Chapter 2 of 8

All 8 chapters
  1. 1Reflections and rotations, readunderstood
  2. 2Translations and combined moves, readunderstood
  3. 3Enlargements, readunderstood
  4. 4Congruent triangles, readunderstood
  5. 5Similar shapes, readunderstood
  6. 6Similar areas and volumes, readunderstood
  7. 7Arcs and sectors, readunderstood
  8. 8Angle rules in circles, readunderstood

M4.4.2 Chapter 2

Translations and combined moves

A shape slides 3 squares right and 2 squares up. How do I write that move with numbers?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Translate a shape by a column vector
  • Describe a translation as a column vector
  • Find one move that does the same as two moves

A translation slides a shape. It does not turn the shape, flip it or change its size. Every point of the shape moves the same distance in the same direction. So the image faces the same way as the shape.

Draw it first

Slide triangle A 3 right and 2 up

  1. A grid with an x axis from 0 to 8 and a y axis from 0 to 6. Step 1: triangle A with corners at (1, 1), (4, 1) and (1, 3), labelled A. Step 2: an arrow from the corner (4, 1) to the right to (7, 1), labelled 3 right, and an arrow from there up to (7, 3), labelled 2 up. Step 3: an arrow from each corner of A to the matching corner of the image. Step 4: the image, triangle A′, with corners at (4, 3), (7, 3) and (4, 5), labelled A′.123456781234560xyA

    1 Triangle A has corners at (1, 1), (4, 1) and (1, 3).

  2. A grid with an x axis from 0 to 8 and a y axis from 0 to 6. Step 1: triangle A with corners at (1, 1), (4, 1) and (1, 3), labelled A. Step 2: an arrow from the corner (4, 1) to the right to (7, 1), labelled 3 right, and an arrow from there up to (7, 3), labelled 2 up. Step 3: an arrow from each corner of A to the matching corner of the image. Step 4: the image, triangle A′, with corners at (4, 3), (7, 3) and (4, 5), labelled A′.123456781234560xy3 right2 upA

    2 Take one corner. Move it 3 squares right, then 2 squares up.

  3. A grid with an x axis from 0 to 8 and a y axis from 0 to 6. Step 1: triangle A with corners at (1, 1), (4, 1) and (1, 3), labelled A. Step 2: an arrow from the corner (4, 1) to the right to (7, 1), labelled 3 right, and an arrow from there up to (7, 3), labelled 2 up. Step 3: an arrow from each corner of A to the matching corner of the image. Step 4: the image, triangle A′, with corners at (4, 3), (7, 3) and (4, 5), labelled A′.123456781234560xy3 right2 upA

    3 Move every other corner the same way. The three arrows are parallel and equally long.

  4. A grid with an x axis from 0 to 8 and a y axis from 0 to 6. Step 1: triangle A with corners at (1, 1), (4, 1) and (1, 3), labelled A. Step 2: an arrow from the corner (4, 1) to the right to (7, 1), labelled 3 right, and an arrow from there up to (7, 3), labelled 2 up. Step 3: an arrow from each corner of A to the matching corner of the image. Step 4: the image, triangle A′, with corners at (4, 3), (7, 3) and (4, 5), labelled A′.123456781234560xy3 right2 upAA′

    4 Join the new corners. This is the image, A′.

We write a slide as a column vector: two numbers in brackets, one above the other. The top number is the move right. The bottom number is the move up. A minus sign means left, for the top number, or down, for the bottom number.

Two column vectors in brackets. The first has 3 on top and 2 below, and is labelled 3 right, 2 up. The second has −2 on top and 1 below, and is labelled 2 left, 1 up.323 right, 2 up−212 left, 1 up
The top number is the move right. The bottom number is the move up.
Two column vectors in brackets. The first has 0 on top and −4 below, and is labelled 4 down. The second has 1 on top and −1 below, and is labelled 1 right, 1 down.0−44 down1−11 right, 1 down
A zero means no move in that direction. A minus sign means left or down.

To translate a point by a vector, add the top number to x and the bottom number to y. A vector says how far to move. It does not say where to land.

A grid with an x axis from 0 to 8 and a y axis from 0 to 7. Triangle A with corners at (4, 1), (7, 1) and (7, 4), labelled A. An arrow from the corner (4, 1) to the corner (1, 3) of the image, labelled 3 left, 2 up. The image, triangle A′, with corners at (1, 3), (4, 3) and (4, 6), labelled A′.1234567812345670xy3 left, 2 upAA′
Triangle A and its image after a slide of 3 left and 2 up.

Worked example

Translate the triangle with corners (4, 1), (7, 1) and (7, 4) by the vector with −3 on top and 2 below.

  1. The top number is −3, so move every corner 3 left. The bottom number is 2, so move every corner 2 up.
  2. The corner (4, 1) goes to (1, 3).
  3. The corner (7, 1) goes to (4, 3).
  4. The corner (7, 4) goes to (4, 6).

Answer: (1, 3), (4, 3) and (4, 6)

Now go the other way. Pick one corner of the shape and its image. The move right is the new x minus the old x. The move up is the new y minus the old y.

A grid with an x axis from 0 to 10 and a y axis from 0 to 8. Triangle A with corners at (1, 7), (4, 7) and (1, 4), labelled A. An arrow from the corner (4, 7) to the right to (9, 7), labelled 5 right, and an arrow from there down to (9, 4), labelled 3 down. The image, triangle A′, has corners at (6, 4), (9, 4) and (6, 1), labelled A′.12345678910123456780xy5 right3 downAA′
The corner (4, 7) moves to (9, 4).

Worked example

The corner (4, 7) of triangle A moves to (9, 4) in the image. Write the translation as a column vector.

  1. Pick one corner and its image. The corner is (4, 7) and its image is (9, 4).
  2. The move right is the new x minus the old x: from 4 to 9 is 5.
  3. The move up is the new y minus the old y: from 7 to 4 is −3. That is 3 down.
  4. The column vector has 5 on top and −3 below.

Answer: 5 on top and −3 below

Two slides in a row make one slide. Add the top numbers and add the bottom numbers. Two moves one after the other are a combined move. Take the vector with 3 on top and 1 below, then the vector with −5 on top and 3 below. Together they make one slide of 2 left and 4 up.

A grid with an x axis from 0 to 10 and a y axis from 0 to 8. Triangle A with corners at (5, 1), (7, 1) and (5, 3), labelled A. Triangle B with corners at (8, 2), (10, 2) and (8, 4), labelled B. Triangle C with corners at (3, 5), (5, 5) and (3, 7), labelled C. An arrow from A to B, labelled first, an arrow from B to C, labelled then, and a dashed arrow from A to C, labelled total.12345678910123456780xyfirstthentotalABC
A moves to B, then B moves to C. One slide takes A straight to C.

Worked example

A shape moves by the vector with 3 on top and 1 below. Then it moves by the vector with −5 on top and 3 below. Find one translation that does the same.

  1. Add the top numbers: 3 and −5 make −2.
  2. Add the bottom numbers: 1 and 3 make 4.
  3. The one move has −2 on top and 4 below.
  4. Check with the corner (5, 1). The first move takes it to (8, 2).
  5. The second move takes (8, 2) to (3, 5). The one move takes (5, 1) to (3, 5) as well.

Answer: −2 on top and 4 below

See it three ways

With things

Slide a book across a table: 3 hand spans to the right and 2 hand spans away from you. Every corner of the book moves the same way.

As a picture

One column vector in brackets with 3 on top and 2 below, labelled 3 right, 2 up.323 right, 2 up

In symbols

The vector with 3 on top and 2 below moves the point (x, y) to (x + 3, y + 2).

Try it

Slide a tracing

You need: squared paper, thin paper to trace on, a pencil.

  1. Draw a small triangle on the squared paper. Trace it on the thin paper.
  2. Slide the tracing 3 squares right and 2 squares up. Keep its edges parallel to the grid lines. Do not turn it.
  3. Press through, or draw round the tracing, to mark the image.
  4. Join each corner of the triangle to the matching corner of the image.

What you will see: The three lines are parallel and have the same length. Every corner moved by the same vector.

If you cannot try it, this is what happens.
A grid with an x axis from 0 to 8 and a y axis from 0 to 7. Triangle A with corners at (1, 1), (4, 1) and (1, 4), labelled A. The image, triangle A′, with corners at (4, 3), (7, 3) and (4, 6), labelled A′. A dashed arrow from each corner of A to the matching corner of A′.1234567812345670xyAA′

A wrong idea many people have

The vector with 3 on top and 2 below moves a shape to the point (3, 2).

Why it feels right: It has the same two numbers as the point, and a point is written with the x number first.

The test that breaks it: Take the corner (1, 1) and the vector with 3 on top and 2 below. Add 3 to x and 2 to y. The corner goes to (4, 3), not to (3, 2).

What is true: A column vector says how far to move: right, then up. It does not say where to land. To find the image, add the vector to each corner.

Key words

translation
a slide of a shape. Every point moves the same distance in the same direction.
column vector
two numbers in brackets, one above the other. The top number is the move right and the bottom number is the move up.
combined move
two moves one after the other. For two translations it is one translation.

Back to your guess

The answer is B: With two numbers in a column: 3 on top and 2 below.

A slide has two parts, a move right and a move up, so it needs two numbers. By agreement the move right goes on top and the move up goes below. One number cannot tell the two parts apart.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Triangle A moves to triangle B by a translation. Which column vector describes the move?

    A grid with an x axis from 0 to 9 and a y axis from 0 to 8. Triangle A with corners at (1, 4), (4, 4) and (1, 7), labelled A, and triangle B with corners at (5, 2), (8, 2) and (5, 5), labelled B.123456789123456780xyAB
    Tap your answer.

    1. Not this one. You wrote the numbers in the wrong order. The top number is the move right, which is 4. The bottom number is the move up or down. Go to M4.4.2. Try another.

    2. Yes, that is right. The corner (1, 4) moves to (5, 2). That is 4 right and 2 down, so the top number is 4 and the bottom number is −2.

    3. Not this one. You described the move from B back to A. The vector must go from the shape to its image. Go to M4.4.2. Try another.

    4. Not this one. The shape moves down, not up. A move down needs a minus sign on the bottom number. Go to M4.4.2. Try another.

    See the answer

    top number 4, bottom number −2

    The corner (1, 4) moves to (5, 2). That is 4 right and 2 down, so the top number is 4 and the bottom number is −2.

    • If you chose A: You wrote the numbers in the wrong order. The top number is the move right, which is 4. The bottom number is the move up or down. Go to M4.4.2.
    • If you chose C: You described the move from B back to A. The vector must go from the shape to its image. Go to M4.4.2.
    • If you chose D: The shape moves down, not up. A move down needs a minus sign on the bottom number. Go to M4.4.2.

  2. The corner (2, −1) is translated by the vector shown. Where does the corner go?

    One column vector in brackets with 4 on top and 3 below, labelled v.43v
    Tap your answer.

    1. Not this one. You added the numbers the wrong way round. The top number, 4, goes with x. The bottom number, 3, goes with y. Go to M4.4.2. Try another.

    2. Not this one. A vector says how far to move, not where to land. Add 4 to the x number and 3 to the y number of the corner. Go to M4.4.2. Try another.

    3. Yes, that is right. Add 4 to x: 2 + 4 = 6. Add 3 to y: −1 + 3 = 2. The corner goes to (6, 2).

    4. Not this one. You took the vector away. A translation by the vector adds its numbers. Go to M4.4.2. Try another.

    See the answer

    (6, 2)

    Add 4 to x: 2 + 4 = 6. Add 3 to y: −1 + 3 = 2. The corner goes to (6, 2).

    • If you chose A: You added the numbers the wrong way round. The top number, 4, goes with x. The bottom number, 3, goes with y. Go to M4.4.2.
    • If you chose B: A vector says how far to move, not where to land. Add 4 to the x number and 3 to the y number of the corner. Go to M4.4.2.
    • If you chose D: You took the vector away. A translation by the vector adds its numbers. Go to M4.4.2.

  3. A shape moves by the vector with 2 on top and 1 below. Then it moves by the vector with −5 on top and 2 below. Which one vector does the same?

    Tap your answer.

    1. Yes, that is right. Add the top numbers: 2 + (−5) = −3. Add the bottom numbers: 1 + 2 = 3.

    2. Not this one. You took the second move away from the first. Two moves in a row are added. Go to M4.4.2. Try another.

    3. Not this one. You multiplied. Two moves in a row are added, not multiplied. Go to M4.4.2. Try another.

    See the answer

    −3 on top, 3 below

    Add the top numbers: 2 + (−5) = −3. Add the bottom numbers: 1 + 2 = 3.

    • If you chose B: You took the second move away from the first. Two moves in a row are added. Go to M4.4.2.
    • If you chose C: You multiplied. Two moves in a row are added, not multiplied. Go to M4.4.2.

  4. A shape moves 5 units left and does not move up or down. Which column vector is that?

    Tap your answer.

    1. Not this one. A move up or down goes on the bottom. This shape moves sideways, so the 5 belongs on top. Go to M4.4.2. Try another.

    2. Not this one. A positive top number is a move right. A move left needs a minus sign. Go to M4.4.2. Try another.

    3. Yes, that is right. The top number is the move right, and left is the minus direction, so it is −5. The bottom number is 0, because there is no move up or down.

    See the answer

    −5 on top, 0 below

    The top number is the move right, and left is the minus direction, so it is −5. The bottom number is 0, because there is no move up or down.

    • If you chose A: A move up or down goes on the bottom. This shape moves sideways, so the 5 belongs on top. Go to M4.4.2.
    • If you chose B: A positive top number is a move right. A move left needs a minus sign. Go to M4.4.2.

  5. A translation moves the point (1, 1) to (4, −1). Where does the same translation move the point (0, 0)?

    Tap your answer.

    1. Not this one. That is where (1, 1) goes. A translation moves each point by the same vector, so (0, 0) lands somewhere else. Go to M4.4.2. Try another.

    2. Not this one. You used the vector the wrong way round. The vector goes from (1, 1) to (4, −1), so it is added, not taken away. Go to M4.4.2. Try another.

    3. Not this one. A translation moves every point, and (0, 0) is not a special point. The origin moves with the rest. Go to M4.4.2. Try another.

    4. Yes, that is right. From (1, 1) to (4, −1) is 3 right and 2 down, so the vector has 3 on top and −2 below. Every point moves by that vector, so (0, 0) goes to (3, −2).

    See the answer

    (3, −2)

    From (1, 1) to (4, −1) is 3 right and 2 down, so the vector has 3 on top and −2 below. Every point moves by that vector, so (0, 0) goes to (3, −2).

    • If you chose A: That is where (1, 1) goes. A translation moves each point by the same vector, so (0, 0) lands somewhere else. Go to M4.4.2.
    • If you chose B: You used the vector the wrong way round. The vector goes from (1, 1) to (4, −1), so it is added, not taken away. Go to M4.4.2.
    • If you chose C: A translation moves every point, and (0, 0) is not a special point. The origin moves with the rest. Go to M4.4.2.

  6. From an earlier chapter

    The point (4, 1) is turned 90° anticlockwise about (0, 0). What is the y number of the image?

    See the answer

    4

    A 90° anticlockwise turn about (0, 0) sends (x, y) to (−y, x). So (4, 1) goes to (−1, 4), and the y number is 4.

  7. The corner (3, 2) is translated by the vector with 4 on top and 5 below. What is the y number of the image?

    See the answer

    7

    Add the bottom number to y: 2 + 5 = 7.

  8. A corner moves from (1, 2) to (6, 9). What is the number on the bottom of the column vector?

    See the answer

    7

    The move up is the new y minus the old y: from 2 to 9 is 7.

  9. A shape moves by the vector with 2 on top and 5 below. Then it moves by the vector with −6 on top and 4 below. What is the number on the bottom of the one vector that does the same?

    See the answer

    9

    Add the bottom numbers: 5 + 4 = 9.

Keep it

Copy this into your exercise book.

A translation slides a shape. The column vector has the move right on top and the move up below. To combine two moves, add the tops and add the bottoms.

Draw: A triangle and its image with an arrow from one corner, and the vector written beside it.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (1.0 MB). It opens on any phone.