Chapter 2: Operations with Integers

Learn Grade 8 Math from beginner foundations through advanced Grade 8 problem solving with detailed explanations, at least five examples per topic, practice exercises, and review questions.

Grade 8Beginner Friendly5+ Examples Per TopicPractice30 Q&A

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What this chapter covers

This chapter contains 13 topics. Technical terms are followed by plain-language meanings where they first appear. Each topic includes at least five worked examples. Coding is included only where it naturally supports the Grade 8 coding expectations.

2.1 Adding positive integers

Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Adding positive integers.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Calculate -7 + (5).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -2

Worked Example 2

Problem: Calculate 4 + (-9).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -5

Worked Example 3

Problem: Calculate -6 + (-3).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -9

Worked Example 4

Problem: Calculate 12 + (-8).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 4

Worked Example 5

Problem: Calculate -15 + (20).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 5

Worked Example 6

Problem: Calculate -11 + (-4).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -15

Worked Example 7

Problem: Calculate 9 + (13).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 22

Worked Example 8

Problem: Calculate -2 + (7).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 5

Worked Example 9

Problem: Calculate 18 + (-25).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -7

Worked Example 10

Problem: Calculate -30 + (12).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -18

Practice exercise

Create one new Grade 8 problem involving Adding positive integers. Show the important steps, include units when needed, and explain how you checked the answer.

2.2 Adding negative integers

Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Adding negative integers.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Calculate -7 + (5).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -2

Worked Example 2

Problem: Calculate 4 + (-9).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -5

Worked Example 3

Problem: Calculate -6 + (-3).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -9

Worked Example 4

Problem: Calculate 12 + (-8).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 4

Worked Example 5

Problem: Calculate -15 + (20).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 5

Worked Example 6

Problem: Calculate -11 + (-4).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -15

Worked Example 7

Problem: Calculate 9 + (13).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 22

Worked Example 8

Problem: Calculate -2 + (7).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 5

Worked Example 9

Problem: Calculate 18 + (-25).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -7

Worked Example 10

Problem: Calculate -30 + (12).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -18

Practice exercise

Create one new Grade 8 problem involving Adding negative integers. Show the important steps, include units when needed, and explain how you checked the answer.

2.3 Adding integers with different signs

Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Adding integers with different signs.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Calculate -7 + (5).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -2

Worked Example 2

Problem: Calculate 4 + (-9).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -5

Worked Example 3

Problem: Calculate -6 + (-3).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -9

Worked Example 4

Problem: Calculate 12 + (-8).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 4

Worked Example 5

Problem: Calculate -15 + (20).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 5

Worked Example 6

Problem: Calculate -11 + (-4).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -15

Worked Example 7

Problem: Calculate 9 + (13).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 22

Worked Example 8

Problem: Calculate -2 + (7).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 5

Worked Example 9

Problem: Calculate 18 + (-25).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -7

Worked Example 10

Problem: Calculate -30 + (12).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -18

Practice exercise

Create one new Grade 8 problem involving Adding integers with different signs. Show the important steps, include units when needed, and explain how you checked the answer.

2.4 Subtracting integers

Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Subtracting integers.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Calculate -7 - (5).

Change subtraction to adding the opposite.
The opposite of 5 is -5.
Now calculate -7 + (-5).

Very beginner explanation: Subtracting a number is the same as adding its opposite.

Answer: -12

Worked Example 2

Problem: Calculate 4 - (-9).

Change subtraction to adding the opposite.
The opposite of -9 is 9.
Now calculate 4 + (9).

Very beginner explanation: Subtracting a number is the same as adding its opposite.

Answer: 13

Worked Example 3

Problem: Calculate -6 - (-3).

Change subtraction to adding the opposite.
The opposite of -3 is 3.
Now calculate -6 + (3).

Very beginner explanation: Subtracting a number is the same as adding its opposite.

Answer: -3

Worked Example 4

Problem: Calculate 12 - (-8).

Change subtraction to adding the opposite.
The opposite of -8 is 8.
Now calculate 12 + (8).

Very beginner explanation: Subtracting a number is the same as adding its opposite.

Answer: 20

Worked Example 5

Problem: Calculate -15 - (20).

Change subtraction to adding the opposite.
The opposite of 20 is -20.
Now calculate -15 + (-20).

Very beginner explanation: Subtracting a number is the same as adding its opposite.

Answer: -35

Worked Example 6

Problem: Calculate -11 - (-4).

Change subtraction to adding the opposite.
The opposite of -4 is 4.
Now calculate -11 + (4).

Very beginner explanation: Subtracting a number is the same as adding its opposite.

Answer: -7

Worked Example 7

Problem: Calculate 9 - (13).

Change subtraction to adding the opposite.
The opposite of 13 is -13.
Now calculate 9 + (-13).

Very beginner explanation: Subtracting a number is the same as adding its opposite.

Answer: -4

Worked Example 8

Problem: Calculate -2 - (7).

Change subtraction to adding the opposite.
The opposite of 7 is -7.
Now calculate -2 + (-7).

Very beginner explanation: Subtracting a number is the same as adding its opposite.

Answer: -9

Worked Example 9

Problem: Calculate 18 - (-25).

Change subtraction to adding the opposite.
The opposite of -25 is 25.
Now calculate 18 + (25).

Very beginner explanation: Subtracting a number is the same as adding its opposite.

Answer: 43

Worked Example 10

Problem: Calculate -30 - (12).

Change subtraction to adding the opposite.
The opposite of 12 is -12.
Now calculate -30 + (-12).

Very beginner explanation: Subtracting a number is the same as adding its opposite.

Answer: -42

Practice exercise

Create one new Grade 8 problem involving Subtracting integers. Show the important steps, include units when needed, and explain how you checked the answer.

2.5 Adding the opposite

Adding the opposite is an important Grade 8 concept in Operations with Integers. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Calculate -7 - (5).

Change subtraction to adding the opposite.
The opposite of 5 is -5.
Now calculate -7 + (-5).

Very beginner explanation: Subtracting a number is the same as adding its opposite.

Answer: -12

Worked Example 2

Problem: Calculate 4 - (-9).

Change subtraction to adding the opposite.
The opposite of -9 is 9.
Now calculate 4 + (9).

Very beginner explanation: Subtracting a number is the same as adding its opposite.

Answer: 13

Worked Example 3

Problem: Calculate -6 - (-3).

Change subtraction to adding the opposite.
The opposite of -3 is 3.
Now calculate -6 + (3).

Very beginner explanation: Subtracting a number is the same as adding its opposite.

Answer: -3

Worked Example 4

Problem: Calculate 12 - (-8).

Change subtraction to adding the opposite.
The opposite of -8 is 8.
Now calculate 12 + (8).

Very beginner explanation: Subtracting a number is the same as adding its opposite.

Answer: 20

Worked Example 5

Problem: Calculate -15 - (20).

Change subtraction to adding the opposite.
The opposite of 20 is -20.
Now calculate -15 + (-20).

Very beginner explanation: Subtracting a number is the same as adding its opposite.

Answer: -35

Worked Example 6

Problem: Calculate -11 - (-4).

Change subtraction to adding the opposite.
The opposite of -4 is 4.
Now calculate -11 + (4).

Very beginner explanation: Subtracting a number is the same as adding its opposite.

Answer: -7

Worked Example 7

Problem: Calculate 9 - (13).

Change subtraction to adding the opposite.
The opposite of 13 is -13.
Now calculate 9 + (-13).

Very beginner explanation: Subtracting a number is the same as adding its opposite.

Answer: -4

Worked Example 8

Problem: Calculate -2 - (7).

Change subtraction to adding the opposite.
The opposite of 7 is -7.
Now calculate -2 + (-7).

Very beginner explanation: Subtracting a number is the same as adding its opposite.

Answer: -9

Worked Example 9

Problem: Calculate 18 - (-25).

Change subtraction to adding the opposite.
The opposite of -25 is 25.
Now calculate 18 + (25).

Very beginner explanation: Subtracting a number is the same as adding its opposite.

Answer: 43

Worked Example 10

Problem: Calculate -30 - (12).

Change subtraction to adding the opposite.
The opposite of 12 is -12.
Now calculate -30 + (-12).

Very beginner explanation: Subtracting a number is the same as adding its opposite.

Answer: -42

Practice exercise

Create one new Grade 8 problem involving Adding the opposite. Show the important steps, include units when needed, and explain how you checked the answer.

2.6 Multiplying integers

Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Multiplying integers.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Calculate (-7) × (5).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: -35

Worked Example 2

Problem: Calculate (4) × (-9).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: -36

Worked Example 3

Problem: Calculate (-6) × (-3).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: 18

Worked Example 4

Problem: Calculate (12) × (-8).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: -96

Worked Example 5

Problem: Calculate (-15) × (20).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: -300

Worked Example 6

Problem: Calculate (-11) × (-4).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: 44

Worked Example 7

Problem: Calculate (9) × (13).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: 117

Worked Example 8

Problem: Calculate (-2) × (7).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: -14

Worked Example 9

Problem: Calculate (18) × (-25).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: -450

Worked Example 10

Problem: Calculate (-30) × (12).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: -360

Practice exercise

Create one new Grade 8 problem involving Multiplying integers. Show the important steps, include units when needed, and explain how you checked the answer.

2.7 Dividing integers

Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Dividing integers.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Calculate (-7) × (5).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: -35

Worked Example 2

Problem: Calculate (4) × (-9).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: -36

Worked Example 3

Problem: Calculate (-6) × (-3).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: 18

Worked Example 4

Problem: Calculate (12) × (-8).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: -96

Worked Example 5

Problem: Calculate (-15) × (20).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: -300

Worked Example 6

Problem: Calculate (-11) × (-4).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: 44

Worked Example 7

Problem: Calculate (9) × (13).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: 117

Worked Example 8

Problem: Calculate (-2) × (7).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: -14

Worked Example 9

Problem: Calculate (18) × (-25).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: -450

Worked Example 10

Problem: Calculate (-30) × (12).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: -360

Practice exercise

Create one new Grade 8 problem involving Dividing integers. Show the important steps, include units when needed, and explain how you checked the answer.

2.8 Integer sign rules

Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Integer sign rules.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Calculate (-7) × (5).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: -35

Worked Example 2

Problem: Calculate (4) × (-9).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: -36

Worked Example 3

Problem: Calculate (-6) × (-3).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: 18

Worked Example 4

Problem: Calculate (12) × (-8).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: -96

Worked Example 5

Problem: Calculate (-15) × (20).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: -300

Worked Example 6

Problem: Calculate (-11) × (-4).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: 44

Worked Example 7

Problem: Calculate (9) × (13).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: 117

Worked Example 8

Problem: Calculate (-2) × (7).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: -14

Worked Example 9

Problem: Calculate (18) × (-25).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: -450

Worked Example 10

Problem: Calculate (-30) × (12).

Multiply the absolute values.
Same signs give a positive answer; different signs give a negative answer.

Very beginner explanation: The sign rule tells the sign; ordinary multiplication gives the size of the answer.

Answer: -360

Practice exercise

Create one new Grade 8 problem involving Integer sign rules. Show the important steps, include units when needed, and explain how you checked the answer.

2.9 Multi-step integer calculations

Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Multi-step integer calculations.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Calculate -7 + (5).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -2

Worked Example 2

Problem: Calculate 4 + (-9).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -5

Worked Example 3

Problem: Calculate -6 + (-3).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -9

Worked Example 4

Problem: Calculate 12 + (-8).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 4

Worked Example 5

Problem: Calculate -15 + (20).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 5

Worked Example 6

Problem: Calculate -11 + (-4).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -15

Worked Example 7

Problem: Calculate 9 + (13).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 22

Worked Example 8

Problem: Calculate -2 + (7).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 5

Worked Example 9

Problem: Calculate 18 + (-25).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -7

Worked Example 10

Problem: Calculate -30 + (12).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -18

Practice exercise

Create one new Grade 8 problem involving Multi-step integer calculations. Show the important steps, include units when needed, and explain how you checked the answer.

2.10 Temperature-change problems

Temperature-change problems is an important Grade 8 concept in Operations with Integers. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Calculate -7 + (5).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -2

Worked Example 2

Problem: Calculate 4 + (-9).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -5

Worked Example 3

Problem: Calculate -6 + (-3).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -9

Worked Example 4

Problem: Calculate 12 + (-8).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 4

Worked Example 5

Problem: Calculate -15 + (20).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 5

Worked Example 6

Problem: Calculate -11 + (-4).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -15

Worked Example 7

Problem: Calculate 9 + (13).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 22

Worked Example 8

Problem: Calculate -2 + (7).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 5

Worked Example 9

Problem: Calculate 18 + (-25).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -7

Worked Example 10

Problem: Calculate -30 + (12).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -18

Practice exercise

Create one new Grade 8 problem involving Temperature-change problems. Show the important steps, include units when needed, and explain how you checked the answer.

2.11 Elevation problems

Elevation problems is an important Grade 8 concept in Operations with Integers. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Calculate -7 + (5).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -2

Worked Example 2

Problem: Calculate 4 + (-9).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -5

Worked Example 3

Problem: Calculate -6 + (-3).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -9

Worked Example 4

Problem: Calculate 12 + (-8).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 4

Worked Example 5

Problem: Calculate -15 + (20).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 5

Worked Example 6

Problem: Calculate -11 + (-4).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -15

Worked Example 7

Problem: Calculate 9 + (13).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 22

Worked Example 8

Problem: Calculate -2 + (7).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 5

Worked Example 9

Problem: Calculate 18 + (-25).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -7

Worked Example 10

Problem: Calculate -30 + (12).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -18

Practice exercise

Create one new Grade 8 problem involving Elevation problems. Show the important steps, include units when needed, and explain how you checked the answer.

2.12 Financial integer problems

Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Financial integer problems.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Calculate -7 + (5).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -2

Worked Example 2

Problem: Calculate 4 + (-9).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -5

Worked Example 3

Problem: Calculate -6 + (-3).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -9

Worked Example 4

Problem: Calculate 12 + (-8).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 4

Worked Example 5

Problem: Calculate -15 + (20).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 5

Worked Example 6

Problem: Calculate -11 + (-4).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -15

Worked Example 7

Problem: Calculate 9 + (13).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 22

Worked Example 8

Problem: Calculate -2 + (7).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 5

Worked Example 9

Problem: Calculate 18 + (-25).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -7

Worked Example 10

Problem: Calculate -30 + (12).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -18

Practice exercise

Create one new Grade 8 problem involving Financial integer problems. Show the important steps, include units when needed, and explain how you checked the answer.

2.13 Checking integer answers

Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Checking integer answers.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Calculate -7 + (5).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -2

Worked Example 2

Problem: Calculate 4 + (-9).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -5

Worked Example 3

Problem: Calculate -6 + (-3).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -9

Worked Example 4

Problem: Calculate 12 + (-8).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 4

Worked Example 5

Problem: Calculate -15 + (20).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 5

Worked Example 6

Problem: Calculate -11 + (-4).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -15

Worked Example 7

Problem: Calculate 9 + (13).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 22

Worked Example 8

Problem: Calculate -2 + (7).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 5

Worked Example 9

Problem: Calculate 18 + (-25).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -7

Worked Example 10

Problem: Calculate -30 + (12).

If signs match, add absolute values and keep the sign.
If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -18

Practice exercise

Create one new Grade 8 problem involving Checking integer answers. Show the important steps, include units when needed, and explain how you checked the answer.

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Chapter 2 Review Questions and Answers

Q1. What is important to remember about Adding positive integers?

Answer: Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Adding positive integers.

Q2. What is important to remember about Adding negative integers?

Q3. What is important to remember about Adding integers with different signs?

Q4. What is important to remember about Subtracting integers?

Answer: Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Subtracting integers.

Q5. What is important to remember about Adding the opposite?

Answer: Adding the opposite is an important Grade 8 concept in Operations with Integers. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q6. What is important to remember about Multiplying integers?

Answer: Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Multiplying integers.

Q7. What is important to remember about Dividing integers?

Answer: Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Dividing integers.

Q8. What is important to remember about Integer sign rules?

Answer: Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Integer sign rules.

Q9. What is important to remember about Multi-step integer calculations?

Q10. What is important to remember about Temperature-change problems?

Answer: Temperature-change problems is an important Grade 8 concept in Operations with Integers. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q11. What is important to remember about Elevation problems?

Answer: Elevation problems is an important Grade 8 concept in Operations with Integers. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q12. What is important to remember about Financial integer problems?

Q13. What is important to remember about Checking integer answers?

Q14. Why should you show your steps?

Answer: Showing steps makes reasoning easier to verify and helps locate mistakes.

Q15. Why is estimation useful?

Answer: Estimation helps you judge whether a calculated answer is reasonable.

Q16. Why do units matter?

Answer: Units tell what a value measures and prevent incompatible quantities from being mixed.

Q17. When should you round?

Answer: Usually round near the end unless the question specifically asks for earlier rounding.

Q18. How can you check an answer?

Answer: Use an inverse operation, substitution, estimation, a graph, or a second method.

Q19. What should you identify first in a word problem?

Answer: Identify what is known, what is unknown, and the mathematical relationship connecting them.

Q20. What makes a final answer complete?

Answer: Give the value, correct units when needed, and a brief interpretation in context.

Q21. What should you do if an answer seems unreasonable?

Answer: Re-read the question and check copied values, signs, units, formulas, and calculations.

Q22. Why are diagrams useful?

Answer: A labelled diagram can reveal relationships among lengths, angles, areas, and coordinates.

Q23. Why are tables useful?

Answer: Tables organize values and help reveal patterns, rates, relationships, and missing information.

Q24. Why should you explain your reasoning?

Answer: Reasoning shows why a method works, not just what answer was obtained.

Q25. How do mistakes help learning?

Answer: Analyzing a mistake identifies the misunderstood rule or step and helps prevent repetition.

Q26. When is a calculator useful?

Answer: A calculator helps with lengthy arithmetic after the mathematical setup is understood and estimated.

Q27. Why compare more than one strategy?

Answer: Different strategies may make a problem easier and provide a way to verify the result.

Q28. What is mathematical communication?

Answer: It is presenting ideas clearly with words, symbols, diagrams, tables, graphs, and justified steps.

Q29. What is mathematical modelling?

Answer: It is representing a real situation with mathematics, testing the model, and revising it if needed.

Q30. Why should assumptions be stated?

Answer: Assumptions show the conditions the model depends on and help readers judge whether it is reasonable.