Grade by Grade
E. Spatial Sense · Spatial Sense: Angles, Triangles, Transformations and Area

Translations and Reflections on a Grid

E1.5About 40 minutes
Learning goals
  • I can translate (slide) a shape on a grid and give the new coordinates of its vertices.
  • I can reflect (flip) a shape in a vertical or horizontal mirror line.
  • I can predict where an image will be before I draw it.
  • I can tell whether an image was made by a translation or a reflection.

Words to know

transformation
A move that changes a shape's position. Translations, reflections and rotations are transformations.
image
The new shape after a transformation. Its vertices are often named with a mark, like A′ (say "A prime").
translation
A slide. Every point moves the same distance in the same direction.
reflection
A flip over a mirror line. Every point ends up the same distance from the line, on the other side.
mirror line
The line a shape is reflected in. It is also called the line of reflection.

Remember from Grade 4

In Grade 4 you translated (slid) and reflected (flipped) shapes on grids. You learned that the image is the same size and shape as the original, so it is congruent. This year you will use coordinates to describe these moves exactly, and predict where the image will land.

Let's start with a game piece

In a board game, you slide your piece 4 spaces right and 2 spaces up. Every corner of the piece moves the same way, and the piece still faces the same direction. That's a translation.

Now think of a mitten and its reflection in a frozen pond. The reflection is a flipped copy. A left mitten's reflection looks like a right mitten! That's a reflection.

Translating with coordinates

To translate a shape, move every vertex the same way:

  • Right means add to the x-number. Left means subtract from it.
  • Up means add to the y-number. Down means subtract from it.

Then join the new vertices in the same order.

0123456789101234567xyCC′ABA′B′4 right, 2 up
Triangle ABC with A (1, 1), B (3, 1) and C (1, 4) slides 4 units right and 2 units up to make triangle A′B′C′ with A′ (5, 3), B′ (7, 3) and C′ (5, 6).
Worked example

Example 1: Translate a triangle

Translate triangle A (1, 1), B (3, 1), C (1, 4) 4 units right and 2 units up. Give the coordinates of the image.

  1. Add 4 to each x-number and add 2 to each y-number.
  2. A (1, 1) → A′ (1 + 4, 1 + 2) = (5, 3)
  3. B (3, 1) → B′ (3 + 4, 1 + 2) = (7, 3)
  4. C (1, 4) → C′ (1 + 4, 4 + 2) = (5, 6)
  5. Check: the image faces the same way and is the same size.

A′ (5, 3), B′ (7, 3), C′ (5, 6)

Reflecting in a mirror line

  1. For each vertex, count how far it is from the mirror line, going straight across to the line.
  2. Count the same distance on the other side. Mark the image vertex.
  3. Join the image vertices.

A point on the mirror line stays where it is. A vertical mirror line has the same x-number all along it, like x = 5. A horizontal mirror line has the same y-number all along it, like y = 4.

0123456789101234567xymirror linex = 5 (dashed)
A triangle with vertices (2, 1), (4, 1) and (2, 4) is reflected in the vertical line x = 5. The image has vertices (8, 1), (6, 1) and (8, 4).
Worked example

Example 2: A vertical mirror line

Reflect the triangle with vertices (2, 1), (4, 1), (2, 4) in the mirror line x = 5.

  1. (2, 1) is 3 units left of the line (5 − 2 = 3). Go 3 units right of the line: 5 + 3 = 8. Image: (8, 1).
  2. (4, 1) is 1 unit left of the line. Go 1 unit right: 5 + 1 = 6. Image: (6, 1).
  3. (2, 4) is 3 units left of the line. Image: (8, 4).
  4. The y-numbers don't change, because the points move straight across.

The image has vertices (8, 1), (6, 1), (8, 4). It faces the opposite way.

0123456781234567xymirror line goesacross at y = 4
A triangle with vertices (1, 5), (4, 5) and (4, 7) is reflected in the horizontal line y = 4. The image has vertices (1, 3), (4, 3) and (4, 1).
Worked example

Example 3: A horizontal mirror line

Reflect the triangle with vertices (1, 5), (4, 5), (4, 7) in the mirror line y = 4.

  1. Now the points move straight down across the line, so the x-numbers stay the same.
  2. (1, 5) is 1 unit above the line. Go 1 unit below: 4 − 1 = 3. Image: (1, 3).
  3. (4, 5) → (4, 3).
  4. (4, 7) is 3 units above the line. Go 3 units below: 4 − 3 = 1. Image: (4, 1).

The image has vertices (1, 3), (4, 3), (4, 1). It is upside down compared with the original.

Which transformation was it?

  • Translation: the image faces the same way. Every vertex moved by the same amount.
  • Reflection: the image faces the opposite way, like a mirror image. Vertices close to the line stay close to it; far ones stay far.
Worked example

Example 4: Predict

A triangle is to the left of a vertical mirror line, with its point aimed right, toward the line. Predict what the reflection will look like.

  1. The image will be on the right side of the line.
  2. It will be the same size and shape.
  3. Its point will aim left, back toward the line, because a reflection faces the opposite way.

On the right of the line, same size, pointing left.

Predict first

Before you draw, close your eyes and picture the image. Where will it be? Which way will it face? Then draw it and compare. Making predictions, and checking them, trains your spatial brain.

Just for fun

Write your name on paper and hold it up to a mirror: the letters flip! That's why some ambulances have AMBULANCE written backwards on the front. In a driver's rear-view mirror, it reflects the right way around.

Hands-on activity

Mirror and slide

You need: grid paper, a ruler, a pencil, scissors (with a grown-up) and a small mirror if you have one (handle glass mirrors carefully).

  1. Draw coordinate axes from 0 to 10 on grid paper. Draw a triangle with vertices (1, 2), (3, 2), (1, 5).
  2. Cut out a congruent paper triangle. Place it on your triangle, then slide it 5 right and 1 up without turning. Trace it. Write the new coordinates.
  3. Draw a dashed mirror line at x = 5. Predict the reflection's coordinates first. Then flip your paper triangle over the line and trace it. Were you right?
  4. If you have a mirror, stand it upright along the dashed line. Does the image in the mirror match your drawing?

(Answers: step 2 gives (6, 3), (8, 3), (6, 6). Step 3 gives (9, 2), (7, 2), (9, 5).)

Math talk: "How can you tell a slide from a flip just by looking?"

Your turn

1

Translate the point (2, 3) 5 units right and 1 unit up. Where does it land?

2

Translate the triangle (1, 1), (4, 1), (1, 3) 3 units right and 4 units up. Give the image's vertices.

3

Reflect the point (3, 6) in the mirror line x = 5.

4

Reflect the point (6, 2) in the mirror line y = 4.

5

Reflect the point (5, 3) in the mirror line x = 5.

6

An image faces the opposite way from the original. Was it translated or reflected?

More practice: warm-up

1

What is another name for a translation?

2

What is another name for a reflection?

3

After a translation, is the image bigger, smaller or the same size?

4

What is another name for the mirror line?

5

Translate (0, 0) 3 units right and 2 units up.

More practice: on your own

1

Reflect the triangle (1, 2), (3, 2), (3, 5) in the mirror line x = 4.

2

A point moves from (2, 5) to (9, 1). Describe the translation. Then use the same translation on (4, 6).

3

A triangle has vertices (2, 2), (5, 2), (5, 4). Its image has vertices (2, 6), (5, 6), (5, 4). What transformation was it?

4

A shape sits far to the right of a vertical mirror line. Predict where its reflection will be.

5

Reflect (1, 1) in the line x = 3. Then translate the image 2 units right. Where is it now?

Challenge

1

You reflect a shape in the line x = 5, then reflect the image in x = 5 again. Where does it end up?

2

Point (2, 3) is reflected and its image is (8, 3). Where is the mirror line?

3

A rectangle with corners (1, 1), (3, 1), (3, 2), (1, 2) is reflected in x = 4. Then the original is translated 4 units right. Compare the two images. Why are they the same?

Check your understanding

Quiz

1.

Translate (4, 2) 3 units left and 5 units up.

2.

Reflect (2, 5) in the mirror line x = 4.

3.

Reflect (3, 1) in the mirror line y = 3.

4.

After which transformation does the image face the same way as the original?

5.

A point is on the mirror line. Where is its reflection?

6.

A triangle (1, 1), (2, 3), (4, 1) is translated so that (1, 1) moves to (6, 2). Where does (4, 1) move?

For parents and teachers

Big idea: The Ontario curriculum (E1.5) asks Grade 5 students to describe and perform translations, reflections and rotations up to 180° on a grid, and predict the results. This lesson covers translations and reflections with coordinates. The next lesson covers rotations.

Common difficulties:

  • Copying the shape across the mirror line without flipping it. That's a translation. Count from the line for each vertex.
  • Moving only some vertices. Every vertex must move.
  • Mixing up which number changes. In a vertical mirror line only x changes; in a horizontal one only y changes.

How to help: Use cut-out shapes so your child can physically slide and flip them. Ask for a prediction before drawing. Supervise if using a glass mirror.