Chapter 47: Translations, Reflections, and Rotations
Learn Grade 7 mathematics from very beginner explanations through worked examples, practice, reasoning, and review.
Chapter Overview
This chapter teaches Translations, Reflections, and Rotations with simple language, step-by-step reasoning, and original worked examples. Technical words are explained in plain language.
Key Technical Terms
- 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.
47.1 Transformation
Transformation is an important Grade 7 idea in Translations, Reflections, and Rotations. 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: Transformation: Identify a correct example.
- State the definition and show why the example fits.
- Show all important reasoning.
Very beginner explanation: Transformation is an important Grade 7 idea in Translations, Reflections, and Rotations. 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: Transformation: Identify a non-example.
- Explain which requirement is missing.
- Show all important reasoning.
Very beginner explanation: Transformation is an important Grade 7 idea in Translations, Reflections, and Rotations. 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: Transformation: Compare two cases.
- State one similarity and one difference.
- Show all important reasoning.
Very beginner explanation: Transformation is an important Grade 7 idea in Translations, Reflections, and Rotations. 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: Transformation: Apply the idea in a real situation.
- Translate the situation into mathematical language.
- Show all important reasoning.
Very beginner explanation: Transformation is an important Grade 7 idea in Translations, Reflections, and Rotations. 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: Transformation: Create and check your own example.
- Use the definition, then verify each condition.
- Show all important reasoning.
Very beginner explanation: Transformation is an important Grade 7 idea in Translations, Reflections, and Rotations. 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 Transformation 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: Transformation 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 Transformation 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: Transformation 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 Transformation 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: Transformation 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 Transformation 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: Transformation 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 Transformation 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: Transformation 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 Transformation. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
47.2 Translation
A translation slides every point of a figure the same distance in the same direction. This topic focuses on Translation .
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: Translate (2,3) by vector (3,-2).
- Add 3 to x.
- Subtract 2 from y.
Very beginner explanation: A translation slides every point of a figure the same distance in the same direction. This topic focuses on Translation .
Answer: (5,1)
Worked Example 2
Problem: Translate (-1,4) by vector (3,-2).
- Add 3 to x.
- Subtract 2 from y.
Very beginner explanation: A translation slides every point of a figure the same distance in the same direction. This topic focuses on Translation .
Answer: (2,2)
Worked Example 3
Problem: Translate (5,-2) by vector (3,-2).
- Add 3 to x.
- Subtract 2 from y.
Very beginner explanation: A translation slides every point of a figure the same distance in the same direction. This topic focuses on Translation .
Answer: (8,-4)
Worked Example 4
Problem: Translate (0,6) by vector (3,-2).
- Add 3 to x.
- Subtract 2 from y.
Very beginner explanation: A translation slides every point of a figure the same distance in the same direction. This topic focuses on Translation .
Answer: (3,4)
Worked Example 5
Problem: Translate (-3,-5) by vector (3,-2).
- Add 3 to x.
- Subtract 2 from y.
Very beginner explanation: A translation slides every point of a figure the same distance in the same direction. This topic focuses on Translation .
Answer: (0,-7)
Worked Example 6
Problem: Find the supplementary angle to 35°.
- Supplementary angles total 180°.
- 180 - 35 = 145.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 145°
Worked Example 7
Problem: Find the supplementary angle to 48°.
- Supplementary angles total 180°.
- 180 - 48 = 132.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 132°
Worked Example 8
Problem: Find the supplementary angle to 67°.
- Supplementary angles total 180°.
- 180 - 67 = 113.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 113°
Worked Example 9
Problem: Find the supplementary angle to 72°.
- Supplementary angles total 180°.
- 180 - 72 = 108.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 108°
Worked Example 10
Problem: Find the supplementary angle to 110°.
- Supplementary angles total 180°.
- 180 - 110 = 70.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 70°
Practice Exercise
Create one new question about Translation. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
47.3 Translation vectors introduction
A translation slides every point of a figure the same distance in the same direction. This topic focuses on Translation vectors 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: Translate (2,3) by vector (3,-2).
- Add 3 to x.
- Subtract 2 from y.
Very beginner explanation: A translation slides every point of a figure the same distance in the same direction. This topic focuses on Translation vectors introduction .
Answer: (5,1)
Worked Example 2
Problem: Translate (-1,4) by vector (3,-2).
- Add 3 to x.
- Subtract 2 from y.
Very beginner explanation: A translation slides every point of a figure the same distance in the same direction. This topic focuses on Translation vectors introduction .
Answer: (2,2)
Worked Example 3
Problem: Translate (5,-2) by vector (3,-2).
- Add 3 to x.
- Subtract 2 from y.
Very beginner explanation: A translation slides every point of a figure the same distance in the same direction. This topic focuses on Translation vectors introduction .
Answer: (8,-4)
Worked Example 4
Problem: Translate (0,6) by vector (3,-2).
- Add 3 to x.
- Subtract 2 from y.
Very beginner explanation: A translation slides every point of a figure the same distance in the same direction. This topic focuses on Translation vectors introduction .
Answer: (3,4)
Worked Example 5
Problem: Translate (-3,-5) by vector (3,-2).
- Add 3 to x.
- Subtract 2 from y.
Very beginner explanation: A translation slides every point of a figure the same distance in the same direction. This topic focuses on Translation vectors introduction .
Answer: (0,-7)
Worked Example 6
Problem: Find the supplementary angle to 35°.
- Supplementary angles total 180°.
- 180 - 35 = 145.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 145°
Worked Example 7
Problem: Find the supplementary angle to 48°.
- Supplementary angles total 180°.
- 180 - 48 = 132.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 132°
Worked Example 8
Problem: Find the supplementary angle to 67°.
- Supplementary angles total 180°.
- 180 - 67 = 113.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 113°
Worked Example 9
Problem: Find the supplementary angle to 72°.
- Supplementary angles total 180°.
- 180 - 72 = 108.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 108°
Worked Example 10
Problem: Find the supplementary angle to 110°.
- Supplementary angles total 180°.
- 180 - 110 = 70.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 70°
Practice Exercise
Create one new question about Translation vectors introduction. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
47.4 Reflection
A reflection flips a figure across a line of reflection. This topic focuses on Reflection .
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: Reflect (2,3) across the x-axis.
- Keep x the same.
- Change the sign of y.
Very beginner explanation: A reflection flips a figure across a line of reflection. This topic focuses on Reflection .
Answer: (2,-3)
Worked Example 2
Problem: Reflect (-1,4) across the x-axis.
- Keep x the same.
- Change the sign of y.
Very beginner explanation: A reflection flips a figure across a line of reflection. This topic focuses on Reflection .
Answer: (-1,-4)
Worked Example 3
Problem: Reflect (5,-2) across the x-axis.
- Keep x the same.
- Change the sign of y.
Very beginner explanation: A reflection flips a figure across a line of reflection. This topic focuses on Reflection .
Answer: (5,2)
Worked Example 4
Problem: Reflect (0,6) across the x-axis.
- Keep x the same.
- Change the sign of y.
Very beginner explanation: A reflection flips a figure across a line of reflection. This topic focuses on Reflection .
Answer: (0,-6)
Worked Example 5
Problem: Reflect (-3,-5) across the x-axis.
- Keep x the same.
- Change the sign of y.
Very beginner explanation: A reflection flips a figure across a line of reflection. This topic focuses on Reflection .
Answer: (-3,5)
Worked Example 6
Problem: Reflect (2,3) across the x-axis.
- Keep x the same.
- Change the sign of y.
Very beginner explanation: Reflection across the x-axis changes vertical position but not horizontal position.
Answer: (2, -3)
Worked Example 7
Problem: Reflect (3,4) across the x-axis.
- Keep x the same.
- Change the sign of y.
Very beginner explanation: Reflection across the x-axis changes vertical position but not horizontal position.
Answer: (3, -4)
Worked Example 8
Problem: Reflect (4,5) across the x-axis.
- Keep x the same.
- Change the sign of y.
Very beginner explanation: Reflection across the x-axis changes vertical position but not horizontal position.
Answer: (4, -5)
Worked Example 9
Problem: Reflect (5,6) across the x-axis.
- Keep x the same.
- Change the sign of y.
Very beginner explanation: Reflection across the x-axis changes vertical position but not horizontal position.
Answer: (5, -6)
Worked Example 10
Problem: Reflect (6,7) across the x-axis.
- Keep x the same.
- Change the sign of y.
Very beginner explanation: Reflection across the x-axis changes vertical position but not horizontal position.
Answer: (6, -7)
Practice Exercise
Create one new question about Reflection. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
47.5 Line of reflection
A reflection flips a figure across a line of reflection. This topic focuses on Line of reflection .
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: Reflect (2,3) across the x-axis.
- Keep x the same.
- Change the sign of y.
Very beginner explanation: A reflection flips a figure across a line of reflection. This topic focuses on Line of reflection .
Answer: (2,-3)
Worked Example 2
Problem: Reflect (-1,4) across the x-axis.
- Keep x the same.
- Change the sign of y.
Very beginner explanation: A reflection flips a figure across a line of reflection. This topic focuses on Line of reflection .
Answer: (-1,-4)
Worked Example 3
Problem: Reflect (5,-2) across the x-axis.
- Keep x the same.
- Change the sign of y.
Very beginner explanation: A reflection flips a figure across a line of reflection. This topic focuses on Line of reflection .
Answer: (5,2)
Worked Example 4
Problem: Reflect (0,6) across the x-axis.
- Keep x the same.
- Change the sign of y.
Very beginner explanation: A reflection flips a figure across a line of reflection. This topic focuses on Line of reflection .
Answer: (0,-6)
Worked Example 5
Problem: Reflect (-3,-5) across the x-axis.
- Keep x the same.
- Change the sign of y.
Very beginner explanation: A reflection flips a figure across a line of reflection. This topic focuses on Line of reflection .
Answer: (-3,5)
Worked Example 6
Problem: Reflect (2,3) across the x-axis.
- Keep x the same.
- Change the sign of y.
Very beginner explanation: Reflection across the x-axis changes vertical position but not horizontal position.
Answer: (2, -3)
Worked Example 7
Problem: Reflect (3,4) across the x-axis.
- Keep x the same.
- Change the sign of y.
Very beginner explanation: Reflection across the x-axis changes vertical position but not horizontal position.
Answer: (3, -4)
Worked Example 8
Problem: Reflect (4,5) across the x-axis.
- Keep x the same.
- Change the sign of y.
Very beginner explanation: Reflection across the x-axis changes vertical position but not horizontal position.
Answer: (4, -5)
Worked Example 9
Problem: Reflect (5,6) across the x-axis.
- Keep x the same.
- Change the sign of y.
Very beginner explanation: Reflection across the x-axis changes vertical position but not horizontal position.
Answer: (5, -6)
Worked Example 10
Problem: Reflect (6,7) across the x-axis.
- Keep x the same.
- Change the sign of y.
Very beginner explanation: Reflection across the x-axis changes vertical position but not horizontal position.
Answer: (6, -7)
Practice Exercise
Create one new question about Line of reflection. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
47.6 Rotation
A rotation turns a figure around a fixed centre. This topic focuses on Rotation .
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: Rotate (2,3) 90° counterclockwise about the origin.
- Use rule (x,y)→(-y,x).
Very beginner explanation: A rotation turns a figure around a fixed centre. This topic focuses on Rotation .
Answer: (-3,2)
Worked Example 2
Problem: Rotate (-1,4) 90° counterclockwise about the origin.
- Use rule (x,y)→(-y,x).
Very beginner explanation: A rotation turns a figure around a fixed centre. This topic focuses on Rotation .
Answer: (-4,-1)
Worked Example 3
Problem: Rotate (5,-2) 90° counterclockwise about the origin.
- Use rule (x,y)→(-y,x).
Very beginner explanation: A rotation turns a figure around a fixed centre. This topic focuses on Rotation .
Answer: (2,5)
Worked Example 4
Problem: Rotate (0,6) 90° counterclockwise about the origin.
- Use rule (x,y)→(-y,x).
Very beginner explanation: A rotation turns a figure around a fixed centre. This topic focuses on Rotation .
Answer: (-6,0)
Worked Example 5
Problem: Rotate (-3,-5) 90° counterclockwise about the origin.
- Use rule (x,y)→(-y,x).
Very beginner explanation: A rotation turns a figure around a fixed centre. This topic focuses on Rotation .
Answer: (5,-3)
Worked Example 6
Problem: Rotate (1,2) 90° counterclockwise about the origin.
- Use the rule (x,y) → (-y,x).
Very beginner explanation: A 90° counterclockwise rotation swaps coordinates and changes the sign of the old y-value.
Answer: (-2, 1)
Worked Example 7
Problem: Rotate (2,3) 90° counterclockwise about the origin.
- Use the rule (x,y) → (-y,x).
Very beginner explanation: A 90° counterclockwise rotation swaps coordinates and changes the sign of the old y-value.
Answer: (-3, 2)
Worked Example 8
Problem: Rotate (3,4) 90° counterclockwise about the origin.
- Use the rule (x,y) → (-y,x).
Very beginner explanation: A 90° counterclockwise rotation swaps coordinates and changes the sign of the old y-value.
Answer: (-4, 3)
Worked Example 9
Problem: Rotate (4,5) 90° counterclockwise about the origin.
- Use the rule (x,y) → (-y,x).
Very beginner explanation: A 90° counterclockwise rotation swaps coordinates and changes the sign of the old y-value.
Answer: (-5, 4)
Worked Example 10
Problem: Rotate (5,6) 90° counterclockwise about the origin.
- Use the rule (x,y) → (-y,x).
Very beginner explanation: A 90° counterclockwise rotation swaps coordinates and changes the sign of the old y-value.
Answer: (-6, 5)
Practice Exercise
Create one new question about Rotation. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
47.7 Centre of rotation
A rotation turns a figure around a fixed centre. This topic focuses on Centre of rotation .
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: Rotate (2,3) 90° counterclockwise about the origin.
- Use rule (x,y)→(-y,x).
Very beginner explanation: A rotation turns a figure around a fixed centre. This topic focuses on Centre of rotation .
Answer: (-3,2)
Worked Example 2
Problem: Rotate (-1,4) 90° counterclockwise about the origin.
- Use rule (x,y)→(-y,x).
Very beginner explanation: A rotation turns a figure around a fixed centre. This topic focuses on Centre of rotation .
Answer: (-4,-1)
Worked Example 3
Problem: Rotate (5,-2) 90° counterclockwise about the origin.
- Use rule (x,y)→(-y,x).
Very beginner explanation: A rotation turns a figure around a fixed centre. This topic focuses on Centre of rotation .
Answer: (2,5)
Worked Example 4
Problem: Rotate (0,6) 90° counterclockwise about the origin.
- Use rule (x,y)→(-y,x).
Very beginner explanation: A rotation turns a figure around a fixed centre. This topic focuses on Centre of rotation .
Answer: (-6,0)
Worked Example 5
Problem: Rotate (-3,-5) 90° counterclockwise about the origin.
- Use rule (x,y)→(-y,x).
Very beginner explanation: A rotation turns a figure around a fixed centre. This topic focuses on Centre of rotation .
Answer: (5,-3)
Worked Example 6
Problem: Rotate (1,2) 90° counterclockwise about the origin.
- Use the rule (x,y) → (-y,x).
Very beginner explanation: A 90° counterclockwise rotation swaps coordinates and changes the sign of the old y-value.
Answer: (-2, 1)
Worked Example 7
Problem: Rotate (2,3) 90° counterclockwise about the origin.
- Use the rule (x,y) → (-y,x).
Very beginner explanation: A 90° counterclockwise rotation swaps coordinates and changes the sign of the old y-value.
Answer: (-3, 2)
Worked Example 8
Problem: Rotate (3,4) 90° counterclockwise about the origin.
- Use the rule (x,y) → (-y,x).
Very beginner explanation: A 90° counterclockwise rotation swaps coordinates and changes the sign of the old y-value.
Answer: (-4, 3)
Worked Example 9
Problem: Rotate (4,5) 90° counterclockwise about the origin.
- Use the rule (x,y) → (-y,x).
Very beginner explanation: A 90° counterclockwise rotation swaps coordinates and changes the sign of the old y-value.
Answer: (-5, 4)
Worked Example 10
Problem: Rotate (5,6) 90° counterclockwise about the origin.
- Use the rule (x,y) → (-y,x).
Very beginner explanation: A 90° counterclockwise rotation swaps coordinates and changes the sign of the old y-value.
Answer: (-6, 5)
Practice Exercise
Create one new question about Centre of rotation. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
47.8 Clockwise rotation
A rotation turns a figure around a fixed centre. This topic focuses on Clockwise rotation .
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: Rotate (2,3) 90° counterclockwise about the origin.
- Use rule (x,y)→(-y,x).
Very beginner explanation: A rotation turns a figure around a fixed centre. This topic focuses on Clockwise rotation .
Answer: (-3,2)
Worked Example 2
Problem: Rotate (-1,4) 90° counterclockwise about the origin.
- Use rule (x,y)→(-y,x).
Very beginner explanation: A rotation turns a figure around a fixed centre. This topic focuses on Clockwise rotation .
Answer: (-4,-1)
Worked Example 3
Problem: Rotate (5,-2) 90° counterclockwise about the origin.
- Use rule (x,y)→(-y,x).
Very beginner explanation: A rotation turns a figure around a fixed centre. This topic focuses on Clockwise rotation .
Answer: (2,5)
Worked Example 4
Problem: Rotate (0,6) 90° counterclockwise about the origin.
- Use rule (x,y)→(-y,x).
Very beginner explanation: A rotation turns a figure around a fixed centre. This topic focuses on Clockwise rotation .
Answer: (-6,0)
Worked Example 5
Problem: Rotate (-3,-5) 90° counterclockwise about the origin.
- Use rule (x,y)→(-y,x).
Very beginner explanation: A rotation turns a figure around a fixed centre. This topic focuses on Clockwise rotation .
Answer: (5,-3)
Worked Example 6
Problem: Rotate (1,2) 90° counterclockwise about the origin.
- Use the rule (x,y) → (-y,x).
Very beginner explanation: A 90° counterclockwise rotation swaps coordinates and changes the sign of the old y-value.
Answer: (-2, 1)
Worked Example 7
Problem: Rotate (2,3) 90° counterclockwise about the origin.
- Use the rule (x,y) → (-y,x).
Very beginner explanation: A 90° counterclockwise rotation swaps coordinates and changes the sign of the old y-value.
Answer: (-3, 2)
Worked Example 8
Problem: Rotate (3,4) 90° counterclockwise about the origin.
- Use the rule (x,y) → (-y,x).
Very beginner explanation: A 90° counterclockwise rotation swaps coordinates and changes the sign of the old y-value.
Answer: (-4, 3)
Worked Example 9
Problem: Rotate (4,5) 90° counterclockwise about the origin.
- Use the rule (x,y) → (-y,x).
Very beginner explanation: A 90° counterclockwise rotation swaps coordinates and changes the sign of the old y-value.
Answer: (-5, 4)
Worked Example 10
Problem: Rotate (5,6) 90° counterclockwise about the origin.
- Use the rule (x,y) → (-y,x).
Very beginner explanation: A 90° counterclockwise rotation swaps coordinates and changes the sign of the old y-value.
Answer: (-6, 5)
Practice Exercise
Create one new question about Clockwise rotation. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
47.9 Counterclockwise rotation
A rotation turns a figure around a fixed centre. This topic focuses on Counterclockwise rotation .
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: Rotate (2,3) 90° counterclockwise about the origin.
- Use rule (x,y)→(-y,x).
Very beginner explanation: A rotation turns a figure around a fixed centre. This topic focuses on Counterclockwise rotation .
Answer: (-3,2)
Worked Example 2
Problem: Rotate (-1,4) 90° counterclockwise about the origin.
- Use rule (x,y)→(-y,x).
Very beginner explanation: A rotation turns a figure around a fixed centre. This topic focuses on Counterclockwise rotation .
Answer: (-4,-1)
Worked Example 3
Problem: Rotate (5,-2) 90° counterclockwise about the origin.
- Use rule (x,y)→(-y,x).
Very beginner explanation: A rotation turns a figure around a fixed centre. This topic focuses on Counterclockwise rotation .
Answer: (2,5)
Worked Example 4
Problem: Rotate (0,6) 90° counterclockwise about the origin.
- Use rule (x,y)→(-y,x).
Very beginner explanation: A rotation turns a figure around a fixed centre. This topic focuses on Counterclockwise rotation .
Answer: (-6,0)
Worked Example 5
Problem: Rotate (-3,-5) 90° counterclockwise about the origin.
- Use rule (x,y)→(-y,x).
Very beginner explanation: A rotation turns a figure around a fixed centre. This topic focuses on Counterclockwise rotation .
Answer: (5,-3)
Worked Example 6
Problem: Rotate (1,2) 90° counterclockwise about the origin.
- Use the rule (x,y) → (-y,x).
Very beginner explanation: A 90° counterclockwise rotation swaps coordinates and changes the sign of the old y-value.
Answer: (-2, 1)
Worked Example 7
Problem: Rotate (2,3) 90° counterclockwise about the origin.
- Use the rule (x,y) → (-y,x).
Very beginner explanation: A 90° counterclockwise rotation swaps coordinates and changes the sign of the old y-value.
Answer: (-3, 2)
Worked Example 8
Problem: Rotate (3,4) 90° counterclockwise about the origin.
- Use the rule (x,y) → (-y,x).
Very beginner explanation: A 90° counterclockwise rotation swaps coordinates and changes the sign of the old y-value.
Answer: (-4, 3)
Worked Example 9
Problem: Rotate (4,5) 90° counterclockwise about the origin.
- Use the rule (x,y) → (-y,x).
Very beginner explanation: A 90° counterclockwise rotation swaps coordinates and changes the sign of the old y-value.
Answer: (-5, 4)
Worked Example 10
Problem: Rotate (5,6) 90° counterclockwise about the origin.
- Use the rule (x,y) → (-y,x).
Very beginner explanation: A 90° counterclockwise rotation swaps coordinates and changes the sign of the old y-value.
Answer: (-6, 5)
Practice Exercise
Create one new question about Counterclockwise rotation. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
47.10 Coordinate rules
Coordinates locate a point as (x, y): horizontal movement first, vertical movement second. This topic focuses on Coordinate rules .
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 rules .
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 rules .
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 rules .
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 rules .
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 rules .
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 rules. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
47.11 Combining transformations
Combining transformations is an important Grade 7 idea in Translations, Reflections, and Rotations. 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: Combining transformations: Identify a correct example.
- State the definition and show why the example fits.
- Show all important reasoning.
Very beginner explanation: Combining transformations is an important Grade 7 idea in Translations, Reflections, and Rotations. 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: Combining transformations: Identify a non-example.
- Explain which requirement is missing.
- Show all important reasoning.
Very beginner explanation: Combining transformations is an important Grade 7 idea in Translations, Reflections, and Rotations. 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: Combining transformations: Compare two cases.
- State one similarity and one difference.
- Show all important reasoning.
Very beginner explanation: Combining transformations is an important Grade 7 idea in Translations, Reflections, and Rotations. 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: Combining transformations: Apply the idea in a real situation.
- Translate the situation into mathematical language.
- Show all important reasoning.
Very beginner explanation: Combining transformations is an important Grade 7 idea in Translations, Reflections, and Rotations. 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: Combining transformations: Create and check your own example.
- Use the definition, then verify each condition.
- Show all important reasoning.
Very beginner explanation: Combining transformations is an important Grade 7 idea in Translations, Reflections, and Rotations. 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 Combining transformations 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: Combining transformations 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 Combining transformations 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: Combining transformations 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 Combining transformations 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: Combining transformations 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 Combining transformations 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: Combining transformations 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 Combining transformations 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: Combining transformations 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 Combining transformations. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
47.12 Symmetry
Symmetry is an important Grade 7 idea in Translations, Reflections, and Rotations. 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: State one defining/property clue for a Square.
- 4 equal sides; 4 right angles
Very beginner explanation: Symmetry is an important Grade 7 idea in Translations, Reflections, and Rotations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: Use the listed property to classify the shape.
Worked Example 2
Problem: State one defining/property clue for a Rectangle.
- opposite sides equal; 4 right angles
Very beginner explanation: Symmetry is an important Grade 7 idea in Translations, Reflections, and Rotations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: Use the listed property to classify the shape.
Worked Example 3
Problem: State one defining/property clue for a Parallelogram.
- opposite sides parallel and equal
Very beginner explanation: Symmetry is an important Grade 7 idea in Translations, Reflections, and Rotations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: Use the listed property to classify the shape.
Worked Example 4
Problem: State one defining/property clue for a Rhombus.
- 4 equal sides; opposite angles equal
Very beginner explanation: Symmetry is an important Grade 7 idea in Translations, Reflections, and Rotations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: Use the listed property to classify the shape.
Worked Example 5
Problem: State one defining/property clue for a Trapezoid.
- at least one pair of parallel sides
Very beginner explanation: Symmetry is an important Grade 7 idea in Translations, Reflections, and Rotations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: Use the listed property to classify the shape.
Worked Example 6
Problem: Explain Symmetry 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: Symmetry 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 Symmetry 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: Symmetry 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 Symmetry 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: Symmetry 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 Symmetry 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: Symmetry 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 Symmetry 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: Symmetry 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 Symmetry. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
47.13 Real-life transformations
Real-life transformations is an important Grade 7 idea in Translations, Reflections, and Rotations. 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: Real-life transformations: Identify a correct example.
- State the definition and show why the example fits.
- Show all important reasoning.
Very beginner explanation: Real-life transformations is an important Grade 7 idea in Translations, Reflections, and Rotations. 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: Real-life transformations: Identify a non-example.
- Explain which requirement is missing.
- Show all important reasoning.
Very beginner explanation: Real-life transformations is an important Grade 7 idea in Translations, Reflections, and Rotations. 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: Real-life transformations: Compare two cases.
- State one similarity and one difference.
- Show all important reasoning.
Very beginner explanation: Real-life transformations is an important Grade 7 idea in Translations, Reflections, and Rotations. 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: Real-life transformations: Apply the idea in a real situation.
- Translate the situation into mathematical language.
- Show all important reasoning.
Very beginner explanation: Real-life transformations is an important Grade 7 idea in Translations, Reflections, and Rotations. 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: Real-life transformations: Create and check your own example.
- Use the definition, then verify each condition.
- Show all important reasoning.
Very beginner explanation: Real-life transformations is an important Grade 7 idea in Translations, Reflections, and Rotations. 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 Real-life transformations 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: Real-life transformations 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 Real-life transformations 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: Real-life transformations 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 Real-life transformations 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: Real-life transformations 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 Real-life transformations 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: Real-life transformations 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 Real-life transformations 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: Real-life transformations 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 Real-life transformations. 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 Transformation . Show your reasoning and check your answer.
- Create and solve a new question about Translation . Show your reasoning and check your answer.
- Create and solve a new question about Translation vectors introduction . Show your reasoning and check your answer.
- Create and solve a new question about Reflection . Show your reasoning and check your answer.
- Create and solve a new question about Line of reflection . Show your reasoning and check your answer.
- Create and solve a new question about Rotation . Show your reasoning and check your answer.
- Create and solve a new question about Centre of rotation . Show your reasoning and check your answer.
- Create and solve a new question about Clockwise rotation . Show your reasoning and check your answer.
- Create and solve a new question about Counterclockwise rotation . Show your reasoning and check your answer.
- Create and solve a new question about Coordinate rules . Show your reasoning and check your answer.
- Create and solve a new question about Combining transformations . Show your reasoning and check your answer.
- Create and solve a new question about Symmetry . 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 Transformation?
Answer: Transformation is an important Grade 7 idea in Translations, Reflections, and Rotations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q2. What is important to remember about Translation?
Answer: A translation slides every point of a figure the same distance in the same direction. This topic focuses on Translation.
Q3. What is important to remember about Translation vectors introduction?
Answer: A translation slides every point of a figure the same distance in the same direction. This topic focuses on Translation vectors introduction.
Q4. What is important to remember about Reflection?
Answer: A reflection flips a figure across a line of reflection. This topic focuses on Reflection.
Q5. What is important to remember about Line of reflection?
Answer: A reflection flips a figure across a line of reflection. This topic focuses on Line of reflection.
Q6. What is important to remember about Rotation?
Answer: A rotation turns a figure around a fixed centre. This topic focuses on Rotation.
Q7. What is important to remember about Centre of rotation?
Answer: A rotation turns a figure around a fixed centre. This topic focuses on Centre of rotation.
Q8. What is important to remember about Clockwise rotation?
Answer: A rotation turns a figure around a fixed centre. This topic focuses on Clockwise rotation.
Q9. What is important to remember about Counterclockwise rotation?
Answer: A rotation turns a figure around a fixed centre. This topic focuses on Counterclockwise rotation.
Q10. What is important to remember about Coordinate rules?
Answer: Coordinates locate a point as (x, y): horizontal movement first, vertical movement second. This topic focuses on Coordinate rules.
Q11. What is important to remember about Combining transformations?
Answer: Combining transformations is an important Grade 7 idea in Translations, Reflections, and Rotations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q12. What is important to remember about Symmetry?
Answer: Symmetry is an important Grade 7 idea in Translations, Reflections, and Rotations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q13. What is important to remember about Real-life transformations?
Answer: Real-life transformations is an important Grade 7 idea in Translations, Reflections, and Rotations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q14. Why should you show your steps?
Answer: Showing steps makes reasoning easier to verify and helps locate mistakes.
Q15. Why is estimation useful?
Answer: Estimation helps you judge whether a 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.