math10_q1_m4_sum_arithmetic_sequence.pdf

10 Mathematics Quarter 1 – Module 4: Sum of Arithmetic Sequence Mathematics – Grade 10 Alternative Delivery Mode Quarter 1 – Module 4: Sum of Arithmetic Sequence First Edition, 2020 COPYRIGHT PAGE 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 module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these 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’s Name: Evelyn N. Ballatong Editor’s Name: Laila B. Kiw-isen Reviewer’s Name: Bryan A. Hidalgo, Heather G. Banagui, Selalyn B. Maguilao, Jim Alberto Management Team: May B. Eclar, PhD Marie Carolyn B. Verano Carmel F. Meris Ethielyn E. Taqued Edgar H. Madlaing Soraya T. Faculo Lydia I. Belingon Printed in the Philippines by: Department of Education – Cordillera Administrative Region Office Address: Wangal, La Trinidad, Benguet Telefax: (074) 422-4074 E-mail Address: [email protected] 10 Mathematics Quarter 1 – Module 4: Sum of Arithmetic Sequence M10AL Ic-2 0 i INTRODUCTORY MESSAGE This module was collaboratively designed, developed and reviewed by educators both from public and private institutions to assist you, the teacher or facilitator in helping the learners meet the standards set by the K to 12 Curriculum while overcoming their personal, social, and economic constraints in schooling. This is a part of the third learning competency in our mathematics 10 curriculum standard hence mastery of the skills is significant to have a smooth progress in the succeeding lessons. For the facilitator: This module is intended to help grade 10 students understand and master the concepts of arithmetic series. Please have patience in assisting the learners accomplish this module. Do not forget to remind the learner to use separate sheets in answering all the activities found in this module. For the learner: Hello learner. I hope you are ready to progress in your Grade 10 Mathematics by accomplishing this learning module. This is designed to provide you with interactive tasks to further develop the desired learning competencies on finding the sum of the terms of an arithmetic sequence. Please read completely the written texts and follow the instructions carefully so that you will be able to get the most of this learning material. We hope that you will enjoy learning. Here is a guide on the parts of the learning modules which you need to understand as you progress in reading and analyzing its content. ICON LABEL DETAIL This will give you an idea of the skills or What I need to know competencies you are expected to learn in the module. This part includes an activity that aims to check what you already know about the What I know lesson to take. If you get all the answers correct (100%), you may decide to skip this module. This is a brief drill or review to help you link What’s in the current lesson with the previous one. In this portion, the new lesson will be introduced to you in various ways such as What’s new a story, a song, a poem, a problem opener, an activity or a situation. This section provides a brief discussion of the What is it lesson. This aims to help you discover and understand new concepts and skills. ii This comprises activities for independent practice to solidify your understanding and What’s more skills of the topic. You may check the answers to the exercises using the Answer Key at the end of the module. This includes questions or blank What I have learned sentence/paragraph to be filled in to process what you learned from the lesson. This section provides an activity which will help What I can do you transfer your new knowledge or skill into real life situations or concerns. This is a task which aims to evaluate your level Assessment of mastery in achieving the learning competency. In this portion, another activity will be given to you to enrich your knowledge or skill of the Additional Activities lesson learned. This also tends retention of learned concepts. This contains answers to all activities in the Answer Key module. At the end of this module you will also find: References This is a list of all sources used in developing this module. The following are some reminders in using this module: 1) Use the module with care. Do not put unnecessary mark/s on any part of the module. Use a separate sheet of paper in answering the exercises. 2) Don’t forget to answer What I Know before moving on to the other activities included in the module. 3) Read the instruction carefully before doing each task. 4) Observe honesty and integrity in doing the tasks and checking your answers. 5) Finish the task at hand before proceeding to the next. 6) Return this module to your teacher/facilitator once you are through with it. If you encounter any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator. Always bear in mind that you are not alone. We hope that through this material, you will experience meaningful learning and gain deep understanding of the relevant competencies. You can do it! iii WHAT I NEED TO KNOW This module was designed and written with you in mind. It is here to help you find the sum of the terms of an arithmetic sequence. The scope of this module permits it to be used in many different learning situations. The lessons are arranged to follow the standard sequence of the course but the pacing in which you read and answer this module will depend on your ability. After going through this module, you are expected to be able to demonstrate knowledge and skill related to sequences and apply these in solving problems. Specifically, you should be able to: a) define arithmetic series, b) find the sum of the first 𝑛 terms of a given arithmetic sequence, and c) solve word problems involving arithmetic series. WHAT I KNOW Find out how much you already know about the topics in this module. Choose the letter of your answer from the given choices. Write your answer on your answer sheet. 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. It is the sum of the terms of a sequence. A) mean B) sequence C) nth term D) series 2. Find the sum of the first ten terms of the arithmetic sequence 4, 10, 16, 22, 28, … A) 310 B) 430 C) 410 D) 390 3. Find the sum of the first 25 terms of the arithmetic sequence 17, 22, 27,32, … A) 1925 B) 1195 C) 1655 D) 1895 1 4. The sum of the first 10 terms of an arithmetic sequence is 530. What is the first term if the last term is 80? A) 40 B) 36 C) 30 D) 26 5. The third term of an arithmetic sequence is −12 and the seventh term is 8. What is the sum of the first 10 terms? A) 5 B) 8 C) 11 D) 15 6. Find the sum of the first 50 terms of the arithmetic sequence if the first term is 21 and the twentieth term is 154. A) 9635 B) 9765 C) 9265 D) 9625 7. Find the sum of the first eighteen terms of the arithmetic sequence whose nth term is 𝑎𝑛 = 15 + 8𝑛. A) 1438 B) 1638 C) 1836 D) 1783 8. The first term of an arithmetic sequence is 5, the last term is 45 and the sum is 275. Find the number of terms. A) 13 B) 10 C) 12 D) 11 9. If the first n terms of the arithmetic sequence 20, 18, 16,... are added, how many of these terms will be added to get a sum of −100? A) 35 B) 25 C) 15 D) 30 10. A worker saves Php 36,000 from his salary this year. If he increases his savings yearly by Php 3,000, how much will be his total savings for 8 years? A) Php 315,000 B) Php 372,000 C) Php 432,000 D) Php 495,000 11. Jane was saving for a pair of shoes. From her weekly allowance, she was able to save Php 10 on the first week, Php 13 on the second, Php 16 on the third week, and so on. If she continued saving in this pattern and made 52 deposits, how much did Jane save? A) Php 3984 B) Php 4568 C) Php 4498 D) Php 5678 12. Mary gets a starting monthly salary of Php 6,000 and an increase of Php 600 annually. How much income did she receive for the first three years? A) Php 276,300 C) Php 637, 300 B) Php 237, 600 D) Php 673, 200 2 13. Mirasol saved 10 pesos on the first day of January, 12 pesos on the second day, 14 pesos on the third day, and so on, up to the last day of the month. How much did Mirasol save at the end of January? A) Php 2 710 B) Php 2 170 C) Php 1 240 D) Php 1 420 14. Mrs. De la Cruz started her business with an income of Php 125,000 for the first year and an increase of Php 5,000 yearly. How much is the total income of Mrs. De la Cruz for 8 years since she started her business? A) Php 1,104,000 C) Php 1,140,000 B) Php 1,410,000 D) Php 2,140,000 15. A hall has 30 rows. Each successive row contains one additional seat. If the first row has 25 seats, how many seats are in the hall? A) 1 185 B) 1 815 C) 1 970 D) 1 780 Finding the Sum of the Lesson First n Terms of an Arithmetic Sequence. WHAT’S IN In the previous module, it was discussed that to find the nth term of a given arithmetic sequence, the formula an = a1 + d(n – 1) can be used. For this module, we will be discussing how to find the sum of the first n terms in an arithmetic sequence. For example, how do we compute the sum of all the terms of each of the following sequences? a) 1, 2, 3,... , 100 b) 5, 10, 15, 20,... , 50 c) −5, −2, 1, 4,... , 31 3 Adding manually the terms of a sequence is manageable when there are only few terms in the sequence. However, if the sequence involves numerous terms, then it is no longer practical to be adding the terms manually. It is a tedious work to do. Thus, this module will present to you a formula that will make the computation easier and faster. WHAT’S NEW To let you experience getting the sum of the terms in a sequence manually, do the following. 1. Find the sum of the first 20 natural numbers. Solution: a. By listing all the natural numbers from 1 to 20 and adding them, we have: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 = 210 b. Thus, the sum of the first 20 natural numbers is 210. 2. Find the sum of all the terms of the sequence: 5, 10, 15, 20, …, 50. Solution: a. By listing all the terms of the sequence and adding them, we have: 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 = 275 b. Thus, the sum of the terms of the sequence is 275. 3. Find the sum −5, −2 , 1, 4,…, 31. Solution: a. By listing all the terms of the sequence and adding them, we have: −5 + ( − 2) + 1 + 4 + 7 + 10 + 13 + 16 + 19 + 22 + 25 + 28 + 31 = 169 b. Thus, the sum of the terms of the sequence is 169. 4 In doing this kind of solution, it is very challenging specially if you are dealing with a sequence that has many terms. For example, finding the sum of the terms of the sequence: 1, 2, 3,... , 10,000. There are 10,000 terms to be added one by one to get their sum. To derive a formula to be used in finding the sum of the terms of an arithmetic sequence, consider the following illustration: The terms of an arithmetic sequence with common difference, 𝑑, are First term 𝑎1 Second term 𝑎1 + 𝑑 Third term 𝑎1 + 2𝑑 Fourth term 𝑎1 + 3𝑑 ⋮ ⋮ 𝑛 term th 𝑎1 + (𝑛 − 1)𝑑 Thus, the sum of the terms, 𝑆𝑛 , is: 𝑆𝑛 = 𝑎1 + (𝑎1 + 𝑑) + (𝑎1 + 2𝑑) + (𝑎1 + 3𝑑) + ⋯ + [𝑎1 + (𝑛 − 1)𝑑] equation 1 1st 2nd 3rd 4th nth The terms of an arithmetic sequence can also be written starting from the nth term and successively subtracting the common difference, 𝑑. Hence, 𝑆𝑛 = 𝑎𝑛 + (𝑎𝑛 − 𝑑) + (𝑎𝑛 − 2𝑑) + (𝑎𝑛 − 3𝑑) + ⋯ + [𝑎𝑛 − (𝑛 − 1)𝑑] equation 2 To find the rule for 𝑆𝑛 , add the two equations: 𝑆𝑛 = 𝑎1 + (𝑎1 + 𝑑) + (𝑎1 + 2𝑑) + (𝑎1 + 3𝑑) + ⋯ + [𝑎1 + (𝑛 − 1)𝑑] + 𝑆𝑛 = 𝑎𝑛 + (𝑎𝑛 − 𝑑) + (𝑎𝑛 − 2𝑑) + (𝑎𝑛 − 3𝑑) + ⋯ + [𝑎𝑛 − (𝑛 − 1)𝑑] 2𝑆𝑛 = (𝑎1 + 𝑎𝑛 ) + (𝑎1 + 𝑎𝑛 ) + (𝑎1 + 𝑎𝑛 ) + (𝑎1 + 𝑎𝑛 ) + ⋯ + (𝑎1 + 𝑎𝑛 ) Notice that all the terms containing d added out. So, 2𝑆𝑛 = 𝑛(𝑎1 + 𝑎𝑛 ) Divide both sides of the equation by two, 5 𝑛(𝑎1 + 𝑎𝑛 ) 𝑆𝑛 = 2 Substituting 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑 to 𝑎𝑛 , will lead to the following formula: 𝑛[𝑎1 + 𝑎1 + (𝑛 − 1)𝑑] 𝑆𝑛 = 2 𝑛[2𝑎1 + (𝑛 − 1)𝑑] 𝑆𝑛 = 2 Thus, the sum of the terms of an arithmetic sequence is 𝑛 𝑆𝑛 = [2𝑎1 + 𝑑(𝑛 − 1)] 2 where: 𝑆𝑛 is the sum of the first n terms 𝑎1 is the first term d is the common difference WHAT IS IT In getting the sum of the terms of an arithmetic sequence. We will be using any of the following the formula: 1) n Sn  a1  a n  if the first and last term are given 2 2) S n  2a1  (n  1)d  n if the last term is not given 2 Example 1. Find the sum of the first 20 natural numbers. Given: 𝑎1 = 1 𝑎𝑛 = 20 𝑛 = 20 𝑆𝑛 = ? Solution: Since the last term is given, we used the following formula: 𝑛 𝑆𝑛 = 2 ( 𝑎1 + 𝑎𝑛 ) 6 Substituting the given values in the formula: 20 𝑆20 = (1 + 20) 2 𝑆20 = 10 ( 21 ) 𝑆20 = 210 ∴ The sum of the first 20 natural numbers is 210. Example 2. Find the sum of the first 16 terms of the arithmetic sequence: 8, 11, 14, 17, 20, … Given: 𝑎1 = 8 𝑛 = 16 𝑑 = 3 𝑆16 = ? Solution: The last term is not given, so we use the formula 𝑛 𝑆𝑛 = 2 [ 2𝑎1 + ( 𝑛 − 1 ) 𝑑 ] Substitute the given values in the formula: 16 𝑆16 = [ 2 ( 8 ) + ( 16 − 1 ) 3 ] 2 = 8 [ 16 + ( 15) 3 ] = 8 ( 16 + 45 ) = 8 ( 61 ) 𝑆16 = 488 ∴ The sum of the first 16 terms of the series is 488. Example 3. If the first n terms of the sequence: 9, 12, 15, 18, … are added, how many terms give a sum of 126? Given: 𝑎1 = 9 𝑆𝑛 = 126 𝑑 = 3 𝑛 = ? Solution: The last term is not given so we use the following formula 𝑛 𝑆𝑛 = [ 2𝑎1 + (𝑛 − 1)𝑑 ] 2 Substituting the given: 𝑛 126 = [2(9) + (𝑛 − 1)3] 2 𝑛 126 = [ 18 + (3𝑛 − 3)] 2 7 252 = 𝑛 [18 + 3𝑛 − 3] 252 = 𝑛 [3𝑛 + 15] 252 = 3𝑛2 + 15𝑛 0 3𝑛2 + 15𝑛 − 252 = 3 3 0 = 𝑛2 + 5𝑛 − 84 by factoring (𝑛 + 12)(𝑛 − 7) = 0 (𝑛 + 12) = 0 (𝑛 − 7) = 0 𝑛 = −12 𝑛=7 Since the domain of a sequence is the set of positive integers, we reject 𝑛 = −12. Hence, we only accept 𝑛 = 7. ∴ The number of terms that will add up to 126 is 7. Example 4. Find the sum of the integers between 1 and 70 that are divisible by 3. Given: 𝑎1 = 3 𝑎𝑛 = 69 𝑑=3 𝑛 =? 𝑆𝑛 =? Solution: a) To solve for 𝑛, use the formula: 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑 Substitute the given values: 69 = 3 + (𝑛 − 1)3 69 = 3 + 3𝑛 − 3 69 = 3𝑛 𝑛 = 23 b) Since we already solved 𝑛, we can now solve for 𝑆𝑛. 𝑛 𝑆𝑛 = [ 2𝑎1 + (𝑛 − 1)𝑑 ] 2 23 𝑆23 = [ 2 ( 3) + ( 23 − 1 ) 3 ] 2 23 𝑆23 = [ 6 + ( 22) 3 ] 2 23 𝑆23 = ( 6 + 66 ) 2 23 𝑆23 = ( 72 ) 2 8 𝑆23 = 828 ∴ The sum of the integers from 1 to 70 that are divisible by 3 is 828. Example 5. The sum of the first 15 terms of an arithmetic sequence is 765. If the first term is 23, what is the common difference? Given: 𝑎1 = 23 𝑛 = 15 𝑆15 = 765 𝑑 =? Solution: 𝑛 𝑆𝑛 = [ 2𝑎1 + (𝑛 − 1)𝑑 ] 2 15 𝑆15 = [2 (23) + (15 − 1)𝑑 ] 2 15 765 = [46 + ( 14) 𝑑 ] 2 1530 = 15 (46 + 14𝑑 ) 1530 = 690 + 210𝑑 210𝑑 = 1530 − 690 210𝑑 = 840 𝑑=4 ∴ The common difference is 4. WHAT’S MORE After knowing all the needed concept in finding the sum of an arithmetic sequence. You are now ready to answer the following exercises: A. Find the indicated partial sum of each arithmetic series. 1) The first 9 terms of 5 + 8 + 11 + ⋯ 2) The first 30 terms of 1 + 3 + 5 + ⋯ 3) The first 14 terms of 6 + 9 + 12 + ⋯ 4) The first 25 terms of 5 + 8 + 11 + ⋯ 5) The first 15 terms of −12 + (−6) + 0 + ⋯ 9 B. Solve for the value of 𝑛. 1) 𝑆𝑛 = −80, 𝑎1 = 10, 𝑎𝑛 = −26, 𝑛 =? 2) 𝑆𝑛 = 50, 𝑎1 = 4, 𝑎𝑛 = 16, 𝑛 =? 3) 𝑆𝑛 = −15, 𝑎1 = 12, 𝑑 = −3, 𝑛 =? 4) 𝑆𝑛 = 180, 𝑎1 = 5, 𝑑 = 5, 𝑛 =? C) Answer what is asked. 1) Find the sum of the first 13 terms of the sequence: −3, −1, 1, 3, … 2) Find the sum of the first 15 terms of the arithmetic sequence: 10, 15, 20, 25, … ? 3) Find the sum of the first 11 terms of the arithmetic sequence: −4, 3, 10, 17, … ? 4) Find the sum of the first 19 terms of the arithmetic sequence: 9, 14, 19, 24, … ? 5) Find the sum of the integers from 8 and 35. 6) Find the sum of all even integers from 10 to 70. 7) Find the sum of all odd integers from 1 to 50. 8) Find the sum of the integers from 20 to 130 and are divisible by 5. 9) If the sum of the first 8 terms of an arithmetic sequence is 172 and its common difference is 3, what is the first term? 10) If the sum of the first 9 terms of an arithmetic sequence is 216 and its first term is 4, what is the common difference? WHAT I HAVE LEARNED To find the sum of the terms of an arithmetic sequence, you can use the following formulae: A. If the first and last terms are given: 𝑛 𝑆𝑛 = (𝑎 + 𝑎𝑛 ) 2 1 where: 𝑆𝑛 is the sum of the first n terms 𝑎1 is the first term 𝑎𝑛 is the last term 10 B. If the last term is not given: 𝑛 𝑆𝑛 = [2𝑎1 + 𝑑(𝑛 − 1)] 2 where: 𝑆𝑛 is the sum of the first n terms 𝑎1 is the first term 𝑑 is the common difference WHAT I CAN DO Read and understand the problems and answer what is asked. 1. Suppose a cinema has 42 rows of seats and there are 20 seats in the first row. Each row after the first row has two more seats than the row that it precedes. How many seats are in the cinema? 2. A 25-layer of logs is being piled to be used on a construction. The uppermost layer is composed of 25 logs, the second upper layer contains 27 logs, and the third upper layer contains 29 logs, and so on. If the pattern continues up to the lowest layer, what is the total number of logs piled for construction? ASSESSMENT Read and analyze each item carefully. Write the letter of the correct answer in a separate paper. 1. Which of the following is a formula for arithmetic series? 1 1 A) 𝑆𝑛 = 2 (𝑎1 + 𝑎𝑛 ) C) 𝑆𝑛 = 2 (𝑎1 + 𝑑) 𝑛 𝑛 B) 𝑆𝑛 = 2 (𝑎1 + 𝑎𝑛 ) D) 𝑆𝑛 = 2 (𝑎1 + 𝑑𝑎𝑛 ) 11 2. Find the sum of the first 12 terms of the arithmetic sequence 4, 11, 18, 25, 32, … A) 610 B) 530 C) 510 D) 410 3. Find the sum of the first 15 terms of the arithmetic sequence 17, 12, 7, 2, … A) 270 B) 287 C) −287 D) −270 4. The sum of the first 12 terms of an arithmetic sequence is 606. What is the first term if the last term is 67? A) 64 B) 54 C) 34 D) 44 5. The second term of an arithmetic sequence is −16 and the eighth term is 8. What is the sum of the first 10 terms? A) −15 B) −20 C) 15 D) 20 6. Find the sum of the first 40 terms of the arithmetic sequence if the first term is 16 and the tenth term is 70. A) 5320 B) 1720 C) 2200 D) 6320 7. Find the sum of the first 15th terms of the arithmetic sequence whose nth term is 𝑎𝑛 = 5 + 3𝑛. A) 870 B) 860 C) 435 D) 430 8. The first term of an arithmetic sequence is 8, the last term is 56 and the sum is 416. Find the number of terms. A) 13 B) 12 C) 11 D) 10 9. If the first n terms of the arithmetic sequence 24, 20, 16,... are added, how many of these terms will be added to get a sum of −60? A) 35 B) 30 C) 25 D) 15 10. A yaya receives a starting annual salary of Php 60,000 with a yearly increase of Php 3600. What is her total income for 5 years? A) Php 672,000 B) Php 552,000 C) Php 276,000 D) Php 336,000 12 11. Jane was saving for a pair of shoes. From her weekly allowance, she was able to save Php 5 on the first week, Php 9 on the second, Php 13 on the third week, and so on. If she continued saving in this pattern and made 43 deposits, how much did Jane save? A) Php 3822 B) Php 3827 C) Php 7644 D) Php 6574 12. Mary gets a starting monthly salary of Php 8,000 and an increase of Php 800 annually. How much income did she receive for the first four years? A) Php 441,600 B) Php 388,800 C) Php 40,000 D) Php 36,800 13. Mirasol saved 8 pesos on the first day of January, 11 pesos on the second day, 14 pesos on the third day, and so on, up to the last day of the month. How much did Mirasol save at the end of January? A) Php 4282 B) Php 4290 C) Php 1643 D) Php 1 590 14. Mrs. De la Cruz started her business with an income of Php 250,000 for the first year and an increase of Php 6,000 yearly. How much is the total income of Mrs. De la Cruz for 6 years since she started her business? A) Php 530,000 C) Php 1,590,000 B) Php 3,180,000 D) Php 1,608,000 15. A hall has 35 rows. Each successive row contains two additional seats. If the first row has 20 seats, how many seats are in the hall? A) 1 080 B) 1 100 C) 1 925 D) 1 890 13 ADDITIONAL ACTIVITY Let us sing the song titled “Twelve Days of Christmas.” Afterwards, answer the question that follows. Verse 1: Five golden rings On the first day of Christmas my true love Four calling birds sent to me Three French hens A partridge in a pear tree Two turtle doves, and a partridge in a pear Verse 2: tree On the second day of Christmas my true Verse 7: love sent to me On the seventh day of Christmas my true Two turtle doves, and a partridge in a pear love sent to me tree Seven swans a - swimming Verse 3: Six geese a - laying On the third day of Christmas my true love Five golden rings sent to me Four calling birds Three French hens Three French hens Two turtle doves, and a partridge in a pear tree Two turtle doves, and a partridge in a pear tree Verse 4: Verse 8: On the fourth day of Christmas my true love sent to me On the eighth day of Christmas my true love sent to me Four calling birds Eight maids a-milking Three French hens Seven swans a - swimming Two turtle doves, and a partridge in a pear tree Six geese a - laying Verse 5: Five golden rings On the fifth day of Christmas my true love Four calling birds sent to me Three French hens Five golden rings Two turtle doves, and a partridge in a pear Four calling birds tree Three French hens Two turtle doves, and a partridge in a pear tree Verse 6: On the six day of Christmas my true love sent to me Six geese a – laying 14 Verse 9: Verse 11: On the ninth day of Christmas my true love On the 11th day of Christmas my true love sent to me sent to me Nine ladies dancing 11 pipers piping Eight maids a-milking 10 lords a-leaping Seven swans a - swimming Nine ladies dancing Six geese a - laying Eight maids a-milking Five golden rings Seven swans a - swimming Four calling birds Six geese a - laying Five golden rings Three French hens Four calling birds Two turtle doves, and a partridge in a pear Three French hens tree Two turtle doves, and a partridge in a pear tree Verse 10: Verse 12: On the tenth day of Christmas my true love On the 12th day of Christmas my true sent to me love sent to me 10 lords a-leaping 12 drummers drumming Nine ladies dancing 11 pipers piping Eight maids a-milking 10 lords a-leaping Seven swans a - swimming Nine ladies dancing Six geese a - laying Eight maids a-milking Five golden rings Seven swans a - swimming Four calling birds Six geese a - laying Three French hens Five golden rings Two turtle doves, and a partridge in a pear Four calling birds tree Three French hens Two turtle doves, and a partridge in a pear tree Summarizing, we have the following: 12 drummers drumming 6 geese – a – laying 11 pipers piping 5 golden rings 10 lords-a-leaping 4 calling birds 9 ladies dancing 3 French hens 8 maids- a – milking 2 turtles doves, and 7 swans-a-swimming A partridge in a pear tree. Question: How many gifts are given after the 12th day of Christmas? 15 16 What I Know What’s More What I Can Do 1. D A. 1. 2562 seats 2. A 1) 153 2. 1225 logs. 3. A 2) 900 4. D 3) 357 Assessment 5. A 4) 1 025 1) B 6. D 5) 450 2) C 7. B B. 3) D 8. D 1) 10 4) C 9. B 2) 5 5) B 10. B 3) 10 6) A 11. C 4) 8 7) C 12. B C) 8) A 13. C 1. 117 9) D 14. C 2. 675 10) D 15. A 3. 341 11) B 4. 1 026 12) A 5. 602 13) C 6. 1 240 14) C 7. 625 15) D 8. 1 725 9. 11 10. 5 Additional Activity  78 gifts ANSWER KEY REFERENCES: Callanta, Melvin M., et al., Mathematics Learner’s Module.Pasig City, 2015 Nivera, Gladys C. and Lapinid, Minie Rose C.,Grade 10 Mathematics: Patterns and Practicalities. Makati City, Don Bosco Press, 2015. 17 For inquiries or feedback, please write or call: Department of Education – Bureau of Learning Resources (DepEd-BLR) Ground Floor, Bonifacio Bldg., DepEd Complex Meralco Avenue, Pasig City, Philippines 1600 Telefax: (632) 8634-1072; 8634-1054; 8631-4985 Email Address: [email protected] * [email protected] 18

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