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Chapter 24: Relations and Tables of Values

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.

24.1 Relations

A relation pairs input values with output values. In this section, the focus is Relations.

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 Relations. Show the important steps, include units when needed, and explain how you checked the answer.

24.2 Input

Input is an important Grade 8 concept in Relations and Tables of Values. 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 Input. Show the important steps, include units when needed, and explain how you checked the answer.

24.3 Output

Output is an important Grade 8 concept in Relations and Tables of Values. 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 Output. Show the important steps, include units when needed, and explain how you checked the answer.

24.4 Independent variable

A variable is a letter or symbol that represents a value that may be unknown or may change. In this section, the focus is Independent variable.

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 Independent variable. Show the important steps, include units when needed, and explain how you checked the answer.

24.5 Dependent variable

A variable is a letter or symbol that represents a value that may be unknown or may change. In this section, the focus is Dependent variable.

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 Dependent variable. Show the important steps, include units when needed, and explain how you checked the answer.

24.6 Tables of values

Tables of values is an important Grade 8 concept in Relations and Tables of Values. 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.

24.7 Finding rules from tables

Finding rules from tables is an important Grade 8 concept in Relations and Tables of Values. 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 Finding rules from tables. Show the important steps, include units when needed, and explain how you checked the answer.

24.8 Writing equations from tables

An equation is a mathematical statement that two expressions have the same value. In this section, the focus is Writing equations from tables.

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: Solve 2x + 5 = 17.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 6

Worked Example 2

Problem: Solve 3x - 4 = 11.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 5

Worked Example 3

Problem: Solve 5x + 7 = 2x + 22.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 5

Worked Example 4

Problem: Solve 4(x + 2) = 24.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 4

Worked Example 5

Problem: Solve 7x - 3 = 4x + 12.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 5

Worked Example 6

Problem: Solve 0.5x + 2 = 7.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 10

Worked Example 7

Problem: Solve 2(x-3)+4=10.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 6

Worked Example 8

Problem: Solve 9 - 2x = 1.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 4

Worked Example 9

Problem: Solve 3x + 8 = 3x + 8.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: all real numbers

Worked Example 10

Problem: Solve 4x + 1 = 4x + 9.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: no solution

Practice exercise

Create one new Grade 8 problem involving Writing equations from tables. Show the important steps, include units when needed, and explain how you checked the answer.

24.9 Completing tables

Completing tables is an important Grade 8 concept in Relations and Tables of Values. 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 Completing tables. Show the important steps, include units when needed, and explain how you checked the answer.

24.10 Comparing relations

A relation pairs input values with output values. In this section, the focus is Comparing relations.

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 Comparing relations. Show the important steps, include units when needed, and explain how you checked the answer.

24.11 Equivalent representations

Equivalent representations is an important Grade 8 concept in Relations and Tables of Values. 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: Equivalent representations: Explain Equivalent representations in one simple sentence.

  1. Look at the words in “Equivalent representations”.
  2. State what the idea is used for.
  3. Give one small mathematical example.

Very beginner explanation: A beginner should first understand the meaning before memorizing a rule.

Answer: Equivalent representations is a Grade 8 math idea used to describe or solve a specific mathematical relationship.

Worked Example 2

Problem: Equivalent representations: A student says, “I can use Equivalent representations without checking units or labels.” Is that a good method?

  1. Read the statement.
  2. Ask whether units, signs, labels, or conditions matter.
  3. Decide whether the method is safe.

Very beginner explanation: Many math mistakes come from using a correct calculation on the wrong quantity.

Answer: No. Important units, labels, signs, and conditions must be checked.

Worked Example 3

Problem: Equivalent representations: What is the first step when solving a problem about Equivalent representations?

  1. Read the whole question.
  2. Underline the known information.
  3. Identify what must be found.

Very beginner explanation: This prevents you from calculating before you understand the problem.

Answer: Identify the given information and the unknown before choosing a rule.

Worked Example 4

Problem: Equivalent representations: After solving a Equivalent representations problem, what should you do before accepting the answer?

  1. Estimate the expected size or direction.
  2. Check units and signs.
  3. Use an inverse method if possible.

Very beginner explanation: Checking is part of solving, not an optional extra.

Answer: Check whether the answer is reasonable and consistent with the problem.

Worked Example 5

Problem: Equivalent representations: Give one way Equivalent representations could appear outside a textbook.

  1. Think about money, measurements, data, maps, geometry, or technology.
  2. Connect the topic to one of those situations.

Very beginner explanation: Connecting math to real situations makes the rule easier to remember.

Answer: Equivalent representations can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.

Worked Example 6

Problem: Equivalent representations: Which representation could help explain Equivalent representations: a table, graph, diagram, equation, or number line?

  1. Choose the representation that makes the relationship easiest to see.
  2. Label it clearly.

Very beginner explanation: Different representations show different features of the same mathematics.

Answer: Use the representation that best matches the problem; more than one may be valid.

Worked Example 7

Problem: Equivalent representations: A student gets an answer for Equivalent representations but cannot explain the steps. What should be improved?

  1. Rewrite the solution one step at a time.
  2. Name the rule used at each important step.

Very beginner explanation: A correct final number without reasoning may hide an error.

Answer: The reasoning should be shown so the solution can be checked.

Worked Example 8

Problem: Equivalent representations: Why can estimation help before a detailed Equivalent representations calculation?

  1. Round or use benchmark values.
  2. Predict the approximate answer.
  3. Compare the exact result with the estimate.

Very beginner explanation: If the exact answer is far from the estimate, recheck the work.

Answer: Estimation gives a target range for the final answer.

Worked Example 9

Problem: Equivalent representations: How can you test whether your rule for Equivalent representations works?

  1. Choose a very small easy example.
  2. Apply the rule.
  3. Check the result another way.

Very beginner explanation: Small examples expose mistakes quickly.

Answer: Test the rule on a simple case whose answer can be verified.

Worked Example 10

Problem: Equivalent representations: How would you teach Equivalent representations to someone seeing it for the first time?

  1. Define the idea.
  2. Show one easy example.
  3. Explain every step.
  4. Then let the learner try a similar problem.

Very beginner explanation: This sequence reduces memorization without understanding.

Answer: Teach meaning first, then a small worked example, then guided practice.

Practice exercise

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

24.12 Real-life relationships

A relation pairs input values with output values. In this section, the focus is Real-life relationships.

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 Real-life relationships. Show the important steps, include units when needed, and explain how you checked the answer.

24.13 Word problems

Word problems is an important Grade 8 concept in Relations and Tables of Values. 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: Word problems: Explain Word problems in one simple sentence.

  1. Look at the words in “Word problems”.
  2. State what the idea is used for.
  3. Give one small mathematical example.

Very beginner explanation: A beginner should first understand the meaning before memorizing a rule.

Answer: Word problems is a Grade 8 math idea used to describe or solve a specific mathematical relationship.

Worked Example 2

Problem: Word problems: A student says, “I can use Word problems without checking units or labels.” Is that a good method?

  1. Read the statement.
  2. Ask whether units, signs, labels, or conditions matter.
  3. Decide whether the method is safe.

Very beginner explanation: Many math mistakes come from using a correct calculation on the wrong quantity.

Answer: No. Important units, labels, signs, and conditions must be checked.

Worked Example 3

Problem: Word problems: What is the first step when solving a problem about Word problems?

  1. Read the whole question.
  2. Underline the known information.
  3. Identify what must be found.

Very beginner explanation: This prevents you from calculating before you understand the problem.

Answer: Identify the given information and the unknown before choosing a rule.

Worked Example 4

Problem: Word problems: After solving a Word problems problem, what should you do before accepting the answer?

  1. Estimate the expected size or direction.
  2. Check units and signs.
  3. Use an inverse method if possible.

Very beginner explanation: Checking is part of solving, not an optional extra.

Answer: Check whether the answer is reasonable and consistent with the problem.

Worked Example 5

Problem: Word problems: Give one way Word problems could appear outside a textbook.

  1. Think about money, measurements, data, maps, geometry, or technology.
  2. Connect the topic to one of those situations.

Very beginner explanation: Connecting math to real situations makes the rule easier to remember.

Answer: Word problems can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.

Worked Example 6

Problem: Word problems: Which representation could help explain Word problems: a table, graph, diagram, equation, or number line?

  1. Choose the representation that makes the relationship easiest to see.
  2. Label it clearly.

Very beginner explanation: Different representations show different features of the same mathematics.

Answer: Use the representation that best matches the problem; more than one may be valid.

Worked Example 7

Problem: Word problems: A student gets an answer for Word problems but cannot explain the steps. What should be improved?

  1. Rewrite the solution one step at a time.
  2. Name the rule used at each important step.

Very beginner explanation: A correct final number without reasoning may hide an error.

Answer: The reasoning should be shown so the solution can be checked.

Worked Example 8

Problem: Word problems: Why can estimation help before a detailed Word problems calculation?

  1. Round or use benchmark values.
  2. Predict the approximate answer.
  3. Compare the exact result with the estimate.

Very beginner explanation: If the exact answer is far from the estimate, recheck the work.

Answer: Estimation gives a target range for the final answer.

Worked Example 9

Problem: Word problems: How can you test whether your rule for Word problems works?

  1. Choose a very small easy example.
  2. Apply the rule.
  3. Check the result another way.

Very beginner explanation: Small examples expose mistakes quickly.

Answer: Test the rule on a simple case whose answer can be verified.

Worked Example 10

Problem: Word problems: How would you teach Word problems to someone seeing it for the first time?

  1. Define the idea.
  2. Show one easy example.
  3. Explain every step.
  4. Then let the learner try a similar problem.

Very beginner explanation: This sequence reduces memorization without understanding.

Answer: Teach meaning first, then a small worked example, then guided practice.

Practice exercise

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

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Chapter 24 Review Questions and Answers

Q1. What is important to remember about Relations?

Answer: A relation pairs input values with output values. In this section, the focus is Relations.

Q2. What is important to remember about Input?

Answer: Input is an important Grade 8 concept in Relations and Tables of Values. 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.

Q3. What is important to remember about Output?

Answer: Output is an important Grade 8 concept in Relations and Tables of Values. 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.

Q4. What is important to remember about Independent variable?

Answer: A variable is a letter or symbol that represents a value that may be unknown or may change. In this section, the focus is Independent variable.

Q5. What is important to remember about Dependent variable?

Answer: A variable is a letter or symbol that represents a value that may be unknown or may change. In this section, the focus is Dependent variable.

Q6. What is important to remember about Tables of values?

Answer: Tables of values is an important Grade 8 concept in Relations and Tables of Values. 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 rules from tables?

Answer: Finding rules from tables is an important Grade 8 concept in Relations and Tables of Values. 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.

Q8. What is important to remember about Writing equations from tables?

Answer: An equation is a mathematical statement that two expressions have the same value. In this section, the focus is Writing equations from tables.

Q9. What is important to remember about Completing tables?

Answer: Completing tables is an important Grade 8 concept in Relations and Tables of Values. 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 Comparing relations?

Answer: A relation pairs input values with output values. In this section, the focus is Comparing relations.

Q11. What is important to remember about Equivalent representations?

Answer: Equivalent representations is an important Grade 8 concept in Relations and Tables of Values. 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 Real-life relationships?

Answer: A relation pairs input values with output values. In this section, the focus is Real-life relationships.

Q13. What is important to remember about Word problems?

Answer: Word problems is an important Grade 8 concept in Relations and Tables of Values. 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.

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.