Mathematics: Generating Patterns and Illustrating Arithmetic Sequences - Grade 10 ADM, Study notes of Mathematics

Download Mathematics: Generating Patterns and Illustrating Arithmetic Sequences - Grade 10 ADM and more Study notes Mathematics in PDF only on Docsity! Mathematics Quarter 1 – Module 1: Generating Patterns and Illustrating Arithmetic Sequence 10 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 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 Printed in the Philippines by Department of Education – Schools Division of Bataan Office Address: Provincial Capitol Compound, Balanga City, Bataan Telefax: (047) 237-2102 E-mail Address: bataan@deped.gov.ph Development Team of the Module Writer: Flordeliza R. Angeles Editor: Nina S. Manuel Reviewer: Sherwin G. Serrano Illustrator: Roden D. De Guzman Layout Artist: Shiela S. Murciano Cover Design: Emmanuel S. Gimena Jr. Management Team: Schools Division Superintendent : Romeo M. Alip, PhD, CESO V OIC-Asst. Schools Division Superintendent: William Roderick R. Fallorin Chief Education Supervisor, CID : Milagros M. Peñaflor, PhD Education Program Supervisor, LRMDS : Edgar E. Garcia, MITE Education Program Supervisor, AP/ADM : Romeo M. Layug Education Program Supervisor, Mathematics: Danilo C. Caysido District Supervisor, Dinalupihan : Rodger R. De Padua, EdD Division Lead Book Designer : Joriel J. Cruz District LRMDS Coordinators, Dinalupihan: Sherwin G. Serrano Regina M. Poli School LRMDS Coordinator : Regina M. Poli School Principal : Lorinda R. Poblete District Lead Layout Artist, Mathematics : Onofre M. Aquino Jr. District Lead Illustrator, Mathematics : Nathaniel C. Sebastian District Lead Evaluator, Mathematics : Marise M. Barlis Rufino V. Rubino iii For the learner: Welcome to the Mathematics – Grade 10 Alternative Delivery Mode (ADM) Module on Generating Patterns and Illustrating Arithmetic Sequence! The hand is one of the most symbolized part of the human body. It is often used to depict skill, action and purpose. Through our hands we may learn, create and accomplish. Hence, the hand in this learning resource signifies that you as a learner is capable and empowered to successfully achieve the relevant competencies and skills at your own pace and time. Your academic success lies in your own hands! This module was designed to provide you with fun and meaningful opportunities for guided and independent learning at your own pace and time. You will be enabled to process the contents of the learning resource while being an active learner. This module has the following parts and corresponding icons: What I Need to Know This will give you an idea of the skills or competencies you are expected to learn in the module. What I Know This part includes an activity that aims to check what you already know about the lesson to take. If you get all the answers correct (100%), you may decide to skip this module. What’s In This is a brief drill or review to help you link the current lesson with the previous one. What’s New In this portion, the new lesson will be introduced to you in various ways such as a story, a song, a poem, a problem opener, an activity or a situation. What is It This section provides a brief discussion of the lesson. This aims to help you discover and understand new concepts and skills. What’s More This comprises activities for independent practice to solidify your understanding and skills of the topic. You may check the answers to the exercises using the Answer Key at the end of the module. What I Have Learned This includes questions or blank sentence/paragraph to be filled in to process what you learned from the lesson. iv What I Can Do This section provides an activity which will help you transfer your new knowledge or skill into real life situations or concerns. Assessment This is a task which aims to evaluate your level of mastery in achieving the learning competency. Additional Activities In this portion, another activity will be given to you to enrich your knowledge or skill of the lesson learned. This also tends retention of learned concepts. Answer Key This contains answers to all activities in the module. At the end of this module you will also find: 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! References This is a list of all sources used in developing this module. 1 What I Need to Know One of the most amazing things we can observe from our environment is that it is full of patterns and sequences. The way petals of flowers are arranged, designs in our floor tiles, the cone of pine tree and even the outside appearance of pineapple fruit, these all exhibit patterns. Around the world, police departments have relied on mathematics in solving some of their cases. Special algorithm can use the information about the past crimes to predict on when or where crimes might occur. It also seemed like earthquakes follow the same pattern just as crimes. An earthquake might trigger an aftershock just like a crime might result to another crime of retaliation. The above scenario presented an ideal chance for the learners to realize that studying patterns are important. This scenario illustrates a sequence. In this learning module, you will know more about sequences and how the concept of a sequence is utilized in our daily lives. In these lessons you will learn to: 1. generate and describe patterns. (M10AL-Ia-1) 2. illustrates an arithmetic sequence. (M10AL-Ib-1) What I Know Directions. Find out how much you already know about the lessons in this module. Choose the letter of the best answer. 1. Complete the following pattern by filling in the blanks and then describe the pattern in words. B B G B G B Y B B G __ G B __ B B __ B G B __ __ a. B, Y, G, Y, B b. G, B, B, Y, G c. Y, Y, G, G, B d. B, G, G, Y, 2. Look at the pattern below. Continue the pattern by filling in the blanks. O, T, T, F, F, S, S, E, N, T, E, T, T, __, __, __, __ a. F, F, T, F b. S, O, F, T c. F, F, S, S d. S, S, F, F Explain how the change is created in the following patterns and sequence. 4 What’s New Below is an activity. In this activity you will work with pattern recognition. Activity 1. Each item below shows a pattern. Take this test as you would take a test in class. Then check your work with the solutions in the answer key at the back matters. 1. What is the next shape? , , , , , , , , , , , , , , __. 2. 0 , 3 , 6 , 9 , 12 , ___ . What is the next number? What is the 10th number? 3. 7 , 3 , -1 , -5 , -9 , __ . What is the next number? What is the 9th number? 4. 1 , 4 , 16 , 64 , ___ . What is the next number? What is the 10th number? 5. 120 , 60 , 30 , 15 , ___ . What is the next number? What is the 7th number? In the next items, draw the fourth object following the pattern. 6. , , , ____________ 7. , , , _____________ 8. , , , _____________ 9. , , , ____________ 10. , , , _______________ How did you find the activity? Were you able to find the patterns and get the next number in the sequence? 5 What is It Let’s now give the formal definition of a sequence. The set of figures and numbers above the given activities are called sequences. A special notation is often used with sequence. Instead of writing 𝑎(3) = 6 to indicate the 3rd term, we write 𝒂𝟑 = 𝟔. This is read as “ 𝑎 𝑠𝑢𝑏 3 𝑒𝑞𝑢𝑎𝑙𝑠 6.” The number 3 is the index because it indicates the position of the term in a sequence. A sequence (of real numbers) is a function whose domain is the finite set {1, 2, 3, . . . 𝑛 } or infinite set { 1 , 2 , 3 , . . . }. Set of ordered pair numbers can also be written in tabular form. Finite set This is a finite sequence that has 5 terms {0, 3, 6, 9, 12}. The pattern used to get the succeeding term is 𝒂𝒏 = 𝟑𝒏 − 𝟑 . (Steps in forming this pattern will be discuss to you later.) Infinite set This is an infinite sequence that has an infinite number of terms denoted by three dots (…), the pattern used to get the succeeding term is 𝑎𝑛 = 3𝑛 + 1 In the next activity, you will learn more about sequences. A general term or nth term will be given to you as a guide to solve the next few terms. Before you proceed, here are the examples. Example 1 Find the first 5 terms of the sequence with the given nth rule as your guide. The first 5 terms (1, 2, 3, 4, 5) are to be substituted one at a time into the nth term. The nth term is 𝒂𝒏 = 𝒏 + 𝟒. . 𝑎𝑛 = 𝑛 + 4 𝑎1 = 1 + 4 𝑎2 = 2 + 4 𝑎2 = 3 + 4 𝑎2 = 4 + 4 𝑎2 = 5 + 4 𝑎1 = 5 𝑎2 = 6 𝑎2 = 7 𝑎2 = 8 𝑎2 = 9 Therefore, the first 5 terms of the sequence using the nth rule 𝒂𝒏 = 𝒏 + 𝟒 are 5, 6, 7, 8, 9. Here is another example: Example 2 Find the first 5 terms of the sequence with the given nth term 𝑎𝑛 = 2𝑛 − 1. Again, substitute (1, 2, 3, 4, 5) one at a time into the nth rule 𝑎𝑛 = 2𝑛 − 1. n 1 2 3 4 5 𝑎𝑛 0 3 6 9 12 n 1 2 3 4 … 𝑎𝑛 4 7 10 13 … n 1 2 3 4 5 𝑎𝑛 5 6 7 8 9 6 𝑎1 = 2(1) − 1 𝑎2 = 2(2) − 1 𝑎3 = 2(3) − 1 𝑎1 = 1 𝑎2 = 4 − 1 𝑎3 = 6 − 1 𝑎2 = 3 𝑎3 = 5 and so on… In this case, the first 5 terms of the sequence using the nth term 𝒂𝒏 = 𝟐𝒏 − 𝟏 are 1, 3, 5, 7,9. Example 3 Find the first five terms of the sequence using the nth rule, 𝑎𝑛 = 8 − 3𝑛 Again, substitute (1, 2, 3, 4, 5) one at a time into the nth rule 𝑎𝑛 = 8 − 3𝑛. 𝑎1 = 8 − 3(1) 𝑎2 = 8 − 3 (2) 𝑎3 = 8 − 3(3) 𝑎1 = 5 𝑎2 = 8 − 6 𝑎3 = 8 − 9 𝑎2 = 2 𝑎3 = −1 And so on… Hence, the first 5 terms of the sequence using the nth term 𝒂𝒏 = 𝟖 − 𝟑𝒏 are 5, 2, -1, -4 and -7. Example 4 Find the first five terms of the sequence using the nth rule, 𝑎𝑛 = 3𝑛 Again, substitute (1, 2, 3, 4, 5) one at a time into the nth rule , 𝑎𝑛 = 3𝑛. 𝑎1 = 31 𝑎2 = 32 𝑎3 = 33 𝑎1 = 3 𝑎2 = 9 𝑎3 = 27 And so on… Thus, the first 5 terms of the sequence using the nth term 𝒂𝒏 = 𝟑𝒏 are 3, 9, 27, 81 and 243. After giving you the these examples, you may now start with the next activity. What’s More 1. Find the first 5 terms (1, 2, 3, 4, 5) of the sequence given the nth term. 1.) 𝑎𝑛 = 𝑛 + 3 2.) 𝑎𝑛 = 3𝑛 − 1 3.) 𝑎𝑛 = 10 − 2𝑛 4.) 𝑎𝑛 = 2𝑛 5.) 𝑎𝑛 = (−3)𝑛 2. Consider the formula 𝒂𝒏 = 𝟒𝒏 − 𝟏. a.) What are the first four terms of the sequence? b.) Evaluate 𝒂𝟐𝟎 and write what it represent. n 1 2 3 4 5 𝑎𝑛 1 3 5 7 9 n 1 2 3 4 5 𝑎𝑛 5 2 -1 -4 -7 N 1 2 3 4 5 𝑎𝑛 3 9 27 81 243 9 4. Formulating a _____________________________________or the nth term is useful because it lets you calculate a specific term without having to calculate all the previous terms. 5. ____________________ is the nth term of a sequence. Complete the steps in finding the nth term of a sequence. 6. Draw the __________________. 7. Find the ___________________________. 8. Multiply the common difference by 𝑛 and add/subtract a particular number to get ______________________. 9. Find the 8th term of the sequence, an = 6n -4 10. Find the generated pattern or the nth term using the table of values below. n 1 2 3 4 5 an -4 -1 2 5 8 What I Can Do Is the meaning of sequence clear to you? Next is analyzing a real-life scenario on sequence. Mr. Ric gave Lia 20 pesos on her first day of cleaning the house, 30 pesos on her second day, 40 pesos on her third day and so on, increasing by 10 pesos each day. a.) Draw a table of values showing the first 5 days of cleaning the house and the amount Lia received each day of cleaning the house. b.) Write a pattern to describe the situation. Let 𝑎𝑛 be the amount received on the nth day. __________________________________________________________________________ c.) How much money will Lia received on the 6th day of cleaning the house? __________________________________________________________________________ d.) How about on 100th day of cleaning, can you tell how much Lia received? __________________________________________________________________________ After analyzing the real-life scenario, did you better understand the key concepts of sequences? Written below is the post assessment. Be able to evaluate your level of mastery about sequences by answering the Post Assessment. 10 Assessment I. Choose the letter that you think best answers the question. 1. What is missing in the sequence ___, 4, 10, 16, ___? a. 2 and 22 b. -2 and 22 c. 2 and 24 d. -2 and 24 2. Give the first three terms of the 𝑛th term 𝑎𝑛 = 3𝑛 + 7. a. 10, 12, 15 b. 10, 13, 16 c. 10, 12, 14 d. 10, 11, 12 3. Formulate the rule for the given sequence, 9, 6, 3, 0, -3. a. 𝑎𝑛 = 12 − 3𝑛 b. 𝑎𝑛 = 3𝑛 + 12 c. 𝑎𝑛 = 3𝑛 + 6 d. 𝑎𝑛 = 6 − 3𝑛 4. What is the next term in the sequence, 4, 7, 10, 13, 16, 19, __? a. 20 b. 21 c. 22 d. 23 5. Find the next term, 7, 15, 23, 31 and the common difference. a. 38; 5 b. 38; 6 c. 39; 7 d. 39; 8 6. What is the nth term of the sequence, 2, 4, 6, 8, …? a. 𝑎𝑛 = 𝑛 + 1 b. 𝑎𝑛 = 𝑛2 + 1 c. 𝑎𝑛 = 2𝑛 d. 𝑎𝑛 = 4𝑛 − 2 7. What are the next two terms in the sequence, 3, 6, 9, 12, ___? a. 14; 16 b. 15; 16 c. 14; 15 d. 15; 18 8. What is the 7th term of the sequence, 4, 10, 16, …? a. 20 b. 30 c. 40 d. 50 9. Formulate the nth term of the sequence, 7, 15, 23, 31 … a. 𝑎𝑛 = 8𝑛 − 1 b. 𝑎𝑛 = 8𝑛 + 1 c. 𝑎𝑛 = 7𝑛 − 1 d. 𝑎𝑛 = 8𝑛 + 1 10. The first 3 terms of the sequence given the rule 𝑎𝑛 = 2𝑛. a. 1, 2, 4 b. 2, 4, 8 c. 2, 6, 8 d. 2, 4, 6 II. On a number square like this one, shade all the multiples of 7. Then answer the questions that follows. 11. What is the 3rd multiple of 7? 12. What is the 10th multiple of 7? 13. What is the 24th multiple of 7? 14. What is the 50th multiple of 7? 15. What is the 100th multiple of 7? 11 Additional Activities A. The Missing Link! Try to fill up the missing numbers below and determine the numerical sequence; 1. 15, 20, 25, 30, 35, 40, ____, ____, 55, ____, 65, 70, 75, ____, 85, ____ The rule for this numerical sequence is: __________ 2. 93, 87, 81, ____, ____, 63, 57, 51, 45, ____, 33, 27, 21, ____, ____, 3 The rule for this numerical sequence is: __________ 3. Generate two numerical sequences starting at zero using the given rules. Then compare and explain the relationship between the two sequences. Add 2: , , , , , Add 8: , , , , , 4. Generate two numerical sequences starting at zero using the given rules. Then compare and explain the relationship between the two sequences. Add 3: , , , , , Add 27: , , , , , B. Decide whether a sequence is arithmetic or not. If it is, find the common difference. Example: 3, 7, 11, 15, 19, . . . a.) arithmetic sequence 𝑑 = 4 1.) 4, 16, 64, 256 2.) 1, 1 2 , 0, - 1 2 3.) 48, 24, 12, 6, 3 4.) 1, 0, -1, -2, -3 5.) 9.5, 7.5, 5.5, 3.5, … 6.) 1, 6, 11, 16, … 7.) 100, 85, 70, 55, … 8.) 4, 4.5, 5, 5.5, 6 … 9.) 3, 6, 12, 24, 48 … 10.) 5, 9, 13, 17, 21 … Compare and explain: Compare and explain: