Chapter 48: Dilations
Learn Grade 7 mathematics from very beginner explanations through worked examples, practice, reasoning, and review.
Chapter Overview
This chapter teaches Dilations with simple language, step-by-step reasoning, and original worked examples. Technical words are explained in plain language.
Key Technical Terms
- Factor (a number that divides another number exactly)
- Dilation (an enlargement or reduction using a scale factor)
- Surface Area (total area of the outside of a 3D object)
- Estimate (a close approximation used to check whether an answer is reasonable)
- Solution (a value or result that satisfies the problem)
- Representation (a way to show mathematics using symbols, diagrams, tables, graphs, or words)
- Reasonableness (whether an answer makes sense in the context of the problem)
How to Learn This Chapter
Read the explanation first, follow the examples one step at a time, cover the answer and retry the problem yourself, then use the practice and review questions to check understanding.
48.1 Meaning of dilation
The mean is the sum of all values divided by the number of values. This topic focuses on Meaning of dilation .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Dilate (2,3) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: The mean is the sum of all values divided by the number of values. This topic focuses on Meaning of dilation .
Answer: (4,6)
Worked Example 2
Problem: Dilate (-1,4) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: The mean is the sum of all values divided by the number of values. This topic focuses on Meaning of dilation .
Answer: (-2,8)
Worked Example 3
Problem: Dilate (5,-2) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: The mean is the sum of all values divided by the number of values. This topic focuses on Meaning of dilation .
Answer: (10,-4)
Worked Example 4
Problem: Dilate (0,6) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: The mean is the sum of all values divided by the number of values. This topic focuses on Meaning of dilation .
Answer: (0,12)
Worked Example 5
Problem: Dilate (-3,-5) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: The mean is the sum of all values divided by the number of values. This topic focuses on Meaning of dilation .
Answer: (-6,-10)
Worked Example 6
Problem: Find the mean of [4, 6, 8].
- Add the values: 18.
- Divide by 3.
Very beginner explanation: The mean is the total shared equally among all values.
Answer: 6
Worked Example 7
Problem: Find the mean of [5, 9, 10, 12].
- Add the values: 36.
- Divide by 4.
Very beginner explanation: The mean is the total shared equally among all values.
Answer: 9
Worked Example 8
Problem: Find the mean of [3, 7, 7, 11].
- Add the values: 28.
- Divide by 4.
Very beginner explanation: The mean is the total shared equally among all values.
Answer: 7
Worked Example 9
Problem: Find the mean of [20, 25, 30].
- Add the values: 75.
- Divide by 3.
Very beginner explanation: The mean is the total shared equally among all values.
Answer: 25
Worked Example 10
Problem: Find the mean of [6, 8, 9, 12, 15].
- Add the values: 50.
- Divide by 5.
Very beginner explanation: The mean is the total shared equally among all values.
Answer: 10
Practice Exercise
Create one new question about Meaning of dilation. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
48.2 Enlargement
Enlargement is an important Grade 7 idea in Dilations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Enlargement: Identify a correct example.
- State the definition and show why the example fits.
- Show all important reasoning.
Very beginner explanation: Enlargement is an important Grade 7 idea in Dilations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 2
Problem: Enlargement: Identify a non-example.
- Explain which requirement is missing.
- Show all important reasoning.
Very beginner explanation: Enlargement is an important Grade 7 idea in Dilations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 3
Problem: Enlargement: Compare two cases.
- State one similarity and one difference.
- Show all important reasoning.
Very beginner explanation: Enlargement is an important Grade 7 idea in Dilations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 4
Problem: Enlargement: Apply the idea in a real situation.
- Translate the situation into mathematical language.
- Show all important reasoning.
Very beginner explanation: Enlargement is an important Grade 7 idea in Dilations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 5
Problem: Enlargement: Create and check your own example.
- Use the definition, then verify each condition.
- Show all important reasoning.
Very beginner explanation: Enlargement is an important Grade 7 idea in Dilations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 6
Problem: Explain Enlargement in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Enlargement becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Enlargement and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Enlargement becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Enlargement using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Enlargement becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Enlargement problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Enlargement becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Enlargement could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Enlargement becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Enlargement. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
48.3 Reduction
Reduction is an important Grade 7 idea in Dilations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Reduction: Identify a correct example.
- State the definition and show why the example fits.
- Show all important reasoning.
Very beginner explanation: Reduction is an important Grade 7 idea in Dilations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 2
Problem: Reduction: Identify a non-example.
- Explain which requirement is missing.
- Show all important reasoning.
Very beginner explanation: Reduction is an important Grade 7 idea in Dilations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 3
Problem: Reduction: Compare two cases.
- State one similarity and one difference.
- Show all important reasoning.
Very beginner explanation: Reduction is an important Grade 7 idea in Dilations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 4
Problem: Reduction: Apply the idea in a real situation.
- Translate the situation into mathematical language.
- Show all important reasoning.
Very beginner explanation: Reduction is an important Grade 7 idea in Dilations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 5
Problem: Reduction: Create and check your own example.
- Use the definition, then verify each condition.
- Show all important reasoning.
Very beginner explanation: Reduction is an important Grade 7 idea in Dilations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 6
Problem: Explain Reduction in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Reduction becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Reduction and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Reduction becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Reduction using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Reduction becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Reduction problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Reduction becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Reduction could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Reduction becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Reduction. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
48.4 Centre of dilation
A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Centre of dilation .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Dilate (2,3) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Centre of dilation .
Answer: (4,6)
Worked Example 2
Problem: Dilate (-1,4) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Centre of dilation .
Answer: (-2,8)
Worked Example 3
Problem: Dilate (5,-2) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Centre of dilation .
Answer: (10,-4)
Worked Example 4
Problem: Dilate (0,6) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Centre of dilation .
Answer: (0,12)
Worked Example 5
Problem: Dilate (-3,-5) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Centre of dilation .
Answer: (-6,-10)
Worked Example 6
Problem: Dilate (2,3) from the origin by scale factor 2.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (4, 6)
Worked Example 7
Problem: Dilate (-1,4) from the origin by scale factor 0.5.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (-0.5, 2)
Worked Example 8
Problem: Dilate (5,-2) from the origin by scale factor 3.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (15, -6)
Worked Example 9
Problem: Dilate (0,6) from the origin by scale factor 1.5.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (0, 9)
Worked Example 10
Problem: Dilate (-3,-5) from the origin by scale factor 0.25.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (-0.75, -1.25)
Practice Exercise
Create one new question about Centre of dilation. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
48.5 Scale factor greater than 1
A factor (a number that divides another number exactly) helps with simplifying, grouping, and fraction work. This topic focuses on Scale factor greater than 1 .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: List the factors of 18.
- Find all whole numbers that divide exactly.
Very beginner explanation: A factor (a number that divides another number exactly) helps with simplifying, grouping, and fraction work. This topic focuses on Scale factor greater than 1 .
Answer: 1, 2, 3, 6, 9, 18
Worked Example 2
Problem: List the factors of 24.
- Find all whole numbers that divide exactly.
Very beginner explanation: A factor (a number that divides another number exactly) helps with simplifying, grouping, and fraction work. This topic focuses on Scale factor greater than 1 .
Answer: 1, 2, 3, 4, 6, 8, 12, 24
Worked Example 3
Problem: List the factors of 36.
- Find all whole numbers that divide exactly.
Very beginner explanation: A factor (a number that divides another number exactly) helps with simplifying, grouping, and fraction work. This topic focuses on Scale factor greater than 1 .
Answer: 1, 2, 3, 4, 6, 9, 12, 18, 36
Worked Example 4
Problem: List the factors of 45.
- Find all whole numbers that divide exactly.
Very beginner explanation: A factor (a number that divides another number exactly) helps with simplifying, grouping, and fraction work. This topic focuses on Scale factor greater than 1 .
Answer: 1, 3, 5, 9, 15, 45
Worked Example 5
Problem: List the factors of 60.
- Find all whole numbers that divide exactly.
Very beginner explanation: A factor (a number that divides another number exactly) helps with simplifying, grouping, and fraction work. This topic focuses on Scale factor greater than 1 .
Answer: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Worked Example 6
Problem: List all positive factors of 18.
- Test whole numbers that divide with no remainder.
Very beginner explanation: A factor divides a number exactly.
Answer: 1, 2, 3, 6, 9, 18
Worked Example 7
Problem: List all positive factors of 24.
- Test whole numbers that divide with no remainder.
Very beginner explanation: A factor divides a number exactly.
Answer: 1, 2, 3, 4, 6, 8, 12, 24
Worked Example 8
Problem: List all positive factors of 30.
- Test whole numbers that divide with no remainder.
Very beginner explanation: A factor divides a number exactly.
Answer: 1, 2, 3, 5, 6, 10, 15, 30
Worked Example 9
Problem: List all positive factors of 42.
- Test whole numbers that divide with no remainder.
Very beginner explanation: A factor divides a number exactly.
Answer: 1, 2, 3, 6, 7, 14, 21, 42
Worked Example 10
Problem: List all positive factors of 60.
- Test whole numbers that divide with no remainder.
Very beginner explanation: A factor divides a number exactly.
Answer: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Practice Exercise
Create one new question about Scale factor greater than 1. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
48.6 Scale factor between 0 and 1
A factor (a number that divides another number exactly) helps with simplifying, grouping, and fraction work. This topic focuses on Scale factor between 0 and 1 .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: List the factors of 18.
- Find all whole numbers that divide exactly.
Very beginner explanation: A factor (a number that divides another number exactly) helps with simplifying, grouping, and fraction work. This topic focuses on Scale factor between 0 and 1 .
Answer: 1, 2, 3, 6, 9, 18
Worked Example 2
Problem: List the factors of 24.
- Find all whole numbers that divide exactly.
Very beginner explanation: A factor (a number that divides another number exactly) helps with simplifying, grouping, and fraction work. This topic focuses on Scale factor between 0 and 1 .
Answer: 1, 2, 3, 4, 6, 8, 12, 24
Worked Example 3
Problem: List the factors of 36.
- Find all whole numbers that divide exactly.
Very beginner explanation: A factor (a number that divides another number exactly) helps with simplifying, grouping, and fraction work. This topic focuses on Scale factor between 0 and 1 .
Answer: 1, 2, 3, 4, 6, 9, 12, 18, 36
Worked Example 4
Problem: List the factors of 45.
- Find all whole numbers that divide exactly.
Very beginner explanation: A factor (a number that divides another number exactly) helps with simplifying, grouping, and fraction work. This topic focuses on Scale factor between 0 and 1 .
Answer: 1, 3, 5, 9, 15, 45
Worked Example 5
Problem: List the factors of 60.
- Find all whole numbers that divide exactly.
Very beginner explanation: A factor (a number that divides another number exactly) helps with simplifying, grouping, and fraction work. This topic focuses on Scale factor between 0 and 1 .
Answer: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Worked Example 6
Problem: List all positive factors of 18.
- Test whole numbers that divide with no remainder.
Very beginner explanation: A factor divides a number exactly.
Answer: 1, 2, 3, 6, 9, 18
Worked Example 7
Problem: List all positive factors of 24.
- Test whole numbers that divide with no remainder.
Very beginner explanation: A factor divides a number exactly.
Answer: 1, 2, 3, 4, 6, 8, 12, 24
Worked Example 8
Problem: List all positive factors of 30.
- Test whole numbers that divide with no remainder.
Very beginner explanation: A factor divides a number exactly.
Answer: 1, 2, 3, 5, 6, 10, 15, 30
Worked Example 9
Problem: List all positive factors of 42.
- Test whole numbers that divide with no remainder.
Very beginner explanation: A factor divides a number exactly.
Answer: 1, 2, 3, 6, 7, 14, 21, 42
Worked Example 10
Problem: List all positive factors of 60.
- Test whole numbers that divide with no remainder.
Very beginner explanation: A factor divides a number exactly.
Answer: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Practice Exercise
Create one new question about Scale factor between 0 and 1. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
48.7 Coordinate dilation from origin
Coordinates locate a point as (x, y): horizontal movement first, vertical movement second. This topic focuses on Coordinate dilation from origin .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Locate (2,3).
- Move 2 units right.
- Move 3 units up.
Very beginner explanation: Coordinates locate a point as (x, y): horizontal movement first, vertical movement second. This topic focuses on Coordinate dilation from origin .
Answer: Quadrant I
Worked Example 2
Problem: Locate (-4,5).
- Move 4 units left.
- Move 5 units up.
Very beginner explanation: Coordinates locate a point as (x, y): horizontal movement first, vertical movement second. This topic focuses on Coordinate dilation from origin .
Answer: Quadrant II
Worked Example 3
Problem: Locate (-3,-2).
- Move 3 units left.
- Move 2 units down.
Very beginner explanation: Coordinates locate a point as (x, y): horizontal movement first, vertical movement second. This topic focuses on Coordinate dilation from origin .
Answer: Quadrant III
Worked Example 4
Problem: Locate (6,-1).
- Move 6 units right.
- Move 1 units down.
Very beginner explanation: Coordinates locate a point as (x, y): horizontal movement first, vertical movement second. This topic focuses on Coordinate dilation from origin .
Answer: Quadrant IV
Worked Example 5
Problem: Locate (0,4).
- Move 0 units right.
- Move 4 units up.
Very beginner explanation: Coordinates locate a point as (x, y): horizontal movement first, vertical movement second. This topic focuses on Coordinate dilation from origin .
Answer: on an axis
Worked Example 6
Problem: Locate (2,3) on the coordinate plane.
- Move horizontally using x.
- Move vertically using y.
- Identify the quadrant or axis.
Very beginner explanation: Ordered pairs are written (x, y): horizontal movement first, vertical movement second.
Answer: Quadrant I
Worked Example 7
Problem: Locate (-4,5) on the coordinate plane.
- Move horizontally using x.
- Move vertically using y.
- Identify the quadrant or axis.
Very beginner explanation: Ordered pairs are written (x, y): horizontal movement first, vertical movement second.
Answer: Quadrant II
Worked Example 8
Problem: Locate (-3,-2) on the coordinate plane.
- Move horizontally using x.
- Move vertically using y.
- Identify the quadrant or axis.
Very beginner explanation: Ordered pairs are written (x, y): horizontal movement first, vertical movement second.
Answer: Quadrant III
Worked Example 9
Problem: Locate (6,-1) on the coordinate plane.
- Move horizontally using x.
- Move vertically using y.
- Identify the quadrant or axis.
Very beginner explanation: Ordered pairs are written (x, y): horizontal movement first, vertical movement second.
Answer: Quadrant IV
Worked Example 10
Problem: Locate (0,4) on the coordinate plane.
- Move horizontally using x.
- Move vertically using y.
- Identify the quadrant or axis.
Very beginner explanation: Ordered pairs are written (x, y): horizontal movement first, vertical movement second.
Answer: on an axis
Practice Exercise
Create one new question about Coordinate dilation from origin. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
48.8 Dilation on a grid
A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Dilation on a grid .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Dilate (2,3) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Dilation on a grid .
Answer: (4,6)
Worked Example 2
Problem: Dilate (-1,4) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Dilation on a grid .
Answer: (-2,8)
Worked Example 3
Problem: Dilate (5,-2) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Dilation on a grid .
Answer: (10,-4)
Worked Example 4
Problem: Dilate (0,6) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Dilation on a grid .
Answer: (0,12)
Worked Example 5
Problem: Dilate (-3,-5) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Dilation on a grid .
Answer: (-6,-10)
Worked Example 6
Problem: Dilate (2,3) from the origin by scale factor 2.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (4, 6)
Worked Example 7
Problem: Dilate (-1,4) from the origin by scale factor 0.5.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (-0.5, 2)
Worked Example 8
Problem: Dilate (5,-2) from the origin by scale factor 3.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (15, -6)
Worked Example 9
Problem: Dilate (0,6) from the origin by scale factor 1.5.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (0, 9)
Worked Example 10
Problem: Dilate (-3,-5) from the origin by scale factor 0.25.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (-0.75, -1.25)
Practice Exercise
Create one new question about Dilation on a grid. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
48.9 Corresponding lengths
Corresponding lengths is an important Grade 7 idea in Dilations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Corresponding lengths: Identify a correct example.
- State the definition and show why the example fits.
- Show all important reasoning.
Very beginner explanation: Corresponding lengths is an important Grade 7 idea in Dilations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 2
Problem: Corresponding lengths: Identify a non-example.
- Explain which requirement is missing.
- Show all important reasoning.
Very beginner explanation: Corresponding lengths is an important Grade 7 idea in Dilations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 3
Problem: Corresponding lengths: Compare two cases.
- State one similarity and one difference.
- Show all important reasoning.
Very beginner explanation: Corresponding lengths is an important Grade 7 idea in Dilations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 4
Problem: Corresponding lengths: Apply the idea in a real situation.
- Translate the situation into mathematical language.
- Show all important reasoning.
Very beginner explanation: Corresponding lengths is an important Grade 7 idea in Dilations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 5
Problem: Corresponding lengths: Create and check your own example.
- Use the definition, then verify each condition.
- Show all important reasoning.
Very beginner explanation: Corresponding lengths is an important Grade 7 idea in Dilations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 6
Problem: Explain Corresponding lengths in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Corresponding lengths becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Corresponding lengths and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Corresponding lengths becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Corresponding lengths using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Corresponding lengths becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Corresponding lengths problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Corresponding lengths becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Corresponding lengths could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Corresponding lengths becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Corresponding lengths. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
48.10 Angles under dilation
An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Angles under dilation .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Dilate (2,3) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Angles under dilation .
Answer: (4,6)
Worked Example 2
Problem: Dilate (-1,4) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Angles under dilation .
Answer: (-2,8)
Worked Example 3
Problem: Dilate (5,-2) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Angles under dilation .
Answer: (10,-4)
Worked Example 4
Problem: Dilate (0,6) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Angles under dilation .
Answer: (0,12)
Worked Example 5
Problem: Dilate (-3,-5) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Angles under dilation .
Answer: (-6,-10)
Worked Example 6
Problem: Dilate (2,3) from the origin by scale factor 2.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (4, 6)
Worked Example 7
Problem: Dilate (-1,4) from the origin by scale factor 0.5.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (-0.5, 2)
Worked Example 8
Problem: Dilate (5,-2) from the origin by scale factor 3.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (15, -6)
Worked Example 9
Problem: Dilate (0,6) from the origin by scale factor 1.5.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (0, 9)
Worked Example 10
Problem: Dilate (-3,-5) from the origin by scale factor 0.25.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (-0.75, -1.25)
Practice Exercise
Create one new question about Angles under dilation. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
48.11 Perimeter under dilation
A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Perimeter under dilation .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Dilate (2,3) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Perimeter under dilation .
Answer: (4,6)
Worked Example 2
Problem: Dilate (-1,4) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Perimeter under dilation .
Answer: (-2,8)
Worked Example 3
Problem: Dilate (5,-2) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Perimeter under dilation .
Answer: (10,-4)
Worked Example 4
Problem: Dilate (0,6) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Perimeter under dilation .
Answer: (0,12)
Worked Example 5
Problem: Dilate (-3,-5) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Perimeter under dilation .
Answer: (-6,-10)
Worked Example 6
Problem: Dilate (2,3) from the origin by scale factor 2.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (4, 6)
Worked Example 7
Problem: Dilate (-1,4) from the origin by scale factor 0.5.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (-0.5, 2)
Worked Example 8
Problem: Dilate (5,-2) from the origin by scale factor 3.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (15, -6)
Worked Example 9
Problem: Dilate (0,6) from the origin by scale factor 1.5.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (0, 9)
Worked Example 10
Problem: Dilate (-3,-5) from the origin by scale factor 0.25.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (-0.75, -1.25)
Practice Exercise
Create one new question about Perimeter under dilation. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
48.12 Area under dilation introduction
A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Area under dilation introduction .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Dilate (2,3) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Area under dilation introduction .
Answer: (4,6)
Worked Example 2
Problem: Dilate (-1,4) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Area under dilation introduction .
Answer: (-2,8)
Worked Example 3
Problem: Dilate (5,-2) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Area under dilation introduction .
Answer: (10,-4)
Worked Example 4
Problem: Dilate (0,6) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Area under dilation introduction .
Answer: (0,12)
Worked Example 5
Problem: Dilate (-3,-5) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Area under dilation introduction .
Answer: (-6,-10)
Worked Example 6
Problem: Dilate (2,3) from the origin by scale factor 2.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (4, 6)
Worked Example 7
Problem: Dilate (-1,4) from the origin by scale factor 0.5.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (-0.5, 2)
Worked Example 8
Problem: Dilate (5,-2) from the origin by scale factor 3.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (15, -6)
Worked Example 9
Problem: Dilate (0,6) from the origin by scale factor 1.5.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (0, 9)
Worked Example 10
Problem: Dilate (-3,-5) from the origin by scale factor 0.25.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (-0.75, -1.25)
Practice Exercise
Create one new question about Area under dilation introduction. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
48.13 Real-life dilation applications
A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Real-life dilation applications .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Dilate (2,3) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Real-life dilation applications .
Answer: (4,6)
Worked Example 2
Problem: Dilate (-1,4) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Real-life dilation applications .
Answer: (-2,8)
Worked Example 3
Problem: Dilate (5,-2) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Real-life dilation applications .
Answer: (10,-4)
Worked Example 4
Problem: Dilate (0,6) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Real-life dilation applications .
Answer: (0,12)
Worked Example 5
Problem: Dilate (-3,-5) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Real-life dilation applications .
Answer: (-6,-10)
Worked Example 6
Problem: Dilate (2,3) from the origin by scale factor 2.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (4, 6)
Worked Example 7
Problem: Dilate (-1,4) from the origin by scale factor 0.5.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (-0.5, 2)
Worked Example 8
Problem: Dilate (5,-2) from the origin by scale factor 3.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (15, -6)
Worked Example 9
Problem: Dilate (0,6) from the origin by scale factor 1.5.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (0, 9)
Worked Example 10
Problem: Dilate (-3,-5) from the origin by scale factor 0.25.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (-0.75, -1.25)
Practice Exercise
Create one new question about Real-life dilation applications. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
Chapter Notes, Practice, and Common Mistakes
Chapter Notes
- Read the complete question before choosing an operation, formula, graph, or model.
- Write technical words together with their meaning until you are comfortable using them.
- Show all important steps so another student can follow your reasoning.
- Keep units, labels, signs, axes, variables, and mathematical symbols clear.
- Estimate before or after calculating when an estimate can help check reasonableness.
- For real-life problems, explain what the final number means in the situation.
Extra Practice
- Create and solve a new question about Meaning of dilation . Show your reasoning and check your answer.
- Create and solve a new question about Enlargement . Show your reasoning and check your answer.
- Create and solve a new question about Reduction . Show your reasoning and check your answer.
- Create and solve a new question about Centre of dilation . Show your reasoning and check your answer.
- Create and solve a new question about Scale factor greater than 1 . Show your reasoning and check your answer.
- Create and solve a new question about Scale factor between 0 and 1 . Show your reasoning and check your answer.
- Create and solve a new question about Coordinate dilation from origin . Show your reasoning and check your answer.
- Create and solve a new question about Dilation on a grid . Show your reasoning and check your answer.
- Create and solve a new question about Corresponding lengths . Show your reasoning and check your answer.
- Create and solve a new question about Angles under dilation . Show your reasoning and check your answer.
- Create and solve a new question about Perimeter under dilation . Show your reasoning and check your answer.
- Create and solve a new question about Area under dilation introduction . Show your reasoning and check your answer.
Common Mistakes
- Assuming a diagram is drawn to scale.
- Confusing radius and diameter.
- Using square units for volume or cubic units for area.
- Applying a scale factor to only one dimension.
- Forgetting the circular bases in cylinder surface area.
30 Review Questions and Answers
Q1. What is important to remember about Meaning of dilation?
Answer: The mean is the sum of all values divided by the number of values. This topic focuses on Meaning of dilation.
Q2. What is important to remember about Enlargement?
Answer: Enlargement is an important Grade 7 idea in Dilations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q3. What is important to remember about Reduction?
Answer: Reduction is an important Grade 7 idea in Dilations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q4. What is important to remember about Centre of dilation?
Answer: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Centre of dilation.
Q5. What is important to remember about Scale factor greater than 1?
Answer: A factor (a number that divides another number exactly) helps with simplifying, grouping, and fraction work. This topic focuses on Scale factor greater than 1.
Q6. What is important to remember about Scale factor between 0 and 1?
Answer: A factor (a number that divides another number exactly) helps with simplifying, grouping, and fraction work. This topic focuses on Scale factor between 0 and 1.
Q7. What is important to remember about Coordinate dilation from origin?
Answer: Coordinates locate a point as (x, y): horizontal movement first, vertical movement second. This topic focuses on Coordinate dilation from origin.
Q8. What is important to remember about Dilation on a grid?
Answer: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Dilation on a grid.
Q9. What is important to remember about Corresponding lengths?
Answer: Corresponding lengths is an important Grade 7 idea in Dilations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q10. What is important to remember about Angles under dilation?
Answer: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Angles under dilation.
Q11. What is important to remember about Perimeter under dilation?
Answer: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Perimeter under dilation.
Q12. What is important to remember about Area under dilation introduction?
Answer: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Area under dilation introduction.
Q13. What is important to remember about Real-life dilation applications?
Answer: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Real-life dilation applications.
Q14. Why should you show your steps?
Answer: Showing steps makes reasoning easier to verify and helps locate mistakes.
Q15. Why is estimation useful?
Answer: Estimation helps you judge whether a final answer is reasonable before accepting it.
Q16. Why do units matter?
Answer: Units tell what a value measures and prevent incompatible quantities from being mixed.
Q17. When should you round?
Answer: Usually round near the end of a calculation unless the question specifically asks for earlier rounding.
Q18. How can you check an answer?
Answer: Use an inverse operation, substitution, estimation, a graph, or a second method.
Q19. What should you identify first in a word problem?
Answer: Identify what is known, what is unknown, and the relationship connecting them.
Q20. What makes a final answer complete?
Answer: Give the value, correct units when needed, and a brief interpretation in context.
Q21. What should you do if an answer seems unreasonable?
Answer: Re-read the question, check copied values, signs, units, formulas, and calculations.
Q22. Why are diagrams useful?
Answer: A labelled diagram can reveal relationships between lengths, angles, areas, and other quantities.
Q23. Why are tables useful?
Answer: Tables organize values and help reveal patterns, relationships, rates, and missing information.
Q24. Why should you explain your reasoning?
Answer: Reasoning shows why a method works, not just what answer you obtained.
Q25. How do mistakes help learning?
Answer: Analyzing a mistake identifies the misunderstood rule or step and helps prevent the same error later.
Q26. How should you study this chapter?
Answer: Review definitions, redo worked examples without looking, practise mixed questions, and explain solutions in your own words.
Q27. When is a calculator useful?
Answer: A calculator is useful for lengthy arithmetic after you understand the mathematical setup and can estimate the expected size of the answer.
Q28. Why compare more than one strategy?
Answer: Different strategies can make a problem easier and provide a way to verify the result.
Q29. What is a mathematical model?
Answer: It is a simplified mathematical representation of a real situation used to analyze, explain, or predict.
Q30. Why should assumptions be stated?
Answer: Assumptions show what conditions the solution depends on and help readers judge whether the model is reasonable.