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10 Mathematics First Quarter – Module 1

GENERATING PATTERNS AND ILLUSTRATING ARITHMETIC SEQUENCE

𝒏 𝒅 π’‚πŸ

𝒂𝒏

Module 1: Generating Patterns and Illustrating Arithmetic Sequence

Mathematics - Grade 10 Alternative Delivery Mode Quarter 1 – Module 1: Generating Patterns and Illustrating Arithmetic Sequence First Edition, 2020 REPUBLIC Act 8293, section 176 states that No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things impose as a condition the payment of royalties.

Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this book are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use this materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them. Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio Development Team of the Module Writer: Roy R. Flores Editor: Jimjun F. Ramas Reviewers: Dr. Anecita U. Mendez (Moderator) Mr. Carmelito M. Lauron Sr. Illustrator and Layout Artist: Myrna P. Soco Management Team Schools Division Superintendent: Dr. Marilyn S. Andales, CESO V Assistant Schools Division Superintendents: Dr. Cartesa M. Perico Dr. Ester A. Futalan Dr. Leah B. Apao Chief, CID: Dr. Mary Ann P. Flores EPS in LRMS: Mr. Isaiash T. Wagas EPS in Math: Dr. Pamela A. Rodemio Printed in the Philippines by: Department of Education, Region VII, Division of Cebu Province Office Address: IPHO Bldg. Sudlon, Lahug, Cebu City Telefax: (032) 255 - 6405 Email Address: [emailprotected]

10 Mathematics First Quarter – Module 1:

GENERATING PATTERNS AND ILLUSTRATING ARITHMETIC SEQUENCE

Module 1: Generating Patterns and Illustrating Arithmetic Sequence

Introductory Message This module is carefully designed to continually facilitate learners to achieve mastery on the Most Essential Learning Competencies and develop their 21st century skills. This module consists of essential components developed appropriately for self-instructional mode of learning. The components come in various developmental purposes that are designed to diagnose (pretest), recall and associate (review), discuss, explain and even provide practice activities, enrichment tasks, assessments and answer keys. Upon taking the pretest, determine whether you need to take or skip this module. At 100% accuracy, you possess the mastery of the topic in the module; hence, you don’t need to take it and you may choose to proceed to the next module. At 99% and below, you are recommended to undertake the module to acquire the necessary skills. Though allowed, adult supervision is limited only to providing assistance in accomplishing this module. It is highly recommended that YOU, the learner, should try to engage independently in doing the different tasks for you to become a critical thinker and problem solver which are the twin goals of Mathematics. May this module be utilized to its fullest extent in the purpose of learning the competencies construed as Most Essential for a learner in this level. God bless and enjoy learning! PAMELA A. RODEMIO Education Program Supervisor - MATH

Module 1: Generating Patterns and Illustrating Arithmetic Sequence

GENERATING PATTERNS AND ILLUSTRATING ARITHMETIC SEQUENCE

MOST ESSENTIAL LEARNING COMPETENCY: ο‚·

generates patterns (M10AL-Ia-1)

ο‚·

illustrates an arithmetic sequence (M10AL-Ib-1)

What I Need To know

Patterns and sequences can be applied in almost all aspects of our lives. We just have to analyze how it can be used in our day-to-day life. Having knowledge about this lessons can give us a different perspective on how things happen in our lives. In this lesson, the learner: ο‚· ο‚· ο‚· ο‚· ο‚· ο‚· ο‚· ο‚· ο‚·

generates pattern of a given sequence finds the next few terms of a sequence illustrates an arithmetic sequence differentiates finite from infinite sequence identifies the rule of a given arithmetic sequence determines the common difference of an arithmetic sequence writes the next few terms of an arithmetic sequence solves problems involving arithmetic sequence with patience relates patterns and sequences to real-life situations

Grade 10 Mathematics

1

Module 1: Generating Patterns and Illustrating Arithmetic Sequence

What I know

Directions: Find out how much you already know about the topics in this module. Choose the letter of the correct answer and write your answers on a separate sheet of paper. Take note of the items that you were not able to answer correctly and find the right answer as you go through this module. 1. What is the next term of the sequence 5z, 8x, 11v,…? A. 14u

B. 14t

C. 15s

D. 15r

2. Which of the following is an infinite sequence? A. 1, 2, 3, 4,…5

B. 2, 4, 8, 16, 32,… C. 20, -15, 10, - 5, 0

D. 6, 2, -2, - 4, - 10

3. The pattern of a certain sequence is β€œdivided by 3”. Which of the following sequences follows the pattern? A. 81, 27, 9, 3, 1

B. 1, 3, 27, 81

C. 3, 6, 9, 12, 15

D. 12, 9, 6, 3, 0

4. Which of the following completes the sequence ___, 2, 4, 6, 8, ___, 12? A. 0 and 10

B. 0 and 14

C. 2 and 10

D. 2 and 12

5. What must be true about an arithmetic sequence whose common difference is negative? A. All the terms in the sequence are positive. B. All the terms in the sequence are negative. C. All the terms in the sequence are increasing. D. All the terms in the sequence are decreasing. 6. Give the common difference of the arithmetic sequence 2, - 5, - 12, - 19. A. – 7

B. 3

C. 7

D. – 3

7. Which of the following has a constant difference? A. 1, 2, 3, 5, 8

B. 64, 32, 16, 8, 4

C. -10, - 6, - 2, 2, 6

D. -2, 4, -8, 16, -32

8. Which rule is applied in getting the values of a in the table below? n

1

2

3

4

5

a

1

4

9

16

25

A. a = n + 1

C. a = 𝑛2

B. a = 2n

D. a = n + 2

9. One of the sequences below is NOT an arithmetic sequence. Which one is it? A. 9, 7, 5 , 3, 1

B. 0, - 3, - 6, 9, 12

C. – 1, - 2, - 3, - 4,- 5

D. Β½, 1, 1Β½, 2, 2 Β½

10. Find t so that 3t + 2, 4t + 3, 6t form an arithmetic sequence. A. 4

B. – 4

Grade 10 Mathematics

C. 5

2

D. – 5

Module 1: Generating Patterns and Illustrating Arithmetic Sequence 1 11. The common difference of an arithmetic sequence is 1 ⁄ 4 . If the first term is ⁄2 , find the 11th term. 3

A. 1 4

C. 2

B. 2

1

D. 3

4

12. A pile of logs has 24 in the first layer, 23 in the second, 22 in the third, and so on. How many logs are there in the 10th layer? A. 16

B. 15

C. 14

D. 13

13. Rona started her blog last April 1, 2020. On her first week she had 8 followers, on the second week she had 15, on the third week she had 22. If her followers grew consistently, how many followers did she have on the 15 th week? A. 95

B. 100

C. 103

D. 106

14. After a knee surgery, your trainer told you to return to your jogging program slowly. He suggested jogging for 12 minutes each day for the first week. Each week thereafter, he suggested that you increase that time by 6 minutes per day. How many weeks will it be before you can jog 1 hour per day? A. 8 weeks

B. 9 weeks

C. 10 weeks

D. 11 weeks

15. Henry started working in 2009 at an annual salary of Php 50 000 and received and increment of Php 15 000 each year. In which year did his annual income reach Php 200 000? A. 2016

B. 2017

Grade 10 Mathematics

C. 2018

3

D. 2019

Module 1: Generating Patterns and Illustrating Arithmetic Sequence

What’s In

What comes next? a)

,

,

,

b)

A, D, G, J, ______

c)

18, 14, 10, 6, _______ ο‚· ο‚· ο‚·

d) 2,

,

4, 8, 16, ____

e)

, _____

What is the next figure in each group? What is your basis to get the next figure? Is there a pattern to determine what comes next?

What’s New

Task 1. Given the following sets of numbers, observe the elements and answer the questions that follow. Set A: {160, 80, 40, 20, 10…} ο‚·

Are the numbers written in specific order?

ο‚·

What is the next number?

ο‚·

Give the pattern.

Set B: {1, 3, 9, 27, 81} ο‚·

Is there a relationship among the elements of the set?

ο‚·

If we will add another number, what is it?

ο‚·

Give the pattern.

Spot the difference(s) between set A and set B.

Grade 10 Mathematics

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Module 1: Generating Patterns and Illustrating Arithmetic Sequence Task 2. Below are squares formed by matchsticks.

Count the number of matchsticks in each figure and record the results in a table. Number of squares

1

2

3

4

5

6

7

8

9

10

Number of matchsticks

Answer the following. 1. 2. 3. 4.

Did you see a pattern on the number of matchsticks used? Is there a constant increase in the number of matchsticks? By how much? Complete the table without actual counting the number of matchsticks. How did you determine the next numbers to complete the table?

What is It

In the first task, the next number in set A is 5 and the pattern is β€œdivided by 2”. In set B, the next number is 243 and the pattern is β€œmultiplied by 3”. The set of numbers 160, 80, 40, 20, 10… and 1, 3, 9, 27, 81 are called sequences. A sequence is usually associated with a pattern or a rule in order to generate the next terms. β€œDivided by 2” and β€œmultiplied by 3” are examples of a rule or pattern of a sequence.

Definition: A sequence or progression is a list of numbers written in a specified order that follows a definite pattern or rule. Each number in the sequence is called term. The first number is called first term, followed by second term and so on.

Sequences are separated into two groups. A finite sequence contains a finite number of terms (with first and last term) while infinite sequence contains an infinite number of terms (with first term but no indicated last term). The three dots mean to continue forward in the pattern established without limit. In task 1, set A is infinite while set B is finite.

Grade 10 Mathematics

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Module 1: Generating Patterns and Illustrating Arithmetic Sequence

Finite Sequence

Infinite Sequence

1, 1, 2, 3, 5

2, 6, 10, 14, …

18, 15, 12, 9,…- 6

11 1 1, , , , … 2 4 8

√2, 2, 2√2, 4

-5, 10, -20, 40,…

In general, the terms of a sequence are written as a1, a2, a3, a4, a5,…, an where n represents the term position or number of terms. For example, a3 is the third term, a5 is the 5th term and an is the nth term. Sometimes a sequence is expressed as equation relating the nth term ( an) and the number of terms (n). Study this the following tables. Table 1. n

1 (a1)

2 (a2)

3 (a3)

4 (a4)

5 (a5)

…

n (an)

an

12 = 1

22 = 4

32 = 9

42 = 16

52 = 25

…

n2

The table shows that the equation of the sequence 1, 4, 9, 16, 25,… is an = n2. This equation can be written in words as: β€œThe nth term of a sequence is the square of the number of terms”. So if 10th term is unknown, we solve this way: π‘Ž10 = 102 or 100

Table 2. No. of Terms (n)

1 (a1)

2 (a2)

3 (a3)

4 (a4)

…

n (an)

nth Term (an)

(3x1) = 3

(3x2) = 6

(3x3) = 9

(3x4) = 12

…

3n

The table shows that the equation of the sequence 3, 6, 9, 12,… is an = 3n This can be translated as: β€œThe nth term of the sequence is thrice the number of terms”. So if the 12th term is unknown, we get: a12 = 3(12) or 36. Remember: To generate the next terms of a sequence, study the given terms and search for the pattern or rule. It could be squaring, adding, subtracting, multiplying, dividing or a combination of operations.

Grade 10 Mathematics

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Module 1: Generating Patterns and Illustrating Arithmetic Sequence In the second task, the complete table is shown below. Number of squares

1

2

3

4

5

6

7

8

9

10

Number of matchsticks

4

7

10

13

16

19

22

25

28 31

Notice that the number of matchsticks 4, 7, 10, 13, ….31 is a sequence whose pattern is β€œplus 3” or β€œincreased by 3”. This means that to get the next term of the sequence we have to increase the preceding term by 3. Take note also that when we subtract any preceding term from the next term the difference is constant. 7 βˆ’ 4 = πŸ‘, 10 βˆ’ 7 = πŸ‘, 13 βˆ’ 10 = πŸ‘, 16 βˆ’ 13 = πŸ‘ and so on. In this case, 3 is the constant difference known as common difference or Sequences with a common difference are called arithmetic sequence.

d.

Definition: An arithmetic sequence or progression is a sequence where every term after the first is obtained by adding a constant called the common difference (d). In general, the common difference (d) of an arithmetic sequence

π‘Ž1 , π‘Ž2 , π‘Ž3 , π‘Ž4 , … π‘Žπ‘› is given by the formula 𝒅 = 𝒂𝒏 βˆ’ π’‚π’βˆ’πŸ where 𝒂𝒏 is the last term and π’‚π’βˆ’πŸ is the previous term. If the common difference between consecutive terms is positive, we say that the sequence is increasing and when the difference is negative we say that the sequence is decreasing.

Remember: To generate the terms of an arithmetic sequence, we take the current term and add the common difference to get to the next terms.

Grade 10 Mathematics

7

Module 1: Generating Patterns and Illustrating Arithmetic Sequence

What’s More (Note: Use a whole sheet of paper to answer activities A to F.) A. Independent Activity 1 Directions: Write F if the sequence is finite or I if the sequence is infinite. 1. 2, 3, 4, 5, …, 10 2. 7, 10, 13, 16, 19, 22, 25 3. 4, 9, 14, 19, … 4. 2, 6, 18, 54 5. 3, 9, 27, 81, …., 729, … B. Independent Assessment 1 Directions: Generate a sequence with five terms given the first term and the pattern. 1. Starts with 5 and the pattern is β€œadd 3” _____, _____, _____, _____, _____ 2. The pattern is β€œdivide by 2 ” and the first term is 480. _____, _____, _____, _____, _____ 3. The rule is β€œmultiply by - 3” and starts from -12. _____, _____, _____, _____, _____ 4. The first term is 2 and the pattern is β€œmultiply by 5 minus 7”. _____, _____, _____, _____, _____ 5. Starts with 112 and the pattern is β€œhalf of the current number plus 8”. _____, _____, _____, _____, _____ C. Independent Activity 2 Directions: Match each sequence with its pattern. Write the letter only. 1) 2) 3) 4) 5)

Sequence 4, 11, 18, 25, … 40, 20, 10, 5, … 100, 96, 92, 88, … 4, 12, 36, 108, … 81, 27, 9, 3

Pattern A. Multiply the previous term by 3. B. Divide the previous term by 2. C. One-third of the previous term. D. Add 7 to the previous term. E. Subtract 4 from the previous term.

D. Independent Assessment 2 D.1. Directions: Give the common difference of the following arithmetic sequences. 1. 7, 14, 21, 28,… 2. 0.50, 0.35, 0.2, 0.05… 3. 10a + 5, 8a + 6, 6a + 7, 4a + 8

Grade 10 Mathematics

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Module 1: Generating Patterns and Illustrating Arithmetic Sequence D. 2. Directions: Given the value of d, write the next 3 terms of the sequence. 4. 24, ______, ______, ______ 5.

1

d = 11

, _____, ______, ______

d=

4

3 4

E. Independent Activity 3 Directions: Choose the letter corresponding to the correct answer. 1. Which of the following is the 10th term of the sequence 50, 42, 34, 26,….? A. – 2

B. – 8

C. – 18

D. – 22

2. The pattern of a certain sequence is β€œtriple it and subtract 15”. If the 2nd term is 45, what is the first term? A. 15

B. 20

C. 25

3. In the arithmetic sequence A.

1 2

B.

1 2

2

3

, 1, , 2, 2

C.

3

D. 30 5 2

1

, what is the common difference? D.

4

3 4

4. Given the sequence 4, 8, 12, 16, …, which of the following is the pattern or rule that describes it? A. multiply by 2

B. divide by 2

C. add 4

5. A sequence is defined by the equation an =

𝟏 𝒏 𝟐

D. subtract 4

βˆ’ πŸ“ where n is the number of

terms. Which one is the value of a10? A. 5

B. 15

C. – 5

D. 0

F. Independent Assessment 3 Directions: Choose the letter corresponding to the correct answer. 1. Which of the following is an arithmetic sequence? A. 1, 2, 3, 5, 8,..

B. 3, -9, 27, -81,...

D.1, - 6, -13, -20

C. 24, 12, 6, 3,…

2. Find the next two terms in the sequence – Β½, - 5/6, - 7/6, ___, ___. A. – 3/2, - 11/6

B. – 5/2, - 2

C. – 3/2, - 5/6 2

D. – 5/2, - 13/6 th

3. A sequence is defined by the equation an = 3n . Find the 12 term. A. 416

B. 432

C. 455

D. 467

4. The next term of a sequence is obtained by multiplying the previous term by (- 5) then add 3. If the previous term is – 12, what is the next term? A. – 20

B. 60

C. 63

D. – 57

5. Find m if m + 5, 4m + 1, and 6m form an arithmetic sequence. A. 3 Grade 10 Mathematics

B. 4

C. 5

9

D. 6

Module 1: Generating Patterns and Illustrating Arithmetic Sequence

What I Have Learned Answer the following. (Note: Write your answers on a separate sheet of paper.) A. Fill in the Blank. Directions: Fill in the blanks with the correct word or words to make the statement complete. 1. A ________ is a list of numbers written in a specified order that follows a definite pattern or rule. 2. A sequence is also called _________. 3. Each number in the sequence is called _____. 4. _______ sequence contains a finite number of terms, while ________ sequence contains an infinite number of terms. 5. In a sequence, n refers to _______ and an ref...

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