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Chapter 16: Proportions, Scale, and Similarity

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 14 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.

16.1 Meaning of proportion

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

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

16.2 Solving proportions

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

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

16.3 Scale factor

A factor (a whole number that divides another number exactly) helps with simplifying fractions and finding common group sizes. In this section, the focus is Scale factor.

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 original length is 6 cm and the scale factor is 0.5. Find the image length.

  1. Multiply the original by 0.5.
  2. 6 × 0.5 = 3.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 3 cm

Worked Example 2

Problem: An original length is 10 cm and the scale factor is 1.5. Find the image length.

  1. Multiply the original by 1.5.
  2. 10 × 1.5 = 15.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 15 cm

Worked Example 3

Problem: An original length is 17 cm and the scale factor is 2. Find the image length.

  1. Multiply the original by 2.
  2. 17 × 2 = 34.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 34 cm

Worked Example 4

Problem: An original length is 23 cm and the scale factor is 2.5. Find the image length.

  1. Multiply the original by 2.5.
  2. 23 × 2.5 = 57.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 57.5 cm

Worked Example 5

Problem: An original length is 20 cm and the scale factor is 3. Find the image length.

  1. Multiply the original by 3.
  2. 20 × 3 = 60.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 60 cm

Worked Example 6

Problem: An original length is 14 cm and the scale factor is 0.5. Find the image length.

  1. Multiply the original by 0.5.
  2. 14 × 0.5 = 7.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 7 cm

Worked Example 7

Problem: An original length is 11 cm and the scale factor is 1.5. Find the image length.

  1. Multiply the original by 1.5.
  2. 11 × 1.5 = 16.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 16.5 cm

Worked Example 8

Problem: An original length is 18 cm and the scale factor is 2. Find the image length.

  1. Multiply the original by 2.
  2. 18 × 2 = 36.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 36 cm

Worked Example 9

Problem: An original length is 27 cm and the scale factor is 2.5. Find the image length.

  1. Multiply the original by 2.5.
  2. 27 × 2.5 = 67.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 67.5 cm

Worked Example 10

Problem: An original length is 16 cm and the scale factor is 3. Find the image length.

  1. Multiply the original by 3.
  2. 16 × 3 = 48.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 48 cm

Practice exercise

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

16.4 Scale drawings

Scale drawings is an important Grade 8 concept in Proportions, Scale, and Similarity. 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 original length is 6 cm and the scale factor is 0.5. Find the image length.

  1. Multiply the original by 0.5.
  2. 6 × 0.5 = 3.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 3 cm

Worked Example 2

Problem: An original length is 10 cm and the scale factor is 1.5. Find the image length.

  1. Multiply the original by 1.5.
  2. 10 × 1.5 = 15.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 15 cm

Worked Example 3

Problem: An original length is 17 cm and the scale factor is 2. Find the image length.

  1. Multiply the original by 2.
  2. 17 × 2 = 34.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 34 cm

Worked Example 4

Problem: An original length is 23 cm and the scale factor is 2.5. Find the image length.

  1. Multiply the original by 2.5.
  2. 23 × 2.5 = 57.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 57.5 cm

Worked Example 5

Problem: An original length is 20 cm and the scale factor is 3. Find the image length.

  1. Multiply the original by 3.
  2. 20 × 3 = 60.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 60 cm

Worked Example 6

Problem: An original length is 14 cm and the scale factor is 0.5. Find the image length.

  1. Multiply the original by 0.5.
  2. 14 × 0.5 = 7.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 7 cm

Worked Example 7

Problem: An original length is 11 cm and the scale factor is 1.5. Find the image length.

  1. Multiply the original by 1.5.
  2. 11 × 1.5 = 16.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 16.5 cm

Worked Example 8

Problem: An original length is 18 cm and the scale factor is 2. Find the image length.

  1. Multiply the original by 2.
  2. 18 × 2 = 36.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 36 cm

Worked Example 9

Problem: An original length is 27 cm and the scale factor is 2.5. Find the image length.

  1. Multiply the original by 2.5.
  2. 27 × 2.5 = 67.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 67.5 cm

Worked Example 10

Problem: An original length is 16 cm and the scale factor is 3. Find the image length.

  1. Multiply the original by 3.
  2. 16 × 3 = 48.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 48 cm

Practice exercise

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

16.5 Maps

Maps is an important Grade 8 concept in Proportions, Scale, and Similarity. 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 original length is 6 cm and the scale factor is 0.5. Find the image length.

  1. Multiply the original by 0.5.
  2. 6 × 0.5 = 3.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 3 cm

Worked Example 2

Problem: An original length is 10 cm and the scale factor is 1.5. Find the image length.

  1. Multiply the original by 1.5.
  2. 10 × 1.5 = 15.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 15 cm

Worked Example 3

Problem: An original length is 17 cm and the scale factor is 2. Find the image length.

  1. Multiply the original by 2.
  2. 17 × 2 = 34.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 34 cm

Worked Example 4

Problem: An original length is 23 cm and the scale factor is 2.5. Find the image length.

  1. Multiply the original by 2.5.
  2. 23 × 2.5 = 57.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 57.5 cm

Worked Example 5

Problem: An original length is 20 cm and the scale factor is 3. Find the image length.

  1. Multiply the original by 3.
  2. 20 × 3 = 60.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 60 cm

Worked Example 6

Problem: An original length is 14 cm and the scale factor is 0.5. Find the image length.

  1. Multiply the original by 0.5.
  2. 14 × 0.5 = 7.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 7 cm

Worked Example 7

Problem: An original length is 11 cm and the scale factor is 1.5. Find the image length.

  1. Multiply the original by 1.5.
  2. 11 × 1.5 = 16.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 16.5 cm

Worked Example 8

Problem: An original length is 18 cm and the scale factor is 2. Find the image length.

  1. Multiply the original by 2.
  2. 18 × 2 = 36.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 36 cm

Worked Example 9

Problem: An original length is 27 cm and the scale factor is 2.5. Find the image length.

  1. Multiply the original by 2.5.
  2. 27 × 2.5 = 67.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 67.5 cm

Worked Example 10

Problem: An original length is 16 cm and the scale factor is 3. Find the image length.

  1. Multiply the original by 3.
  2. 16 × 3 = 48.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 48 cm

Practice exercise

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

16.6 Floor plans

Floor plans is an important Grade 8 concept in Proportions, Scale, and Similarity. 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 original length is 6 cm and the scale factor is 0.5. Find the image length.

  1. Multiply the original by 0.5.
  2. 6 × 0.5 = 3.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 3 cm

Worked Example 2

Problem: An original length is 10 cm and the scale factor is 1.5. Find the image length.

  1. Multiply the original by 1.5.
  2. 10 × 1.5 = 15.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 15 cm

Worked Example 3

Problem: An original length is 17 cm and the scale factor is 2. Find the image length.

  1. Multiply the original by 2.
  2. 17 × 2 = 34.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 34 cm

Worked Example 4

Problem: An original length is 23 cm and the scale factor is 2.5. Find the image length.

  1. Multiply the original by 2.5.
  2. 23 × 2.5 = 57.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 57.5 cm

Worked Example 5

Problem: An original length is 20 cm and the scale factor is 3. Find the image length.

  1. Multiply the original by 3.
  2. 20 × 3 = 60.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 60 cm

Worked Example 6

Problem: An original length is 14 cm and the scale factor is 0.5. Find the image length.

  1. Multiply the original by 0.5.
  2. 14 × 0.5 = 7.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 7 cm

Worked Example 7

Problem: An original length is 11 cm and the scale factor is 1.5. Find the image length.

  1. Multiply the original by 1.5.
  2. 11 × 1.5 = 16.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 16.5 cm

Worked Example 8

Problem: An original length is 18 cm and the scale factor is 2. Find the image length.

  1. Multiply the original by 2.
  2. 18 × 2 = 36.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 36 cm

Worked Example 9

Problem: An original length is 27 cm and the scale factor is 2.5. Find the image length.

  1. Multiply the original by 2.5.
  2. 27 × 2.5 = 67.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 67.5 cm

Worked Example 10

Problem: An original length is 16 cm and the scale factor is 3. Find the image length.

  1. Multiply the original by 3.
  2. 16 × 3 = 48.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 48 cm

Practice exercise

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

16.7 Finding actual dimensions

Finding actual dimensions is an important Grade 8 concept in Proportions, Scale, and Similarity. 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: Finding actual dimensions: Explain Finding actual dimensions in one simple sentence.

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

Worked Example 2

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

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

Worked Example 6

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

16.8 Finding drawing dimensions

Finding drawing dimensions is an important Grade 8 concept in Proportions, Scale, and Similarity. 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: Finding drawing dimensions: Explain Finding drawing dimensions in one simple sentence.

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

Worked Example 2

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

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

Worked Example 6

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

16.9 Enlargement

Enlargement is an important Grade 8 concept in Proportions, Scale, and Similarity. 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 original length is 6 cm and the scale factor is 0.5. Find the image length.

  1. Multiply the original by 0.5.
  2. 6 × 0.5 = 3.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 3 cm

Worked Example 2

Problem: An original length is 10 cm and the scale factor is 1.5. Find the image length.

  1. Multiply the original by 1.5.
  2. 10 × 1.5 = 15.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 15 cm

Worked Example 3

Problem: An original length is 17 cm and the scale factor is 2. Find the image length.

  1. Multiply the original by 2.
  2. 17 × 2 = 34.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 34 cm

Worked Example 4

Problem: An original length is 23 cm and the scale factor is 2.5. Find the image length.

  1. Multiply the original by 2.5.
  2. 23 × 2.5 = 57.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 57.5 cm

Worked Example 5

Problem: An original length is 20 cm and the scale factor is 3. Find the image length.

  1. Multiply the original by 3.
  2. 20 × 3 = 60.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 60 cm

Worked Example 6

Problem: An original length is 14 cm and the scale factor is 0.5. Find the image length.

  1. Multiply the original by 0.5.
  2. 14 × 0.5 = 7.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 7 cm

Worked Example 7

Problem: An original length is 11 cm and the scale factor is 1.5. Find the image length.

  1. Multiply the original by 1.5.
  2. 11 × 1.5 = 16.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 16.5 cm

Worked Example 8

Problem: An original length is 18 cm and the scale factor is 2. Find the image length.

  1. Multiply the original by 2.
  2. 18 × 2 = 36.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 36 cm

Worked Example 9

Problem: An original length is 27 cm and the scale factor is 2.5. Find the image length.

  1. Multiply the original by 2.5.
  2. 27 × 2.5 = 67.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 67.5 cm

Worked Example 10

Problem: An original length is 16 cm and the scale factor is 3. Find the image length.

  1. Multiply the original by 3.
  2. 16 × 3 = 48.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 48 cm

Practice exercise

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

16.10 Reduction

Reduction is an important Grade 8 concept in Proportions, Scale, and Similarity. 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 original length is 6 cm and the scale factor is 0.5. Find the image length.

  1. Multiply the original by 0.5.
  2. 6 × 0.5 = 3.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 3 cm

Worked Example 2

Problem: An original length is 10 cm and the scale factor is 1.5. Find the image length.

  1. Multiply the original by 1.5.
  2. 10 × 1.5 = 15.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 15 cm

Worked Example 3

Problem: An original length is 17 cm and the scale factor is 2. Find the image length.

  1. Multiply the original by 2.
  2. 17 × 2 = 34.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 34 cm

Worked Example 4

Problem: An original length is 23 cm and the scale factor is 2.5. Find the image length.

  1. Multiply the original by 2.5.
  2. 23 × 2.5 = 57.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 57.5 cm

Worked Example 5

Problem: An original length is 20 cm and the scale factor is 3. Find the image length.

  1. Multiply the original by 3.
  2. 20 × 3 = 60.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 60 cm

Worked Example 6

Problem: An original length is 14 cm and the scale factor is 0.5. Find the image length.

  1. Multiply the original by 0.5.
  2. 14 × 0.5 = 7.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 7 cm

Worked Example 7

Problem: An original length is 11 cm and the scale factor is 1.5. Find the image length.

  1. Multiply the original by 1.5.
  2. 11 × 1.5 = 16.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 16.5 cm

Worked Example 8

Problem: An original length is 18 cm and the scale factor is 2. Find the image length.

  1. Multiply the original by 2.
  2. 18 × 2 = 36.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 36 cm

Worked Example 9

Problem: An original length is 27 cm and the scale factor is 2.5. Find the image length.

  1. Multiply the original by 2.5.
  2. 27 × 2.5 = 67.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 67.5 cm

Worked Example 10

Problem: An original length is 16 cm and the scale factor is 3. Find the image length.

  1. Multiply the original by 3.
  2. 16 × 3 = 48.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 48 cm

Practice exercise

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

16.11 Similar figures

Similar figures have the same shape and proportional corresponding lengths. In this section, the focus is Similar figures.

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 original length is 6 cm and the scale factor is 0.5. Find the image length.

  1. Multiply the original by 0.5.
  2. 6 × 0.5 = 3.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 3 cm

Worked Example 2

Problem: An original length is 10 cm and the scale factor is 1.5. Find the image length.

  1. Multiply the original by 1.5.
  2. 10 × 1.5 = 15.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 15 cm

Worked Example 3

Problem: An original length is 17 cm and the scale factor is 2. Find the image length.

  1. Multiply the original by 2.
  2. 17 × 2 = 34.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 34 cm

Worked Example 4

Problem: An original length is 23 cm and the scale factor is 2.5. Find the image length.

  1. Multiply the original by 2.5.
  2. 23 × 2.5 = 57.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 57.5 cm

Worked Example 5

Problem: An original length is 20 cm and the scale factor is 3. Find the image length.

  1. Multiply the original by 3.
  2. 20 × 3 = 60.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 60 cm

Worked Example 6

Problem: An original length is 14 cm and the scale factor is 0.5. Find the image length.

  1. Multiply the original by 0.5.
  2. 14 × 0.5 = 7.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 7 cm

Worked Example 7

Problem: An original length is 11 cm and the scale factor is 1.5. Find the image length.

  1. Multiply the original by 1.5.
  2. 11 × 1.5 = 16.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 16.5 cm

Worked Example 8

Problem: An original length is 18 cm and the scale factor is 2. Find the image length.

  1. Multiply the original by 2.
  2. 18 × 2 = 36.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 36 cm

Worked Example 9

Problem: An original length is 27 cm and the scale factor is 2.5. Find the image length.

  1. Multiply the original by 2.5.
  2. 27 × 2.5 = 67.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 67.5 cm

Worked Example 10

Problem: An original length is 16 cm and the scale factor is 3. Find the image length.

  1. Multiply the original by 3.
  2. 16 × 3 = 48.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 48 cm

Practice exercise

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

16.12 Corresponding lengths

Corresponding lengths is an important Grade 8 concept in Proportions, Scale, and Similarity. 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 original length is 6 cm and the scale factor is 0.5. Find the image length.

  1. Multiply the original by 0.5.
  2. 6 × 0.5 = 3.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 3 cm

Worked Example 2

Problem: An original length is 10 cm and the scale factor is 1.5. Find the image length.

  1. Multiply the original by 1.5.
  2. 10 × 1.5 = 15.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 15 cm

Worked Example 3

Problem: An original length is 17 cm and the scale factor is 2. Find the image length.

  1. Multiply the original by 2.
  2. 17 × 2 = 34.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 34 cm

Worked Example 4

Problem: An original length is 23 cm and the scale factor is 2.5. Find the image length.

  1. Multiply the original by 2.5.
  2. 23 × 2.5 = 57.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 57.5 cm

Worked Example 5

Problem: An original length is 20 cm and the scale factor is 3. Find the image length.

  1. Multiply the original by 3.
  2. 20 × 3 = 60.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 60 cm

Worked Example 6

Problem: An original length is 14 cm and the scale factor is 0.5. Find the image length.

  1. Multiply the original by 0.5.
  2. 14 × 0.5 = 7.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 7 cm

Worked Example 7

Problem: An original length is 11 cm and the scale factor is 1.5. Find the image length.

  1. Multiply the original by 1.5.
  2. 11 × 1.5 = 16.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 16.5 cm

Worked Example 8

Problem: An original length is 18 cm and the scale factor is 2. Find the image length.

  1. Multiply the original by 2.
  2. 18 × 2 = 36.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 36 cm

Worked Example 9

Problem: An original length is 27 cm and the scale factor is 2.5. Find the image length.

  1. Multiply the original by 2.5.
  2. 27 × 2.5 = 67.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 67.5 cm

Worked Example 10

Problem: An original length is 16 cm and the scale factor is 3. Find the image length.

  1. Multiply the original by 3.
  2. 16 × 3 = 48.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 48 cm

Practice exercise

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

16.13 Perimeter and scale

Perimeter is the total distance around a two-dimensional figure. In this section, the focus is Perimeter and scale.

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 perimeter of a rectangle 5 cm by 2 cm.

  1. Use P = 2(l+w).
  2. P = 2(5+2).

Very beginner explanation: Perimeter is the total distance around the outside of a figure.

Answer: 14 cm

Worked Example 2

Problem: Find the perimeter of a rectangle 6 cm by 3 cm.

  1. Use P = 2(l+w).
  2. P = 2(6+3).

Very beginner explanation: Perimeter is the total distance around the outside of a figure.

Answer: 18 cm

Worked Example 3

Problem: Find the perimeter of a rectangle 7 cm by 4 cm.

  1. Use P = 2(l+w).
  2. P = 2(7+4).

Very beginner explanation: Perimeter is the total distance around the outside of a figure.

Answer: 22 cm

Worked Example 4

Problem: Find the perimeter of a rectangle 8 cm by 5 cm.

  1. Use P = 2(l+w).
  2. P = 2(8+5).

Very beginner explanation: Perimeter is the total distance around the outside of a figure.

Answer: 26 cm

Worked Example 5

Problem: Find the perimeter of a rectangle 9 cm by 6 cm.

  1. Use P = 2(l+w).
  2. P = 2(9+6).

Very beginner explanation: Perimeter is the total distance around the outside of a figure.

Answer: 30 cm

Worked Example 6

Problem: Find the perimeter of a rectangle 10 cm by 7 cm.

  1. Use P = 2(l+w).
  2. P = 2(10+7).

Very beginner explanation: Perimeter is the total distance around the outside of a figure.

Answer: 34 cm

Worked Example 7

Problem: Find the perimeter of a rectangle 11 cm by 8 cm.

  1. Use P = 2(l+w).
  2. P = 2(11+8).

Very beginner explanation: Perimeter is the total distance around the outside of a figure.

Answer: 38 cm

Worked Example 8

Problem: Find the perimeter of a rectangle 12 cm by 9 cm.

  1. Use P = 2(l+w).
  2. P = 2(12+9).

Very beginner explanation: Perimeter is the total distance around the outside of a figure.

Answer: 42 cm

Worked Example 9

Problem: Find the perimeter of a rectangle 13 cm by 10 cm.

  1. Use P = 2(l+w).
  2. P = 2(13+10).

Very beginner explanation: Perimeter is the total distance around the outside of a figure.

Answer: 46 cm

Worked Example 10

Problem: Find the perimeter of a rectangle 14 cm by 11 cm.

  1. Use P = 2(l+w).
  2. P = 2(14+11).

Very beginner explanation: Perimeter is the total distance around the outside of a figure.

Answer: 50 cm

Practice exercise

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

16.14 Area and scale

Area measures the amount of two-dimensional space inside a figure and uses square units. In this section, the focus is Area and scale.

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 area of a rectangle 6 cm by 3 cm.

  1. Use A = length × width.
  2. A = 6×3.

Very beginner explanation: Area measures the space inside a 2D figure and uses square units.

Answer: 18 cm²

Worked Example 2

Problem: Find the area of a rectangle 7 cm by 4 cm.

  1. Use A = length × width.
  2. A = 7×4.

Very beginner explanation: Area measures the space inside a 2D figure and uses square units.

Answer: 28 cm²

Worked Example 3

Problem: Find the area of a rectangle 8 cm by 5 cm.

  1. Use A = length × width.
  2. A = 8×5.

Very beginner explanation: Area measures the space inside a 2D figure and uses square units.

Answer: 40 cm²

Worked Example 4

Problem: Find the area of a rectangle 9 cm by 6 cm.

  1. Use A = length × width.
  2. A = 9×6.

Very beginner explanation: Area measures the space inside a 2D figure and uses square units.

Answer: 54 cm²

Worked Example 5

Problem: Find the area of a rectangle 10 cm by 7 cm.

  1. Use A = length × width.
  2. A = 10×7.

Very beginner explanation: Area measures the space inside a 2D figure and uses square units.

Answer: 70 cm²

Worked Example 6

Problem: Find the area of a rectangle 11 cm by 8 cm.

  1. Use A = length × width.
  2. A = 11×8.

Very beginner explanation: Area measures the space inside a 2D figure and uses square units.

Answer: 88 cm²

Worked Example 7

Problem: Find the area of a rectangle 12 cm by 9 cm.

  1. Use A = length × width.
  2. A = 12×9.

Very beginner explanation: Area measures the space inside a 2D figure and uses square units.

Answer: 108 cm²

Worked Example 8

Problem: Find the area of a rectangle 13 cm by 10 cm.

  1. Use A = length × width.
  2. A = 13×10.

Very beginner explanation: Area measures the space inside a 2D figure and uses square units.

Answer: 130 cm²

Worked Example 9

Problem: Find the area of a rectangle 14 cm by 11 cm.

  1. Use A = length × width.
  2. A = 14×11.

Very beginner explanation: Area measures the space inside a 2D figure and uses square units.

Answer: 154 cm²

Worked Example 10

Problem: Find the area of a rectangle 15 cm by 12 cm.

  1. Use A = length × width.
  2. A = 15×12.

Very beginner explanation: Area measures the space inside a 2D figure and uses square units.

Answer: 180 cm²

Practice exercise

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

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Chapter 16 Review Questions and Answers

Q1. What is important to remember about Meaning of proportion?

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

Q2. What is important to remember about Solving proportions?

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

Q3. What is important to remember about Scale factor?

Answer: A factor (a whole number that divides another number exactly) helps with simplifying fractions and finding common group sizes. In this section, the focus is Scale factor.

Q4. What is important to remember about Scale drawings?

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

Q5. What is important to remember about Maps?

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

Q6. What is important to remember about Floor plans?

Answer: Floor plans is an important Grade 8 concept in Proportions, Scale, and Similarity. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q7. What is important to remember about Finding actual dimensions?

Answer: Finding actual dimensions is an important Grade 8 concept in Proportions, Scale, and Similarity. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q8. What is important to remember about Finding drawing dimensions?

Answer: Finding drawing dimensions is an important Grade 8 concept in Proportions, Scale, and Similarity. 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.

Q9. What is important to remember about Enlargement?

Answer: Enlargement is an important Grade 8 concept in Proportions, Scale, and Similarity. 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 Reduction?

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

Q11. What is important to remember about Similar figures?

Answer: Similar figures have the same shape and proportional corresponding lengths. In this section, the focus is Similar figures.

Q12. What is important to remember about Corresponding lengths?

Answer: Corresponding lengths is an important Grade 8 concept in Proportions, Scale, and Similarity. 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.

Q13. What is important to remember about Perimeter and scale?

Answer: Perimeter is the total distance around a two-dimensional figure. In this section, the focus is Perimeter and scale.

Q14. What is important to remember about Area and scale?

Answer: Area measures the amount of two-dimensional space inside a figure and uses square units. In this section, the focus is Area and scale.

Q15. Why should you show your steps?

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

Q16. Why is estimation useful?

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

Q17. Why do units matter?

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

Q18. When should you round?

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

Q19. How can you check an answer?

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

Q20. What should you identify first in a word problem?

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

Q21. What makes a final answer complete?

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

Q22. What should you do if an answer seems unreasonable?

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

Q23. Why are diagrams useful?

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

Q24. Why are tables useful?

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

Q25. Why should you explain your reasoning?

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

Q26. How do mistakes help learning?

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

Q27. When is a calculator useful?

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

Q28. Why compare more than one strategy?

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

Q29. What is mathematical communication?

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

Q30. What is mathematical modelling?

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