Translations and combinations of different transformations - Teacher Slides.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Translations and combinations of different transformations

Transformations and constructions

Lesson 4 of 8

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up

Transformations and constructions

Lesson 4 of 8

Answer each one, then check.

1

What is the reflection of (3, 1) in the y-axis?

2

What is the image of (2, 3) under a 180° rotation about the origin?

3

What does congruent mean?

4

Work out (5, 2) + (3, −4) as coordinates.

5

What is a scale factor?

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up - Answers

Transformations and constructions

Lesson 4 of 8

1

What is the reflection of (3, 1) in the y-axis?

(−3, 1)

2

What is the image of (2, 3) under a 180° rotation about the origin?

(−2, −3)

3

What does congruent mean?

Same shape and size

4

Work out (5, 2) + (3, −4) as coordinates.

(8, −2)

5

What is a scale factor?

The number lengths are multiplied by

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Transformations and constructions

Lesson 4 of 8

1

Translate a shape using a column vector.

2

Describe a translation with a vector.

3

Carry out and describe combinations of transformations.

4

Find the single transformation equivalent to a combination.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Transformations and constructions

Lesson 4 of 8

THE KEY IDEA

A translation slides every point of a shape the same distance in the same direction.

It is described by a column vector: the top number is the move right (negative means left), the bottom number is the move up (negative means down).

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Translation

Transformations and constructions

Lesson 4 of 8

Every point moves the same way.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Translating a Shape

Transformations and constructions

Lesson 4 of 8

Triangle A has vertices (1, 1), (1, 3) and (3, 1). Translate it by the vector beginpmatrix 4 \ −3 endpmatrix.

1

Add 4 to each x-coordinate (move right)

1 + 4 = 5

2

Subtract 3 from each y-coordinate (move down)

1 − 3 = −2

3

Apply it to every vertex

(1, 1) → (5, −2), (1, 3) → (5, 0), (3, 1) → (7, −2)

ANSWER

The image has vertices (5, −2), (5, 0) and (7, −2).

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Describing a Translation

Transformations and constructions

Lesson 4 of 8

Point P is at (2, 5) and its image is at (−1, 2). Describe the translation.

1

Change in x

−1 − 2 = −3: 3 to the left

2

Change in y

2 − 5 = −3: 3 down

3

Write as a vector

beginpmatrix −3 \ −3 endpmatrix

ANSWER

A translation by the vector beginpmatrix −3 \ −3 endpmatrix.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

PART TWO

Combining Transformations

Do them one after the other.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Combination of Two Transformations

Transformations and constructions

Lesson 4 of 8

Apply the first, then the second to the image.

Do them in order

The order can change the answer, so work through the steps carefully.

Label each image

Name the shapes A, B and C to keep track.

Find the single equivalent

Look for the one transformation that takes the original straight to the final image.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Useful Combinations

Transformations and constructions

Lesson 4 of 8

Two reflections in parallel lines

Equivalent to a translation.

Two reflections in lines that cross

Equivalent to a rotation about the crossing point.

Two rotations about the same centre

Equivalent to one rotation about that centre.

Two translations

Equivalent to one translation: add the vectors.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Describing a Combination

Transformations and constructions

Lesson 4 of 8

Triangle A has vertices (1, 1), (1, 3), (3, 1). Triangle B is A reflected in the y-axis. Triangle C is B reflected in the x-axis. Describe the single transformation that maps A onto C.

1

B has vertices

(−1, 1), (−1, 3), (−3, 1)

2

C is B reflected in the x-axis

(−1, −1), (−1, −3), (−3, −1)

3

Compare A and C

Every coordinate has changed sign

4

Identify

A half turn about the origin

ANSWER

A rotation of 180° about (0, 0).

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Transformations and constructions

Lesson 4 of 8

Translation

A slide of every point by the same vector.

Column vector

A pair of numbers, one above the other, giving a move right and up.

Combination

Two or more transformations done one after the other.

Single transformation

One transformation that does the same job as a combination.

Invariant point

A point that does not move under a transformation.

Image

The shape after a transformation.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Transformations and constructions

Lesson 4 of 8

YOUR TASK

Slide and Flip

12 minutes

Shape A has vertices (1, 1), (3, 1), (1, 2). Translate A by beginpmatrix 2 \ 3 endpmatrix to get B, then reflect B in the y-axis to get C. Write the vertices of B and C, and describe fully the single transformation that maps B onto C.

1

Do the translation first.

2

Reflect the image.

3

Compare with the original.

WHAT A GOOD ANSWER SHOWS

B has vertices (3, 4), (5, 4), (3, 5). C has vertices (−3, 4), (−5, 4), (−3, 5). B maps onto C by a reflection in the y-axis (the line x = 0).

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Transformations and constructions

Lesson 4 of 8

Translate using a column vector.

Describe a translation with a vector.

Add vectors for two translations.

Do a combination in the right order.

Label images clearly.

Find a single equivalent transformation.

Identify invariant points.

Describe every transformation fully.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Translation vector: right/left on top, up/down underneath.

Two translations add.

Work through combinations one step at a time.

Always describe fully: name plus details.

EXAM FOCUS

Triangle B is the reflection of triangle A in the y-axis. Triangle C is the reflection of triangle B in the x-axis. Describe fully the single transformation that maps triangle A onto triangle C. (3 marks)

Find the coordinates of A and C and compare them. Every sign changing points to a half turn about the origin.

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Exam Practice: Translations and Combinations

Transformations and constructions

Lesson 4 of 8

Answer all questions. Show your working. · 30 minutes

Question 1 · 2 marks · Non-calculator

Triangle A has vertices (1, 1), (1, 3) and (3, 1). It is translated by the vector beginpmatrix 4 \ −3 endpmatrix. Write down the…

Question 2 · 2 marks · Non-calculator

The point P(2, 5) is mapped to P'(−1, 2) by a translation. Write the translation as a column vector.

Question 3 · 4 marks · Non-calculator

The diagram shows triangles A, B and C. (a) Describe fully the single transformation that maps A onto B. (b) Describe fully the single…

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Translations and Combinations (cont.)

Transformations and constructions

Lesson 4 of 8

Question 4 · 3 marks · Non-calculator

Using the diagram in the last question, describe fully the single transformation that maps triangle A onto triangle C.

Question 5 · 2 marks · Non-calculator

A shape is translated by beginpmatrix 3 \ 2 endpmatrix and then by beginpmatrix −5 \ 4 endpmatrix. Write down the single translation that…

Question 6 · 3 marks · Non-calculator

The shape P is reflected in the line x = 1 and then the image is reflected in the line x = 4. Describe fully the single transformation that…

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · 2 marks · Non-calculator

Transformations and constructions

Lesson 4 of 8

“

Triangle A has vertices (1, 1), (1, 3) and (3, 1). It is translated by the vector beginpmatrix 4 \ −3 endpmatrix. Write down the coordinates of the vertices of the image.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · mark scheme

Transformations and constructions

Lesson 4 of 8

2 marks available. Award a mark for each point made.

Two vertices correct

B1

All three correct

B1

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Question 2 · 2 marks · Non-calculator

Transformations and constructions

Lesson 4 of 8

“

The point P(2, 5) is mapped to P'(−1, 2) by a translation. Write the translation as a column vector.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · mark scheme

Transformations and constructions

Lesson 4 of 8

2 marks available. Award a mark for each point made.

One number correct

M1

Both correct

A1

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Question 3 · 4 marks · Non-calculator

Transformations and constructions

Lesson 4 of 8

The diagram shows triangles A, B and C. (a) Describe fully the single transformation that maps A onto B. (b) Describe fully the single transformation that maps B onto C. (4 marks)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · mark scheme

Transformations and constructions

Lesson 4 of 8

4 marks available. Award a mark for each point made.

(a) Reflection

B1

(a) The y-axis, or x = 0

B1

(b) Reflection

B1

(b) The x-axis, or y = 0

B1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · 3 marks · Non-calculator

Transformations and constructions

Lesson 4 of 8

“

Using the diagram in the last question, describe fully the single transformation that maps triangle A onto triangle C.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 3 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · mark scheme

Transformations and constructions

Lesson 4 of 8

3 marks available. Award a mark for each point made.

Rotation

B1

180°

B1

Centre (0, 0)

B1

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Question 5 · 2 marks · Non-calculator

Transformations and constructions

Lesson 4 of 8

“

A shape is translated by beginpmatrix 3 \ 2 endpmatrix and then by beginpmatrix −5 \ 4 endpmatrix. Write down the single translation that does the same job.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · mark scheme

Transformations and constructions

Lesson 4 of 8

2 marks available. Award a mark for each point made.

Adding the vectors

M1

beginpmatrix −2 \ 6 endpmatrix

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · 3 marks · Non-calculator

Transformations and constructions

Lesson 4 of 8

“

The shape P is reflected in the line x = 1 and then the image is reflected in the line x = 4. Describe fully the single transformation that is equivalent to these two reflections.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 3 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · mark scheme

Transformations and constructions

Lesson 4 of 8

3 marks available. Award a mark for each point made.

Translation

B1

Distance 6 (twice the gap between the lines)

M1

beginpmatrix 6 \ 0 endpmatrix

A1

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