This is the Grade 6 Mathematics self-learning module for Quarter 1, Module 11: “Differentiating Terminating from Repeating Non-Terminating Decimal Quotients.” It was made by DepEd Region VI 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 tell whether a division gives a decimal that stops or one that repeats forever.
What learners will learn
By the end of the module, learners are expected to:
- Differentiate terminating from repeating, non-terminating decimal quotients (M6NS-Ii-119)
The module builds on the earlier lessons on dividing decimals. It has one lesson, with the same title as the module.
Terminating and repeating decimals, explained
The module opens with a word problem. At a school meet at Mabuhay Elementary School, the girls’ team won 1 out of 3 games (1/3) in badminton, and the boys’ team won 3 out of 4 games (3/4) in volleyball. The sports coordinator turns both results into decimals by long division.
| Team | Division | Quotient | What happens to the remainder | Type |
|---|---|---|---|---|
| Boys | 3 ÷ 4 | 0.75 | It becomes 0, so the division stops | Terminating |
| Girls | 1 ÷ 3 | 0.333… | It stays 1 no matter how many zeros you add | Repeating non-terminating |
Key ideas
- A terminating decimal quotient comes from a division that stops with a zero remainder. Example: 3 ÷ 4 = 0.75.
- A repeating non-terminating decimal quotient comes from a division that never ends because there is always a remainder. One digit or a group of digits in the decimal part repeats without end. Example: 1 ÷ 3 = 0.333…
- A bar is written over the repeating digit or digits to show that they repeat.
More examples from the module
| Given | Quotient | Type |
|---|---|---|
| 2/10 | 0.2 | Terminating |
| 8 ÷ 12 | 0.666… | Repeating non-terminating |
| 10 out of 100 | 0.1 | Terminating |
| 5 ÷ 40 | 0.125 | Terminating |
| 8/10 | 0.8 | Terminating |
| 4/6 | 0.666… | Repeating non-terminating |
A simple way to check: divide using long division and watch the remainder. If it reaches zero, the decimal terminates. If the same remainder keeps coming back, the digits will repeat forever.
Parts of the module
- What I Know (pretest): 10 division items; write T for terminating or NT for repeating non-terminating.
- What’s In: a review that turns fractions such as 2/4 and 4/9 into decimals and divides numbers such as 79 ÷ 3 and 185 ÷ 11.
- What’s New and What is It: the school meet problem and the explanation of both kinds of quotients.
- What’s More: five divisions (such as 3 ÷ 15 and 9 ÷ 12) to solve and mark as terminating or repeating non-terminating.
- What I Have Learned: a summary of the key ideas.
- What I Can Do: five true-or-false statements; draw a star if correct and a triangle if not.
- Assessment: 10 more T or NT division items.
- Additional Activities: five fractions, such as 15/20 and 6/11, to write as decimals and classify.
- Answer Key for all the activities and tests.
How to use this module
- Learners: Start with the pretest to see what you already know, then go through the parts in order. Write all answers on a separate sheet of paper and do not mark the module. Check your work with the answer key only after you finish, and be honest with yourself.
- Parents and guardians: Help your child do each long division on paper and look at the remainder together. Ask the teacher if your child gets stuck.
- Teachers: The Assessment works as the post-test. Before giving the pretest key to learners, check it: the printed key for the pretest matches the Assessment key, so items 5 (5 ÷ 10 = 0.5), 7 (1 ÷ 11 = 0.0909…), and 8 (10 ÷ 50 = 0.2) are marked wrongly. In What’s More, item 2 (4 ÷ 45) equals 0.0888…, not 0.88 as printed.
Frequently asked questions
What grade and quarter is this module for?
It is for Grade 6 Mathematics, Quarter 1, Module 11. It covers the competency M6NS-Ii-119.
What is the difference between a terminating and a repeating non-terminating decimal?
A terminating decimal comes from a division that stops with a zero remainder, like 3 ÷ 4 = 0.75. A repeating non-terminating decimal comes from a division that never ends, so a digit or group of digits repeats, like 1 ÷ 3 = 0.333…
Why is a bar placed over some digits?
The bar shows which digit or digits repeat without end in a repeating non-terminating decimal, such as the 3 in 0.333…
Does the module have an answer key?
Yes. It has answers for the pretest, review, activities, assessment, and additional activities. Note that a few pretest answers and one What’s More answer are printed incorrectly, so check them with long division.
Full text of the module
The text below is a transcription of the PDF above, as printed.
Mathematics
Quarter 1 – Module 11: Differentiating Terminating from Repeating Non-Terminating Decimal Quotients
[illustration: students reading and studying outdoors and at a table, a teacher helping a learner, a globe, and a child working by lamplight]
CO_Q1_Mathematics6_Module11
Mathematics – Grade 6
Alternative Delivery Mode Quarter 1 – Module 11: Differentiating Terminating from Repeating Non-Terminating Decimal Quotients First Edition, 2020
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Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosda5do M. San Antonio
Printed in the Philippines by ________________________ Department of Education – Region VI Office Address: Duran Street, Iloilo City_________________________
Telefax: (033)336-2816, (033)509-7653___________________ E-mail Address: region6@deped.gov.ph_________________________
-Development Team of the Module
Authors: Lenie Rose E. Tabifranca and Realyn F. Crispe
Editors: Rebecca O. Billones, Ma. Portia G. Galanto, and Collin G. Sales
Reviewers: Rosemarie D. Aclan and Collin G. Sales
Illustrators: Lenie Rose E. Tabifranca and Realyn F. Crispe
Layout Artist: Felizardo S. Valdez III
Management Team: Ramir B.Uytico Pedro T. Escobarte, Jr. Elena P. Gonzaga Donald T. Genine Ma. Roselyn J. Palcat Nordy D. Siason, Jr. Rowena S. Carillo Rosemarie D. Aclan Arthur J. Cotimo Felizardo S. Valdez III Marve E. Gelera
Mathematics
Quarter 1 – Module 11: Differentiating Terminating from Repeating Non-Terminating Decimal Quotients
6
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 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, section icon]
This module was designed and written with you in mind. It is here to help you master the skills in differentiating terminating from repeating, non-terminating decimal quotients. The scope of this module allows you to use it in many different learning situations. The language used recognizes your diverse vocabulary level. The lessons are arranged to follow the standard sequence of your course. But the order in which you read them can be changed to match with the textbook you are now using.
After going through this module, you are expected to differentiate terminating from repeating, non-terminating decimal quotients. (M6NS-Ii-119)
What I Know
[illustration: pencil, section icon]
Solve for the quotient. Write T if the quotient is a terminating decimal and NT if repeating non-terminating. Write your answers on your answer sheet.
1.) 3 ÷ 4 2.) 5 ÷ 6 3.) 8 ÷ 12 4.) 7 ÷ 8 5.) 5 ÷ 10 6.) 9 ÷ 15 7.) 1 ÷ 11 8.) 10 ÷ 50 9.) 3 ÷ 16 10.) 4 ÷ 5
Lesson 1: Differentiating Terminating from Repeating Non-Terminating Decimal Quotients
In the previous lessons you have learned how to divide decimals. This time we will focus on how to differentiate terminating from repeating non-terminating decimal quotients.
What’s In
[illustration: puzzle pieces, section icon]
A. Find the value of N. Find the decimal equivalent of the following fractions. Write your answers on a piece of paper.
1.) 2 / 4 = N 2.) 4 / 9 = N
B. Find the value of N. Find the quotient. Write your answers on a separate piece of paper.
3.) 79 ÷ 3 = N 4.) 96 ÷ 9 = N 5.) 185 ÷ 11 = N
What’s New
[illustration: seedling, section icon]
Read the problem below.
Mabuhay Elementary School conducted school meet. Girls’ team won 1 out of 3 or 1/3 game in Badminton and Boys’ team won 3 out of 4 or 3/4 game in volleyball. The School Sports Coordinator converted the averages of each team in decimal form.
What is It
[illustration: magnifying glass, section icon]
This is how he converted the averages of each team into decimal form.
[chart: long division; Boys’ Team and Girls’ Team, shown side by side]
Boys’ Team
0.75
4 ) 3.00
- 0
----
30
- 28
----
20
- 20
----
0
Girls’ Team
0.333 (bar over 333)
3 ) 1.00
- 0
----
10
- 9
----
10
- 9
----
1
- A terminating decimal quotient results when the division process terminates or stops with a zero remainder.
- What is the average of the boys’ team in decimal form? The average of boys’ team in decimal is 0.75
- The quotient 0.75 is an example of a terminating decimal.
- What is the remainder? The remainder is 0
- What is the average of the girls’ team in decimal form? The average of the girls’ team in decimal is 0.333.
- What is the remainder? (It has always a remainder of 1 no matter how many zeros you will add in the dividend. The division process never ends and the digit 3 in the quotient is repeating.)
transcriber’s note: in the Girls’ Team quotient, the bar is printed over the digits 333, marking them as repeating; the bar is shown here in words.
The quotient $0.\overline{333}$ is a repeating non-terminating decimal.
A repeating non-terminating decimal quotient results when the division process never ends. It has always a remainder. A remainder is a digit or group of digits in the decimal part that keeps repeating itself without end. A bar is placed on top of the repeating digit.
Study more examples of terminating and repeating non-terminating decimal quotients.
| Given | Quotient | Terminating or repeating non-terminating decimal |
|---|---|---|
| 1.) $\frac{2}{10}$ | 0.2 | terminating decimal |
| 2.) 8 ÷ 12 | $0.\overline{66}$ | repeating non-terminating decimal |
| 3.) 10 out of 100 | 0.1 | terminating decimal |
| 4.) 40 ⟌ 5 | 0.125 | terminating decimal |
What’s More
Find the quotient. Differentiate whether the quotient is terminating or repeating non-terminating decimal by putting a check (/) under the corresponding column. Write your answers in your answer sheet.
| Equation | Quotient | Terminating | Repeating Non-terminating |
|---|---|---|---|
| Ex. 5 ÷ 15 = | $0.\overline{33}$ | / | |
| 1.) 3 ÷ 15 = | |||
| 2.) 4 ÷ 45 = | |||
| 3.) 9 ÷ 12 = | |||
| 4.) 8 ÷ 12 = | |||
| 5.) 2 ÷ 4 = |
What I Have Learned
How do you differentiate terminating decimal quotient from non-terminating/repeating decimal quotient?
- A terminating decimal quotient results when the division process terminates or comes to an end with a zero remainder.
- A repeating non-terminating decimal quotient results when the division process never ends. It has always a remainder.
- A Remainder is a digit or group of digits in the decimal part that keeps repeating itself without end.
- A bar is placed above the repeated digit/s.
What I Can Do
Draw a star ☆ if the statement is correct and triangle △ if not. Write your answers on your answer sheet.
1.) The quotient of 20 divided by 30 is a repeating non-terminating decimal. 2.) When you divide 7 by 30, the quotient is a terminating decimal. 3.) The decimal value of $\frac{3}{5}$ is a repeating non-terminating decimal. 4.) Nine divided by ten is equal to 0.9, which is a terminating decimal. 5.) The equivalent value of $\frac{8}{11}$ in decimal form is $0.\overline{72}$, which is a terminating decimal.
Assessment
Solve for the quotient. Write T if the quotient is a terminating decimal and NT if repeating non-terminating. Write your answers on your answer sheet.
1.) 3 ÷ 20 2.) 5 ÷ 30 3.) 8 ÷ 12 4.) 10 ÷ 40 5.) 8 ÷ 9 6.) 1 ÷ 5 7.) 2 ÷ 8 8.) 7 ÷ 12 9.) 3 ÷ 8 10.) 1 ÷ 20
Additional Activities
Give the equivalent value in decimal form. Tell whether the quotient is terminating or repeating non-terminating decimal. Write your answers on your answer sheet.
Example: $\frac{8}{10}$ = 0.8 – terminating decimal $\frac{4}{6}$ = $0.\overline{66}$ – repeating non-terminating decimal
1.) $\frac{15}{20}$ = 2.) $\frac{6}{11}$ = 3.) $\frac{3}{16}$ = 4.) $\frac{4}{16}$ = 5.) $\frac{2}{12}$ =
Answer Key
[transcriber’s note: this page is printed upside down; the columns below are given in reading order, with each column’s entries listed top to bottom]
What I Know
- T
- NT
- NT
- T
- NT
- T
- T
- NT
- T
- T
What’s In
- 0.5
- $0.\overline{4}$
- $26.\overline{33}$
- $10.\overline{66}$
- $16.\overline{81}$
[transcriber’s note: the bars on the repeating digits in items 2 to 5 are as read from the image and may not be exact]
What’s More
- 0.2 – Terminating
- 0.88 – Repeating
- 0.75 – Terminating
Non Terminating - 0.66 – Repeating Non
Terminating - 0.5 – Terminating
[transcriber’s note: the words “Non Terminating” and “Terminating” appear alone on their own lines in this column]
What I Can Do
- ☆
- △
- △
- ☆
- △
Assessment
- T
- NT
- NT
- T
- NT
- T
- T
- NT
- T
- T
Additional Activities
- 0.75 – Terminating
- 0.54 – Repeating Non
Terminating - 0.1875 – Terminating
Terminating - 0.25 – Terminating
- 0.16 – Repeating Non
Terminating
[transcriber’s note: the word “Terminating” appears alone on its own line in items 2 and 3 of this column]
Reference
- Most Essential Learning Competency (MELC) in Mathematics 6
For inquiries or feedback, please write or call:
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Email Address: blr.lrqad@deped.gov.ph * blr.lrpd@deped.gov.ph
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