This is the Mathematics Grade 6 Quarter 2 Module 14: Dividing Integers, a self-learning module (Alternative Delivery Mode, First Edition 2020) from DepEd Region VI. It is for Grade 6 learners studying at home or in class. After working through it, learners can divide positive and negative integers using counters and number lines, apply the sign rules for division, and solve word problems involving division of integers.
On this page 7 sections
What learners will learn
After going through this module, learners are expected to:
- Describe and interpret division of integers using materials such as algebra tiles, counters, chips, and cards (M6NS-IIh-155).
- Perform division of integers (M6NS-IIi-156).
- Solve routine and non-routine problems involving division of integers using appropriate strategies and tools (M6NS-IIj-157).
Lessons in this module
The module has three lessons. Its introduction says “two lessons” but lists three.
- Lesson 1: Dividing Integers with Like Signs. Learners divide two positive or two negative integers using counters and a number line, then learn the sign rules.
- Lesson 2: Dividing Integers with Unlike Signs. Learners divide a positive integer by a negative one, or the other way around, and find that the quotient is negative.
- Lesson 3: Solving Routine and Non-routine Problems Involving Division of Integers Using Appropriate Strategies and Tools. Learners use Polya’s problem-solving steps and strategies such as working backward to solve word problems.
Key ideas: dividing integers
Division is the inverse of multiplication. The dividend is the number being divided, and the divisor is the number that divides. You divide integers the same way you divide whole numbers. The extra step is choosing the correct sign for the answer.
Using counters
The module uses positive and negative counters in three steps:
- Take counters equal to the dividend. For +18 ÷ +3, take 18 positive counters.
- Draw boxes equal to the divisor. Here you draw 3 boxes.
- Share the counters equally among the boxes. The number of counters in each box is the quotient: +6.
When the divisor is negative, the number of sets is negative. You cannot draw “negative boxes,” so the module tells you to flip every counter to its opposite side.
- For –18 ÷ –3, each of the 3 boxes first holds 6 negative counters. Flipping them gives 6 positive counters, so –18 ÷ –3 = +6.
- For +9 ÷ –3, each box first holds 3 positive counters. Flipping them gives 3 negative counters, so +9 ÷ –3 = –3.
Using a number line
Start at 0 and move to the dividend. For –10, move 10 units to the left. Then split that distance into as many equal parts as the divisor. The module’s example is a temperature drop of 10°C over 2 hours: –10 ÷ +2 = –5, so the temperature dropped 5°C each hour. In another example, JR bikes 18 km north in 3 hours: +18 ÷ +3 = +6, or 6 km each hour.
Sign rules
| Signs | Quotient | Example from the module |
|---|---|---|
| (+) ÷ (+) | Positive | +18 ÷ +3 = +6 |
| (–) ÷ (–) | Positive | –10 ÷ –2 = +5 |
| (+) ÷ (–) | Negative | +24 ÷ –4 = –6 |
| (–) ÷ (+) | Negative | –50 ÷ +5 = –10 |
In short, when the two signs are the same, the quotient is positive. When the signs are different, the quotient is negative.
Key ideas: solving word problems
Routine problems are problems where the method is easy to see. Solve them with Polya’s four steps: Understand, Plan, Solve, and Check and Look Back.
The module’s example: Jessa owes ₱360, and she and her two sisters will share the debt.
- The debt is written as –360, and the 3 persons as +3.
- So –360 ÷ +3 = –120, and each person pays ₱120.
- To check: –120 × 3 = –360.
Non-routine problems are unfamiliar problems with no ready-made method. They can be solved with strategies such as Guess and Check, Using Models, Acting Out, Logical Reasoning, and Working Backwards.
The module’s example: an integer is multiplied by –5, then divided by +3, then increased by +4, and the result is –6. To work backward, reverse each operation:
- (–6) – (+4) = –10
- (–10) × (+3) = –30
- (–30) ÷ (–5) = +6
The unknown integer is +6. Checking it in the original problem gives –6, as expected.
How to use this module
Each lesson follows the same parts:
- What I Know (pretest)
- What’s In (review)
- What’s New (an opening problem)
- What is It (discussion)
- What’s More (practice)
- What I Have Learned (summary)
- What I Can Do (application)
- Assessment
- Additional Activities
An answer key follows each lesson.
- Learners: Answer every item on a separate answer sheet, and do not write on the module. Keep positive and negative counters nearby for the counter activities. Check your work with the answer key only after you finish, and be honest with yourself.
- Parents: Help your child find simple counters, such as two-colored chips or cards. Ask your child to explain why each quotient is positive or negative.
- Teachers: The pretest shows which learners already know the rules. The Assessment and Additional Activities in each lesson can serve as checks of learning. Before sharing the answer keys, review them. For example, in Lesson 1’s What I Know, the answers printed for items 6 and 7 appear to be swapped.
Frequently asked questions
What grade and quarter is this Dividing Integers module for?
It is for Grade 6 Mathematics, Quarter 2, Module 14. It was published as a First Edition in 2020 by DepEd Region VI.
What are the rules for dividing integers?
If the dividend and divisor have the same sign, the quotient is positive. If they have different signs, the quotient is negative. For example, –10 ÷ –2 = +5, while +24 ÷ –4 = –6.
How do you show a negative divisor with counters?
Share the counters equally into as many boxes as the divisor shows. Then flip every counter to its opposite side, since you cannot draw negative boxes.
Does the module have an answer key?
Yes. Each of the three lessons ends with an answer key for its pretest, activities, assessment, and additional activities.
What is the difference between routine and non-routine problems?
Routine problems have an easy-to-see method and can be solved with Polya’s four steps. Non-routine problems are unfamiliar and need strategies such as guess and check or working backwards.
Full text of the module
The text below is a transcription of the PDF above, as printed.
Mathematics Quarter 2 – Module 14: Dividing Integers
6
[illustration: students studying outdoors and indoors with a teacher, a globe, and a lamp]
Mathematics – Grade 6 Alternative Delivery Mode Quarter 2 – Module 14: Dividing Integers 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 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 – 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 F. Tabifranca, Realyn F. Crispe and Collin G. Sales
Editors: Ma. Portia G. Galanto and Rebecca O. Billones
Reviewers: Elleda E. de la Cruz and Rosemarie D. Aclan
Illustrators: Lenie Rose F. Tabifranca, Realyn F. Crispe, Marina B. Damasco, and Collin G. Sales
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
6
Mathematics
Quarter 2 – Module 14: Dividing Integers
[illustration: bullseye target with an arrow]
What I Need to Know
This module was designed and written with you in mind. It is here to help you master Dividing Integers. The scope of this module permits it to be used in many different learning situations. The language used recognizes your diverse vocabulary as student. The lessons are arranged to follow the standard sequence of the course. But the order in which you read them can be changed to correspond with the textbook you are now using.
The module is divided into two lessons, namely:
- Lesson 1 – Dividing Integers with Like Signs
- Lesson 2 – Dividing Integers with Unlike Signs
- Lesson 3 – Solving Routine and Non-routine Problems Involving Division of Integers Using Appropriate Strategies and Tools
After going through this module, you are expected to:
- describe and interpret division of integers using materials such as algebra tiles, counters, chips, and cards (M6NS-IIh-155);
- perform division of integers (M6NS-IIi-156); and,
- solve routine and non-routine problems involving division of integers using appropriate strategies and tools (M6NS-IIj-157).
What I Know
A. Use the number line to solve for the quotient of the following integers. Write your answers on your answer sheet.
[chart: number line; untitled, printed under instruction A. Labels: -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12]
+10 ÷ +5 = _____
+9 ÷ +3 = _____
B. Use your positive and negative counters to divide the following integers. Write your answers with correct sign on your answer sheet.
+16 ÷ +2 = _____
-18 ÷ -3 = _____
-80 ÷ -4 = _____
C. Below are division equations involving integers. Find the value of N by dividing each item. Write your answers on your answer sheet.
+36 ÷ +6 = N
+72 ÷ +8 = N
-6 ÷ -3 = N
-39 ÷ -13 = N
+45 ÷ +15 = N
Lesson 1: Dividing Integers with Like Signs
Division is the inverse of multiplication. It is the process of finding the quotient of two numbers, the dividend which is the number to be divided and the divisor, the number that divides. Dividing integers follow the same steps in dividing whole numbers. But, in dividing integers, you must be careful of the sign to be used in your final answer. In this lesson, you will learn how to divide integers with like signs.
What’s In
Use the given illustrations below to solve for the quotient of the following numbers. Write your answers on your answer sheet.
[illustration: five boxes, each holding two cowboy-style hats, ten hats in total]
- 10 ÷ 5 = _____
[illustration: three rows of four red blocks each, twelve blocks in total]
- 12 ÷ 3 = _____
[illustration: two boxes, one with four yellow smiley faces and one with four yellow smiley faces, eight in total]
- 8 ÷ 2 = _____
[illustration: four boxes, each holding four cupcakes with frosting, sixteen in total]
- 16 ÷ 4 = _____
[illustration: three boxes, each holding eight green trees, twenty-four in total]
- 24 ÷ 3 = _____
What’s New
Read and study the problem below.
JR traveled with his bicycle a distance of 18 kilometers in 3 hours going north at a constant speed. How many kilometers did he travel in 1 hour?
What is It
To determine the number of kilometers JR traveled per hour, we are going to divide 18 by 3.
Eighteen kilometers going north is represented by the integer positive eighteen (+18).
Three hours is represented by the integer positive three (+3).
So our equation will be +18 ÷ +3 = N
Now, let us visualize +18 ÷ +3 = N using counters.
To perform the division operation of integers using counters, follow the suggested steps below:
Step 1: Get the exact number of counters to represent your dividend. Get the 18 positive counters. Place them on your table.
[illustration: a row of eighteen positive counters on a table]
Step 2: Draw boxes to represent the number of sets to where the counters will be distributed evenly. Draw 3 boxes.
[illustration: a row of eighteen positive counters with arrows pointing down to three empty boxes]
The quotient is the number of equal counters in each set.
[illustration: three boxes, each holding six positive counters]
How many counters did you distribute to all the sets? (18) In to how many sets did you distribute the counters evenly? (3) How many counters do you have in each set?
[illustration: a box holding six positive counters] 6 positive counters
Do these counters in each set have the same sign? (Yes) What is the sign? (Positive)
Therefore, if you divide +18 by +3 you will get the quotient of +6.
+18 ÷ +3 = +6
Going back to the problem, JR traveled 6 kilometers per hour going north (+6).
We can also visualize how to divide +18 by +3 using a number line.
Step 1: From 0, locate +18 on the number line by moving 18 units to the right of 0.
[chart: number line; untitled, printed under Step 1. A bracket labeled +18 spans from 0 to a marked point at +18. Labels: -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18]
Step 2: Divide the total number of units (18) which is the dividend into 3 equal parts since our divisor is +3.
[chart: number line; untitled, printed under Step 2. A bracket labeled 18 ÷ 3 spans from 0 to +18, with red marks at 0, +6, +12, and +18. Labels: -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18. Values are not labeled on the chart beyond the printed tick labels.]
Each part represents the distance traveled in one hour and the number of units in each part represents the number of kilometers JR traveled every hour. See (the) illustration below.
[chart: number line; distance traveled by JR over three hours] Labels above the arrows: “distance traveled after 1hour”, “distance traveled after 2hours”, “distance traveled after 3 hours”. Number line labels: -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9+10 +11+12+13+14+15+16+17+18 Brackets below: “6 units”, “6 units”, “6 units”.
On the first hour, JR traveled a distance of 6 kilometers, another 6 kilometers during the second hour and then 6 kilometers during the third hour. Hence, he traveled a distance of 6 kilometers every hour.
Therefore +18 ÷ +3 = +6
How about if we change the signs of our given to both negative, what will be the sign of the quotient? Let us try this one.
-18 ÷ -3 = N
Step 1: Get the exact number of counters to represent your dividend. Get the 18 negative counters. Place them on your table.
[illustration: 18 negative counters laid out on a table]
Step 2: Draw boxes to represent the number of sets to where the counters will be distributed evenly. Draw 3 boxes.
[illustration: 18 negative counters above three empty boxes]
Step 3: Perform the operation and write your answer. Division means to divide the dividend equally to sets (divisor). Distribute your 18 negative counters evenly to 3 sets.
[illustration: 18 negative counters with arrows pointing down into three empty boxes]
The quotient is the number of equal counters in each set.
[illustration: three boxes, each holding six negative counters]
However, in this case the divisor is negative. It means that the number of sets is negative. It is difficult indeed to illustrate 3 negative boxes or 3 negative sets. So, instead, you will just get the opposite of 3 positive sets to show the 3 negative sets. To illustrate 3 negative sets, you need to flip to the opposite side all your 3 positive sets to show its opposite. In this case you cannot flip your box drawing so you just flip to the opposite side all the counters in every set.
If you flip the counters, negative side of the counter will become positive. See the following illustration.
[illustration: “3 Negative sets (before you flip)”: three boxes, each holding six negative counters]
[illustration: “3 Negative sets (after you flipped)”: three boxes, each holding six positive counters]
Notice that your negative counters in 3 negative sets after you flipped change their opposite sign which is positive. This is how to illustrate the negative set (divisor).
How many counters did you distribute to all the sets? (18) In to how many sets did you distribute the counters evenly? (3) How many counters do you have in each set?
[illustration: six positive counters in a box] (6 positive counters)
Do these counters in each set have the same sign? (Yes) What is the sign? (Positive) Therefore, if you divide -18 by -3 you will get the quotient of +6 or
-18 ÷ -3 = +6
Remember: In dividing, if your divisor which represents the number of groups/sets is negative, you need to flip to the opposite side all the counters in each set to illustrate the opposite of positive set/group which is the negative set/group.
Here is another example for you to study.
-10 ÷ -2 = N
Step 1: Get the exact number of counters to represent your dividend. Get the 10 negative counters. Place them on your table.
[illegible]
[illustration: 10 negative counters in a box]
Step 2: Draw boxes to represent the number of sets to where the counters will be distributed evenly. Draw 2 boxes.
[illustration: 10 negative counters above two empty boxes]
Step 3: Perform the operation and write your answer. Division means to divide the dividend equally to sets (divisor). Distribute your 10 negative counters evenly to 2 sets.
[illustration: 10 negative counters with arrows pointing down into two empty boxes]
The quotient is the number of equal counters in each set.
[illustration: two boxes, each holding five negative counters]
However, in this case the divisor is negative. It means that number of sets is negative. It is difficult indeed to illustrate 2 negative boxes or 2 negative sets. So, instead, you will just get the opposite of 2 positive sets to show the 2 negative sets. To illustrate 2 negative sets, you need to flip to the opposite side all your 2 positive sets to show its opposite. In this case you cannot flip your box drawing so you just flip to the opposite side all the counters in every set.
If you flip the counters, negative side of the counter will become positive. See the following illustration.
[illustration: “2 Negative sets (before you flip)”: two boxes, each holding five negative counters]
[illustration: “2 Negative sets (after you flipped)”: two boxes, each holding five positive counters]
Notice that your negative counters in 2 negative sets after you flipped change their opposite sign which is positive. This is how to illustrate the negative set (divisor). How many counters did you distribute to all the sets? (10) In to how many sets did you distribute the counters evenly? (2) How many counters do you have in each set?
[illustration: five positive counters in a box] (5 positive counters)
Do these counters in each set have the same sign? (Yes) What is the sign? (Positive)
Therefore, if you divide -10 by -2 you will get the quotient of +5.
10 ÷ 2 = 5
After visualizing how to divide integers using counters and number line, you will now learn the rules in dividing integers with like signs.
Dividing integers is just like dividing whole numbers. In dividing integers with like signs, the following rules are applied.
Rule 1. The quotient of two positive integers is a positive integer.
(+) ÷ (+) = (+) Example: (+18) ÷ (+3) = (+6)
Rule 2. The quotient of two negative integers is a positive integer.
(-) ÷ (-) = (+) Example: (-10) ÷ (-2) = (+5)
Point to Remember In short, the quotient of two integers having the same signs is a positive integer.
Study more examples below.
Rule Applied
- +36 ÷ +6 = +6 Rule 1
- –50 ÷ –10 = +5 Rule 2
- +35 ÷ +5 = +7 Rule 1
- –24 ÷ –3 = +8 Rule 2
- +15 ÷ +3 = +5 Rule 1
What’s More
A. Using your positive and negative counters, divide the given integers. Write your answers with correct sign on your answer sheet.
- +24 ÷ +6 =
- –8 ÷ –4 =
- +15 ÷ +3 =
B. Complete the table by solving for the quotient of the given mathematical expressions. Write the quotient on the 2nd column and the rule applied on the third column. Write your answers on your answer sheet
| Mathematical Expressions | Quotient | Rule applied |
|---|---|---|
| Examples: +30 ÷ +10 –15 ÷ –3 | +3 +5 | Rule 1 Rule 2 |
| 4) +64 ÷ +8 | ||
| 5) –24 ÷ –6 | ||
| 6) +60 ÷ +12 |
What I Have Learned
In dividing integers with like signs,
- the quotient of two positive integers is a positive integer ( + ) ÷ ( + ) = ( + ); and,
- the quotient of two negative integers is a positive integer ( – ) ÷ ( – ) = ( + ).
Point to Remember
- The quotient of two integers having the same signs is a positive integer.
What I Can Do
A. Find the value of N by dividing the following integers using counters or number line. Write your answers on your answer sheet.
- –35 ÷ –7 = N
- +28 ÷ +2 = N
- –18 ÷ –3 = N
B. Give the quotient of the following by applying the rules in dividing integers with like signs. Write your answers on your answer sheet.
- +45 ÷ +15 = N
- –150 ÷ –15 = N
C. Solve the problem below. Show your solution on your answer sheet and label your final answer.
Hans received ₱300 for helping his aunt in her sari-sari store for 6 hours. If his aunt pays him in an hourly basis, how much did he earn in an hour?
Assessment
A. Use the number line to solve for the quotient of the following integers. Write your answers with correct sign on your answer sheet.
[chart: number line; Assessment A, first line] –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14
- +12 ÷ +2 = _____
[chart: number line; Assessment A, second line] –5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +21 +22
- +20 ÷ +4 = _____
B. Use your positive and negative counters to divide the following integers. Write your answers with correct sign on your answer sheet.
- –18 ÷ –6 = _____
- –21 ÷ –3 = _____
- –16 ÷ –2 = _____
C. Find the value of N by dividing the following integers. Write your answers on your answer sheet.
–50 ÷ –5 = N
+64 ÷ +8 = N
–36 ÷ –9 = N
–81 ÷ –9 = N
Marites deposited a total amount of ₱96 000 in the bank for a year. If she deposited same amount each month, how much is her monthly deposit? Write an integer to represent her monthly deposit.
Additional Activities
A. Put the correct sign to make the mathematical sentence true. Write your answers on your answer sheet.
- _____90 ÷ –30 = +3
- +120 ÷ +10 = _____12
- _____20 ÷ –5 = +4
- _____45 ÷ _____15 = +3
- +20 ÷ _____2 = +10
B. Solve the following problems. Show the solutions on your answer sheet and label your final answers.
Ninety-six people have been invited to a birthday party. The caterer is arranging the tables. Each table can accommodate 8 persons. How many tables are needed?
A bus can hold 80 passengers. If there are 20 rows of seats in the bus, how many seats are in each row?
Answer Key
What I Know
A.
- +2
- +3
B. 3) +8 4) +6 5) +20
C. 6) +9 7) +6 8) +2 9) +3 10) +3
What’s More
A.
- +4
- +2
- +5
B. 4) +8 Rule 1 5) +4 Rule 2 6) +5 Rule 1
Assessment
A.
- +6
- +5
B. 3) +3 4) +7 5) +8
C. 6) +10 7) +8 8) +4 9) +9 10) +₱8 000
What’s In
- 2
- 4
- 4
- 4
- 8
What I Can Do
A.
- +5
- +14
- +6
B. 4) +3 5) +10
C. 6) ₱50
Additional Activities
A.
- –90
- +12
- –20
- –45 ÷ –15 or +45 ÷ +15
- +2
B. 6) 12 tables 7) 4 seats
What is It
To solve for the number of degrees the temperature dropped each hour, we are going to divide the decrease in the temperature which is 10˚C by the number of hours (2). 10˚C which is the decrease in the temperature is represented by the integer negative ten (–10).
2-hour period is represented by the integer positive two (+2).
So our equation will be –10 ÷ +2 = N
Using counters to visualize division of integers with unlike signs, follow these steps.
Step 1: Get the exact number of counters to represent your dividend. Get the 10 negative counters. Place them on your table.
[illustration: ten negative counters placed in a box representing a table]
Step 2: Draw boxes to represent the number of sets to where the counters will be distributed evenly. Draw 2 boxes.
[illustration: ten negative counters above two empty boxes, with dotted arrows from the counters pointing into the boxes]
The quotient is the number of equal counters in each set.
[illustration: two boxes, each holding five negative counters]
How many counters did you distribute to all the sets? (10) In to how many sets did you distribute the counters evenly? (2) How many counters do you have in each set? (5 negative counters)
[illustration: five negative counters in a box] 5 negative counters
Do these counters in each set have the same sign? (Yes)
What is the sign? (Negative)
Since there are 5 negative counters in each set, therefore if you divide –10 by +2 you will get a quotient of –5.
Going back to the problem, the temperature dropped by 5˚C every hour (–5).
Using a number line to visualize division of integers with unlike signs, follow these steps.
Step 1: From 0, locate –10 on the number line by moving ten units to the left of 0.
[chart: number line; locating –10 by moving ten units to the left of 0] Labels: –12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11+12+13 Bracket label: –10
Step 2: Divide the total number of units (10 units) which is the dividend into 2 equal parts since our divisor is +2.
[chart: number line; dividing –10 into 2 equal parts] Labels: –12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11+12+13 Bracket label: –10 ÷ 2
Each part represents the decrease in the temperature in one hour and the number of units in each part represents the number of degrees the temperature dropped every hour. See illustration below.
[chart: number line; decrease in temperature shown in two equal parts] Labels: decrease in temperature after 2 hours; decrease in temperature after 1hour Number line labels: –12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11+12+13 Bracket labels: 5 units; 5 units
On the first hour, the temperature dropped by 5 degrees Celsius (–5˚C) and on the second hour, it dropped again by 5 degrees Celsius (–5˚C). Therefore, the temperature dropped by 5˚C every hour.
Dropped by 5˚C is represented by the integer negative 5 or (–5).
Therefore, –10 ÷ +2 = –5
Here is another example for you to study.
+9 ÷ –3 = N
(Page 19 is the same as the earlier attachment page; see the transcription above.)
(Page 20 is the same as the earlier attachment page; see the transcription above.)
Step 1: Get the exact number of counters to represent your dividend. Get the 9 positive counters. Place them on your table.
[illustration: nine positive counters in a row inside a box]
Step 2: Draw boxes to represent the number of sets to where the counters will be distributed evenly. Draw 3 boxes.
[illustration: nine positive counters above three empty boxes]
Step 3: Perform the operation and write your answer. Division means to divide the dividend equally to sets (divisor). Distribute your 9 positive counters evenly to 3 sets.
[illustration: nine positive counters with arrows pointing down to three empty boxes]
The quotient is the number of equal counters in each set.
[illustration: three boxes, each holding three positive counters]
As discussed in the previous lesson, when the divisor is negative, you are going to flip the counters their opposite side. From 3 sets of positive counters, this becomes 3 sets of negative counters as shown below.
[chart: before-and-after diagram; “3 Positive Sets (before you flip)” shows three boxes, each holding three positive counters, with an arrow pointing to “3 Positive Sets (after you flipped)”, which shows three boxes, each holding three negative counters]
Since there are 3 negative counters in each set after you flipped, therefore, +9 ÷ –3 = –3.
After visualizing division of integers with unlike signs using counters, are you now ready to solve mathematical equations on dividing integers with unlike signs?
To solve for the quotient of two integers, we need to follow the rule in dividing integers with unlike signs.
Rule: In dividing integers with unlike signs, divide as whole numbers and consider the signs of the dividend and the divisor. Thus, the quotient of a negative and a positive integer is always negative integer. In symbol: ( +) ÷ ( –) = ( –) or ( –) ÷ ( +) = ( –)
Study more examples below.
- What is the quotient of –50 ÷ +5 ?
Solution: –50 ÷ +5 ? = –10 - Divide +24 by –4
Solution: +24 ÷ –4 = –6
What’s More
A. Use your counters to solve for the quotient. Write your answers on your answer sheet.
- –18 ÷ +9 =
- +10 ÷ –2 =
B. Give the quotient of the following integers applying the rule in dividing integers with unlike signs. Write your answers on your answer sheet.
- –15 ÷ +3 =
- +14 ÷ –2 =
- –9 ÷ +9 =
What I Have Learned
In dividing positive and negative integers, the quotient is always a negative integer. In symbol: ( + ) ÷ ( –) = ( –) or ( –) ÷ ( + ) = ( –)
What I Can Do
A. Divide the following integers. Write your answers on your answer sheet.
- –40 ÷ +8 =
- +28 ÷ –2 =
- –18 ÷ +2 =
- –45 ÷ +15 =
- –100 ÷ +5 =
Assessment
A. The following multiple-choice items are for you to answer. Some items use illustration for your reference. Choose the letter of the correct answer and write your answers on your answer sheet.
- Lance has 18 negative counters. He decided to distribute them evenly into 6 groups. What is the correct number sentence for this situation?
- A. –18 + +6 = –12
- B. –18 – +6 = –12
- C. –18 ÷ +6 = –3
- D. –18 x +6 = –108
- EJ wants to distribute 9 negative counters in every set he made. He has prepared 3 sets to be filled in. What number sentence is he illustrating?
- A. –9 x +3 = –27
- B. –9 + +3 = –6
- C. –9 – +3 = –12
- D. –9 ÷ +3 = –3
- What is the correct number sentence for the illustration given below?
[illustration: six negative counters above two empty ovals, with arrows pointing down from the counters to the ovals]
- A. –6 + +2 = N
- B. –6 ÷ +2 = N
- C. –6 x +2 = N
- D. –6 – +2 = N
- What is the correct answer for item number 3?
- A. –4
- B. –3
- C. –12
- D. –8
- Which is the correct illustration for –8 ÷ +4 = N?
- A. [illustration: eight positive counters above four empty boxes]
- B. [illustration: eight negative counters above four empty boxes]
- C. [illustration: eight negative counters above two empty boxes]
- D. [illustration: eight positive counters above two empty boxes]
B. Find the value of N by dividing the following integers. Write your answers on your answer sheet.
- –70 ÷ +5 = N
- +88 ÷ –8 = N
- –63 ÷ +9 = N
- –81 ÷ +3 = N
- –72 ÷ +8 = N
Additional Activities
A. Find the missing integers. Write your answers on your answer sheet.
- _____ ÷ +8 = –4
- +40 ÷ _____ = –10
- –28 ÷ +7 = _____
- +42 ÷ –2 = _____
- _____ ÷ +9 = –6
[transcriber’s note: this page is printed upside down, and its page number is 23 in the source. The answer key table is transcribed in upright reading order.]
Answer Key
What I Know
A.
- C
- B
- D
- B
- B
B. 6) –3 7) –24 8) –7 9) –4 10) –8
Assessment
A.
- C
- D
- B
- B
- B
B. 6) –14 7) –11 8) –7 9) –27 10) –9
What’s More
A.
- –2
- –5
B. 3) –5 4) –7 5) –1
What’s In
- +4
- +10
- +4
- +10
- +20
Additional Activities
A.
- –32
- –4
- –4
- –21
- –54
What I Can Do
A.
- –5
- –14
- –9
- –3
- –20
What I Know
Solve the following problems. Write the number sentence and show your solutions. Write your answers on your answer sheet.
- There were 30 tables arranged in rows of 6 in a birthday party catered by Rita’s Catering Services. Jess paid a total of ₱900 for the rental of tables. How much did Jess pay for each table? Write an integer to represent the amount he paid.
- At a constant speed, a submarine descended a distance of 320 feet in 4 minutes. How many feet did it travel every minute? Write an integer to represent the distance travelled every minute.
- I am an integer greater than –20 but less than +20. If you divide me by +5, my quotient is –3. Who am I?
- When an integer is subtracted by –5 then the result is multiplied by +3 and then the product is divided by –6, the result is –10. What is the integer?
- Give the sum of the averages of the following sets of integers.
–7, –2, +10, +5, +4 and +7, +3, –9, –12, +21
Lesson 3: Solving Routine and Non-Routine Problems Involving Division of Integers Using Appropriate Strategies and Tools
From the previous lessons, you learned how to visualize division of integers using counters and number lines as well as how to solve it directly by applying the rules. In this lesson, you will learn the skill in solving word problems involving division of integers with like and unlike signs.
What’s In
Give the quotient of the following integers. Write your answers on your answer sheet.
- –36 ÷ +9 = _____
- –15 ÷ –3 = _____
- +72 ÷ +8 = _____
- +60 ÷ –20 = _____
- –18 ÷ –3 = _____
What’s New
Jessa owes her friend ₱360 for a pair of pants. If her two sisters will help her pay off her debt, how much will each of them share to pay her debt?
What is It
In order to solve the problem, we are going to follow the steps in problem solving.
1) Understand ➢ What is asked in the problem? The amount each of them will share to pay off the debt. ➢ What facts are needed to solve the problem? ₱360- debt or we can also represent it by the integer –360 3 persons- will share to pay off the debt (+3)
2) Plan ➢ What operation will be used to solve the problem? Division ➢ What is the number sentence? –360 ÷ +3 = N
3) Solve –360 ÷ +3 = –120 The amount to be shared can be represented by the integer –120. Knowing that each of the girls will equally share to pay the debt amount of ₱360, which is represented as –360 ÷ +3 = –120, therefore each of them will share ₱120.
4) Check and look back –120 x 3 = –360
The first problem presented is an example of a routine problem. Routine problems are types of problems wherein you can easily identify the solution to solve them. After reading the situation, the solver clearly sees the method/process to be used and the answer to the problem.
Here is another problem that will challenge you more.
Read and analyze the problem. Then study the solution below.
When an integer is multiplied by –5, then the product is divided by +3 and then the quotient is added by +4, the answer is –6. What is the integer?
In order to solve the problem, we have to consider the following steps:
Understand ➢ What is asked in the problem? The unknown integer ( input )
Plan
➢ Strategy to solve the problem: Working Backward Strategy In working backward strategy, we will start with the output or final result to the input or missing given by reversing all operations used.
Solve
This is our original equation.
−[ [ ? ] x ( −5 ) ] ÷ ( +3 ) + ( +4 ) = ( −6 )
In working backward strategy, our equation becomes
? = −[ [ ( −6 ) − ( +4 ) ] x ( +3 ) ] ÷ ( −5 )
Step 1: Subtract positive 4 from negative 6 which is negative 10. ( −6 ) − ( +4 ) = ( −10 )
Step 2: Multiply the difference, negative 10, by positive 3 which is negative 30. ( −10 ) x ( +3 ) = ( −30 )
Step 3: Divide the product, negative 30, by negative 5. ( −30 ) ÷ ( −5 ) = ( +6 )
Therefore, the unknown integer is ( +6 ).
Check: To check, substitute positive 6 which is the output when you used the working backward strategy to the input in the original equation. Then, evaluate.
( +6 ) x ( −5 ) ÷ ( +3 ) + ( +4 ) = ( −6 ) ( −30 ) ÷ ( +3 ) + ( +4 ) = ( −6 ) ( −10 ) + ( +4 ) = ( −6 ) ( −6 ) = ( −6 )
The second problem presented is an example of a non-routine problem. Non-routine problems are unfamiliar problems wherein the solver has no readily available solution to them. The method/process and the answer are not clearly seen the first glance. Non routine problems can be solved using different strategies such as Guess and Check, Using Models, Acting Out, Logical Reasoning, Working Backwards and others.
The second problem presented uses working backward strategy.
What’s More
Solve the following problems. Show your solutions and answers on your answer sheet.
- Mico’s scores during the Mathlympics Da Math Competition are as follows: +5, –8, –3, +13, +20 and –9. What was his average score?
- Blessie harvested 70 kg of avocados. She gave 16kg to her cousin and divided equally the remaining kilograms of avocados to her 9 friends. How many kilograms of avocados did she give to each of her friends?
- Billy’s Restaurant recorded a profit of ₱8 450 on the first week, ₱6 570 on the second week, and ₱5 320 on the third week of March. Unluckily, he lost ₱4 285 on the fourth week of the same month because of the lockdown due to COVID-19 and paid each of his three helpers at ₱2 500 from his remaining profit. How much is Billy’s profit for the month?
What I Have Learned
Routine problems are problems having readily available solutions. The methods/processes can easily be identified. We can solve routine problems using Polya’s steps in problem solving as follows:
- Understand
- Plan
- Solve
- Check and Look Back
Non-routine problems are problems that do not have readily available procedure to solve them. Non routine problems can be solved using different strategies such as Guess and Check, Using Models, Acting Out, Logical Reasoning, Working Backwards and others.
What I Can Do
Solve the following problems. Show your solutions and answers on your answer sheet.
- Ralf has twenty-five ₱5 coins in his purse. If his younger brother took 15 of the coins and divided the remaining coins to his 5 nephews, how much did each of his nephews receive?
- When an integer is divided by +25 and the result is multiplied by –5 more than +3, the result is +10 more than twice seven. What is the integer?
- A 10-kilogram bag of flour is to be repacked into smaller bags weighing 3 kilograms each. If there are 3 bags of flour, how many smaller packs can you make?
Assessment
Solve the following problems. Show your solutions and write your answers on your answer sheet.
- Carlo missed a total of 45 points in the subjects he took test. If he took 9 subjects and got the same number of wrong answers in each subject, how many items did he miss in each subject?
- Jessa withdrew ₱1 000 from her savings account in the first week. During the second week, she deposited ₱6 000 in the same account and withdrew ₱8 000 the third week. She then tripled her money in the fourth week and resulted to ₱4 500. How much money does she have at the start?
- Fe baked 200 cookies in the first day, 140 on the second day and an additional of 150 cookies on the third day. She divided them equally into packs containing 14 cookies per pack and sold each pack for ₱ 30. How much did Fe earn in three days?
- An integer is divided by –6. The result is then multiplied by half of –6 and finally added to +15. This gives the result of +105. What is the integer?
- You are challenged by a friend to get the difference between the average of the following sets of integers: | −3, −9, −13, −27, −12 | and | −2, −9, −10, −4, −5 | What will be your answer?
Additional Activities
Solve the following problems. Show your solutions and write your answers on your answer sheet.
- With the amount he received from the government’s aid during the pandemic, Mike ventured into business and started with a small capital. On the first week, he gained ₱500. On the second and third week, he lost ₱ 400 and ₱250 respectively. On the fourth week, he gained ₱900 and his money doubled after the fifth week and now has ₱6 000. How much money did he use for his capital?
- Ken thinks of two integers that when he divides the larger number by the smaller number will give him the quotient of –4 and when he multiplies will give him the product of –100. What are the two integers?
- In a certain barangay, there are 3 000 food packs to be distributed to the victims of earthquake. If there are 600 households in the barangay, how many food packs will each household receive?
- After reaching the peak of the mountain, a group of mountaineers decided to go down from the top due to thick fog. If they walk at a constant speed of 1 kilometer per 30 minutes and reached the base of the mountain after 4 hours, how many kilometers did they walk going down?
- In a spelling contest, positive points are given for every correct answer and negative points for every wrong answers. Thea’s scores after 4 rounds are as follows: +15, –11, –9, and +25. What is her average score?
Answer Key
| What I Know | What’s More | Assessment |
|---|---|---|
| 1) 30, +30 2) 80 feet, −80 3) −15 4) +15 5. +4 | 1) +3 2) 6kg 3) ₱8 555 | 1) 5 item 2) ₱3 056 3) ₱1 050 4) −300 5) −8 |
| What’s In | What I Can Do | Additional Activities |
|---|---|---|
| 1) −4 2) +5 3) +9 4) −3 5) +6 | 1) ₱10 2) +180 3) 10 smaller packs | 1) ₱2 250 2) −20, +5 or +20, −5 3) 5 food packs 4) 8 kilometers 5) +5 |
Reference
Most Essential Learning Competencies (MELC) in Mathematics 6
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
Leave a Reply