This is the Grade 6 Mathematics Quarter 2 Module 11, Adding Integers, a self-learning module from the Department of Education Region VI under the Alternative Delivery Mode. It is for Grade 6 learners studying at home or in class, and for the teachers and parents who guide them. After working through it, learners can add integers with like and unlike signs using counters and number lines, and use addition of integers to solve word problems.
On this page 8 sections
What learners will learn
The module states that, after going through it, learners are expected to:
- describe and interpret addition of integers using materials such as algebra tiles, counters, chips, and cards (M6NS-IIh-155);
- perform addition of integers (M6NS-IIi-156); and
- solve routine and non-routine problems involving addition of integers using appropriate strategies and tools (M6NS-IIj-157).
Lessons in this module
- Lesson 1 – Adding Integers with Like Signs: adding two positive integers or two negative integers with counters and number lines.
- Lesson 2 – Adding Integers with Unlike Signs: adding a positive and a negative integer by pairing counters or moving both ways on a number line.
- Lesson 3 – Solving Routine and Non-Routine Problems Involving Addition of Integers Using Appropriate Strategies and Tools: using the Understand, Plan, Solve, Check steps and strategies such as the block model.
Each lesson has its own pretest (What I Know), review (What’s In), worked examples (What is It), practice, summary (What I Have Learned), Assessment, Additional Activities, and Answer Key.
Key ideas: adding integers with like signs
Addition means to put together or to combine two or more quantities. The module shows two tools:
- Counters: combine positive counters with positive counters, or negative counters with negative counters, then count them. For +5 + +4, combining 5 positive counters and 4 positive counters gives 9 positive counters, so the sum is +9.
- Number line: start at 0 and move right for positive numbers, left for negative numbers. For -5 + -4, move 5 units left, then 4 more units left, landing on -9.
| Example | Same sign? | Sum |
|---|---|---|
| +5 + +4 | Yes | +9 |
| +3 + +7 | Yes | +10 |
| -5 + -4 | Yes | -9 |
| -3 + -7 | Yes | -10 |
Rule: the sum of two positive integers is always positive, and the sum of two negative integers is always negative. Add them like whole numbers, and the sum takes the sign of both addends.
Key ideas: adding integers with unlike signs
When the signs are different, the counters cannot simply be combined. Pair one positive counter with one negative counter; each pair has a value of zero and is no longer counted. The counters left without a pair give the sum.
For +6 + -3, there are 3 pairs, and 3 positive counters are left over, so the sum is +3. On a number line, start at 0, move 6 units right, then 3 units left, landing on +3.
Rule: to add integers with unlike signs:
- disregard the signs of the two addends;
- get the difference; and
- give the sum the sign of the bigger number.
The module’s practice example: +44 + -28 = +16.
Key ideas: solving word problems
For routine problems, the module uses four steps: Understand (what is asked, what facts are given), Plan (choose the operation and a strategy), Solve, and Check.
- Elevator problem: Edna gets on at the ninth floor, goes down two floors, then three more. Written as +9 + -2 + -3 = +4, so she gets off on the 4th floor. Check: +9 + -2 = +7, then +7 + -3 = +4.
- Consecutive integers problem (block model): the first seven of fourteen consecutive positive integers add up to 161. Draw blocks, remove the twenty-one 1s (161 – 21 = 140), and divide by the 7 unknown units (140 ÷ 7 = 20). The first seven integers are 20 to 26, so the last seven are 27 to 33, and their sum is 210.
For non-routine problems, the module lists these strategies: draw a diagram, picture, or model; find a pattern; make a table or an organized list; work backward; and simplify the problem. Practice problems include temperatures, bank deposits and withdrawals, weight gained and lost, and a frog climbing out of a pit.
How to use this module
- Learners: start each lesson with What I Know to check what you already know, then go through the parts in order. Draw your counters and number lines in your notebook, and write your answers on a separate answer sheet, not on the module. Use the Answer Key only after you finish each activity, and be honest when checking.
- Parents: let your child use real counters, such as bottle caps or paper chips marked plus and minus, to model the problems. If your child gets stuck, help them read the instructions again or contact the teacher.
- Teachers: each lesson’s Assessment and Additional Activities can serve as checks for the three competencies. For word problems, ask learners to show their Understand, Plan, Solve, and Check steps and label their final answers, as the module asks.
Frequently asked questions
What grade and quarter is this Adding Integers module for?
It is the Grade 6 Mathematics module for Quarter 2, numbered Module 11.
What lessons are in the module?
It has three lessons: adding integers with like signs, adding integers with unlike signs, and solving routine and non-routine problems involving addition of integers.
Does the module have an answer key?
Yes. Each lesson ends with its own answer key covering the pretest, activities, and assessment.
How do you add integers with unlike signs?
Disregard the signs, find the difference between the two numbers, and give the answer the sign of the bigger number. For example, +6 + -3 = +3.
What competencies does the module cover?
It covers M6NS-IIh-155 (describing addition of integers with materials such as counters), M6NS-IIi-156 (performing addition of integers), and M6NS-IIj-157 (solving problems involving addition of integers).
Full text of the module
The text below is a transcription of the PDF above, as printed.
Mathematics Quarter 2 – Module 11: Adding Integers
6
[illustration: Filipino students and a teacher reading, writing, and studying outdoors and indoors, with a globe on a table]
CO_Q2_Mathematics6_Module11
Mathematics – Grade 6 Alternative Delivery Mode Quarter 2 – Module 11: Adding Integers First Edition, 2020
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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: Undersecretary: Assistant Secretary:
Development Team of the Module
Authors: Charlyn Joy Alingalan Estimera and Collin G. Sales
Editors: Rebecca O. Billones, Ma. Portia Gencianeo Galanto and Collin Gange Sales
Reviewers: Elleda Ebreo de la Cruz, Rosemarie Dicen Aclan, and Collin Gange Sales
Illustrator: Marina Berden Damasco
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
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_________________________
Mathematics Quarter 2 – Module 11: Adding Integers
What I Need to Know
[illustration: bullseye target with an arrow]
This module was designed and written with you in mind. It is here to help you master the skills in adding integers. The scope of this module permits it to be used in many different learning situations. The language used recognizes your diverse vocabulary level. 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 three lessons, namely:
- Lesson 1 – Adding Integers with Like Signs
- Lesson 2 – Adding Integers with Unlike Signs
- Lesson 3 – Solving Routine and Non-Routine Problems Involving Addition of Integers Using Appropriate Strategies and Tools
After going through this module, you are expected to:
- describe and interpret addition of integers using materials such as algebra tiles, counters, chips, and cards (M6NS-IIh-155);
- perform addition of integers (M6NS-IIi-156); and,
- solve routine and non-routine problems involving addition of integers using appropriate strategies and tools (M6NS-IIj-157).
What I Know
[illustration: pencil icon beside the part title]
A. Choose the letters of the correct answers and write them on your answer sheet. (For item 1, refer to Illustration I.)
[illustration: Illustration I, two groups of positive counters shown as plus signs, three in the first group and five in the second]
Illustration I
What is the appropriate number sentence for the Illustration I above?
a. +3 x -5 = 15
b. +3 ÷ -5 = -2
c. +3 + +5 = +8
d. +3 – -5 = +7On the table, you placed 6 negative counters and 3 negative counters. What is the value of the counters on the table?
a. -9
b. -3
c. -1
d. 0Which of the following statement is true?
a. The sum of two negative integers is a positive integer.
b. The sum of positive integer and negative integer is zero.
c. The sum of two different integers can never be a negative integer.
d. The sum of a negative integer and a positive integer is always a positive integer.
(For items 4 and 5 refer to illustration II.)
[chart: number line; Illustration II]
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10
transcriber’s note: arrows drawn above the number line show two jumps that start at 0 and move to the left.
Using the illustration above, what addition sentence can you make?
a. -6 + -9 = N
b. +6 + +9 = N
c. -6 + -3 = N
d. -6 + +3 = NWhat is the sum for the addition sentence in illustration II?
a. -15
b. -9
c. -3
d. +3
B. Below are expressions involving integers. Find the sum of each item by drawing your positive and negative counters or number lines in your notebook. Then, write your answers on your answer sheet.
- +14 + +12 =
- -33 + -15 =
- +24 + +11 =
- -13 + -12 =
- +21 + +11 =
Lesson 1: Adding Integers with Like Signs
This module will help you realize how operations on integers work. It aims to build basic concepts on how to add integers using positive and negative counters and number lines. Learning about these concepts are equally exciting and challenging, right? So, ready your working space and counters to make this lesson more meaningful.
What’s In
[illustration: puzzle-piece icon beside the part title]
Write the addition sentence for each model. Then find the sum. Write your answers on your answer sheet.
[chart: bar model; item 1]
Bracket labeled ? above three boxes: 26, 54, 37
[chart: number line; item 2]
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
transcriber’s note: arrows drawn above the number line show jumps that start at 0 and land at 4, 9, and 16.
[chart: bar model; item 3]
Bracket labeled ? above two boxes: 65, 72
[illustration: cylinders grouped as one addend, cubes grouped as a second addend, followed by an equals sign]
[illustration: crescent shapes grouped as one addend, four-point star shapes grouped as a second addend, followed by an equals sign]
What’s New
Read and analyze the problem.
The temperature on top of a mountain is 5˚C at 10 p.m. During the next hours, the temperature fell further by 4˚C. What is the temperature at 2 a.m.?
What is It
Before working on the given problem above, let us explore how to add integers with the same sign.
Is the sum of two positive integers positive or negative?
How about the sum of two negative integers? What do you think?
Adding Positive Integers
Example 1: +5 + +4 = _____
- Using counters
[chart: counter diagram; Example 1]
Combine 5 positive counters with 4 positive counters. What is the total?
+5 + +4 = +9
How many counters are there in all? (9)
Do they have the same sign? (Yes)
What is the sign of the counters? (positive)
- Using a number line
[chart: number line; Example 1]
Start at 0. Move 5 units to the right. Then move 4 units to the right.
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10
+5 + +4 = +9
Example 2: +3 + +7 = _____
- Using counters
[chart: counter diagram; Example 2]
Add 7 positive counters to 3 positive counters. What is the total?
+3 + +7 = +10
- Using a number line
[chart: number line; Example 2]
Start at 0. Move 3 units to the right. Then move 7 units to the right.
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
+3 + +7 = +10
How about if you have to combine negative integers? What is their sum?
Adding Negative Integers
Example 3: -5 + -4 = _____
- Using counters
[chart: counter diagram; Example 3]
Combine 5 negative counters with 4 negative counters. What is the total?
-5 + -4 = -9
How many counters are there in all? (9)
Do they have the same sign? (Yes)
What is the sign of the counters? (negative)
- Using a number line
[chart: number line; Example 3]
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10
-5 + -4 = -9
• Using a number line
[chart: number line; Example 3: -5 + -4 = -9] Callouts: “Start at 0. Move 5 units to the left.” and “Then move 4 units to the left.” Arrow labels: -5 and -4 Number line labels: -10, -9 (circled), -8, -7, -6, -5, -4, -3, -2, -1, 0, +1, +2, +3, +4, +5, +6, +7, +8, +9, +10 -5 + -4 = -9
Example 4: -3 + -7 = _____
• Using counters
[chart: counters; Example 4: -3 + -7] Callout: “Add 7 negative counters to 3 negative counters” Callout: “What is the total?” Values: -3 | + | -7 | = | -10
• Using a number line
[chart: number line; Example 4: -3 + -7 = -10] Callouts: “Then move 7 units to the left.” and “Start at 0. Move 3 units to the left.” Arrow labels: -7 and -3 Number line labels: -10 (circled), -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, +1, +2, +3, +4, +5, +6, +7, +8, +9, +10 -3 + -7 = -10
Now, let us use the previous examples to complete the following table.
| Example | Same Sign? (Yes/No) | Sum | Sign (+/-) |
|---|---|---|---|
| 1 | +5 + +4 | Yes | +9 |
| 2 | +3 + +7 | Yes | +10 |
| 3 | -5 + -4 | Yes | -9 |
| 4 | -3 + -7 | Yes | -10 |
What do you notice?
When two positive integers are added together, the result is a positive sum. When two negative integers are combined, the result is a negative sum.
What’s More
Here are some items for you to practice on. Draw a number line in your notebook and find each sum. Write your answers with correct sign on your answer sheet.
Example: +35 + +25 =
[chart: number line; Example: +35 + +25 = +60] Number line labels: -20, -15, -10, -5, 0, +5, +10, +15, +20, +25, +30, +35, +40, +45, +50, +55, +60, +65, +70, +75, +80 +35 + +25 = +60
- -8 + -7 =
- +12 + +9 =
- -2 + -6 =
- +24 + +16 =
- +65 + +10 =
What I Have Learned
Addition means to put together or to combine two or more quantities.
- The sum of two positive integers is always positive.
- The sum of two negative integers is always negative.
When adding two integers with the same sign, add like whole numbers and the sum follows the sign of both addends.
What I Can Do
Find each sum. Write your answers on your answer sheet.
- -21 + -5 =
- +47 + +15 =
- -31 + -16 =
- -12 + -85 =
- +15 + +48 =
- +27 + +42 =
- -15 + -32 =
- +56 + +29 =
- -97 + -28 =
- +46 + +72 =
Assessment
A. Choose the letter of the correct answer. Write the answers on your answer sheet.
- What is the value of the counters in the diagram below?
[illustration: two groups of positive counters in one row, separated by a gap] [transcriber’s note: the counters are not counted in this transcription]
- A. 0
- B. +5
- C. +17
- D. +20
- Which of the following addition sentences will give you a positive sum?
- A. -7 + -8 = N
- B. -7 + +8 = N
- C. +7 + -8 = N
- D. +7 + +8 = N
- What is the value of -6 + -5?
- A. -11
- B. -1
- C. 0
- D. +11
- Using your number line, what is the value of N in +20 + +5 = N?
- A. 0
- B. -5
- C. +5
- D. +25
- Which of the following statements is FALSE?
- A. Adding two negative integers together sometimes gives a positive sum.
- B. Two positive integers added together always results in a positive sum.
- C. Two negative integers combined is equal to a negative sum.
- D. The sum of two integers with like signs follows the same sign as the addends.
B. Add the following integers. Write your answers on your answer sheet.
- +6 + +9 =
- -19 + -16 =
- +13 + +11 =
- -38 + -14 =
- +16 + +23 =
Additional Activities
A. Make your own addition sentence with the following information, then find the sum. Write your answers on your answer sheet.
- [illustration: two groups of positive counters in one row, separated by a gap]
- [chart: number line; no title printed]
Number line labels: -48, -44, -40, -36, -32, -28, -24, -20, -16, -12, -8, -4, 0, +4, +8, +12, +16, +20, +24, +28, +32
Two arcs are drawn from 0 leftward to -28 and then to -44. - [illustration: left group of negative counters, a blank frame in the middle, and a right group of negative counters, with an arrow to a blank box]
[transcriber’s note: the counters and the blank boxes are not counted in this transcription]
B. Find each sum.
- +12 + +5 =
- -64 + -11 =
- +5 + +14 =
- -16 + -11 =
- -16 + -2 =
C. Read and solve. Write your solutions on your answer sheet and label the final answers.
- Find three consecutive positive integers whose sum is 147.
- A submarine submerged 563 feet below sea level. If it submerged another 65 feet, what was the depth of the submarine now?
Answer Key
[transcriber’s note: this page is printed upside down; the answers below were read after rotating the page 180 degrees, and the columns are in the order they appear on the page]
What I Know
A.
- C
- A
- B
- C
- B
B. 6) 26 + 7) 48 – 8) 35 + 9) 25 – 10) 32 +
What’s In
- 26 + 54 + 37 = 117
- 4 + 5 + 7 = 16
- 65 + 72 = 137
- 19 + 15 = 34
- 21 + 6 = 27
Assessment
A.
- C
- D
- A
- D
- A
B. 6) 15 + 7) 35 – 8) 24 + 9) 52 – 10) 39 +
Additional Activities
A.
- +11 + +10 = +21
- -28 + -16 = -44
- -5 + -11 = -16
B. 4) 17 + 5) 75 – 6) 19 + 7) 27 – 8) 18 –
C. 9) 50 + 49, + 48, + 10) – 628 feet – depth of the submarine
What’s More
- 15 –
- 21 +
- 8 –
- 40 +
- 75 +
What I Can Do
A.
- 26 –
- 62 +
- 47 –
- 97 –
- 63 +
- 69 +
- 47 –
- 85 +
- 125 –
- 118 +
What I Know
A. The following multiple-choice items are for you to answer. Some items use illustration for your reference. Choose the letters of the correct answers and write them on your answer sheet.
[illustration: Illustration I; three positive counters and five negative counters in one row] [illustration: Illustration II; positive counters and negative counters in one row]
- In Illustration I, positive counters are combined with negative counters. What is the appropriate number sentence for this?
- A. +3 x -5 = 15
- B. +3 + -5 = -2
- C. +3 ÷ -5 = -2
- D. +3 – -5 = +7
- What is the answer to the illustration II shown above?
- A. -2
- B. 0
- C. +2
- D. -14
- Which of the following statement is FALSE?
- A. If one positive counter is paired with one negative counter, the result is zero.
- B. In each set of counters, if there are more positive counters, the answer is positive.
- C. In adding integers with unlike sign, the sum is always negative.
- D. If the number of positive and negative counter is equal, then the set is zero.
- In which set of counters will give you a negative sum?
- A. [illustration: top, two positive and one negative counter; bottom, one positive and two negative counters]
- B. [illustration: top, three positive counters; bottom, two positive and one negative counter]
- C. [illustration: top, three positive counters; bottom, one positive and two negative counters]
- D. [illustration: top, two positive and one negative counter; bottom, three negative counters]
- What is the sum if -31 added to +53?
- A. -22
- B. 0
- C. +22
- D. +84
B. Below are expressions involving integers. Find the sum of each item by using your positive and negative counters or number lines in your notebook. Then, write your answers on your answer sheet.
- +16 + -18 =
- -17 + +9 =
- +19 + -10 =
- -82 + +45 =
- +34 + -11 =
Lesson 2 Adding Integers with Unlike Signs
You have done a good job in developing your skills in adding integers with like signs. This time, you will move forward to adding integers with unlike signs.
What’s In
Find each sum. Write your answers on your answer sheet.
- +26 + +15
- -38 + -7
- +19 + +13
- -42 + -25
- +84 + +21
What’s New
Read and analyze the story problem below.
From their house, Joyce went to visit her friend in Barangay Cansilayan. She rode a motorcycle 6 kilometers north. Then she rode a tricycle 3 kilometers going back south to her Tita’s place to get her book. How far is Joyce now from their house?
What is It
Furthermore, what will happen if you combine a positive integer and a negative integer?
What do you think will happen to their sum? Will it be a positive or negative integer?
Let’s try to study the following situation.
Example 1: +6 + -3 = ______
- Using counters
In the illustration below, there are positive and negative counters in the set. Take note that we cannot simply combine them together for not all of them have the same signs. You have to pair first a positive and a negative counter for their value is zero and they will not be counted anymore. Then, add the remaining counters with no pair of opposite sign to get the sum of the combined counters.
[chart: counters; Example 1: +6 + -3 = +3] Callout: “Pair one positive counter with one negative counter.” Callout: “What is the total?” Values: +6 | + | -3 | = | +3
How many pairs of positive and negative counters are there? (3pairs) How many counters do not have pairs of opposite sign? (3) What is the sign of the counters with no pair of opposite sign? (positive)
We can also illustrate the given problem by using a number line.
[chart: number line; Example 1: +6 + -3 = +3] Callouts: “Start at 0. Move 6 units to the right.” and “Then move 3 units to the left.” Arrow labels: +6 and -3 Number line labels: -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, +1, +2, +3, +4, +5, +6, +7, +8, +9, +10 +6 + -3 = +3
What do you notice?
In adding integers with unlike sign, the signs are first disregarded and the difference between the numbers is obtained. Then, the sign of the greater number is affixed to the answer.
What’s More
In your notebook, use a number line or positive/negative counters to find the sum of each given item. Write your answers with correct sign on your answer sheet.
Example: +44 + -28 = +16
[chart: number line; example +44 + -28 = +16, labels as printed: -32 -28 -24 -20 -16 -12 -8 -4 0 +4 +8 +12 +16 (circled) +20 +24 +28 +32 +36 +40 +44 +48]
- -25 + +20 =
- +12 + -6 =
- -36 + +12 =
- -15 + +7 =
- +28 + -4 =
What I Have Learned
Addition means to put together or to combine two or more quantities.
To add integers having unlike signs:
- disregard the signs of the two addends;
- obtain the difference; and,
- affix the sign of the bigger number to the sum.
What I Can Do
A. Find each sum. Write your answers on your answer sheet.
- +15 + -5 =
- -18 + +3 =
- -21 + +13 =
- +12 + -5 =
- +8 + -2 =
- +26 + -18 =
- +52 + -17 =
- -91 + +24 =
- +9 + -45 =
- +14 + -27 =
Assessment
A. The following multiple-choice items are for you to answer. Some items use illustration for your reference. Choose the letters of the correct answers and write them on your answer sheet.
Which of the following addition sentences is TRUE?
a. -8 + -4 = +16
b. +7 + -9 = +2
c. -9 + +3 = -6
d. +8 + +9 = +1Using your counters, what is the value of N in -8 + +3 = N?
a. -11
b. -5
c. +5
d. +11What is the value of -3 + +3?
a. -6
b. -3
c. 0
d. +6What is the value of the counters in the rectangle below?
[illustration: rectangle holding a row of positive counters and a row of negative counters]
a. -1 b. -5 c. +5 d. +11
- Using the number line below, what is the sum if positive 5 is added to negative 3?
[chart: number line with arc from 0 to +2 and a dashed arc; labels as printed: -10 -9 -8 -7 -6 -5 -4 -3 -2 +1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10]
a. -8 b. -2 c. +2 d. +8
transcriber’s note: the number line label after -2 is printed as “+1”, and the number line is printed without a label reading -1; reproduced as printed.
B. Find each sum. Write your answers on your answer sheet.
- +26 + -5 =
- +56 + -18 =
- -33 + +15 =
- -67 + +19 =
- -7 + +29 =
Additional Activities
A. Copy and complete the table on your answer sheet. Give the answer of the empty cells below by finding the sum of the number in the first row and the number in the first column.
| + | -9 | -6 | -3 | 0 | +3 | +6 | +9 |
|---|---|---|---|---|---|---|---|
| +9 | 0 | +3 | +15 | ||||
| +6 | +3 | ||||||
| +3 | +3 | ||||||
| 0 | -9 | +3 | +9 | ||||
| -3 | -9 | +6 | |||||
| -6 | -12 | 0 | |||||
| -9 | -3 | 0 |
transcriber’s note: the dark-shaded cells of the original table are left empty here, and some cell boundaries in the original were hard to distinguish, so the placement of the empty and shaded cells may not be exact.
B. Read, analyze, and solve. Write the solutions on your answer sheet and label your final answers.
- A bird flew 100 meters above the ground and then went 79 meters down. How far from the ground is the bird now?
- Mirasol had ₱800.00. She spent ₱455.00 for her groceries. How much money does she have now?
Answer Key
transcriber’s note: this page is printed upside down, and its three columns were read after rotating the page. The “What I Know” items and the completed table in Additional Activities A could not be read with confidence, so they are marked below.
What I Know
A. 1) B 2) B 3) C 4) D 5) C [?]
B. 6) -2 [?] 7) -8 [?] 8) +9 [?] 9) -37 [?] 10) +23 [?]
What’s In
- +41
- -45
- +32
- -67
- +105
What’s More
- -5
- +6
- -24
- -8
- +24
What I Can Do
A. 1) +10 2) -15 3) -8 4) +7 5) +6 6) +8 7) +35 8) -67 9) -36 10) -13
Assessment
A. 1) C 2) B 3) C 4) A 5) C
B. 1) +21 2) +38 3) -18 4) -48 5) +22
Additional Activities
A. [illegible]
transcriber’s note: the completed table in the key is printed upside down and its cell values could not be read reliably.
B. 1) +21 meters – bird’s distance from the ground 2) ₱345.00 – Grace’s amount of money left
transcriber’s note: the key labels item 2 as “Grace’s,” while the question names Mirasol; reproduced as printed.
What I Know
The temperature in Canada was +5˚C in the morning. The temperature in the afternoon dropped to -8˚C. What was the temperature reading in the afternoon?
A. Choose the letter of the correct answer. Write your answers on your answer sheet.
(Study the given facts below for items 1 to 3.)
What is asked in the given problem above?
a. The temperature reading in the afternoon.
b. The temperature reading in the morning.
c. The total temperature reading in Canada.
d. The temperature reading.What operation is needed to solve the problem?
a. Multiplication
b. Addition
c. Addition and Division
d. Subtraction and MultiplicationWhat will be the temperature reading in the afternoon?
a. -13˚C
b. -3˚C
c. +3˚C
d. +13˚C
(For items 4 to 6.) Use number line to illustrate the given problem.
On her dance rehearsal, Sivera moves 2 steps forward, 5 steps backward, 3 steps forward and another 4 steps forward. Find the total steps that Sivera moves from her original place?
What is asked in the given problem above?
a. The total steps that Sivera moves from her original place.
b. The total steps that she moves forward.
c. The total steps that she moves backward.
d. The steps that she moves altogether.What operation is needed to solve the problem?
a. Addition
b. Division
c. Subtraction
d. MultiplicationWhat is the answer to the problem?
a. +5
b. +4
c. +3
d. 0
The cat sits on the branch of the tree which is 9 feet above the ground and then jumps 4 feet down. How far from the ground is the cat now?
B. For items 7 to 10, read, analyze and solve the given problem below. Write your answers on your answer sheet.
Understand
- What is asked in the problem?
- What is/are the given facts?
Plan
- What operation will be used?
Solve
- Solution
Lesson 3: Solving Routine and Non-Routine Problems Involving Addition of Integers
You already have the concept of integers in the previous lessons. The skills you have learned in adding integers will become more useful if you can apply these in solving word problems.
What’s In
Find the sum of the integers below. Use any of the strategies you have learned from the previous lesson. Write your answers on your answer sheet.
+84 + +21 =
+52 + +13 =
-40 + -23 =
+36 + -18 =
-18 + +9 =
What’s New
Read and analyze the problems below.
Problem No. 1 Edna gets in the elevator on the ninth floor. The elevator goes down two floors and stops. It then continues to go down three more floors where Edna gets off. In what floor did she get off the elevator?
Problem No. 2 Fourteen consecutive positive integers are given. The sum of the first seven positive integers is 161. What is the sum of the last seven positive integers?
What is It
What will you do to solve these problems? The following steps will surely help you. Let’s try this one!
Problem No. 1
Edna gets in the elevator on the ninth floor. The elevator goes down two floors and stops. It then continues to go down three more floors where Edna gets off. In what floor did she get off the elevator?
Understand:
What is asked?
The floor number that Edna got off the elevator.
What facts are given?
- ninth floor from the ground +9 ◄ the floor that she gets in first
- two floors down which can be presented as -2
- three more floors where she gets off means -3
Plan: What operation will be used to solve the problem? Decide for a strategy.
Strategy: The problem can be solved using counters and a number line.
Solve: Show your solution.
Applying the strategy:
To solve for the floor that Edna gets off, refer to the number line below.
[chart: number line; curved arrows above the line labeled -3, -2, and +9; tick labels: -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10; the tick +4 is circled]
Therefore, Edna is in the 4th floor now.
Another way to solve the problem is by using our counters.
[chart: counters diagram; a box of nine positive counters (+9), an empty box, two negative counters and three negative counters, labeled “Pair one positive counter with one negative counter.”; a result box of positive and negative counters labeled “What is the total?”; equation: +9 + -2 + -3 = +4]
Check: To check your answer, verify if the sum of the given integers is correct by doing this.
First +9 + -2 = +7. Then, +7 + -3 = +4.
Thus, our answer is correct.
Problem No. 2
Fourteen consecutive positive integers are given. The sum of the first seven positive integers is 161. What is the sum of the last seven positive integers?
Understand:
What is asked?
The sum of the last seven positive integers.
What facts are given?
14 consecutive positive integers
161 – sum of the first seven positive integers
Plan: What operations will be used to solve the problem? Decide for a strategy.
Strategy: The problem can be solved using the Block Model Strategy.
Solve: Show your solution.
Applying the strategy:
Step 1: Draw blocks to represent the first seven positive integers whose sum is 161. Observe the block models below representing the first seven consecutive positive integers. Knowing that the given positive integers are consecutive, it is understood that the second integer is 1 greater than the first integer. The third integer is 1 greater than the second integer. The fourth integer is also 1 greater than the third integer, so on and so forth until you reach the seventh integer.
[chart: block model; rows A to G, each row an unlabeled blank block followed by one more unit block of 1 than the row above (row A none, row B one 1, row G six 1s); a bracket on the right labeled 161]
Step 2: Notice that we have seven equal unknown units without value. The rest have values of 1. To get the value of each unknown unit, we have to take away all the 1’s and subtract it from 161 which is the sum of the first seven integers.
[chart: block model; same rows A to G as in Step 1, with the 1s enclosed in a dashed box of 21 units; bracket labeled 161 – 21 = 140; arrow below the dashed box labeled 21]
Step 3: Based on the illustration, there are twenty-one 1’s. So, if you subtract 21 from 161, the result is 140. Then, divide 140 by 7 equal unknown units. Therefore, each unknown unit is equal to 20.
[chart: block model; rows A to G with a dashed box around the seven unknown units; labels: 161 – 21 = 140; 140 ÷ 7 = 20; bracket labeled 161; arrow labeled “7 unknown units”]
Step 4: To get the value of every integer, it shows that the first integer is equal to 20. The second integer is 20 plus 1. The third integer is 20 plus 1, plus 1. The fourth integer is 20 plus 1, plus 1, plus 1, and, so on and so forth until the seventh integer.
[chart: block model with values; bracket labeled 161 on the right]
| Row | Blocks | Value |
|---|---|---|
| A | 20 | = 20 |
| B | 20 1 | = 21 |
| C | 20 1 1 | = 22 |
| D | 20 1 1 1 | = 23 |
| E | 20 1 1 1 1 | = 24 |
| F | 20 1 1 1 1 1 | = 25 |
| G | 20 1 1 1 1 1 1 | = 26 |
Now, you have the first seven integers which are 20, 21, 22, 23, 24, 25, and 26. You already have the first seven integers. However, you are looking for the sum of the last seven integers. Knowing that they are consecutive integers, so you could say that the next seven integers are 27, 28, 29, 30, 31, 32, and 33.
Step 5: To get the sum of the last seven integers, add 27, 28, 29, 30, 31, 32, and 33.
27 + 28 + 29 + 30 + 31 + 32 + 33 = 210
Therefore, the sum of the last seven integers is 210
Check: To check your answer, verify if the sum of the first seven positive integers is equal to 161.
20 + 21 + 22 + 23 + 24 + 25 + 26 = 161
Then, what are the succeeding seven positive integers from the set of integers above?
27, 28, 29, 30, 31, 32, 33
Therefore, we are looking for the sum of the 27, 28, 29, 30, 31, 32, 33 which is 210.
What’s More
Read, analyze and solve the problem. Write your solutions on your answer sheet and label the final answers.
Dinah has an initial deposit of ₱2 535.00 in her bank account. In April, she deposited ₱5 639.00 and withdrawn ₱7 585.00 in May. Then in June, she deposited again ₱1 590.00. Is the initial deposit bigger that her ending balance? By how much? Express the difference as integer.
Understand What is asked in the problem?
The difference of the initial deposit and her ending balance.
What facts are given?
₱2 535.00 – initial deposit ₱5 639.00 deposited in April
₱7 585.00 – withdrawn in May ₱1 590.00 deposited in June
Plan:
What operations will be used?
What is the number sentence?
Solve:
Solution:
Check:
What I Have Learned
- In solving routine problems involving addition of integers using appropriate strategies and tools we have the following steps:
- Understand
- Plan
- Solve
- Check
- In solving non-routine problems, we apply different strategies:
- Draw a diagram, Picture or Model
- Find a Pattern
- Make a Table or an Organized List
- Work Backward
- Simplify the Problem
What I Can Do
Read, analyze and solve. Write the solutions on your answer sheet and label your final answers.
Noel lost 2 kilograms when he was admitted in the hospital. After three months of recovery, he gained 6 kilograms. He enrolled to a fitness program and lost 3 kilograms. Did he finally loss or gain weight? By how much?
Last Monday, Nenebeth went to the department store to buy her laptop. She was on the eight floor when she decided to go to the tenth floor to pay her bills in the customer service center. She then went down eight floors, and went up three more floors. Where is she now?
At 8 a.m., the temperature in Baguio City was 12˚C. By 3 p.m., it had risen by 4˚C. What was the new temperature reading in Baguio City?
I owed ₱5 000.00 in my credit card. I paid ₱4 500.00 for my balance and then spent ₱3 250.00. How much is my standing balance?
Mrs. Lorena bought apples which cost ₱560.00 from a wholesaler and sold them in her fruit stand. On the first day, her sales was ₱420.00; and on the second day, ₱450.00. But on the third day, she lost ₱150.00 because some of the apples were already rotten. Considering the sales for three days, did Mrs. Lorena gain or loss? By how much?
A frog fell in a 2-meter pit. Every day it struggles to climb 50 centimeters in the morning but it slides down by 20 centimeters at night. After how many days will the frog get out of the pit?
The sum of three negative integers is -132. The largest negative is three times than the smallest negative integer. One of the integers is -36. Find the value of the largest negative integer.
A motorcycle is located 20 kilometers north of San Miguel. If it travelled 15 kilometers north then 25 kilometers south, how far from San Miguel was the motorcycle at the end of its travel.
[transcriber’s note: the three problems above have no printed numbers; they appear after item 5 of “What I Can Do” and are repeated in the Assessment]
Assessment
A. Choose the letter of the correct answer. Write your answers on your answer sheet.
(Study the given facts for items 1 to 3.)
A frog fell in a 2-meter pit. Every day it struggles to climb 50 centimeters in the morning but it slides down by 20 centimeters at night. After how many days will the frog get out of the pit?
What is asked in the problem?
- A. How many days can the frog get out of the pit?
- B. What is the total distance that the frog climb in the morning?
- C. What is the total distance that the frog slides down at night?
- D. How many meter is the pit?
What operations will be used to solve the problem?
- A. Addition and Multiplication
- B. Addition and Subtraction
- C. Addition and Division
- D. Subtraction and Multiplication
After how many days will the frog get out of the pit?
- A. 6 days
- B. 5 days
- C. 4 days
- D. 3 days
(For items 4 and 5.)
The sum of three negative integers is -132. The largest negative is three times than the smallest negative integer. One of the integers is -36. Find the value of the largest negative integer.
What is asked in the given problem above?
- A. The sum of three negative integers.
- B. The value of the smallest negative integer.
- C. The value of the largest negative integer.
- D. The value of the second integer.
What is the value of the largest negative integer?
- A. -36
- B. -44
- C. -72
- D. -132
B. For items 6 to 10, read, analyze and solve the given problem below. Write your answers on your answer sheet.
A motorcycle is located 20 kilometers north of San Miguel. If it travelled 15 kilometers north then 25 kilometers south, how far from San Miguel was the motorcycle at the end of its travel.
Understand
- What is asked in the problem?
- What is/are the given facts?
Plan
- What operation to be used?
- What is the number sentence?
Solve
- Solution
Additional Activities
Read, analyze, and solve. Write the solutions on your answer sheet and label your final answers.
A, B, and C are three positive integers. B and C is equal to 47. The difference of A and B is 4 where B is greater than A. The difference of B and C is 3 where C is greater than B. What is the sum of A and B?
A monkey sits on the twig of the tree that is 10 feet above the ground. He swings up 3 feet then jumps down 5 feet. How far off the ground is the monkey now?
Jeffrey drew 17 negative counters in a piece of paper. LJ drew more counters and they both noticed that the value of their drawing becomes zero. What counters did LJ draw?
Vanissa engaged in an online business to help support her family. On the first month, she gains ₱15 560.00, and ₱13 720.00 for the second month. But on the third and fourth month she losses ₱8 720.00 and ₱7 360.00 respectively. Did she gain or loss in her business? By how much?
Jose saved ₱3 665.00 from his daily allowance. He spent ₱ 230.00 to buy a pair of shorts and ₱250.00 to buy a shirt. How much of his money was left?
Answer Key
[transcriber’s note: this answer key is printed as a table, upside down on the page. Its three columns are read here as What I Know, What’s More, and Assessment. Several computation fragments are partly illegible and are marked [?].]
| What I Know | What’s More | Assessment |
|---|---|---|
| What I Know A. 1) A 2) B 3) B 4) A 5) A 6) B B. 7) The distance of the cat from the ground. 8) 9 feet above the ground and 4 feet down 9) Addition 10) +9 + -4 = +5 feet from the ground What’s In 1) 105 + 2) 65 + 3) 63 – 4) 18 + 5) 9- | What’s More 1) addition and subtraction 2) N=[(₱2 535.00 + ₱1 590.00) – ₱7 585.00] + ₱639.00 [?] 3) N=[(₱2 535.00 + ₱1 590.00) – ₱7 585.00] + ₱639.00 [?] Ending – Initial deposit balance=N 4) N=₱2 535.00 – ₱2 179.00 [?] N=₱356.00 [?] What I Can Do 1) Noel losses 1 kilogram. 2) fifth floor 3) 16˚C 4) ₱3 750.00 – ₱160.00 [?] 5) Gains, ₱160.00 | Assessment A. 1) A 2) B 3) A 4) C 5) C B. 6) The distance travelled by the motorcycle from San Miguel to the end of its travel. 7) 20 kilometers north of San Miguel, 15 kilometers, 25 kilometers 8) Addition 9) N = +20 + +15 + -25 [?] 10) N = +35 + -25 [?] N = +10 Additional Activities 1) +40 2) 8 feet above the ground 3) +17 4) She gains ₱13 200.00 5) ₱3 185.00 money left |
References
- DepEd Most Essential Learning Competencies (MELCs) in Mathematics for Grade 6
- Perez, M., Placer, D, Burgos, J., 2016, 21st Century Mathletes 6, Book Media Press, Inc., 152-157.
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