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Chapter 60: Comprehensive Grade 8 Review and Mathematical Processes

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 24 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.

60.1 Scientific-notation review

Scientific-notation review is an important Grade 8 concept in Comprehensive Grade 8 Review and Mathematical Processes. 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: Scientific-notation review: Explain Scientific-notation review in one simple sentence.

  1. Look at the words in “Scientific-notation review”.
  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: Scientific-notation review is a Grade 8 math idea used to describe or solve a specific mathematical relationship.

Worked Example 2

Problem: Scientific-notation review: A student says, “I can use Scientific-notation review 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: Scientific-notation review: What is the first step when solving a problem about Scientific-notation review?

  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: Scientific-notation review: After solving a Scientific-notation review 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: Scientific-notation review: Give one way Scientific-notation review 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: Scientific-notation review can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.

Worked Example 6

Problem: Scientific-notation review: Which representation could help explain Scientific-notation review: 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: Scientific-notation review: A student gets an answer for Scientific-notation review 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: Scientific-notation review: Why can estimation help before a detailed Scientific-notation review 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: Scientific-notation review: How can you test whether your rule for Scientific-notation review 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: Scientific-notation review: How would you teach Scientific-notation review 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 Scientific-notation review. Show the important steps, include units when needed, and explain how you checked the answer.

60.2 Rational-number review

A rational number (a number that can be written as a fraction of integers with a nonzero denominator) includes integers, fractions, terminating decimals, and repeating decimals. In this section, the focus is Rational-number review.

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 the ratio 4:6.

  1. Find the GCF: 2.
  2. Divide both terms by 2.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both parts by the same common factor.

Answer: 2:3

Worked Example 2

Problem: Simplify the ratio 8:12.

  1. Find the GCF: 4.
  2. Divide both terms by 4.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both parts by the same common factor.

Answer: 2:3

Worked Example 3

Problem: Simplify the ratio 15:25.

  1. Find the GCF: 5.
  2. Divide both terms by 5.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both parts by the same common factor.

Answer: 3:5

Worked Example 4

Problem: Simplify the ratio 21:28.

  1. Find the GCF: 7.
  2. Divide both terms by 7.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both parts by the same common factor.

Answer: 3:4

Worked Example 5

Problem: Simplify the ratio 18:30.

  1. Find the GCF: 6.
  2. Divide both terms by 6.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both parts by the same common factor.

Answer: 3:5

Worked Example 6

Problem: Simplify the ratio 12:20.

  1. Find the GCF: 4.
  2. Divide both terms by 4.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both parts by the same common factor.

Answer: 3:5

Worked Example 7

Problem: Simplify the ratio 9:15.

  1. Find the GCF: 3.
  2. Divide both terms by 3.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both parts by the same common factor.

Answer: 3:5

Worked Example 8

Problem: Simplify the ratio 16:24.

  1. Find the GCF: 8.
  2. Divide both terms by 8.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both parts by the same common factor.

Answer: 2:3

Worked Example 9

Problem: Simplify the ratio 25:35.

  1. Find the GCF: 5.
  2. Divide both terms by 5.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both parts by the same common factor.

Answer: 5:7

Worked Example 10

Problem: Simplify the ratio 14:49.

  1. Find the GCF: 7.
  2. Divide both terms by 7.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both parts by the same common factor.

Answer: 2:7

Practice exercise

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

60.3 Fraction decimal percent review

A fraction (a numerator over a denominator) can represent part of a whole, a quotient, a ratio, or a point on a number line. In this section, the focus is Fraction decimal percent review.

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: Write 1/2 as a decimal.

  1. Divide numerator 1 by denominator 2.
  2. 1 ÷ 2 = 0.5000.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.5

Worked Example 2

Problem: Write 2/3 as a decimal.

  1. Divide numerator 2 by denominator 3.
  2. 2 ÷ 3 = 0.6667.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.6667

Worked Example 3

Problem: Write 3/5 as a decimal.

  1. Divide numerator 3 by denominator 5.
  2. 3 ÷ 5 = 0.6000.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.6

Worked Example 4

Problem: Write 5/8 as a decimal.

  1. Divide numerator 5 by denominator 8.
  2. 5 ÷ 8 = 0.6250.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.625

Worked Example 5

Problem: Write 7/10 as a decimal.

  1. Divide numerator 7 by denominator 10.
  2. 7 ÷ 10 = 0.7000.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.7

Worked Example 6

Problem: Write 4/9 as a decimal.

  1. Divide numerator 4 by denominator 9.
  2. 4 ÷ 9 = 0.4444.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.4444

Worked Example 7

Problem: Write 5/6 as a decimal.

  1. Divide numerator 5 by denominator 6.
  2. 5 ÷ 6 = 0.8333.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.8333

Worked Example 8

Problem: Write 3/4 as a decimal.

  1. Divide numerator 3 by denominator 4.
  2. 3 ÷ 4 = 0.7500.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.75

Worked Example 9

Problem: Write 2/5 as a decimal.

  1. Divide numerator 2 by denominator 5.
  2. 2 ÷ 5 = 0.4000.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.4

Worked Example 10

Problem: Write 7/12 as a decimal.

  1. Divide numerator 7 by denominator 12.
  2. 7 ÷ 12 = 0.5833.

Very beginner explanation: A fraction bar means division, so numerator ÷ denominator gives the decimal form.

Answer: 0.5833

Practice exercise

Create one new Grade 8 problem involving Fraction decimal percent review. Show the important steps, include units when needed, and explain how you checked the answer.

60.4 Integer and exponent review

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

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

60.5 Proportion review

A proportion is an equation showing that two ratios are equal. In this section, the focus is Proportion review.

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 4/6 = x/15.

  1. Cross multiply or use equivalent ratios.
  2. 6x = 4 × 15.
  3. x = 60 ÷ 6.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 10

Worked Example 2

Problem: Solve 8/12 = x/20.

  1. Cross multiply or use equivalent ratios.
  2. 12x = 8 × 20.
  3. x = 160 ÷ 12.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 13.3333

Worked Example 3

Problem: Solve 15/25 = x/25.

  1. Cross multiply or use equivalent ratios.
  2. 25x = 15 × 25.
  3. x = 375 ÷ 25.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 15

Worked Example 4

Problem: Solve 21/28 = x/30.

  1. Cross multiply or use equivalent ratios.
  2. 28x = 21 × 30.
  3. x = 630 ÷ 28.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 22.5

Worked Example 5

Problem: Solve 18/30 = x/35.

  1. Cross multiply or use equivalent ratios.
  2. 30x = 18 × 35.
  3. x = 630 ÷ 30.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 21

Worked Example 6

Problem: Solve 12/20 = x/40.

  1. Cross multiply or use equivalent ratios.
  2. 20x = 12 × 40.
  3. x = 480 ÷ 20.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 24

Worked Example 7

Problem: Solve 9/15 = x/45.

  1. Cross multiply or use equivalent ratios.
  2. 15x = 9 × 45.
  3. x = 405 ÷ 15.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 27

Worked Example 8

Problem: Solve 16/24 = x/50.

  1. Cross multiply or use equivalent ratios.
  2. 24x = 16 × 50.
  3. x = 800 ÷ 24.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 33.3333

Worked Example 9

Problem: Solve 25/35 = x/55.

  1. Cross multiply or use equivalent ratios.
  2. 35x = 25 × 55.
  3. x = 1375 ÷ 35.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 39.2857

Worked Example 10

Problem: Solve 14/49 = x/60.

  1. Cross multiply or use equivalent ratios.
  2. 49x = 14 × 60.
  3. x = 840 ÷ 49.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 17.1429

Practice exercise

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

60.6 Algebra and equation review

An equation is a mathematical statement that two expressions have the same value. In this section, the focus is Algebra and equation review.

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

60.7 Formula review

A formula is a rule written with symbols to show how quantities are related. In this section, the focus is Formula review.

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

60.8 Pattern and relation review

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

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

60.9 Coding review

Coding review is an important Grade 8 concept in Comprehensive Grade 8 Review and Mathematical Processes. 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: Coding review: Explain Coding review in one simple sentence.

  1. Look at the words in “Coding review”.
  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: Coding review is a Grade 8 math idea used to describe or solve a specific mathematical relationship.

Worked Example 2

Problem: Coding review: A student says, “I can use Coding review 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: Coding review: What is the first step when solving a problem about Coding review?

  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: Coding review: After solving a Coding review 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: Coding review: Give one way Coding review 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: Coding review can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.

Worked Example 6

Problem: Coding review: Which representation could help explain Coding review: 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: Coding review: A student gets an answer for Coding review 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: Coding review: Why can estimation help before a detailed Coding review 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: Coding review: How can you test whether your rule for Coding review 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: Coding review: How would you teach Coding review 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 Coding review. Show the important steps, include units when needed, and explain how you checked the answer.

60.10 Mathematical modelling review

Mathematical modelling represents a real situation with mathematics, tests the model, and improves it when needed. In this section, the focus is Mathematical modelling review.

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: Predict taxi cost: use the model cost = 4 + 2.5d with input 8.

  1. Identify the input variable.
  2. Substitute 8 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 24

Worked Example 2

Problem: Predict fundraiser revenue: use the model revenue = 120 + 15s with input 10.

  1. Identify the input variable.
  2. Substitute 10 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 270

Worked Example 3

Problem: Predict distance: use the model distance = 60t with input 2.5.

  1. Identify the input variable.
  2. Substitute 2.5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 150

Worked Example 4

Problem: Predict savings: use the model savings = 200 + 50m with input 6.

  1. Identify the input variable.
  2. Substitute 6 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 500

Worked Example 5

Problem: Predict temperature trend: use the model temp = 18 - 1.5h with input 4.

  1. Identify the input variable.
  2. Substitute 4 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 12

Worked Example 6

Problem: Predict plant height: use the model height = 5 + 2w with input 7.

  1. Identify the input variable.
  2. Substitute 7 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 19

Worked Example 7

Problem: Predict rental cost: use the model cost = 30 + 8d with input 5.

  1. Identify the input variable.
  2. Substitute 5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 70

Worked Example 8

Problem: Predict page count: use the model pages = 12r with input 9.

  1. Identify the input variable.
  2. Substitute 9 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 108

Worked Example 9

Problem: Predict points: use the model points = 6g + 3 with input 4.

  1. Identify the input variable.
  2. Substitute 4 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 27

Worked Example 10

Problem: Predict water use: use the model litres = 20 + 4m with input 5.

  1. Identify the input variable.
  2. Substitute 5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 40

Practice exercise

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

60.11 Scatter-plot review

Scatter-plot review is an important Grade 8 concept in Comprehensive Grade 8 Review and Mathematical Processes. 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: Data pairs are (1,4), (2,6), (3,8). Describe the overall trend.

  1. Compare how y changes as x increases.
  2. Decide whether the association is generally positive, negative, or absent.

Very beginner explanation: Scatter plots show relationships between two numerical variables, but correlation alone does not prove causation.

Answer: A trend can be described from the direction and strength of the point pattern.

Worked Example 2

Problem: Data pairs are (1,5), (2,9), (3,10). Describe the overall trend.

  1. Compare how y changes as x increases.
  2. Decide whether the association is generally positive, negative, or absent.

Very beginner explanation: Scatter plots show relationships between two numerical variables, but correlation alone does not prove causation.

Answer: A trend can be described from the direction and strength of the point pattern.

Worked Example 3

Problem: Data pairs are (1,3), (2,7), (3,7). Describe the overall trend.

  1. Compare how y changes as x increases.
  2. Decide whether the association is generally positive, negative, or absent.

Very beginner explanation: Scatter plots show relationships between two numerical variables, but correlation alone does not prove causation.

Answer: A trend can be described from the direction and strength of the point pattern.

Worked Example 4

Problem: Data pairs are (1,20), (2,25), (3,30). Describe the overall trend.

  1. Compare how y changes as x increases.
  2. Decide whether the association is generally positive, negative, or absent.

Very beginner explanation: Scatter plots show relationships between two numerical variables, but correlation alone does not prove causation.

Answer: A trend can be described from the direction and strength of the point pattern.

Worked Example 5

Problem: Data pairs are (1,6), (2,8), (3,9). Describe the overall trend.

  1. Compare how y changes as x increases.
  2. Decide whether the association is generally positive, negative, or absent.

Very beginner explanation: Scatter plots show relationships between two numerical variables, but correlation alone does not prove causation.

Answer: A trend can be described from the direction and strength of the point pattern.

Worked Example 6

Problem: Data pairs are (1,2), (2,4), (3,4). Describe the overall trend.

  1. Compare how y changes as x increases.
  2. Decide whether the association is generally positive, negative, or absent.

Very beginner explanation: Scatter plots show relationships between two numerical variables, but correlation alone does not prove causation.

Answer: A trend can be described from the direction and strength of the point pattern.

Worked Example 7

Problem: Data pairs are (1,10), (2,11), (3,12). Describe the overall trend.

  1. Compare how y changes as x increases.
  2. Decide whether the association is generally positive, negative, or absent.

Very beginner explanation: Scatter plots show relationships between two numerical variables, but correlation alone does not prove causation.

Answer: A trend can be described from the direction and strength of the point pattern.

Worked Example 8

Problem: Data pairs are (1,1), (2,3), (3,5). Describe the overall trend.

  1. Compare how y changes as x increases.
  2. Decide whether the association is generally positive, negative, or absent.

Very beginner explanation: Scatter plots show relationships between two numerical variables, but correlation alone does not prove causation.

Answer: A trend can be described from the direction and strength of the point pattern.

Worked Example 9

Problem: Data pairs are (1,8), (2,8), (3,8). Describe the overall trend.

  1. Compare how y changes as x increases.
  2. Decide whether the association is generally positive, negative, or absent.

Very beginner explanation: Scatter plots show relationships between two numerical variables, but correlation alone does not prove causation.

Answer: A trend can be described from the direction and strength of the point pattern.

Worked Example 10

Problem: Data pairs are (1,15), (2,18), (3,21). Describe the overall trend.

  1. Compare how y changes as x increases.
  2. Decide whether the association is generally positive, negative, or absent.

Very beginner explanation: Scatter plots show relationships between two numerical variables, but correlation alone does not prove causation.

Answer: A trend can be described from the direction and strength of the point pattern.

Practice exercise

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

60.12 Probability review

Probability measures likelihood from 0 to 1, or 0% to 100%. In this section, the focus is Probability review.

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: Find the probability to roll an even number on a fair die.

  1. Favourable outcomes = 3.
  2. Total equally likely outcomes = 6.
  3. P = 3/6.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 2

Problem: Find the probability to flip heads on a fair coin.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 2.
  3. P = 1/2.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 3

Problem: Find the probability to draw red from 5 red and 3 blue marbles.

  1. Favourable outcomes = 5.
  2. Total equally likely outcomes = 8.
  3. P = 5/8.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 5/8

Worked Example 4

Problem: Find the probability to choose a vowel from A,B,C,E.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 4.
  3. P = 2/4.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 5

Problem: Find the probability to spin section 1 on four equal sections.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 4.
  3. P = 1/4.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/4

Worked Example 6

Problem: Find the probability to roll a number greater than 4 on a die.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 6.
  3. P = 2/6.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/3

Worked Example 7

Problem: Find the probability to draw blue from 2 blue and 6 green counters.

  1. Favourable outcomes = 2.
  2. Total equally likely outcomes = 8.
  3. P = 2/8.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/4

Worked Example 8

Problem: Find the probability to choose an odd number from 1–10.

  1. Favourable outcomes = 5.
  2. Total equally likely outcomes = 10.
  3. P = 5/10.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 9

Problem: Find the probability to flip tails.

  1. Favourable outcomes = 1.
  2. Total equally likely outcomes = 2.
  3. P = 1/2.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/2

Worked Example 10

Problem: Find the probability to choose a multiple of 3 from 1–12.

  1. Favourable outcomes = 4.
  2. Total equally likely outcomes = 12.
  3. P = 4/12.
  4. Simplify if possible.

Very beginner explanation: Probability is favourable outcomes divided by total equally likely outcomes.

Answer: 1/3

Practice exercise

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

60.13 Pythagorean theorem review

The Pythagorean theorem relates the side lengths of a right triangle: a² + b² = c², where c is the hypotenuse. In this section, the focus is Pythagorean theorem review.

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: A right triangle has legs 3 and 4. Find the hypotenuse.

  1. c² = 3² + 4².
  2. c² = 25.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 5

Worked Example 2

Problem: A right triangle has legs 5 and 12. Find the hypotenuse.

  1. c² = 5² + 12².
  2. c² = 169.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 13

Worked Example 3

Problem: A right triangle has legs 6 and 8. Find the hypotenuse.

  1. c² = 6² + 8².
  2. c² = 100.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 10

Worked Example 4

Problem: A right triangle has legs 8 and 15. Find the hypotenuse.

  1. c² = 8² + 15².
  2. c² = 289.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 17

Worked Example 5

Problem: A right triangle has legs 7 and 24. Find the hypotenuse.

  1. c² = 7² + 24².
  2. c² = 625.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 25

Worked Example 6

Problem: A right triangle has legs 9 and 12. Find the hypotenuse.

  1. c² = 9² + 12².
  2. c² = 225.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 15

Worked Example 7

Problem: A right triangle has legs 12 and 16. Find the hypotenuse.

  1. c² = 12² + 16².
  2. c² = 400.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 20

Worked Example 8

Problem: A right triangle has legs 20 and 21. Find the hypotenuse.

  1. c² = 20² + 21².
  2. c² = 841.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 29

Worked Example 9

Problem: A right triangle has legs 10 and 24. Find the hypotenuse.

  1. c² = 10² + 24².
  2. c² = 676.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 26

Worked Example 10

Problem: A right triangle has legs 15 and 20. Find the hypotenuse.

  1. c² = 15² + 20².
  2. c² = 625.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 25

Practice exercise

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

60.14 Angle-property review

An angle is formed by two rays meeting at a vertex and is measured in degrees. In this section, the focus is Angle-property review.

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: Find the supplementary angle to 35°.

  1. Supplementary angles total 180°.
  2. 180° - 35° = 145°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 145°

Worked Example 2

Problem: Find the supplementary angle to 48°.

  1. Supplementary angles total 180°.
  2. 180° - 48° = 132°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 132°

Worked Example 3

Problem: Find the supplementary angle to 67°.

  1. Supplementary angles total 180°.
  2. 180° - 67° = 113°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 113°

Worked Example 4

Problem: Find the supplementary angle to 72°.

  1. Supplementary angles total 180°.
  2. 180° - 72° = 108°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 108°

Worked Example 5

Problem: Find the supplementary angle to 110°.

  1. Supplementary angles total 180°.
  2. 180° - 110° = 70°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 70°

Worked Example 6

Problem: Find the supplementary angle to 25°.

  1. Supplementary angles total 180°.
  2. 180° - 25° = 155°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 155°

Worked Example 7

Problem: Find the supplementary angle to 58°.

  1. Supplementary angles total 180°.
  2. 180° - 58° = 122°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 122°

Worked Example 8

Problem: Find the supplementary angle to 83°.

  1. Supplementary angles total 180°.
  2. 180° - 83° = 97°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 97°

Worked Example 9

Problem: Find the supplementary angle to 95°.

  1. Supplementary angles total 180°.
  2. 180° - 95° = 85°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 85°

Worked Example 10

Problem: Find the supplementary angle to 120°.

  1. Supplementary angles total 180°.
  2. 180° - 120° = 60°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 60°

Practice exercise

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

60.15 Measurement and digital-units review

Measurement and digital-units review is an important Grade 8 concept in Comprehensive Grade 8 Review and Mathematical Processes. 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: Classify this data set: [4, 6, 8].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Worked Example 2

Problem: Classify this data set: [5, 9, 10, 12].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Worked Example 3

Problem: Classify this data set: [3, 7, 7, 11].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Worked Example 4

Problem: Classify this data set: [20, 25, 30].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Worked Example 5

Problem: Classify this data set: [6, 8, 9, 12, 15].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Worked Example 6

Problem: Classify this data set: [2, 4, 4, 5, 20].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Worked Example 7

Problem: Classify this data set: [10, 11, 12, 13, 14].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Worked Example 8

Problem: Classify this data set: [1, 3, 5, 7, 9].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Worked Example 9

Problem: Classify this data set: [8, 8, 8, 9, 10].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Worked Example 10

Problem: Classify this data set: [15, 18, 21, 24, 27].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Practice exercise

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

60.16 Financial-literacy review

Financial-literacy review is an important Grade 8 concept in Comprehensive Grade 8 Review and Mathematical Processes. 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: An item costs $50.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 50.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $6.50

Worked Example 2

Problem: An item costs $100.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 100.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $13.00

Worked Example 3

Problem: An item costs $75.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 75.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $9.75

Worked Example 4

Problem: An item costs $120.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 120.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $15.60

Worked Example 5

Problem: An item costs $200.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 200.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $26.00

Worked Example 6

Problem: An item costs $150.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 150.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $19.50

Worked Example 7

Problem: An item costs $250.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 250.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $32.50

Worked Example 8

Problem: An item costs $80.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 80.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $10.40

Worked Example 9

Problem: An item costs $300.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 300.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $39.00

Worked Example 10

Problem: An item costs $60.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 60.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $7.80

Practice exercise

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

60.17 Problem solving

Problem solving is an important Grade 8 concept in Comprehensive Grade 8 Review and Mathematical Processes. 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: Problem solving: Explain Problem solving in one simple sentence.

  1. Look at the words in “Problem solving”.
  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: Problem solving is a Grade 8 math idea used to describe or solve a specific mathematical relationship.

Worked Example 2

Problem: Problem solving: A student says, “I can use Problem solving 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: Problem solving: What is the first step when solving a problem about Problem solving?

  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: Problem solving: After solving a Problem solving 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: Problem solving: Give one way Problem solving 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: Problem solving can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.

Worked Example 6

Problem: Problem solving: Which representation could help explain Problem solving: 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: Problem solving: A student gets an answer for Problem solving 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: Problem solving: Why can estimation help before a detailed Problem solving 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: Problem solving: How can you test whether your rule for Problem solving 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: Problem solving: How would you teach Problem solving 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 Problem solving. Show the important steps, include units when needed, and explain how you checked the answer.

60.18 Reasoning and proving

Reasoning and proving is an important Grade 8 concept in Comprehensive Grade 8 Review and Mathematical Processes. 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: Reasoning and proving: Explain Reasoning and proving in one simple sentence.

  1. Look at the words in “Reasoning and proving”.
  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: Reasoning and proving is a Grade 8 math idea used to describe or solve a specific mathematical relationship.

Worked Example 2

Problem: Reasoning and proving: A student says, “I can use Reasoning and proving 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: Reasoning and proving: What is the first step when solving a problem about Reasoning and proving?

  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: Reasoning and proving: After solving a Reasoning and proving 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: Reasoning and proving: Give one way Reasoning and proving 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: Reasoning and proving can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.

Worked Example 6

Problem: Reasoning and proving: Which representation could help explain Reasoning and proving: 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: Reasoning and proving: A student gets an answer for Reasoning and proving 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: Reasoning and proving: Why can estimation help before a detailed Reasoning and proving 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: Reasoning and proving: How can you test whether your rule for Reasoning and proving 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: Reasoning and proving: How would you teach Reasoning and proving 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 Reasoning and proving. Show the important steps, include units when needed, and explain how you checked the answer.

60.19 Reflecting

Reflecting is an important Grade 8 concept in Comprehensive Grade 8 Review and Mathematical Processes. 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: Reflecting: Explain Reflecting in one simple sentence.

  1. Look at the words in “Reflecting”.
  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: Reflecting is a Grade 8 math idea used to describe or solve a specific mathematical relationship.

Worked Example 2

Problem: Reflecting: A student says, “I can use Reflecting 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: Reflecting: What is the first step when solving a problem about Reflecting?

  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: Reflecting: After solving a Reflecting 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: Reflecting: Give one way Reflecting 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: Reflecting can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.

Worked Example 6

Problem: Reflecting: Which representation could help explain Reflecting: 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: Reflecting: A student gets an answer for Reflecting 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: Reflecting: Why can estimation help before a detailed Reflecting 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: Reflecting: How can you test whether your rule for Reflecting 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: Reflecting: How would you teach Reflecting 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 Reflecting. Show the important steps, include units when needed, and explain how you checked the answer.

60.20 Selecting tools and strategies

A rate compares quantities measured in different units. In this section, the focus is Selecting tools and strategies.

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: A quantity of 80 units is used over 2 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 80 ÷ 2 = 40.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 40 per unit

Worked Example 2

Problem: A quantity of 120 units is used over 3 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 120 ÷ 3 = 40.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 40 per unit

Worked Example 3

Problem: A quantity of 190 units is used over 4 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 190 ÷ 4 = 47.5.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 47.5 per unit

Worked Example 4

Problem: A quantity of 250 units is used over 5 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 250 ÷ 5 = 50.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 50 per unit

Worked Example 5

Problem: A quantity of 220 units is used over 2 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 220 ÷ 2 = 110.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 110 per unit

Worked Example 6

Problem: A quantity of 160 units is used over 3 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 160 ÷ 3 = 53.3333.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 53.3333 per unit

Worked Example 7

Problem: A quantity of 130 units is used over 4 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 130 ÷ 4 = 32.5.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 32.5 per unit

Worked Example 8

Problem: A quantity of 200 units is used over 5 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 200 ÷ 5 = 40.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 40 per unit

Worked Example 9

Problem: A quantity of 290 units is used over 2 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 290 ÷ 2 = 145.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 145 per unit

Worked Example 10

Problem: A quantity of 180 units is used over 3 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 180 ÷ 3 = 60.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 60 per unit

Practice exercise

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

60.21 Connecting

Connecting is an important Grade 8 concept in Comprehensive Grade 8 Review and Mathematical Processes. 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: Connecting: Explain Connecting in one simple sentence.

  1. Look at the words in “Connecting”.
  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: Connecting is a Grade 8 math idea used to describe or solve a specific mathematical relationship.

Worked Example 2

Problem: Connecting: A student says, “I can use Connecting 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: Connecting: What is the first step when solving a problem about Connecting?

  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: Connecting: After solving a Connecting 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: Connecting: Give one way Connecting 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: Connecting can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.

Worked Example 6

Problem: Connecting: Which representation could help explain Connecting: 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: Connecting: A student gets an answer for Connecting 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: Connecting: Why can estimation help before a detailed Connecting 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: Connecting: How can you test whether your rule for Connecting 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: Connecting: How would you teach Connecting 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 Connecting. Show the important steps, include units when needed, and explain how you checked the answer.

60.22 Representing

Representing is an important Grade 8 concept in Comprehensive Grade 8 Review and Mathematical Processes. 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: Representing: Explain Representing in one simple sentence.

  1. Look at the words in “Representing”.
  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: Representing is a Grade 8 math idea used to describe or solve a specific mathematical relationship.

Worked Example 2

Problem: Representing: A student says, “I can use Representing 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: Representing: What is the first step when solving a problem about Representing?

  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: Representing: After solving a Representing 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: Representing: Give one way Representing 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: Representing can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.

Worked Example 6

Problem: Representing: Which representation could help explain Representing: 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: Representing: A student gets an answer for Representing 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: Representing: Why can estimation help before a detailed Representing 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: Representing: How can you test whether your rule for Representing 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: Representing: How would you teach Representing 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 Representing. Show the important steps, include units when needed, and explain how you checked the answer.

60.23 Communicating

Communicating is an important Grade 8 concept in Comprehensive Grade 8 Review and Mathematical Processes. 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: Communicating: Explain Communicating in one simple sentence.

  1. Look at the words in “Communicating”.
  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: Communicating is a Grade 8 math idea used to describe or solve a specific mathematical relationship.

Worked Example 2

Problem: Communicating: A student says, “I can use Communicating 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: Communicating: What is the first step when solving a problem about Communicating?

  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: Communicating: After solving a Communicating 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: Communicating: Give one way Communicating 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: Communicating can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.

Worked Example 6

Problem: Communicating: Which representation could help explain Communicating: 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: Communicating: A student gets an answer for Communicating 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: Communicating: Why can estimation help before a detailed Communicating 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: Communicating: How can you test whether your rule for Communicating 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: Communicating: How would you teach Communicating 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 Communicating. Show the important steps, include units when needed, and explain how you checked the answer.

60.24 Final mixed practice

Final mixed practice is an important Grade 8 concept in Comprehensive Grade 8 Review and Mathematical Processes. 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: Final mixed practice: Explain Final mixed practice in one simple sentence.

  1. Look at the words in “Final mixed practice”.
  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: Final mixed practice is a Grade 8 math idea used to describe or solve a specific mathematical relationship.

Worked Example 2

Problem: Final mixed practice: A student says, “I can use Final mixed practice 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: Final mixed practice: What is the first step when solving a problem about Final mixed practice?

  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: Final mixed practice: After solving a Final mixed practice 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: Final mixed practice: Give one way Final mixed practice 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: Final mixed practice can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.

Worked Example 6

Problem: Final mixed practice: Which representation could help explain Final mixed practice: 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: Final mixed practice: A student gets an answer for Final mixed practice 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: Final mixed practice: Why can estimation help before a detailed Final mixed practice 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: Final mixed practice: How can you test whether your rule for Final mixed practice 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: Final mixed practice: How would you teach Final mixed practice 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 Final mixed practice. Show the important steps, include units when needed, and explain how you checked the answer.

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Chapter 60 Review Questions and Answers

Q1. What is important to remember about Scientific-notation review?

Answer: Scientific-notation review is an important Grade 8 concept in Comprehensive Grade 8 Review and Mathematical Processes. 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.

Q2. What is important to remember about Rational-number review?

Answer: A rational number (a number that can be written as a fraction of integers with a nonzero denominator) includes integers, fractions, terminating decimals, and repeating decimals. In this section, the focus is Rational-number review.

Q3. What is important to remember about Fraction decimal percent review?

Answer: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, a ratio, or a point on a number line. In this section, the focus is Fraction decimal percent review.

Q4. What is important to remember about Integer and exponent review?

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

Q5. What is important to remember about Proportion review?

Answer: A proportion is an equation showing that two ratios are equal. In this section, the focus is Proportion review.

Q6. What is important to remember about Algebra and equation review?

Answer: An equation is a mathematical statement that two expressions have the same value. In this section, the focus is Algebra and equation review.

Q7. What is important to remember about Formula review?

Answer: A formula is a rule written with symbols to show how quantities are related. In this section, the focus is Formula review.

Q8. What is important to remember about Pattern and relation review?

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

Q9. What is important to remember about Coding review?

Answer: Coding review is an important Grade 8 concept in Comprehensive Grade 8 Review and Mathematical Processes. 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 Mathematical modelling review?

Answer: Mathematical modelling represents a real situation with mathematics, tests the model, and improves it when needed. In this section, the focus is Mathematical modelling review.

Q11. What is important to remember about Scatter-plot review?

Answer: Scatter-plot review is an important Grade 8 concept in Comprehensive Grade 8 Review and Mathematical Processes. 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 Probability review?

Answer: Probability measures likelihood from 0 to 1, or 0% to 100%. In this section, the focus is Probability review.

Q13. What is important to remember about Pythagorean theorem review?

Answer: The Pythagorean theorem relates the side lengths of a right triangle: a² + b² = c², where c is the hypotenuse. In this section, the focus is Pythagorean theorem review.

Q14. What is important to remember about Angle-property review?

Answer: An angle is formed by two rays meeting at a vertex and is measured in degrees. In this section, the focus is Angle-property review.

Q15. What is important to remember about Measurement and digital-units review?

Answer: Measurement and digital-units review is an important Grade 8 concept in Comprehensive Grade 8 Review and Mathematical Processes. 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.

Q16. What is important to remember about Financial-literacy review?

Answer: Financial-literacy review is an important Grade 8 concept in Comprehensive Grade 8 Review and Mathematical Processes. 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.

Q17. What is important to remember about Problem solving?

Answer: Problem solving is an important Grade 8 concept in Comprehensive Grade 8 Review and Mathematical Processes. 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.

Q18. What is important to remember about Reasoning and proving?

Answer: Reasoning and proving is an important Grade 8 concept in Comprehensive Grade 8 Review and Mathematical Processes. 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.

Q19. Why should you show your steps?

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

Q20. Why is estimation useful?

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

Q21. Why do units matter?

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

Q22. When should you round?

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

Q23. How can you check an answer?

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

Q24. What should you identify first in a word problem?

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

Q25. What makes a final answer complete?

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

Q26. What should you do if an answer seems unreasonable?

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

Q27. Why are diagrams useful?

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

Q28. Why are tables useful?

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

Q29. Why should you explain your reasoning?

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

Q30. How do mistakes help learning?

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