of all easy to difficult sums and once the teacher is assured that students are capable of independent work, homework should be handed out. 8. SUPPLEMENTARY WORK
A project or assignment can be organized. It can be group work or individual research to complement and build on what the students have already learnt in the class. • e students can be organized in groups of threes and assigned to make net diagrams for the shapes discussed. • ey can do this work at home and conduct a presentation in class. 9. EVALUATION
As stated earlier, it is an integral teaching tool. e evaluation has to be ongoing while doing a topic and also as an ‘end of topic’ formal test. • Students can be handed a worksheet on day 3 covering the volume of a cube and cuboid. It should be a 15-minute quiz and self / peer corrected in class. • Similarly, a quiz worksheet on the volume and surface area of a cylinder can be handed out. • A formal test of 20 marks can be given to the students at the end of the chapter. viii
Chapter 1 Elementary Set Concepts This chapter has been treated as a beginners’ attempt to understand sets. Introduce this topic by making sets of students wearing glasses or boys wearing glasses, or girls with glasses, etc. This is more of an activity-based chapter. KEYWORDS / TERMINOLOGIES
sets, well defined, elements of the set, roster form, member of (∈), is not a member of (∉) Girls
Boys
Spectacled
Spectacled
IMPROVISE / SUPPLEMENT & ASSOCIATION
Similarly, you could talk about a party and create sets within this set up. For example, set of colours: people in red, blue, various colours; set of people: children, boys, girls, elders; set of food items: cupcakes, drinks, sandwiches, etc. You need to correlate this chapter to real-life situations at the introductory stage, and then make sets of your own. SUGGESTED TIME LIMIT
This chapter should not take more than two lessons.
Chapter 2 Types of Sets Usage of mathematical terms and symbols are taught in this chapter. The teacher should be well-versed in the mathematical language used in this chapter. 2
KEYWORDS / TERMINOLOGIES
Finite, infinite, empty sets, and equivalent sets are terminologies that should not be treated just at that. An effort should be made to correlate these terms to real-life situations. IMPROVISE / SUPPLEMENT
Students could be asked to do a 5-minute brainstorming quiz of creating as many, finite, infinite, and empty sets. Using numbers as members of a particular set should be the next step. ASSOCIATION
In improvising, the students’ knowledge will be tested. They should have revised the terminologies of prime, composite, whole, and natural numbers, integers, etc., learnt earlier. EQUATIONS / RULES / LAWS
Table on page 7 should be memorized by the students. FREQUENTLY MADE MISTAKES
Differentiating between proper subsets and subsets sometimes becomes confusing to students. But if explained with the help of examples, concepts can be understood and retained. IMPROVISE / SUPPLEMENT
Examples may include these: • Giving an equal number of the same sweets to Wasif and Sara to form a subset. • Ahmed and Saad could be given sweets in a way so as to form a proper subset. BACKTRACK QUESTIONS
In exercise 2b, students are now encouraged to use symbols and write in mathematical language. Earlier they were asked to read and comprehend the terminologies. Test 1 could be given as a marked assignment. 3
SUGGESTED TIME LIMIT
5–6 lessons
Chapter 3 Natural Numbers and Whole Numbers This chapter is essentially a chapter on numbers and the significance of place value. Students should be introduced to the terminologies of whole and natural numbers because they form the basis of place value. The definitions explain the properties and thus differentiate between the two. Natural numbers begin from one, and whole numbers begin from zero. The concepts of FINITE and INFINITE should be reinforced using set notation and, less than and greater than signs. Example
All natural numbers < 10 {1, 2, 3, 4 . . . . . 9} This is a finite set. Similarly, All whole numbers >10 {11, 12 13 . . . . . . } This is an infinite set. ASSOCIATION
After the students have learnt definitions, they must be introduced to the ‘Pakistani System of Numeration’ and ‘International System of Numeration’. The comparison table given on page 17 should be memorized by the students. A horizontal table can also be introduced to the students as given below: PAKISTANI SYSTEM OF NUMERATION
Arabs; Ten Crores, Crores; Ten Lac, Lac; TTh, Th; H, T, U
4
INTERNATIONAL SYSTEM OF NUMERATION
HM, TM, M; HTh, TTh, Th; H, T, U The periods for Pakistani and International place values are different; these are denoted by commas. This will enable the students to differentiate between the two systems easily and they will be able to express a set of numbers according to their place value in two different ways. Example
4,56,78,034
four crores, fifty-six lacs, seventy-eight thousand, and thirty-four or
45,678,034
forty-five million, six hundred and seventy-eight thousand, and thirty-four
Activity Exercise 3
It is suggested that questions 1 and 2 be done immediately after introducing the natural and whole number terminologies. The activity could be done for 10 minutes as an oral quiz in the class. The students may be divided into groups and marks could be given for the correct answers. After about 2 or 3 oral classes for place value conversion, questions 3, 4, 5, and 6 should then be attempted by students in their exercise books. Expanded notation can be explained with the help of a place-value table.
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Example
245678 TM M HTh TTh Th H T U 2 4 5, 6 7 8 2 is in the hundred thousands place. 2 0 0, 0 0 0 4 is in the ten thousands place. 4 0, 0 0 0 5 is in the thousands place. 5, 0 0 0 6 is in the hundreds place. 6 0 0 7 is in the tens place. 7 0 8 is in the units place. 8 To check, ask the students to add vertically and see if they get the given number. +
6
200,000 40,000 5,000 600 70 8 245,678
FREQUENTLY MADE MISTAKES
Students are often confused using the Pakistani and International tables simultaneously. The periods with commas should be emphasized and with a lot of practise quizzes, they should not face any problems. Activity
Palindromes are numbers that are the same but expressed and sequenced differently. The activity on page 17 can be done, and the students should be encouraged to make their own palindromes. Question number 9 is one such fun question. Encourage students to make as many numbers and applaud the student who gets the most number of formations. SUGGESTED TIME FRAME
This chapter should be completed in 5 to 6 lessons.
Chapter 4 Whole Numbers and the Number Ray In this chapter, number rays are explained, which is a transition to number lines. The four operations, DMAS (÷, ×, +, –) are taught using the number ray. Diagrams on pages 20 / 21 should be drawn on the board by the teacher and each student should be encouraged to answer questions individually. Activity
Students should be encouraged to make a model number line with strings and flashcards to be put up on the classroom soft board. Every time they solve a sum using the four operations, they can look up and correlate mentally. Soon they would be able to visualize without the help of the model. Another interesting game that could be played would be on the lines of hopscotch; students are asked to draw a number line on the floor and then each student is asked to attempt a sum using the four operations by hopping on the number line. This should not take more than 10 minutes of the class period.
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IMPORTANT It is very important that the students become extremely skilled at using the number line as they would soon be doing algebra which employs the same concept. Properties of zero, one, addition, subtraction, and division form the basis of DMAS. It should be emphasized to the students that the order of the operation does matter. Question 7 of Exercise 4 highlights this fact. FREQUENTLY MADE MISTAKES
The students tend to solve these sums from left to right and do not follow the correct order of operations. The teacher should emphasize the necessity of this practice. MATH PROJECT / RESEARCH
The students could be given a Maths project. For example; ‘History of how the Arabic Numbers evolved’, could be explored and research papers written. When and ‘how’ zero became a significant number, could form a part of this research. Please inform the students that ‘arab’ is an Urdu word written as , and it represents a billion in the International Numeration system. It is not to be confused with ‘Arab’ which represents a native of Arabia. The Pythagorean Doctrine on page 24 could be highlighted and the students encouraged to read about this great Mathematician. Views can be exchanged in class. The students could make a portfolio / folder on various maths projects that they do throughout the term. This will be a treasure chest of mathematical facts for each student to refer back to whenever needed. Exercise 4 should be done in the students’ exercise books. Extended / supplementary practice sums should be handed out as worksheets as the students would need plenty of practice in this chapter. SUGGESTED TIME LIMIT
As this chapter is very brief, 2 to 3 lessons should be enough for it. The properties need not be learnt by the students but they should be understood, with the help of examples, on the classroom board. 8
Chapter 5 Factors and Multiples Chapter 5 is an introduction to chapters 6 and 7. These 3 chapters are interconnected, and can be done at a stretch, and treated as one big chapter comprising of the topics of factors, multiples, LCM, and HCF. In chapter 5, the definitions of factors and multiples are discussed and the differences between prime numbers and composite numbers are highlighted. After this, the test of divisibility becomes an integral part to the text being taught. KEYWORDS / TERMINOLOGIES
Dividend, divisor, quotient, and remainder are terms that the students are familiar with. Factors and multiples are concepts that can be explained while dividing and when the remainder is a zero. The divisor is a factor and the dividend is the multiple of the divisor. Example
When 56 is divided by 7, the remainder is zero. 7 is a factor of 56, and 56 is a multiple of 7. However, actual definitions should be done thoroughly in class and students asked to note these in their exercise books. Prime, composite, twin-prime, prime-triplet, factors, and multiples are some of the terminologies used in this chapter. The students should be encouraged to work with examples of each term to understand the difference. FLOW CHARTS / DIAGRAMS
Students should be asked to make factor trees. Example
35
/ \ 5 7 TABLES
The students should be explained the Test of Divisibility Table on pages 30 and 31 with the help of examples and then asked to memorize them. 9
Examples
1. 940 4 — since the last digit is even, test of divisibility of 2 applies 2. 115 2 — since the last digit is even, test of divisibility of 2 applies 1 + 1 + 5 + 2 = 9 — the sum adds up to a multiple of 3, thus test of divisibility of 3 also applies 3. 450 5 — since the last digit is 5, test of divisibility of 5 applies Exercise 5a
Questions 1 to 6 deal with definitions and terms learnt earlier. Questions 7 to 12 deal with tests of divisibility. Encourage the students to memorize the tests and employ them mentally. The test of divisibility of 11 is a little difficult for the students to do mentally. Activity
The sieve of Eratosthenes is an amazing activity and a supplementary tool to understand the concept of prime and composite numbers. As mentioned in the text, it is the same as sifting the chaff from wheat. The numbers 1 to 100 should be written on a graph paper and pasted in the exercise books. The steps given below should be followed. Step 1: Remove 1 as it does not have 2 factors because all prime numbers have two factors; one and the number itself. One has only one factor itself and therefore it is not a prime number. Step 2: Remove all multiples of 2. Step 3: Remove all multiples of 3. Step 4: Remove all multiples of 5. Step 5: Remove all multiples of 7. The numbers left after this procedure are all prime numbers. Exercise 5b
Ask the students to do the sums in their exercise books. A mental maths quiz may be done in class.
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FREQUENTLY MADE MISTAKES
Students generally assume that 1 is a prime number. Sometimes they forget the tests of divisibility and divide, to figure out the answer. The teacher should insist that the students mention the test of divisibility as a statement. SUGGESTED TIME LIMIT
This topic should not take more than 2 to 3 lessons.
Chapter 6 Factorization This chapter deals with the breaking down of composite numbers into their factors. The condition is that they should be prime factors and not composite numbers. Students are already familiar with these terminologies and can easily apply them here. Example
120 / \ 2 × 60 / \ 2 × 2 × 30 / \ 2 × 2 × 2 × 15 / \ 2×2×2× 3 ×5
120 = 2 × 2 × 2 × 3 × 5 = 23 × 3 × 5
This is a good method to teach the students how to deal with prime factorization. They should be told that the composite numbers should be broken down till they cannot be broken down any further. This will naturally lead to prime factorization by division. The examples given on page 39 could be done on the classroom board. The teacher is expected to give more examples on the board, using the format given in the book. Another important point to be stressed is the prime index. If the same factor is more than one, the power is introduced. 11
The Highest Common Factor (HCF) or the Greatest Common Divisor (GCD) can be taught by the prime factorization method. The important point to note is that the smallest common powers are selected and the product is the HCF. HCF by the division method is also known as the Euclid method. The given set of numbers are divided simultaneously till one is obtained as the remainder. The last divisor is the HCF. Word problems based on HCF are given at the end of Exercise 6. A classroom activity could be done whereby the concept of HCF is reinforced. One such activity could be the students making their own word problems related to HCF. This will ensure that the concept of HCF is fully understood. KEYWORDS / TERMINOLOGIES
Prime factorization, power notation, highest common factor, and greatest common divisor are the terms introduced in this chapter. Exercise 6
The students should be asked to attempt the sums by using the correct format. SUGGESTED TIME LIMIT
The topic should cover more than 3 to 4 lessons.
Chapter 7 Least Common Multiples This chapter is a continuation of the previous chapter. Students were earlier asked to list all the factors and now they are to do the same with multiples. Since the list of multiples is an infinite set, the first common multiple is identified to be the answer. It can be explained that since factors are a finite set, the selection of the largest value is possible. But since multiples are infinite, the smallest value is selected which is common to the given numbers as it is an infinite set.
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RECALL
It is important to go back to concepts done earlier so that different mathematical facts can be inter-linked. The students tend to understand concepts better when they learn to relate the previously learnt concepts with the new ones. EQUATIONS / METHODOLOGY
There are two ways of finding the LCM: 1. LCM by the prime-index notation / factorization method 2. LCM by the division method LCM by the prime-index notation method This should be taught first. Example
/
24
\ 2 × 12 / / \ 2 × 2 × 6 / / / \ 2 × 2 × 2 × 3 24 = 2 × 2 × 2 ×3 This when expressed as prime-index notation becomes this: 23 × 3 To find the LCM of any set of numbers, ask the students to express the numbers as prime-index notation first, and then select the largest powered prime numbers and their products. For example: The LCM of 36 and 54 36 = 22 × 32 54 = 33 × 2 13
Ask the students to circle the largest powered numbers and then find their product. 22 × 33 = 2 × 2 × 3 × 3 × 3 = 108 LCM by the division method This is a more organized and error-free method. The steps given below should be followed when using this method. 1. Only prime numbers to be used while dividing. 2. Divide all the numbers in every step. 3. Bring the numbers down if not divisible. 4. Continue dividing till you have 1s (ones) left. 5. Tests of divisibility should be used to calculate the factors. Example: 2 2 2 2 3 3 3 7
42, 21, 21, 21, 21, 7, 7, 7, 1,
48, 24, 12, 6, 3, 1, 1, 1, 1,
54 27 27 27 27 9 3 1 1
LCM = 24 × 33 × 7 = 3024 SUGGESTED TIME LIMIT
Not more than 3 or 4 lessons.
Chapter 8 Integers This chapter is linked to what the students have learnt in chapter 4. They were introduced to number rays and whole numbers. In this chapter students are taken to the next level of integers and the number line. 14
KEY FACTS
The concept of integers is important; it has to be explained well as it forms the basis of algebra also. Negative whole numbers should be explained using day-to-day situations. For example: • Ahmad gives 4 sweets to Sara; he now has 4 sweets less. This can be written as –4. • The temperature in Kashmir fell below zero to –5°C. • My account has a balance of rupees –800. Once the concept of negative numbers is introduced relating to daily life, the number line can be explained. KEY WORDS / TERMINOLOGIES
Integers make up whole numbers that are both negative and positive. Zero is also an integer but it is neither positive nor negative. FREQUENTLY MADE MISTAKES
Students need to know that the bigger the number, the smaller the value of the negative number. e.g. –4 is greater than –6. As we move along the number line, the values increase as they are positive values but moving towards the left, negative values decrease. This fact will help them arrange numbers in the right order. Exercise 8
This exercise tests the students on the concepts taught. More sums should be given for reinforcement. Activity
The fact mentioned on the side bar on page 55 can be supplemented by asking the students to make a time line on any historical fact, e.g. they can be asked to make a time line on the Greeks and the Romans. The concept of and can be correlated to positive and negative numbers. SUGGESTED TIME LIMIT
This is a very brief topic; 2 lessons can be spent on this concept. 15
Chapter 9 Operations on Integers Explaining the four fundamental operations with the number line is necessary. The basic rule governing integers is understood well with the help of the number line. Example
4 – 6 = –2 –6 – 4 = –10 –6 × –4 = + 24 Activity
Make a number line on the floor of the classroom or maybe on the school ground. Give the students, instructions for the game like, ‘if a negative value is given, jump to the left and if the number is positive jump to the right’. The students will soon enough conclude that if –7 and +3 is given they will end up on number –4. Similarly –3 and –2 will get them to –5. RULES / LAWS
After 10 minutes of this activity, point out the rules stated on pages 58, 59, and 61. These should be copied as bullet points in the exercise books. The teacher can also put the rules up on the soft board in the classroom. This will provide good visual recall for the students. RECALL
The students could be asked to revise the rules of BODMAS. If not introduced earlier, explain it in the following way. B is for brackets. O is for of (multiplication). D is for division. M is for multiplication. A is for addition. S is for subtraction. The order of the operations is demarcated by this rule. Students should avoid solving the sums from left to right. 16
There are three types of brackets and are to be written in the order given below. box brackets [ ] curly brackets { } round brackets ( ) [ { ( ) } ] These brackets are attempted inside out. Example
[{4 + (6 – 3)} ÷ 10] Solve the round brackets first, [{4 + 3} ÷ 10] then the curly brackets, [7 ÷ 10] and lastly the box brackets. = 0.7 Answer Exercise 9
Sufficient practice is provided in this exercise. Question 2 not only tests the students on the rules given in this chapter but also for BODMAS. Questions 13, 14, and 15 are word problems which will enhance the students understanding of integers and the rules governing them. Activity
You can read out the story on page 63 and see who understands how the astrologer was more of a mathematician. The least able students might need an explanation. You can ask the students to try making a game on mathematical mind teasers. Activity
You will need two dice, on one of them mark minus signs with a permanent marker. Students take turns in rolling both the dice simultaneously; they will then have to work out the answer each time. Each student will take five turns. The total score will be tabulated. The highest scorer will be the winner. 17
Example
If a students rolls a dice and gets 4 on the first dice and –3 on the second dice, he will mentally work out the answer as this: +4 – 3 = +1 SUGGESTED TIME LIMIT
This is an important topic. Rules need to be revised time and again. A minimum of 4 to 5 lessons should be given to this topic.
Chapter 10 Square roots of Whole Numbers This topic can be introduced with the help of geometry. Activity
Ask the students to cut out squares of different sizes. Once they have pasted these squares on their exercise books, ask them to measure the sides. They will realize that each square side is the same value, e.g. 7 by 7. KEYWORDS / TERMINOLOGIES
Square numbers are the product of the same number. The square root is the number itself, e.g. 7 × 7 = 49 The product is the ‘square’. The multiplicand is the square root of the square number. √ is the sign for square root. √49 = 7 Square root values can be positive and negative. The multiplication rule states that the product of two negative numbers is always positive. METHODOLOGY
To determine the square root of a value by prime factorization and prime index notation Both these methods have been used earlier to calculate the LCM and the HCF. In calculating the square root of a number, the students have to be taught the concept of pairs. 18
Example
√53361 = √3 × 3 × 7 × 7 × 11 × 11 Whether the students employ the prime factorization method or the division method, pairs have to be made. One value of each pair is then multiplied to find the square root. IMPROVISE / SUPPLEMENT
Example 4 on page 68 tests the students learning by asking them to figure out not only the LCM but also to find out whether the answer is a square number or not. They realize that since a pair is missing, they have to introduce a number to form one. This is clearly demonstrated in examples 3 and 4. FREQUENTLY MADE MISTAKES
The students, while either employing the prime factorization method or division method, tend to use composite numbers instead of prime numbers. EXERCISE 10
This exercise tests the students for the concept in this chapter. Extra practice may be given, as it is true that practice makes perfect. SUGGESTED TIME LIMIT
This topic requires not more than three lessons.
Chapter 11 Ratio and Proportion This chapter introduces the topic of comparisons and uses real-life examples to help explain the topic. Activity
While comparing any two quantities, students must be told that they can only compare them if the values given relate to the same concept. Example
Weight cannot be compared with height. Age cannot be compared with time. 19
Similarly, if the values are the same the units have to be the same as well. Example
Weight given in kilograms cannot be compared to grams. Length given in metres cannot be compared to centimetres. DEFINITION
Ratio can therefore be explained as a comparison between two values of the same unit. METHODOLOGY Rules
This topic can be explained on the basis of the following rules: 1. Ratios are expressed for the same quantity. 2. Ratios have the same units. 3. Ratios are never in the form of fractions. They need to be multiplied by their LCM to remove the fraction. 4. The first ratio is always the new value, thus it is always written as— new : old Exercise 11a
Students must know all the basic rules of ratios in order to solve this exercise. Word problems are a good way of enhancing the concepts the students learn. Activity
The teacher could tell the students to create their own ratio list. • Ratio of the weights of each partner • Ratio of the number of students of class 6 with class 7 • Ratio of money spent on lunch between 2 or 3 students FREQUENTLY MADE MISTAKES
Students generally make mistakes in selecting the factor that would reduce the ratio.
20
Example
21 : 28 The factor would be 7, and 3 : 4 is the reduced ratio. KEYWORDS / TERMINOLOGIES
Terms used in this topic are ‘mean’ and ‘continued’ proportion. Cross products are actually a ratio form of equivalent fractions. If we cross multiply two ratios, the products will be equal. 5 9
25 45
5 × 45 = 25 × 9 225 = 225 Exercise 11b
This exercise covers all concepts taught on the topic of ratio and proportion. The actual volume and weight of an object can be calculated if the ratio and the actual value of the other are given. IMPROVISE / SUPPLEMENT
Extra practice is required which could either be given as a marked classwork assignment or homework. Activity
The teacher could choose a baking activity with the students in school or ask them to do it at home under the supervision of an elder. The recipe to bake a batch of 10 cookies is as follows: 10 oz. of flour 1 egg 5 oz. of sugar 1 tsp. of baking powder 2 tbsp. of butter Ask the students to bake 30 cookies. They will have to multiply all proportions by 3 and then get the desired quantities. Emphasis should be given to the fact that ingredients were in a ratio to each other and that they were proportionally increased. 21
SUGGESTED TIME LIMIT
This is a very important topic and the teacher must at least give 4 lessons on it.
Chapter 12 Day-to-Day Arithmetic KEYWORDS / TERMINOLOGIES
Unitary method is a new term for students; however, they have informally done this concept in some of the word problems in various exercises in the book. It could be described as the method of calculating the value of ‘one’ object. Once this has been calculated, any set of values can be found out. ASSOCIATION
It is important that students are able to relate real-life experiences to mathematics. This is done very easily in arithmetic. The teacher can give them a shopping activity with a shopping list, as follows: 5 packets of orange juice 10 cans of beans 12 sausages The price list is this: 3 packets of orange juice cost Rs 36 2 cans of beans cost Rs 48 5 sausages cost Rs 75 The students will be asked to find the individual value of each item first and then the total cost. The teacher will also ask the students to create their own word problem by going to a supermarket, noting down all the deals available and then finding the cost of each item. Example
BUMPER SALE: ‘3 PACKETS OF BISCUITS FOR THE PRICE OF 2’ ‘BUY 1 GET 1 FREE’ These examples and practical experiences are important to the students’ understanding of core concepts of arithmetic. 22
Exercise 12a
This exercise consists of word problems. Solving these will increase the understanding of the students when dealing with real-life situations. PERCENTAGES
This topic deals with the following core concepts: • How to convert a percentage to a fraction. • How to convert a fraction or a decimal into a percentage. • Unitary method involving percentages. • Increase or decrease in percentages; if the percentage value is calculated, it can be easily subtracted or added to the original value. Example
Ahmed earns Rs 10,000 per month. If his salary is increased by 10 % what would his new salary be? Original salary: Rs 10,000 Salary increase: 10% Increased value: 10% of 10,000 = Rs 1000 New salary = 10,000 + 1000 = Rs 11,000 CONCEPT LIST
The students should be told that per cent means out of 100, so in order to find the percentage of a value it has to be multiplied by 100. Similarly, if a percentage has to be converted to a fraction or decimal it has to be divided by 100. Exercise 12b
This exercise covers all concepts; however, more practice as homework would be advisable. SUGGESTED TIME LIMIT
This is a very important topic and the teacher must spend at least 1 lesson for teaching the concept of conversions and 2–4 lessons to teach the concepts of unitary method, percentage and solving word problems related to them. The best way to know whether any learning is taking place is to take short, 5-minute quizzes. 23
Chapter 13 Profit and Loss FORMULAE
Profit = SP – CP Loss = CP – SP It is important to note that when there is a profit, the sale price is more than the cost price. Similarly, when there is a loss, the cost price is more than the sale price. profit
Profit % = CP × 100 Loss % = loss CP × 100 The students must know the formulae given above when solving the exercises in this chapter. After relating the use of these equations to dayto-day examples, the teacher could ask the students to make a note of these formulae in their exercise books. IMPROVISE / SUPPLEMENT
Students can make their own hypothetical (imagined) situations and create a business plan. This will be an interesting exercise for them. They should be asked to form groups and present their plan in class. FREQUENTLY MADE MISTAKES
Activities and real-life examples are important for teaching this topic. If these are not given, students may not understand the formulae and learn them by rote which would not be helpful in solving sums at a higher level. The examples on pages 89–91 and sums from exercise 13 could be used as real-life examples. Exercise 13
The exercise is based on word problems. The students must be introduced to these by using simple data, for example: Profit = Rs 10 CP = Rs 200 Find the profit %. 24
SUGGESTED TIME LIMIT
This topic should not take more than 2 lessons.
Chapter 14 Simple Interest This is a technical topic. The students may already be familiar with it as leasing of household goods and cars, and borrowing of money is common nowadays. The teacher can easily make them relate to it by arranging a slide presentation on a bank or maybe even arrange a banker as a guest speaker to explain simple interest and how it is calculated. This kind of interactive approach now exists in various curricula and the schools understand the far-reaching impact of this approach. FORMULAE / EQUATIONS SIMPLE INTEREST = PRT 100 where:
P denotes the principal amount. R denotes the rate per year. T denotes the time in years.
AMOUNT = PRINCIPAL + SIMPLE INTEREST
If the interest is given and alternate values are asked, the formula will have to be manipulated. T = SIP××100 R R = SIP××100 R Activity
Ask the students to cut out newspaper advertisements of different banks offering loans. Pin these up on the classroom soft board and compare the offers. ASSOCIATION / RECALL
It is important that students do a lot of mixed worksheets on ratios, percentages, profit and loss, and simple interest sums. This will help them sort through the data and apply the right formula to solve the sums. 25
SUGGESTED TIME LIMIT
By taking the activities into account, this topic should not take more than 4 lessons.
Chapter 15 Introduction to Algebra This is an introductory chapter on algebra which is an important topic in mathematics. Not only is it a transition from arithmetic, but the arithmetic is also enhanced with algebraic knowledge. METHODOLOGY
The fact that in algebra quantities are replaced with letters and symbols, is an important point. This could be introduced with the help of an activity where the resources are some fruit. Example
Take 4 apples and 5 oranges. If we use x to represent apples and y to represent oranges, and the total number of fruit as z , we can write this in an algebraic expression as given below: x + y = z or 4+5=9 Now if the apples cost Rs 2 each and the oranges Rs 3 each, what would the total cost z of the fruit be? 2x + 3 y = z is the expression for the cost of apples and oranges, where x and y are known and so the cost of the fruit can be calculated. KEYWORDS / TERMINOLOGIES
The following terms are introduced in this chapter. Variables: these represent quantities or objects as mentioned earlier. Coefficient: this is the number with a variable. It is always placed before the variable. Constant: this is a number without a variable. Terms: a variable, with its coefficient, forms a term. Terms can be like or unlike. Algebraic expression: this means the combination of terms and a constant. 26
Example
4x + 3 y + 7 4 and 3 are coefficients. x and y are variables. 7 is the constant. Exercise 15
Give exercises 15.1 and 15.2 to the students before they are introduced to key algebraic terminologies. Exercise 15.3 should follow as they test the students’ comprehension skills. BACKTRACK
The teacher can give algebraic expressions and ask the students to make statements. SUGGESTED TIME LIMIT
This chapter should not need more than 2 lessons.
Chapter 16 Operations Using Algebra RULES AND LAWS
The first and foremost rule in algebra is that only like terms (having the same symbols or letters) can be added or subtracted. Unlike terms are left just as they are. Example
4xy and 5xy are like terms and can be added: 4xy and 5xy = 9xy 4x and 5 y are unlike terms and cannot be added as the variables x and y are different. 4x 2 and 5x 3 are also unlike terms and cannot be added as the variables, although the same, have different powers. Once the students are able to identify like and unlike terms, they can apply the four operation rules introduced earlier in the number line chapter.
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ASSOCIATION
Rules for addition and subtraction are the following: a. + and + = addition with the + sign a + 2a + 6a = 9a b. – and – = addition with the – sign –6b – 4b = –10b c. + and – = subtraction with the sign of the bigger term 7c – 5c = 2c or –7c + 5c = –2c Please make the student understand that the absence of a minus sign before a variable means that there is a ‘+’ sign. These rules have been introduced earlier using the number line. Similarly, the brackets are attempted in the sum in the order: ( ), { }, and [ ]. The vinculum ^a + bh, which is a horizontal bar above the term, has to be attempted before the brackets. A minus sign before the brackets signifies that the entire term has to be subtracted from the term before it. In algebra, whenever a term or expression has to be subtracted, the signs change. So if the sign before a term is ‘+’, it becomes minus and vice versa. Examples 1 to 5 on pages 104 to 106 and examples 1 and 2 on pages 108 and 109, respectively, should be solved by the teacher on the class board. Activity
This chapter cannot be done with materials, so one can make it fun and interactive by doing a quiz on the board. The class can be divided into 3 to 4 groups and sums solved by the students on the board. By adapting this method, the teacher will be using peer learning and teaching. Students tend to grasp concepts quickly while working in groups. Exercises 16a, 16b, and 16c
These exercises are based on all the rules mentioned above. Additional reinforcement exercises could be given as homework. FREQUENTLY MADE MISTAKES
Students often forget to change the signs of the entire expression when subtracting. They sometimes forget the order of the brackets thereby completely changing the expression. Identification of like terms is quite easy until 28
powers are introduced. The students should be able to differentiate that the same powers and not the same coefficients, identify like terms. SUGGESTED TIME LIMIT
This chapter should take at least 5 to 6 lessons for the students to understand the topic thoroughly.
Chapter 17 Linear Equations This chapter finally links algebra to other areas of mathematics. Forming equations is an important step which is needed in both arithmetic and geometry. METHODOLOGY
The students need to understand that in an equation, the left hand side is equal to the right hand side. LHS = RHS Similarly, being able to translate the English language to mathematical language is important. Example
Ali is 12 years older than Myra. If Ali’s age is represented by x and Myra’s by y then: x = 12 + y In order for the ages of Myra and Ali to be made equal, 12 years was added to Myra’s age. This can also be written as this: x – 12 = y The 12 on LHS is with a minus sign and on the RHS, it has a + sign. This example will help the teacher to explain the concept of transposing in later classes. Rules
After explaining the concept of balancing equations, the teacher should summarize the topic with the following rules: 29
• • •
A + (plus) term transposes to the other side as a – (minus) sign term. A coefficient in the numerator will go on the other side as a denominator. All variable terms are always collected on the LHS and the constants on the RHS.
Example
4x + 6 = 7 4x + 6 – 6 = 7 – 6
(Transposing)
4x = 1
(Transposing)
4x × 14 = 1 × 14 x = 14 Answer Exercise 17
This exercise involves making equations and solving them, and it also includes word problems related to real-life situations. The students should not face a problem provided the earlier two stages are done systematically. SUGGESTED TIME LIMIT
5 to 6 lessons would be required for this topic.
Chapter 18 Basic Concepts of Geometry This chapter introduces a very important mathematical concept: the spatial and geometrical aspects of mathematics. The teacher will be required to do a lot of hands-on activities in the class with the students. The first and foremost step would be to introduce the students to the instruments of the geometry box. KEYWORDS / TERMINOLOGIES
The terms mentioned in this chapter are line segment, line and ray, plane, collinear points, point of concurrency, and point of intersection 30
METHODOLOGY
In order to introduce the concepts in this chapter, the teacher should employ hands-on material. Once the concepts are introduced, the teacher should encourage the students to use geometrical instruments for work in their exercise books. They should be able to draw diagrams and label them. Activity
To explain the difference between a line, a ray, and a line segment, the teacher can use a thread with knots at each end to represent a line segment; for a ray, the teacher can hold the thread and tell the students to observe that it continues indefinitely at the other end. Similarly, a line can be represented with a knot on one end and the thread going indefinitely from the other end. Now moving on to work in the exercise books, the students are asked to draw lines, rays, and line segments with a ruler and a pencil. The pencil must be sharp as the end-point has to be the smallest point possible. Tell the students not to make big dots for end-points. When introducing planes, ask the students to point out any object resembling a cube or a cuboid in the classroom. For example, a textbook can represent a cuboid. Planes are defined as flat surfaces; it may be pointed out that surfaces can also be curved like that of a cylinder or a sphere. LAWS AND PROPERTIES
Proper definitions and properties should be noted in the exercise books by the students for future use. The properties are written in bold in the textbook on pages 122, 123, 124, and 125. Exercise 18
Before doing this exercise, the teacher should ensure that the students have learnt the definitions and properties well. This can be done by 5-minute pop quizzes before the beginning of each class. Also, reinforce the importance of making good diagrams as this is the basis of formal geometry.
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SUGGESTED TIME LIMIT
Although the concepts are not too many, the students need to be given enough time to develop new skills of drawing diagrams. 5 to 6 lessons should be enough to familiarize the students with this chapter.
Chapter 19 Construction of Line Segments This chapter deals with basic construction of line segments. The geometrical instruments were introduced in the previous chapter. Recall the instruments so that the students are able to identify the instruments by their names. The handling of these instruments is very important. The teacher should observe each student and guide them to the correct way of using the instruments. Stress should be laid on the fact that the diagrams must be neat, clear, and well-labelled drawings. METHODOLOGY
Construction of line segments and adding or subtracting measures of line segments are the topics explained in this chapter. After attempting the mentioned skills, congruency of line segments is introduced. Activity
If we replace the ruler with a measuring tape and ask the students to measure the dimensions of their classroom desks, it will be an interesting activity. Give them a scale to start with. 20 cm is equivalent to 1 cm. The students can then measure the dimensions and convert to the scale and reproduce a drawing of the rectangular plane of the desks. This gives a hands-on experience to the students when they can correlate the concepts learnt, to practical life. The teacher can then teach the concept of congruency with the two sets of line segments drawn to form a rectangle. Similarly it can be emphasized that line segments were used to construct the plane in the above activity as they had definite measurements and end-points.
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Exercise 19
This exercise is an extension of the activity on page 32. Please emphasize the use of correct symbols as there are different symbols for the line segment, line, and ray. FREQUENTLY MADE MISTAKES
Students tend to use unsharpened pencils and therefore get their measurements incorrect by a few millimetres. Remind them to always use a well-sharpened pencil. SUGGESTED TIME LIMIT
This chapter should not take more than 4 lessons.
Chapter 20 Angles Angles have been introduced to the students in earlier classes. The fact to stress upon now is that two rays form an angle when they meet at a point. This point of intersection is called the vertex of the angle. C
A
B
Rays AB and AC form an angle. The point of intersection is at ‘A’. ‘A’ is thus the vertex of ∠BAC. METHODOLOGY
This chapter deals with the construction and measurement of angles. Various types of angles are also mentioned. Activity
It would be interesting if the teacher could engage the students in identifying angles in different real-life objects.
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Examples
• •
The length and breadth of a table forms right angles. The upper and lower jaws of a crocodile form an angle when it opens its mouth to catch its prey. • The door of the classroom when open, forms an angle with its frame. • A pair of scissors when open to cut something, forms an angle. • The leaf stalk, forms an angle with the stem. Ask the students to make a list of examples they can think of and then share with the class. KEYWORDS / TERMINOLOGIES
Acute, obtuse, right, reflex, straight, and complete angles The teacher should make the students write the correct definitions in their exercise books. Pairs of angles: linear, adjacent, complementary, supplementary, and vertically opposite angles. It is important that students make diagrams to match these definitions in their exercise books. FREQUENTLY MADE MISTAKES
Students tend to read the degrees from the wrong end of the protractor, thus it should be emphasized that the reading should be done from the side that the ray of the angle lies on. Also, if the students make a mistake they should immediately be able to identify it. For example, an acute angle would be drawn as an obtuse angle or the other way around. Exercises 20a, 20b, 20c
All these exercises need to be done one at a time, with the teacher explaining the requirements carefully. This might be a relatively easy concept to teach, but it is important that students develop construction skills as early as possible. SUGGESTED TIME LIMIT
Each exercise should take at least 2 lessons. Supplementary exercises could be given for homework as some students may be slow in construction and therefore practice would help them.
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Chapter 21 Triangles ASSOCIATION
This chapter is a continuation of the earlier chapters. The students should be able to easily follow this concept after completing the topics of line segments and angles. METHODOLOGY
The teacher should highlight the definition of a triangle and its parts, e.g. 3 line segments and 3 angles. There are three types of triangles according to the length of the sides, and three according to the measurement of the angles. KEYWORDS / TERMINOLOGIES
Scalene, isosceles, and equilateral angles are classified according to the length of the sides. Acute, right, and obtuse-angled triangles are classified according to measurement of the angles. Exterior and opposite-interior angles are terminologies that the students not only have to be well-versed with, but also be able to identify them. Activity
An interesting activity for students would be to ask them to make cutouts of the 6 types of triangles from a graph paper and paste these in their exercise books. The property, that the sum of the angles of a triangle equals 180°, can be proved by simply asking the students to take a triangle and cut out its end-vertices and paste them on a line; they will clearly form a 180° angle. 80°
Cut here. 50°
50°
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80° 50°
50°
They can repeat this activity with the second property; the exterior angles equal the sum of the two opposite-interior angles. The exterior angle can be cut out and then the two interior angles pasted on top of it. When the students observe these properties practically they are able to understand them better. Similarly, the triangle-inequality property can also be done practically and easily verified by the students. They can measure any two sides and prove them to be larger than the third side. This property can be extended by pointing out that the longer side is always across the larger angle and vice versa. C
6 cm
A
5 cm
10 cm
B
Exercise 21a
This exercise deals with the different types of triangles and their properties. Exercises 21a.4 and 21a.5 are interesting as they deal with composite shapes. The teacher should not help the students on these two questions and let them take their time in identifying them. Generally, the smarter students are quick to identify the triangles, thus robbing the other students of discovering it themselves. Exercise 21b
This exercise involves recall of the concept of ratio as the angles in exercises 21b.5 and 21b.6 are given in the form of ratios. It may be suggested that the students are given extra practice by drawing triangles on the board and engaging them in oral class quizzes.
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SUGGESTED TIME LIMIT
This chapter has a lot of properties related to triangles and many relevant hands-on activities. The teacher should spend at least 4 lessons on this chapter.
Chapter 22 Circles This is a relatively easier chapter than the previous one. The students must learn the parts of a circle, and how to construct it, using a compass. METHODOLOGY
The steps given for the construction of a circle are relatively simple. The teacher should make sure that the students hold the compass by its tip and not its arms. The terminologies of the parts of the circle could be new for the students. They could create a section of keywords in their exercise books for easy reference. KEYWORDS / TERMINOLOGIES
Parts of a circle are simple to understand and the difference between a chord, diameter, and radius should be explained well. A radius touches the circumference of the circle only at one point whereas a diameter does so at the two opposite ends. An arc is a portion of the circumference of a circle between any two points. B arc D
chord diameter A radius
C E
• • • •
Radius touches the circle at ‘F’. Chord touches the circle at ‘B’ and ‘C’. Diameter touches the circle at ‘D’ and ‘E’. Arc is between the points ‘B’ and ‘D’, ‘C’ and ‘E’, ‘E’ and ‘F’, etc.
F
The radius and diameter are constant values as the radius is the measure from the centre to the circumference of the circle whereas the diameter is the measure of the circle across. The chord touches the circumference at any two points but does not pass through the centre. 37
A semicircle is half a circle formed by a diameter, and a quadrant is quarter of a circle formed by two radii. An arc and two radii form major or minor sectors. The sectors are a certain fractional value of the entire circle. The circumference of a circle is actually its perimeter and the circular measure of its boundary. semicircle
quadrant sector
Exercise 22a
This exercise should be done in the exercise books in class. Supplementary work must be given for homework. Exercise 22b
This exercise deals with the bisection of a line segment, angle, and perpendicular lines. Steps of construction given in the textbook should be followed by the students though they need not write them after every construction. They can copy it once in their exercise books. SUGGESTED TIME LIMIT
This chapter can be covered in 4 lessons.
Chapter 23 Mensuration: Area This chapter is a transition from one-dimensional figures to two- and three- dimensional figures. The teacher must help the students make a shift from geometry to mensuration. The chapter deals with the concept of measurement of shapes.
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EQUATIONS / RULES
This chapter deals with the area of squares and rectangles. Area = length × breadth A=L×B Similarly, the area of square is Area = Side × Side A=S×S A = S2 These formulae are quite simple and once the students learn them, they can manipulate them easily to find various unknown quantities. Example L = A B Activity
Area of irregular shapes does not have a formula; instead, a graph paper is used to count the squares to get the area. An interesting activity could be to ask the students to create shapes of given areas. They can be asked to make irregular polygons etc. Assign objects present in the classroom to groups of students. They measure the dimensions in centimetres with a measuring tape and note the values and find the area of each object; the objects can be anything like tables, blackboard, chairs, etc. Ask the students to find the area of all the faces of the object and find the total surface area. This activity will also lead to advanced concepts to be taught in higher classes. Exercise 23a and 23b
Exercise 23a.1 can be done in the textbook; the remaining questions are to be done in the exercise books. The conversion table on page 163 is a must for the students to learn. The teacher has to make sure that the students write this in their exercise books and learn it by heart. The length and breadth of a square or a rectangle should be in the same units for any formula to work. 39
Example
Find the area of a rectangle with the following dimensions: Length 5 cm; breadth 2 m We cannot put different units into the formula; we must first covert m into cm. Therefore breadth = 2 × 100 = 200 cm A = LB A = 5 × 200 = 1000 cm2 AREA OF BORDERS
This concept extends the concept of area. If we have to find the border area, we have to find the external and the internal area. Area of the border = external Area – internal Area The external area is found by using the outer dimensions and the internal area is found by using the inner dimensions. Exercise 23c
Help the students complete these exercises. SUGGESTED TIME LIMIT
This chapter should take at least 6 lessons. It lays the foundation to major mensuration concepts. Students generally find this topic difficult to understand in the later classes; therefore it is important that they are able to visualize the spatial concept well.
Chapter 24 Mensuration: Volume This chapter deals with the concept of the volume of cubes and cuboids. METHODOLOGY
Before giving the formulae to the students, the teacher should make the students feel the dimensions of the cubes and cuboids and identify the edges, faces, and vertices of the objects. This will then lead to the length and breadth of the cubes and cuboids. The difference between a 2-dimensional and 3-dimensional figure is the height. The moment a rectangular / square face acquires depth or height it becomes a 3-dimensional figure. 40
All 3-dimensional figures occupy space. Cubes and cuboids can be brought to the class to show that they can be both, solid or hollow. The hollow space can be used to hold any other material like water or salt and can be used as containers. L
L
B
B
H
H Cube
Cuboid
e length, breadth, and height are all equal L=B=H
Length, breadth, and height are not equal L≠B≠H
CONCEPT LIST
The conversion table on page 173 is important and the students should learn it and also the conversion of cubic metres to litres: 1 cubic metre = 1000 litres 1 litre = 1000 cubic centimetre. Volume of a cuboid = length × breadth × height Total surface area of a cuboid = 2 LB + 2 BH + 2 LH Volume of a cube = L × B × H Since all dimensions of a cube are uniform, breadth and height are replaced with length. Therefore, volume of a cube = L3 Total surface area of a cube = 6L2 Activity
The identification of dimensions is very important as the students need to substitute the values in the formulae. The best way to ensure that the students understand this is to make net diagrams of cube and cuboid. The teacher should make the net diagrams on a worksheet and ask the students to cut and fold them to form the shapes. 41
Cube (Net diagram) Since all sides are equal, the faces are all equal in area and dimensions.
Cuboid (Net diagram) Since dimensions are different, 2 faces each are the same in area and dimensions.
Exercise 24
This exercise is quite easy as it tests the students’ knowledge of the formulae and substitution of the values in the formula. FREQUENTLY MADE MISTAKES
The teacher should ensure that the students do the correct conversions of units and then put the values in. Example If the volume is in litres, it should first be changed to cm 3. The students tend to put in the centimetre value with it, which of course tends to go wrong.
SUGGESTED TIME LIMIT
The formulae given in this chapter must be taught one at a time and the lesson should not take more than six lessons.
Chapter 25 Graphs This chapter takes the students to a separate branch of mathematics called Statistics. It deals with the collection, organization, and interpretation of data. 42
METHODOLOGY
At this level the students need not know the different ways of data collection. The text in the book restricts itself to pictograms and bar graphs. This chapter is more fun and play than mathematical calculations. CONCEPT LIST
The students should be explained what pictograms and bar graphs are, and how to represent information on a scale. Symbols are important while representing data on a pictograph. Example
Suppose there is a data about 500 cars. You would not draw 500 cars. Instead, a single car drawn can represent 500 cars. Half a car drawn would be 250 cars. If two and a half cars are drawn, it would mean 1250 cars. Bar graphs can be both horizontal and vertical. Again the concept of scale is important. A better way of explaining scale would be to link it with RATE. If 2 writing tables cost Rs 5000, then one writing table would cost Rs 2500. A centimetre in a bar graph can represent a grading of Rs 2500.
s R n i t n u o m A
7500 5000 2500 1 2 3 Number of writing tables
The key point that should be stressed is that the bars are always of equal width and the gaps between them are also uniform. Exercise 25
At this level, the students should only be tested on their ability to understand graphs, and Exercise 25 serves this purpose.
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