INSTRUCTIONAL PLAN (IP) Subject: Grade 10 Mathematics Topic: Sequences Subtopics: Arithmetic Sequences
G O A L S
Module No: 1
Standards Content: The learner demonstrates understanding of the ke y concepts of sequences, polynomials, and polynomial equations.
Performance: The learner is able to formulate and solve problems involving sequences, polynomials, and polynomial equations in different disciplines through appropriate and accurate representations. Learning Competencies: 1. The learner generates patterns. 2. The learner illustrates an arithmetic sequence.
A S S E S S M E N T
Assessment Plan Pre- Assessment ( Please Please refer to Learner’s Module pp.4-8) Module pp.4-8) Formative Assessment ( Please Please see the Evaluation section of the lesson plan) plan) Summative Assessment ( Please Please refer to Teacher’s Module pp.32-36 ) Student’s Self -assessment -assessment Journal Writing: Describing difficulties encountered. Time/ Duration
D E L I V E R Y
INSTRUCTIONAL SEQUENCE: The Lesson Plan/Strategy
S.M.A.R.T. Objectives: 1. Generate and describe patterns using symbols and mathematical expressions; 2. Find the next few terms and the nth term of the given sequence; 3. Define and describe an arithmetic sequence. 4. Enumerate the next few terms and the nth term of an arithmetic sequence; 5. Insert means between two given terms of an arithmetic a rithmetic sequence; 6. Find the sum of the first n terms of an arithmetic sequence; 7. Pose and solve problems involving arithmetic sequence. Procedure (4 A’s) Priming/Activating Prior Knowledge I. Find and Using a Pattern Find a pattern for each sequence. 1. 3, 6, 12, 24, … 2. 1, 2, 4, 7, 11, 16, 22, … 3. Monday, Tuesday, Wednesday, … 4. 5, 10, 20, 40, … 5. 3, 33, 333, 3333, 33333, … II. Find a pattern for each sequence. Use the pattern to show the next two terms. 1. 1, -1, 2, -2, -2, 3, … 2. 15, 12, 9, 6, … 3. 81, 27, 9, 3, … 4. J, F, M, A, M, … 5. 1, ½ , ¼ , 1⁄8, …
III. Draw the next figure. 1.
2.
3.
Activities I.
Term After Term (Tag Team) Find the first 5 terms of the sequence given the nth term. 1. = + 4 2. = 2 − 1 3. = 1 2 − 3 4. = 3 5. = (−2) How did you find the activity? Was it easy to find the first 5 terms of each sequence? II.
Getting to Know You (Group of 6) What is the nth term of the each sequence? 1. 3, 4, 5, 6, 7, … 2. 3, 5, 7, 9, 11, … 3. 2, 4, 8, 16, 32, … 4. -1, 1, -1, 1, -1, … 5.
1, , , , 5 , …
III. What do we have in common? (Group of 3, Part 1) (Material needed: matchsticks) 1. Form squares using matchsticks. 2. Count the number of matchsticks in each square and record your observations as No. of squares 1 2 3 4 5 6 7 8 9 No. of matchsticks 4 7 10 13 16 19 22 25 28 (Individual, Part 2) a1 a2 a3 a4 a5 a6 a7 a8 a9 4 4+3 4+3(2) 4+3(3) 4+3(4) 4+3(5) 4+3(6) 4+3(7) 4+3(8) IV. What is missing? A. Find the missing term in each arithmetic sequence. (Group) 1. 3, 12, 21, ___, ___, ___ 2. 8, 3, -2, ___, ___, ___ 3. 5, 12, ___, 26, ___ 4. 2, ___, 20, 29, ___ 5. ___, 4, 10, 16, ___ 6. 17, 14, ___, ___, ___ B. Supply the missing term in each sequence. (Individual) 1. 4, ___, ___, 19, 24 2. ___, ___, ___, 8, 12, 16 3. -1, ___, ___, ___, 31, 39 4. 13, ___, ___, ___, -11, -17 C. Find three terms between 2 and 34 of an arithmetic sequence.
2, a2, a3, a4, 34
6 = 34, = 2 34=2+4 =8 2, 10, 18, 26, 34 Find four terms between 5 and 25. V. Summing Up What is the sum of the terms of each finite sequence? 1. 1, 4, 7, 10 2. 3, 5, 7, 9, 11 3. 10, 5, 0, -5, -10, -15 4. 81, 64, 47, 30, 13, -4 5. -2, -5, -8, -11, -14, -17
Analysis Refer to Activity III (Continuation, Part 2) a. Is there a pattern in the number of matchsticks used? Describe it. b. How is each number of matchsticks term found? c. What is the difference between any two consecutive terms? Refer to Activity IV a. Were you able to find the missing terms in A? In B? In C? b. How did you find the missing terms?
Abstraction A sequence is a function of observed collection of objects that allows repetition. The members in a sequence are called terms or elements. The sequence in Activity III shows addition of con stants after a given term, thus it is called an arithmetic sequence. The constant is called the common difference. Arithmetic Sequence can also be called Arithmetic Progression. Notation: In general, arithmetic sequence can be written as
= , + , + 2, + 3, … As a rule, when finding the nth term,
= + ( − 1 ) The sum of n terms in a sequence divided by the number of terms is the arithmetic mean.
Application (Think – Pair – Share) What is 1 + 2 + 3 + …. + 50 + 51 + … + 99 + 100? Karl Friedrich Gauss found the sum of the integers within second. How did he do it? Will you be able to beat him? Determine the answer to the given series. Then look for a partner to compare answers. Discuss with him/her the technique in getting the answer quickly. Then with your partner, answer the questions that follow: 1. What is the sum of each of the pairs 1 and 100, 2 and 99, 3 and 98, …? 2. How many pairs are there in question 1? 3. From your answer in 1 and 2, how did you get the sum of the integers? 4. What is the sum of the integers from 1 to 100? Evaluation: Please see attached worksheet Closure Part I. Beyond the Basics Do any two of the following.
1. Mathematically speaking, the next term of a sequence cannot be determined by giving only the first finite number. Explain this fact by giving an example. 2. Make a concept map for arithmetic sequence. 3. Using the formula for arithmetic sequence, = + ( − 1), give problems where the unknown value is
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and show how each can be found. 4. What should be the value of x so that x + 2, 3x – 2, 7x – 12 will form an arithmetic sequence? Justify your answer. 5. Find the value of x when the arithmetic mean of x+2 and 4x+5 is 3x+2. Part II. For self-assessment: Write a reflection about the lesson. Is the lesson difficult? If yes, which part was difficult for you?
M A S T E R Y
No. of Learners within Mastery Level 75-79 % (D) : _______ 80-84 % (AP) : _______ 85-89 % (P) : _______ 90 % and above A): _______ No. of Learners Needing Remediation/Reinforcement 74 % and below : ________ Remarks: ____________________________________________________ End of Lesson 1
