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Chapter 23: Patterns Involving 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
Estimated reading time0% read
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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.

23.1 Repeating patterns

A pattern follows a rule that can be used to predict missing or future terms. In this section, the focus is Repeating patterns.

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: Continue the pattern 2, 5, 8, ... for two more terms.

  1. Find the common difference: 3.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 11, 14

Worked Example 2

Problem: Continue the pattern 5, 9, 13, ... for two more terms.

  1. Find the common difference: 4.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 17, 21

Worked Example 3

Problem: Continue the pattern 10, 8, 6, ... for two more terms.

  1. Find the common difference: -2.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 4, 2

Worked Example 4

Problem: Continue the pattern 1.5, 2, 2.5, ... for two more terms.

  1. Find the common difference: 0.5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 3, 3.5

Worked Example 5

Problem: Continue the pattern 20, 17.5, 15, ... for two more terms.

  1. Find the common difference: -2.5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 12.5, 10

Worked Example 6

Problem: Continue the pattern 7, 13, 19, ... for two more terms.

  1. Find the common difference: 6.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 25, 31

Worked Example 7

Problem: Continue the pattern -3, 2, 7, ... for two more terms.

  1. Find the common difference: 5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 12, 17

Worked Example 8

Problem: Continue the pattern 0.5, 1.75, 3, ... for two more terms.

  1. Find the common difference: 1.25.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 4.25, 5.5

Worked Example 9

Problem: Continue the pattern 12, 9, 6, ... for two more terms.

  1. Find the common difference: -3.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 3, 0

Worked Example 10

Problem: Continue the pattern 100, 110, 120, ... for two more terms.

  1. Find the common difference: 10.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 130, 140

Practice exercise

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

23.2 Growing integer patterns

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

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).

  1. If signs match, add absolute values and keep the sign.
  2. 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).

  1. If signs match, add absolute values and keep the sign.
  2. 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).

  1. If signs match, add absolute values and keep the sign.
  2. 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).

  1. If signs match, add absolute values and keep the sign.
  2. 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).

  1. If signs match, add absolute values and keep the sign.
  2. 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).

  1. If signs match, add absolute values and keep the sign.
  2. 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).

  1. If signs match, add absolute values and keep the sign.
  2. 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).

  1. If signs match, add absolute values and keep the sign.
  2. 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).

  1. If signs match, add absolute values and keep the sign.
  2. 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).

  1. If signs match, add absolute values and keep the sign.
  2. 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 Growing integer patterns. Show the important steps, include units when needed, and explain how you checked the answer.

23.3 Shrinking integer patterns

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

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).

  1. If signs match, add absolute values and keep the sign.
  2. 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).

  1. If signs match, add absolute values and keep the sign.
  2. 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).

  1. If signs match, add absolute values and keep the sign.
  2. 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).

  1. If signs match, add absolute values and keep the sign.
  2. 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).

  1. If signs match, add absolute values and keep the sign.
  2. 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).

  1. If signs match, add absolute values and keep the sign.
  2. 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).

  1. If signs match, add absolute values and keep the sign.
  2. 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).

  1. If signs match, add absolute values and keep the sign.
  2. 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).

  1. If signs match, add absolute values and keep the sign.
  2. 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).

  1. If signs match, add absolute values and keep the sign.
  2. 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 Shrinking integer patterns. Show the important steps, include units when needed, and explain how you checked the answer.

23.4 Arithmetic sequences

Arithmetic sequences is an important Grade 8 concept in Patterns Involving 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: Continue the pattern 2, 5, 8, ... for two more terms.

  1. Find the common difference: 3.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 11, 14

Worked Example 2

Problem: Continue the pattern 5, 9, 13, ... for two more terms.

  1. Find the common difference: 4.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 17, 21

Worked Example 3

Problem: Continue the pattern 10, 8, 6, ... for two more terms.

  1. Find the common difference: -2.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 4, 2

Worked Example 4

Problem: Continue the pattern 1.5, 2, 2.5, ... for two more terms.

  1. Find the common difference: 0.5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 3, 3.5

Worked Example 5

Problem: Continue the pattern 20, 17.5, 15, ... for two more terms.

  1. Find the common difference: -2.5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 12.5, 10

Worked Example 6

Problem: Continue the pattern 7, 13, 19, ... for two more terms.

  1. Find the common difference: 6.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 25, 31

Worked Example 7

Problem: Continue the pattern -3, 2, 7, ... for two more terms.

  1. Find the common difference: 5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 12, 17

Worked Example 8

Problem: Continue the pattern 0.5, 1.75, 3, ... for two more terms.

  1. Find the common difference: 1.25.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 4.25, 5.5

Worked Example 9

Problem: Continue the pattern 12, 9, 6, ... for two more terms.

  1. Find the common difference: -3.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 3, 0

Worked Example 10

Problem: Continue the pattern 100, 110, 120, ... for two more terms.

  1. Find the common difference: 10.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 130, 140

Practice exercise

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

23.5 Positive common difference

Positive common difference is an important Grade 8 concept in Patterns Involving 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: Continue the pattern 2, 5, 8, ... for two more terms.

  1. Find the common difference: 3.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 11, 14

Worked Example 2

Problem: Continue the pattern 5, 9, 13, ... for two more terms.

  1. Find the common difference: 4.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 17, 21

Worked Example 3

Problem: Continue the pattern 10, 8, 6, ... for two more terms.

  1. Find the common difference: -2.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 4, 2

Worked Example 4

Problem: Continue the pattern 1.5, 2, 2.5, ... for two more terms.

  1. Find the common difference: 0.5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 3, 3.5

Worked Example 5

Problem: Continue the pattern 20, 17.5, 15, ... for two more terms.

  1. Find the common difference: -2.5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 12.5, 10

Worked Example 6

Problem: Continue the pattern 7, 13, 19, ... for two more terms.

  1. Find the common difference: 6.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 25, 31

Worked Example 7

Problem: Continue the pattern -3, 2, 7, ... for two more terms.

  1. Find the common difference: 5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 12, 17

Worked Example 8

Problem: Continue the pattern 0.5, 1.75, 3, ... for two more terms.

  1. Find the common difference: 1.25.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 4.25, 5.5

Worked Example 9

Problem: Continue the pattern 12, 9, 6, ... for two more terms.

  1. Find the common difference: -3.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 3, 0

Worked Example 10

Problem: Continue the pattern 100, 110, 120, ... for two more terms.

  1. Find the common difference: 10.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 130, 140

Practice exercise

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

23.6 Negative common difference

Negative common difference is an important Grade 8 concept in Patterns Involving 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: Continue the pattern 2, 5, 8, ... for two more terms.

  1. Find the common difference: 3.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 11, 14

Worked Example 2

Problem: Continue the pattern 5, 9, 13, ... for two more terms.

  1. Find the common difference: 4.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 17, 21

Worked Example 3

Problem: Continue the pattern 10, 8, 6, ... for two more terms.

  1. Find the common difference: -2.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 4, 2

Worked Example 4

Problem: Continue the pattern 1.5, 2, 2.5, ... for two more terms.

  1. Find the common difference: 0.5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 3, 3.5

Worked Example 5

Problem: Continue the pattern 20, 17.5, 15, ... for two more terms.

  1. Find the common difference: -2.5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 12.5, 10

Worked Example 6

Problem: Continue the pattern 7, 13, 19, ... for two more terms.

  1. Find the common difference: 6.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 25, 31

Worked Example 7

Problem: Continue the pattern -3, 2, 7, ... for two more terms.

  1. Find the common difference: 5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 12, 17

Worked Example 8

Problem: Continue the pattern 0.5, 1.75, 3, ... for two more terms.

  1. Find the common difference: 1.25.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 4.25, 5.5

Worked Example 9

Problem: Continue the pattern 12, 9, 6, ... for two more terms.

  1. Find the common difference: -3.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 3, 0

Worked Example 10

Problem: Continue the pattern 100, 110, 120, ... for two more terms.

  1. Find the common difference: 10.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 130, 140

Practice exercise

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

23.7 Finding missing terms

A term is one part of an algebraic expression separated by addition or subtraction signs. In this section, the focus is Finding missing terms.

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: Simplify 3x + 2x.

  1. 3x and 2x are like terms.
  2. Add the coefficients: 3 + 2 = 5.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 5x

Worked Example 2

Problem: Simplify 7y - 4y + 3.

  1. 7y and -4y are like terms.
  2. Combine them; keep the constant 3.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 3y + 3

Worked Example 3

Problem: Simplify 4(a + 3).

  1. Multiply 4 by a.
  2. Multiply 4 by 3.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 4a + 12

Worked Example 4

Problem: Simplify 2(3x - 5) + x.

  1. Distribute 2.
  2. 2(3x - 5)=6x-10.
  3. Combine 6x+x.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 7x - 10

Worked Example 5

Problem: Simplify 5m + 8 - 2m - 3.

  1. Combine variable terms.
  2. Combine constant terms.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 3m + 5

Worked Example 6

Problem: Simplify -3(2p + 4).

  1. Multiply -3 by both terms.
  2. Keep signs carefully.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: -6p - 12

Worked Example 7

Problem: Simplify 6x + 4 + x - 9.

  1. Combine x-terms.
  2. Combine constants.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 7x - 5

Worked Example 8

Problem: Simplify 0.5x + 1.5x.

  1. Both terms have x.
  2. Add decimal coefficients 0.5 + 1.5.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 2x

Worked Example 9

Problem: Simplify 3(2a + 1) - a.

  1. Distribute 3.
  2. Combine 6a-a.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 5a + 3

Worked Example 10

Problem: Simplify 8q - 2(q + 3).

  1. Distribute -2.
  2. Combine 8q-2q.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 6q - 6

Practice exercise

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

23.8 Pattern rules

A pattern follows a rule that can be used to predict missing or future terms. In this section, the focus is Pattern 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: Continue the pattern 2, 5, 8, ... for two more terms.

  1. Find the common difference: 3.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 11, 14

Worked Example 2

Problem: Continue the pattern 5, 9, 13, ... for two more terms.

  1. Find the common difference: 4.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 17, 21

Worked Example 3

Problem: Continue the pattern 10, 8, 6, ... for two more terms.

  1. Find the common difference: -2.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 4, 2

Worked Example 4

Problem: Continue the pattern 1.5, 2, 2.5, ... for two more terms.

  1. Find the common difference: 0.5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 3, 3.5

Worked Example 5

Problem: Continue the pattern 20, 17.5, 15, ... for two more terms.

  1. Find the common difference: -2.5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 12.5, 10

Worked Example 6

Problem: Continue the pattern 7, 13, 19, ... for two more terms.

  1. Find the common difference: 6.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 25, 31

Worked Example 7

Problem: Continue the pattern -3, 2, 7, ... for two more terms.

  1. Find the common difference: 5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 12, 17

Worked Example 8

Problem: Continue the pattern 0.5, 1.75, 3, ... for two more terms.

  1. Find the common difference: 1.25.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 4.25, 5.5

Worked Example 9

Problem: Continue the pattern 12, 9, 6, ... for two more terms.

  1. Find the common difference: -3.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 3, 0

Worked Example 10

Problem: Continue the pattern 100, 110, 120, ... for two more terms.

  1. Find the common difference: 10.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 130, 140

Practice exercise

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

23.9 Recursive rules

Recursive rules is an important Grade 8 concept in Patterns Involving 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: Continue the pattern 2, 5, 8, ... for two more terms.

  1. Find the common difference: 3.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 11, 14

Worked Example 2

Problem: Continue the pattern 5, 9, 13, ... for two more terms.

  1. Find the common difference: 4.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 17, 21

Worked Example 3

Problem: Continue the pattern 10, 8, 6, ... for two more terms.

  1. Find the common difference: -2.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 4, 2

Worked Example 4

Problem: Continue the pattern 1.5, 2, 2.5, ... for two more terms.

  1. Find the common difference: 0.5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 3, 3.5

Worked Example 5

Problem: Continue the pattern 20, 17.5, 15, ... for two more terms.

  1. Find the common difference: -2.5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 12.5, 10

Worked Example 6

Problem: Continue the pattern 7, 13, 19, ... for two more terms.

  1. Find the common difference: 6.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 25, 31

Worked Example 7

Problem: Continue the pattern -3, 2, 7, ... for two more terms.

  1. Find the common difference: 5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 12, 17

Worked Example 8

Problem: Continue the pattern 0.5, 1.75, 3, ... for two more terms.

  1. Find the common difference: 1.25.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 4.25, 5.5

Worked Example 9

Problem: Continue the pattern 12, 9, 6, ... for two more terms.

  1. Find the common difference: -3.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 3, 0

Worked Example 10

Problem: Continue the pattern 100, 110, 120, ... for two more terms.

  1. Find the common difference: 10.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 130, 140

Practice exercise

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

23.10 Explicit rules introduction

Explicit rules introduction is an important Grade 8 concept in Patterns Involving 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: Continue the pattern 2, 5, 8, ... for two more terms.

  1. Find the common difference: 3.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 11, 14

Worked Example 2

Problem: Continue the pattern 5, 9, 13, ... for two more terms.

  1. Find the common difference: 4.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 17, 21

Worked Example 3

Problem: Continue the pattern 10, 8, 6, ... for two more terms.

  1. Find the common difference: -2.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 4, 2

Worked Example 4

Problem: Continue the pattern 1.5, 2, 2.5, ... for two more terms.

  1. Find the common difference: 0.5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 3, 3.5

Worked Example 5

Problem: Continue the pattern 20, 17.5, 15, ... for two more terms.

  1. Find the common difference: -2.5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 12.5, 10

Worked Example 6

Problem: Continue the pattern 7, 13, 19, ... for two more terms.

  1. Find the common difference: 6.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 25, 31

Worked Example 7

Problem: Continue the pattern -3, 2, 7, ... for two more terms.

  1. Find the common difference: 5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 12, 17

Worked Example 8

Problem: Continue the pattern 0.5, 1.75, 3, ... for two more terms.

  1. Find the common difference: 1.25.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 4.25, 5.5

Worked Example 9

Problem: Continue the pattern 12, 9, 6, ... for two more terms.

  1. Find the common difference: -3.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 3, 0

Worked Example 10

Problem: Continue the pattern 100, 110, 120, ... for two more terms.

  1. Find the common difference: 10.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 130, 140

Practice exercise

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

23.11 Tables of values

Tables of values is an important Grade 8 concept in Patterns Involving 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: Continue the pattern 2, 5, 8, ... for two more terms.

  1. Find the common difference: 3.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 11, 14

Worked Example 2

Problem: Continue the pattern 5, 9, 13, ... for two more terms.

  1. Find the common difference: 4.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 17, 21

Worked Example 3

Problem: Continue the pattern 10, 8, 6, ... for two more terms.

  1. Find the common difference: -2.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 4, 2

Worked Example 4

Problem: Continue the pattern 1.5, 2, 2.5, ... for two more terms.

  1. Find the common difference: 0.5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 3, 3.5

Worked Example 5

Problem: Continue the pattern 20, 17.5, 15, ... for two more terms.

  1. Find the common difference: -2.5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 12.5, 10

Worked Example 6

Problem: Continue the pattern 7, 13, 19, ... for two more terms.

  1. Find the common difference: 6.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 25, 31

Worked Example 7

Problem: Continue the pattern -3, 2, 7, ... for two more terms.

  1. Find the common difference: 5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 12, 17

Worked Example 8

Problem: Continue the pattern 0.5, 1.75, 3, ... for two more terms.

  1. Find the common difference: 1.25.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 4.25, 5.5

Worked Example 9

Problem: Continue the pattern 12, 9, 6, ... for two more terms.

  1. Find the common difference: -3.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 3, 0

Worked Example 10

Problem: Continue the pattern 100, 110, 120, ... for two more terms.

  1. Find the common difference: 10.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 130, 140

Practice exercise

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

23.12 Visual patterns

A pattern follows a rule that can be used to predict missing or future terms. In this section, the focus is Visual patterns.

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: Continue the pattern 2, 5, 8, ... for two more terms.

  1. Find the common difference: 3.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 11, 14

Worked Example 2

Problem: Continue the pattern 5, 9, 13, ... for two more terms.

  1. Find the common difference: 4.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 17, 21

Worked Example 3

Problem: Continue the pattern 10, 8, 6, ... for two more terms.

  1. Find the common difference: -2.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 4, 2

Worked Example 4

Problem: Continue the pattern 1.5, 2, 2.5, ... for two more terms.

  1. Find the common difference: 0.5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 3, 3.5

Worked Example 5

Problem: Continue the pattern 20, 17.5, 15, ... for two more terms.

  1. Find the common difference: -2.5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 12.5, 10

Worked Example 6

Problem: Continue the pattern 7, 13, 19, ... for two more terms.

  1. Find the common difference: 6.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 25, 31

Worked Example 7

Problem: Continue the pattern -3, 2, 7, ... for two more terms.

  1. Find the common difference: 5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 12, 17

Worked Example 8

Problem: Continue the pattern 0.5, 1.75, 3, ... for two more terms.

  1. Find the common difference: 1.25.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 4.25, 5.5

Worked Example 9

Problem: Continue the pattern 12, 9, 6, ... for two more terms.

  1. Find the common difference: -3.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 3, 0

Worked Example 10

Problem: Continue the pattern 100, 110, 120, ... for two more terms.

  1. Find the common difference: 10.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 130, 140

Practice exercise

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

23.13 Real-life integer patterns

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

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).

  1. If signs match, add absolute values and keep the sign.
  2. 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).

  1. If signs match, add absolute values and keep the sign.
  2. 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).

  1. If signs match, add absolute values and keep the sign.
  2. 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).

  1. If signs match, add absolute values and keep the sign.
  2. 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).

  1. If signs match, add absolute values and keep the sign.
  2. 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).

  1. If signs match, add absolute values and keep the sign.
  2. 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).

  1. If signs match, add absolute values and keep the sign.
  2. 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).

  1. If signs match, add absolute values and keep the sign.
  2. 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).

  1. If signs match, add absolute values and keep the sign.
  2. 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).

  1. If signs match, add absolute values and keep the sign.
  2. 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 Real-life integer patterns. Show the important steps, include units when needed, and explain how you checked the answer.

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Chapter 23 Review Questions and Answers

Q1. What is important to remember about Repeating patterns?

Answer: A pattern follows a rule that can be used to predict missing or future terms. In this section, the focus is Repeating patterns.

Q2. What is important to remember about Growing integer patterns?

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 Growing integer patterns.

Q3. What is important to remember about Shrinking integer patterns?

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 Shrinking integer patterns.

Q4. What is important to remember about Arithmetic sequences?

Answer: Arithmetic sequences is an important Grade 8 concept in Patterns Involving 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.

Q5. What is important to remember about Positive common difference?

Answer: Positive common difference is an important Grade 8 concept in Patterns Involving 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 Negative common difference?

Answer: Negative common difference is an important Grade 8 concept in Patterns Involving 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.

Q7. What is important to remember about Finding missing terms?

Answer: A term is one part of an algebraic expression separated by addition or subtraction signs. In this section, the focus is Finding missing terms.

Q8. What is important to remember about Pattern rules?

Answer: A pattern follows a rule that can be used to predict missing or future terms. In this section, the focus is Pattern rules.

Q9. What is important to remember about Recursive rules?

Answer: Recursive rules is an important Grade 8 concept in Patterns Involving 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.

Q10. What is important to remember about Explicit rules introduction?

Answer: Explicit rules introduction is an important Grade 8 concept in Patterns Involving 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 Tables of values?

Answer: Tables of values is an important Grade 8 concept in Patterns Involving 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 Visual patterns?

Answer: A pattern follows a rule that can be used to predict missing or future terms. In this section, the focus is Visual patterns.

Q13. What is important to remember about Real-life integer patterns?

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 Real-life integer patterns.

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.