Combine translations, reflections, and rotations on a grid

Ontario Grade 6 Mathematics

Spatial Sense

Slide, flip, and turn a shape in the order given

Imagine a small paper triangle on a square grid. You can slide it, flip it over a line, or turn it around a point. Each move changes where the triangle is or which way it faces. To combine moves, finish one move before starting the next. The new position after one move becomes the starting position for the following move. In this lesson, you will use labelled corners and grid squares to keep track of the changes.

What you will learn

Identify translations, reflections, and rotations on a grid.
Track each vertex through a sequence of moves.
Describe and check the final position of a shape after combined transformations.

Start with the grid and the three moves

A grid is made of evenly spaced horizontal and vertical lines. A point is a location where grid lines meet. We can name a point with an ordered pair, such as

(2,1)

. The first number tells how far right or left the point is from the centre line. The second tells how far up or down it is. The centre point, where the two centre lines meet, is called the origin. A vertex is a corner of a shape.

A translation is a slide. Every point moves the same number of grid spaces in the same direction. The shape does not turn or flip. For example, if a shape is translated two spaces right and one space down, move each of its vertices two spaces right and then one space down.

A reflection is a flip across a line. That line is called the mirror line. Each vertex moves to the other side of the mirror line, the same distance away. A vertex on the mirror line stays where it is.

A rotation is a turn around a fixed point, called the centre of rotation. A quarter-turn is one of four equal turns in a full turn. A half-turn is two quarter-turns. The direction matters: clockwise turns the way the hands of a clock move; counterclockwise turns the other way.

A translation slides, a reflection flips, and a rotation turns.
A vertex is a shape’s corner. Label vertices so you can follow them.
Use the grid spaces to show each move.

Combine moves one at a time

Read the whole instruction before moving the shape. If it says “translate, then reflect,” do the translation first. Do not start the reflection from the original position. Start it from the translated position.

A useful method is to trace the shape on paper or draw it on a grid. Mark and label every vertex. After the first move, plot and label the new positions. Then use those positions for the next move. Keep the same vertex name throughout, such as

A

for the original corner, A' after the first move, and A'' after the second move.

For a rotation, keep the centre of rotation fixed. Turn the shape around that point in the direction and amount named. For a reflection, use the stated mirror line. Naming the line and the centre makes the directions clear.

Each move must be done to every vertex. Join the moved vertices in the same order as before. This gives the new shape. Check the result by looking at the path of each labelled vertex and asking whether it followed every instruction.

The order of the moves matters.
After each move, use the new location of every vertex.
Keep vertex labels and join the corners in the same order.

Use a drawing to check your work

A grid drawing is a record of where the shape goes. For a translation, check that each vertex moved the same way. For example, if one vertex moves three spaces right and one space up, every other vertex must do the same.

For a reflection, check the mirror line. A point and its image should be on opposite sides of the line and equally far from it. For a rotation, check that the shape has turned around the stated centre in the stated direction. Do not change the centre while turning.

A coordinate list can help when a drawing has several vertices or several moves. Write down the original locations, then the locations after each move. Coordinates are names for grid points; they do not replace the need to follow the stated movement.

If the final drawing looks wrong, check the first move before changing the last one. A small mistake early in the sequence affects every move that follows.

Check all vertices, not only one corner.
Use the correct mirror line or rotation centre.
Check the first move if the final position seems wrong.

Build a clear solution

Before moving a shape, mark its vertices and note the starting locations. Read the moves in order. After each move, draw the shape again or record where each labelled vertex lands. This makes it easier to see whether you have used the newest position.

You can explain your work in words: name the move, state its direction or line, and say where the labelled vertices go. For example, say that a triangle was translated right, then reflected over a named line. A different order describes a different sequence, so include the word “then.”

When you practise, begin with one move and check it. Then try two moves. For a longer sequence, keep the same careful routine: complete one move, record the new shape, and continue. You do not need a special shortcut. The grid and the vertex labels show what happens.

Name each move and give its direction, line, or centre.
Record the shape after every move.
Use the grid to explain how each vertex travelled.

Worked example

Translate, then reflect

Triangle

ABC

has vertices

A(1,1)

,

B(3,1)

, and

C(1,2)

. Translate it two spaces right and one space up. Then reflect it across the vertical axis. Find its final vertices.

Final triangle

The labelled points show the final vertices after the two stated moves.

Translate the triangle
Move each vertex two grid spaces right and one space up. For example, $A(1,1)$ lands at $(3,2)$ . Apply the same movement to the other two vertices.
$A'(3,2),\ B'(5,2),\ C'(3,3)$
Reflect the new triangle
Now use the translated points. The vertical axis is the mirror line. Each point lands the same distance on the other side of that line, at the same height.
$A''(-3,2),\ B''(-5,2),\ C''(-3,3)$

Answer: The final vertices are

A''(-3,2)

,

B''(-5,2)

, and

C''(-3,3)

.

Check: For instance,

A

first moves from

(1,1)

to

(3,2)

. Its reflection is three spaces left of the vertical axis at the same height,

( -3,2)

. The other vertices follow the same two moves.

Worked example

Rotate, then translate

Triangle

PQR

has vertices

P(1,0)

,

Q(2,0)

, and

R(1,1)

. Turn it a quarter-turn counterclockwise around the origin. Then translate it three spaces right. Find its final vertices.

Final triangle

The labelled points show the final vertices after turning and translating.

Turn around the origin
Picture each point travelling a quarter-turn counterclockwise around the origin. For example, $P(1,0)$ turns to $(0,1)$ . The other vertices turn around the same centre in the same direction.
$P'(0,1),\ Q'(0,2),\ R'(-1,1)$
Translate the turned triangle
Move every new point three spaces right. The point $R'(-1,1)$ , for example, lands at $(2,1)$ . Make the same move for P' and Q'.
$P''(3,1),\ Q''(3,2),\ R''(2,1)$

Answer: The final vertices are

P''(3,1)

,

Q''(3,2)

, and

R''(2,1)

.

Check: The turned points are

(0,1)

,

(0,2)

, and

(-1,1)

. Each then moves three spaces right, giving the final points shown.

Worked example

Follow three moves in order

Triangle

XYZ

has vertices

X(1,1)

,

Y(2,1)

, and

Z(1,2)

. Translate it one space left. Reflect it across the vertical axis. Then turn it a quarter-turn counterclockwise around the origin. Find its final vertices.

Final triangle

The labelled points show the final vertices after all three moves.

Translate left
Move each point one space left. The points become $(0,1)$ , $(1,1)$ , and $(0,2)$ , in the order $X,Y,Z$ .
$X'(0,1),\ Y'(1,1),\ Z'(0,2)$
Reflect across the vertical axis
Use the translated points as the starting positions. The first and third points lie on the vertical axis, so they stay there. The second point moves the same distance to the left side.
$X''(0,1),\ Y''(-1,1),\ Z''(0,2)$
Turn counterclockwise
Turn the reflected triangle a quarter-turn counterclockwise around the origin. Follow each reflected point, not its original position.
$X'''(-1,0),\ Y'''(-1,-1),\ Z'''(-2,0)$

Answer: The final vertices are

X'''(-1,0)

,

Y'''(-1,-1)

, and

Z'''(-2,0)

.

Check: The three moves were completed in order. The points on the vertical axis stayed in place during the reflection, then moved during the rotation.

Common mistakes and how to avoid them

Doing the moves in a different order from the instruction.

Correction: Read the complete sequence first. Finish one move before beginning the next.

Moving only one vertex during a translation.

Correction: A translation moves every vertex the same number of spaces in the same direction.

Using the original shape as the starting point for every move.

Correction: After each move, use the new position of the shape.

Reflecting over the wrong line or turning around the wrong point.

Correction: Check the named mirror line or centre of rotation before moving the shape.

Lesson summary

A translation slides a shape, a reflection flips it over a line, and a rotation turns it around a fixed point.
Label vertices and track each one on the grid.
For combined moves, follow the stated order and use each new position for the next move.

Check your understanding

Question 1

Point

(2,1)

is translated one space left and then reflected across the vertical axis. Where does it finish?

$(1,1)$
$(-1,1)$
$(1,-1)$
$(-3,1)$

Show answer and explanation

(-1,1)

The point first moves to

(1,1)

. It then reflects to the same height on the other side of the vertical axis, at

(-1,1)

.

Question 2

A triangle is turned a quarter-turn counterclockwise around the origin. One vertex starts at

(1,2)

and lands at

(-2,1)

. Which description matches the move?

A quarter-turn counterclockwise around the origin
A quarter-turn clockwise around the origin
A reflection across the vertical axis
A translation two spaces left

Show answer and explanation

A quarter-turn counterclockwise around the origin

The stated counterclockwise quarter-turn sends the point at

(1,2)

to

(-2,1)

. The other choices describe different moves.

Question 3

A shape is translated two spaces right and one space down. Which movement should each vertex make?

Two spaces right and one space down
Two spaces left and one space up
Two spaces right and one space up
One space right and two spaces down

Show answer and explanation

Two spaces right and one space down

A translation moves every vertex the same distance and direction, so each vertex moves two spaces right and one space down.

Key terms

Grid: A set of evenly spaced lines used to show positions and movements.
Ordered pair: Two numbers that name a point’s position on a grid.
Vertex: A corner of a shape.
Translation: A slide that moves every point the same distance in the same direction.
Reflection: A flip across a line called the mirror line.
Rotation: A turn around a fixed point.
Centre of rotation: The fixed point around which a shape turns.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 6 Mathematics, expectation E1.4. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

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E1.4 · Combine translations, reflections, and rotations on a grid

Ontario Grade 6 Mathematics

Slide, flip, and turn a shape in the order given

What you will learn

Start with the grid and the three moves

Combine moves one at a time

Use a drawing to check your work

Build a clear solution

Translate, then reflect

Rotate, then translate

Follow three moves in order

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 6 Math

About this lesson and its review