This is the Grade 5 Mathematics Self-Learning Module (SLM) for Quarter 3, Module 13: Formulating the Rule in Finding the Next Term in a Sequence. It was developed by DepEd Region VIII as part of the Alternative Delivery Mode (First Edition, 2020) for learners studying at home or in class. After working through it, learners should be able to look at a list of numbers, figure out the rule behind it, and use that rule to find the next terms.
On this page 7 sections
What learners will learn
The module states one learning objective. After going through it, learners are expected to:
- formulate the rule in finding the next term in a sequence.
Along the way, learners also practice using patterns to make predictions and solving word problems with sequences.
Lesson in this module
The module has one lesson, Lesson 1: Sequences. It defines a sequence, a term, and a rule. It then shows how to find the rule of number sequences that add, subtract, multiply, or combine two steps, and applies this to real-life problems.
Key ideas: sequences, terms, and rules
- A sequence is an ordered list of numbers (or objects) in a defined or logical order.
- Each number in the sequence is a term. In 3, 5, 7, 9, … the 1st term is 3, the 2nd term is 5, and so on.
- The three dots (…) are read as “and so on”. They mean the pattern continues.
- A rule is the pattern or way to find the value of each term. For 3, 5, 7, 9, … the rule is “add 2”: 3 + 2 = 5, 5 + 2 = 7, 7 + 2 = 9.
To find the next term, you study the given terms, find the rule, and apply it.
Examples from the module
| Sequence | Rule | Next three terms |
|---|---|---|
| 3, 6, 9, 12, … | Add 3 to the term before it | 15, 18, 21 |
| 1, 4, 9, 16, … | Square the counting numbers (1 x 1, 2 x 2, 3 x 3, …) | 25, 36, 49 |
| 1, 2, 4, 7, … | After 1 and 2, add the previous two numbers, then add 1 | 12, 20, 33 |
| 1, 2, 4, 8, 16, … | Multiply the previous term by 2 | 32, 64, 128 |
Some rules use two steps. For example, 7, 15, 31, 63, … follows “multiply by 2 and add 1”, and 14, 41, 122, 365, … follows “multiply by 3 and subtract 1”.
Using a table for word problems
The module shows that a table helps you see a pattern in a word problem. Mang Juan harvested 50 kilograms of lanzones in the first year and 5 kilograms more each year after that. Listing the years gives 50, 55, 60, 65, 70, 75, 80, 85, so he harvested 85 kilograms in the 8th year. Later problems use the same idea with words in a poem, gallery visitors, pencils in boxes, an allowance, and daily savings.
Parts and activities of the module
| Part | What learners do |
|---|---|
| What I Know | 10-item multiple-choice pretest |
| What’s In | Find the next three terms of five sequences |
| What’s New | Complete the lanzones harvest table |
| What is It | Study the examples, then do “Fill Me In”: find the next three terms and the rule |
| What’s More | “Match Me” (match sequences to rules), “Complete Me” (build four terms from a first term and a rule), “Define Me” (write the rule) |
| What I Have Learned | True or False, correcting the underlined part of false statements |
| What I Can Do | Five word problems on sequences |
| Assessment | 10-item multiple-choice post-test, including a shape pattern |
| Additional Activities | Find the missing term and the rule of five sequences |
An answer key for every part is at the end of the module.
How to use this module
- Start with the pretest (What I Know) to see what you already know.
- Go through the parts in order, from What’s In to Additional Activities.
- Write all answers on a separate sheet of paper. Do not write on the module.
- Read each set of directions carefully before answering.
- Answer the Assessment at the end to check your learning, then check your work with the answer key honestly.
For learners: When you get stuck, write out how each term changes from the one before it (plus, minus, times). That change is usually the rule. If you still have trouble, ask your teacher or facilitator.
For parents: You can help by asking your child to explain the rule out loud and to check it on every term, not only the first two.
For teachers: In the pretest, item 10 (1, 3, 5, … roses) gives 9 roses on the fifth day, which is choice C, but the printed answer key lists A. You may want to point this out to learners before they check their answers.
Frequently asked questions
What grade and quarter is this module for?
It is for Grade 5 Mathematics, Quarter 3. It is Module 13 in the series.
What is a rule in a sequence?
A rule is the pattern or way to find the value of each term. For example, the rule of 3, 5, 7, 9, … is add 2.
Does the module have an answer key?
Yes. The answer key at the end covers the pretest, every activity, the assessment, and the additional activities. Note that the key for pretest item 10 lists A, but the pattern 1, 3, 5, 7, 9 gives 9 roses, which is choice C.
How many lessons does the module have?
It has one lesson, Lesson 1: Sequences, with a pretest, several activities, word problems, a post-test, and additional activities.
What do the three dots in a sequence mean?
The three dots (…) are read as and so on. They mean the pattern continues forward.
Full text of the module
The text below is a transcription of the PDF above, as printed.
Quarter 3 – Module 13: Formulating the Rule in Finding the Next Term in a Sequence
Mathematics
[illustration: teacher and students at a table, students reading outdoors under a tree, a boy working by lamplight, and a globe]
CO_Q3_Mathematics 5_ Module 13
Mathematics – Grade 5 Alternative Delivery Mode Quarter 3 – Module 13: Formulating the Rule in Finding the Next Term in a 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
Development Team of the Module
Writers: Teodora Coreal, Danilo S. Jadulco Editors: Nicanor S. Montad Jr., Glendo Carido Reviewers: Renato S. Cagomoc, Rolando Lacbo, Joshua Sherwin T. Lim Illustrator: Manuel Y. Sapinit Jr. Layout Artist: Joey Sustitudo, Emmanuel S. Gimena Jr. Management Team: Ma. Gemma M. Ledesma, Arnulfo R. Balane, Rosemarie M. Guino, Joy B. Bihag, Ryan R. Tiu, Sarah S. Cabaluna, Thelma Cabadsan-Quitalig, Elena S. De Luna, Renato S. Cagomoc, Noel E. Sagayap, Geraldine P. Sumbise, Joshua Sherwin T. Lim
Printed in the Philippines by ________________________
Department of Education – Region VIII
Office Address: DepEd Regional Office No. 8, Candahug, Palo, Leyte Telefax: (053)-832-2997 E-mail Address: region8@deped.gov.ph
5
Mathematics
Quarter 3 – Module 13: Formulating the Rule in Finding the Next Term in a Sequence
Introductory Message
This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson.
Each SLM is composed of different parts. Each part shall guide you step-by-step as you discover and understand the lesson prepared for you.
Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these.
In addition to the material in the main text, Notes to the Teacher are also provided to our facilitators and parents for strategies and reminders on how they can best help you on your home-based learning.
Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task.
If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator.
Thank you.
What I Need to Know
[illustration: target with arrow]
This module was designed and written with you in mind. It is here to help you master the concepts on sequences. The lessons are arranged to follow the standard sequence of the course.
Hi, mathletes! In this module, you are going to test your ability to formulate the rule in finding the next term in a sequence. Knowledge and skills in patterns and sequences enable you to make predictions and create rules. You will also learn to use different problem-solving strategies to solve math problems easier.
After going through this module, you are expected to:
- formulate the rule in finding the next term in a sequence.
What I Know
[illustration: pencil]
Hello, everyone! Patterns and sequences can be found in so many designs and objects in real life such as carpets, necklaces, table runners, floor tiles, and many others. Before proceeding to the next page, answer the test below to find out how well you know the topics to be discussed in this module.
Multiple Choice
Directions: Read each item carefully. Choose the letter of the best answer. Write your answers on a separate sheet of paper.
What is the next number in the sequence 2, 3, 4, 5, …?
A. 6 B. 7 C. 8 D. 9Find the rule in the sequence 10, 20, 30, 40, 50.
A. Add 2 B. Multiply by 2 C. Add 10 D. Multiply by 53, 10, 17, 24, 31, 38,… what is the next number in the sequence?
A. 43 B. 44 C. 45 D. 46What is the missing value in this sequence: 2, 5, 8, 11, 14, ____?
A. 15 B. 16 C. 17 D. 18What is the next number in the sequence 2, 6, 18, 54…?
A. 108 B. 152 C. 162 D. 216What are the next two terms in the sequence 2, 10, 18, ____, ____?
A. 20, 28 B. 22, 30 C. 24, 32 D. 26, 34Which among the sequences has the pattern of squaring the natural numbers?
A. 1, 2, 3, 4, 5 B. 2, 4, 6, 8, 10 C. 1, 6, 11, 16, 21 D. 1, 4, 9, 16, 25What is the next number in the sequence 1, 4, 13, 40,…, if the rule for the sequence is “multiply the previous term by 3 and add 1”.
A. 131 B. 121 C. 111 D. 91What are the missing numbers in the sequence 2, ____, 10, 14, ____?
A. 4, 20 B. 6, 18 C. 5, 17 D. 8, 20Five days before Valentine’s Day, Arnel decided to give Maria flowers according to a certain pattern. On the first day, he sent one red rose; on the second day, three red roses; on the third day, 5 red roses and on so on. How many roses did Maria receive during the fifth day?
A. 7 B. 8 C. 9 D. 10
Lesson 1 Sequences
[illustration: puzzle pieces icon]
What’s In
A sequence is an ordered list of numbers. Each number in the sequence is called a term. The three dots (…) mean to continue forward in the pattern. To fill in the missing numbers or symbols in a given sequence, we need to find out the rule or pattern for generating the next term.
[chart: diagram; SEQUENCE: 3, 5, 7, 9, … with arrows labeled 1st term (3), 2nd term (5), 3rd term (7), 4th term (9); note: “Three dots – this is read as ‘and so on’”]
RULE: A sequence usually has a rule, which is a pattern or a way to find the value of each term.
Example: {3, 5, 7, 9, …} Starts at 3 and goes up 2 every time.
RULE: Add 2. 3 + 2 = 5 5 + 2 = 7 7 + 2 = 9
Let me see if you can identify the rule and find the next term in a sequence.
Directions: Find the next three terms in the sequence. Number 1 is done for you.
- 10,15, 20,25, 30, 35, 40
- 3, 5, 7, 9, ____, ____, ____
- 1, 3, 9, 27, ____, ____, ____
- 2, 5, 8, 11, ____, ____, ____
- 1, 8, 27, ____, ____, ____
What’s New
[illustration: sprout icon]
Study the situation below.
For the first year, Mang Juan harvested 50 kilograms of lanzones. If every year, he harvested 5 more kilograms than the previous year, how many kilograms of lanzones did he harvest on the 8th year?
REMEMBER: In finding the next term in a sequence, we will look for a pattern. Let us construct a table to look at the pattern in the sequence of numbers and fill in the unknown parts of the table.
| Sequence (Year) | Number of Kilograms of Lanzones |
|---|---|
| 1st year | 50 |
| 2nd year | 50+5=55 |
| 3rd year | 55+5=60 |
| 4th year | |
| 5th year | |
| 6th year | |
| 7th year | |
| 8th year |
[photo: man in a hat harvesting lanzones from a tree, with baskets of fruit on the ground]
So, how many kilograms of lanzones did Mang Juan harvest on the 8th year? Yes. He harvested 85 kilograms of lanzones.
Very good. You did it right!
What is It
There are problems in mathematics that can be solved by observing sequences and patterns. A sequence is a list of numbers or objects in a defined or logical order. Patterns and repetitive sequences can be found in nature, shapes, events, sets of numbers and anywhere. Studying sequences is not that difficult. You simply need to analyze the given terms and identify the rule for generating the next term in the sequence.
Study the table below:
| Sequence | Rule | Next Three Terms |
|---|---|---|
| a) 3, 6, 9, 12,… | Every term after the first is obtained by adding 3 to the number preceding it. 0 + 3= 3; 3 + 3= 6; 6 + 3= 9, … | 15, 18, 21 |
| b) 1,4,9, 16,… | Multiply the counting numbers by itself, that is, square the counting numbers. 1 x 1=1; 2 x 2= 4; 3 x 3= 9, … | 25, 36, 49 |
| c) 1, 2, 4, 7,… | After 1 and 2, add the previous two numbers, then plus 1 1+2+1= 4; 2+4+1= 7 | 12, 20, 33 |
| d) 1, 2, 4, 8, 16,… | Multiply the previous term by 2. 1 x 2=2; 2 x 2=4; 4 x 2=8; 8 x 2=16, … | 32, 64, 128 |
Did you understand the lesson? Very good. Try to answer the activity below.
Activity: Fill Me In
Directions: Find the next three terms in each sequence. Then, write the rule in finding the next term. Write your answer on separate sheet of paper.
| Sequence | Next Three Terms | Rule |
|---|---|---|
| 1) 3,6,12,24, … | ||
| 2) 2, 9, 16, 23, … | ||
| 3) 53, 46, 39, 32, … | ||
| 4) 5, 12, 26, 54, … | ||
| 5) 5, 20, 50,110, … |
What’s More
In this section, you will use the concepts and examples that were presented earlier to answer the different activites provided.
Activity 1: Match Me
Directions: Match the sequence in Column A to the rule that generates the terms of the sequence in Column B. Write your answer on a separate sheet of paper.
| Column A (Sequence) | Column B (Rules) |
|---|---|
| 1) 8, 11, 14, 17, … | A. Multiply by 3 and subtract 1 |
| 2) 12, 24, 48, 96, … | B. Subtract by 7 |
| 3) 7, 15, 31, 63, … | C. Multiply by 2 and add 1 |
| 4) 56, 49., 42, 35, … | D. Multiply by 2 |
| 5) 14, 41, 122, 365, … | E. Add 3 |
Activity 2: Complete Me
Directions: Given the first term and the rule, make a sequence consisting of four (4) terms. Write your answer on a separate sheet of paper.
| First Term | Rule | First 4 terms of the Sequence |
|---|---|---|
| 1) 2 | Add 4 and minus 3 | |
| 2) 3 | Multiply by 2 and subtract 1 | |
| 3) 4 | Add 3 and minus 2 | |
| 4) 5 | Subtract 2 and plus 5 | |
| 5) 1 | Add 1 times 2 |
Activity 3: Define Me
Directions: Give the pattern or rule for generating each sequence. Write your answer on a separate sheet of paper.
- 1, 4, 7, 10, 13,… Rule: _____________________
- 3, 8, 18, 38, 78,… Rule: _____________________
- 60, 56, 52, 48, 44,… Rule: _____________________
- 2, 8, 32, 128, 512,… Rule: _____________________
- 2, 9, 44, 219, 1094,… Rule: _____________________
Congratulations! You made it this far.
What I Have Learned
To check how well you understood the lesson in this module, answer the activities below.
Activity: True or False
Directions: Read each statement carefully. Write True if the statement is correct. If the statement is incorrect, write False and change the underlined word, number, or symbol to make it correct. Write your answer on a separate sheet of paper.
_________1) In the sequence 3, 7, 15 and 31, you have to multiply by 2 and add 1 to get the next term which is 63.
_________2) The first term in a sequence is 4. If the rule is “add 5 and subtract by 2”, the first 5 terms of this sequence are 4, 7, 10, 13, and 15”.
_________3) A pattern is a list of numbers or objects in a defined or logical order.
_________4) 4, 8, 12, 16 and 20 form a sequence. 8 is called the second term.
_________5) The rule of the sequence 4, 9, 19, 40, 79… is “multiply by 2 and add 1”.
What I Can Do
The concept of sequences can also be used in solving word problems. Below is an activity that needs application of the knowledge you acquired in the lesson.
Activity:
Directions: Read each problem carefully. Answer it to the best that you can and write your answers on separate sheet of paper.
Nicole is writing a poem. She writes 7 words on the first line, 14 words on the second line, 28 words on the third line, and 56 words on the fourth line. If this pattern continues, how many words will Nicole write on the fifth line?
Cindy is making a tally of the people visiting her art gallery. She noticed that 25 people visited her art gallery on the first day, 35 people visited on the second day, and 45 people visited on the third day. Given the same pattern, on what day did 65 people visit her art gallery?
Iriss sorted pens in small boxes. He put 1 pencil in the first box, 4 pencils in the 2nd box, 16 pencils in the 3rd box, and 64 pencils in the 4th box. Following the same pattern, how many pencils did Iriss place in the 5th box?
Chandy was given an allowance of P750. He spent P100 on the 1st day, P150 on the 2nd day, P200 on the 3rd day and so on. In how many days will his allowance last?
Christian saved Php 14 on Saturday, Php 17 on Sunday, Php 20 on Monday, Php 23 on Tuesday. If this pattern continues, how much will he save next Sunday?
Assessment
At this point, since the lesson is already done and its concept was discussed exhaustively with different activities, it is time to assess the knowledge you gained from this module.
Multiple Choice
Directions: Read each statement carefully. Choose the letter of your best answer and write the chosen letter on a separate sheet of paper.
- What are the next shapes?
[illustration: row of alternating squares and triangles followed by two blank boxes] _____, ____
- A. [illustration: square and triangle]
- B. [illustration: triangle and square]
- C. [illustration: two triangles]
- D. [illustration: two squares]
- What is the next term in the sequence 42, 40, 38, …?
- A. 36
- B. 35
- C. 34
- D. 33
- 5, 10, 15, 20, 25, … What is the next number in the sequence?
- A. 28
- B. 30
- C. 32
- D. 35
- By observing the pattern, what is the missing value?
5, 7, ____, 11, 13
- A. 6
- B. 8
- C. 9
- D. 10
- What is the next number in the sequence 0, 4, 8, 12, 16, ____?
- A. 32
- B. 28
- C. 24
- D. 20
- Determine the next term in the sequence 1, 10, 100, 1 000, ____?
- A. 10 000
- B. 100 000
- C. 1 000 000
- D. 10 000 000
- What is the sequence formed having the rule of multiplying by 2?
- A. 1, 2, 3, 4, 5
- B. 2, 4, 6, 8, 10
- C. 1, 2, 4, 8, 16
- D. 2, 6, 10, 14, 18
- Which sequence is generated by the rule “multiply the first term by 2 and add 1”?
- A. 1, 3, 7, 15
- B. 3, 6, 9, 12
- C. 2, 4, 6, 8
- D. 0, 4, 8, 12
- What numbers will complete the sequence 0, ____, 8, 15, 24, ____?
- A. 2, 28
- B. 5, 30
- C. 6, 34
- D. 3, 35
- Renalyn’s weekly savings from her school allowance are as follows: P10 on the first week, P15 on the second week, P20 on the third week, and so on. How much will she save on the fifth week?
- A. P 25
- B. P30
- C. P35
- D. P40
Additional Activities
To enhance your knowledge and skills on this lesson, answer the activity below.
Find the missing term and give the pattern rule of the following sequences.
- 14, 17, 20, 23, _____ [transcriber’s note: the numbers are printed in a row of boxes, with the last box blank]
Rule: _______________
- 24, 30, 36, 42, _____
Rule: _______________
- 132, 121, 110, 99, _____
Rule: _______________
- 5, 10, 20, 40, _____
Rule: _______________
- 4, 19, 94, 469, _____
Rule: _______________
Answer Key
[transcriber’s note: this page is printed upside down; the entries below are transcribed in the order they read when the page is turned upright]
What’s In
- 30, 35, 40
- 11, 13, 15
- 81, 243, 729
- 14, 17, 20
- 64, 125, 216
What’s New
65, 70, 75, 80, 85
What I Know
- A
- C
- C
- C
- C
- D
- D
- B
- B
- A
What is It
- 48, 96, 192 Multiply by 2
- 30, 37, 44 Add 7
- 25, 18, 11 Subtract 7
- 110, 222, 446 Multiply by 2 and add 2 or Add 1 and multiply by 2
- 230, 470, 950 Add 5 and multiply by 2
What’s More
Match Me: 1. E 2. D 3. C 4. B 5. A
Complete Me:
- 2, 3, 4, 5
- 3, 5, 9, 17
- 4, 5, 6, 7
- 5, 8, 11, 14
- 1, 4, 10, 22
Define Me:
- Add 3
- Multiply by 2 and add 2 or add 1 multiply by 2
- Minus 4
- Multiply by 4
- Multiply by 5 and minus 1
What I Have Learned
- True
- False, 16
- False, Sequence
- True
- False, 39
What I Can Do
- 112 words
- 5th day
- 256 pencils
- 4 days
- P38
Assessment
- B
- A
- B
- C
- D
- A
- C
- A
- D
- B
Additional Activities
- 26 Add 3
- 48 Add 6
- 88 Subtract 11
- 80 Multiply by 2
- 2344 Multiply by 5 and minus 1
References
21ST Century Mathletes, Textbook, Grade 6
Manuel T Kota, et al. Soaring 21ST Century Mathematics. Phoenix Publishing House, Grade 5, 2005
Lesson Guides in Elem. Math, Grade 5
Angelina P. Lumbre, Alvin C Ursua, Donnel P. Placer, Jaime R. Burgos, 21st Century Mathletes. (2016). Vibal Group Inc.
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: blr.lrqad@deped.gov.ph * blr.lrpd@deped.gov.ph
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