Grade by Grade
E. Spatial Sense · Angles, Rectangles, Grids and Measurement

Translations: Sliding Points and Shapes on a Grid

E1.2 E1.3 A1About 40 minutes
Learning goals
  • I can describe the translation that moves a point from one coordinate to another.
  • I can find where a point lands after a translation.
  • I can translate a shape on a grid by moving each vertex.
  • I can predict what a shape will look like after a translation.

Words to know

translation
A slide. Every point of a shape moves the same distance in the same direction.
image
The new shape or point after a move. We often name it with a little mark, like A′ (say "A prime").
unit
One space between two grid lines.
vertex
A corner point of a shape. More than one are called vertices.
predict
Say what you think will happen before you do it.

Let's start with a hockey puck

A hockey player taps a puck along the ice. The puck slides. It doesn't flip over and it doesn't spin around. It just moves to a new spot.

In math, a slide is called a translation. A translation moves a point or a shape right or left and up or down.

Describing a translation between two points

To describe how a point moves from one place to another on a coordinate grid, compare the numbers:

  1. Compare the x-numbers (the first numbers). If x gets bigger, the point moved right. If x gets smaller, it moved left.
  2. Compare the y-numbers (the second numbers). If y gets bigger, the point moved up. If y gets smaller, it moved down.
  3. The difference tells how many units.
0123456712345xyAA′A (1, 1) toA′ (5, 3)x: 1 to 5= 4 righty: 1 to 3= 2 up
Point A at (1, 1) slides to A′ at (5, 3). That is 4 units right and 2 units up.
Worked example

Example 1: Describe the move

A point moves from (2, 5) to (6, 1). Describe the translation.

  1. x-numbers: 2 to 6. It got bigger by 6 − 2 = 4. So the point moved 4 units right.
  2. y-numbers: 5 to 1. It got smaller by 5 − 1 = 4. So the point moved 4 units down.

The translation is 4 units right and 4 units down.

Worked example

Example 2: Where does it land?

A snowball is at (7, 4). It slides 3 units left and 2 units up. What are its new coordinates?

  1. Left changes x. Left makes x smaller: 7 − 3 = 4.
  2. Up changes y. Up makes y bigger: 4 + 2 = 6.

The snowball lands at (4, 6).

Translating a shape

To slide a whole shape, move every vertex the same way. Then join the new vertices in the same order.

The image is the same size and the same shape, and it faces the same way. A translation never flips or turns a shape.

01234567812345xyABCA′B′C′Every vertexmoves 4 rightand 1 up.
Triangle ABC slides 4 units right and 1 unit up to make triangle A′B′C′. A (1, 1) goes to A′ (5, 2), B (3, 1) goes to B′ (7, 2), and C (1, 3) goes to C′ (5, 4).
Worked example

Example 3: Translate a rectangle

A rectangle has vertices (1, 1), (4, 1), (4, 3) and (1, 3). Translate it 2 units right and 3 units up. Find the new vertices.

  1. Add 2 to every x-number (right 2) and add 3 to every y-number (up 3).
  2. (1, 1) → (3, 4)
  3. (4, 1) → (6, 4)
  4. (4, 3) → (6, 6)
  5. (1, 3) → (3, 6)
  6. Check: the new rectangle is still 3 units across and 2 units up, just like the first one.

The image has vertices (3, 4), (6, 4), (6, 6) and (3, 6).

Predict, then check

Before you draw, predict: "The shape will end up up and to the right, and it will look exactly the same." After you draw, check: Is the image the same size? Is it facing the same way? Did every vertex move the same number of units?

Math talk: keep going when it's tricky

If your image looks stretched or squished, don't give up! It means one vertex moved the wrong amount. Check each vertex one at a time. Finding your own mistake is a great math skill.

Just for fun

Q: Why are translations the calmest moves on the grid?

A: The shape just slides over. No flipping, no turning, so it never gets dizzy!

Hands-on activity

Slide a cracker

You need: grid paper (or draw a grid with 2 cm squares), a pencil, a ruler and a small flat object with straight edges, such as a square cracker, a sticky note or a game tile.

  1. Draw axes and number them 0 to 10.
  2. Place your object so its corners sit on grid points. Trace around it. Write the coordinates of each corner.
  3. A grown-up calls out a translation, like "3 right, 2 up". Predict where the object will end up and point to the spot.
  4. Slide the object (don't turn it!). Trace it again and write the new coordinates.
  5. Compare the old and new coordinates. What happened to each x-number? Each y-number?
  6. Switch roles: you call out a translation and the grown-up slides.

Math talk: "How could you find the new coordinates without sliding the object at all?"

01234567812345xySTUV
Points S, T, U and V for the questions below.

Your turn

1

Describe the translation from S to T.

2

Describe the translation from T to V.

3

Describe the translation from V to U.

4

Describe the translation from U to S.

5

Point (2, 3) slides 4 units right and 1 unit up. Where does it land?

6

Point (8, 6) slides 5 units left and 6 units down. Where does it land?

More practice: warm-up

1

What is another word for a translation?

2

Moving right makes the x-number bigger or smaller?

3

Moving down makes the y-number bigger or smaller?

4

Point (1, 1) slides 2 units up. Where does it land?

5

Does a shape change size when it is translated?

More practice: on your own

1

Describe the translation from (0, 2) to (5, 7).

2

Describe the translation from (9, 4) to (3, 4).

3

A triangle has vertices (1, 2), (3, 2) and (2, 5). Translate it 4 units right. What are the new vertices?

4

A point moved 3 units right and 2 units down and landed at (5, 1). Where did it start?

5

Noor slid a shape and the image is facing the opposite way. Was it a translation? Explain.

Challenge

1

Point A is at (2, 2). It slides 3 right and 1 up, then slides again 2 right and 4 up. Where does it end? What single translation would do the same thing?

2

A square has vertices (1, 1), (3, 1), (3, 3) and (1, 3). Ella wants to slide it so that one corner lands on (6, 4) and the square stays in the grid. Give one translation that works and the new vertices.

3

Can a translation move point (4, 2) to (1, 6) and point (5, 2) to (2, 5) at the same time? Explain.

Check your understanding

Quiz

1.

A point moves from (1, 2) to (4, 2). Describe the translation.

2.

Point (5, 5) slides 2 units left and 3 units down. Where does it land?

3.

What stays the same when a shape is translated?

4.

A point moves from (6, 1) to (2, 4). Describe the translation.

5.

To translate a shape, you...

6.

Which one is a translation?

For parents and teachers

Big idea: The Ontario curriculum asks Grade 4 students to describe the translations that move a point from one coordinate to another (E1.2) and to describe and perform translations on a grid and predict the results (E1.3). The key insight is that moving right or left changes only the first number (x), and moving up or down changes only the second number (y).

Common difficulties:

  • Mixing up which number changes. Right/left changes x; up/down changes y.
  • Moving only one corner, which stretches the shape. Have your child move and check each vertex one at a time.
  • Counting the starting line as 1. Count the jumps between lines, not the lines.

How to help: Use grid paper and a small object such as a cracker or game tile. Call out slides and have your child predict before moving. Encourage them to check their own work: "Is the image the same size and facing the same way?"