E. Spatial Sense · Spatial Sense: Shapes, Grids, Angles, Area and Surface Area

Combining Transformations and Predicting Results

E1.4 A1About 45 minutes

Learning goals

I can do two or more transformations in a row and give the coordinates of the final image.
I can show that the order of transformations can change the result.
I can describe a combination of transformations that moves a shape from one place to another.

Words to know

combination of transformations
Two or more transformations done one after another. The image from the first step is the starting shape for the next step.

double prime (A″)
The label for a vertex after two transformations. Say "A double prime".

predict
To say what will happen before you do it, using what you know.

orientation
Which way a shape faces. A reflection reverses it; a translation or a rotation does not.

Let's start with a quilt

Quilters and beadworkers often make a whole design from one small shape. They slide it, flip it and turn it, again and again. Each piece of the pattern is the same shape after a combination of transformations.

Today you'll do transformations one after another. The rule is simple: the image from step 1 becomes the starting shape for step 2. We label the first image A′ ("A prime") and the second image A″ ("A double prime").

A navy and cream quilt covered in pinwheel blocks made from right triangles. — Every triangle in this quilt is the same shape. Look for the ones that were slid, flipped or turned to build each pinwheel. AI-generated image

Triangle ABC (A (1, 1), B (4, 1), C (1, 3)) is reflected in the x-axis to make A′B′C′ (dashed, in quadrant IV), then reflected in the y-axis to make A″B″C″ at (−1, −1), (−4, −1) and (−1, −3), in quadrant III.

Worked example

Example 1: Two reflections

Reflect triangle ABC (A (1, 1), B (4, 1), C (1, 3)) in the x-axis, then reflect the image in the y-axis.

Step 1, x-axis: y changes sign. A′ (1, −1), B′ (4, −1), C′ (1, −3).
Step 2, y-axis: x changes sign. A″ (−1, −1), B″ (−4, −1), C″ (−1, −3).
Look: both signs changed. That's exactly what a 180° rotation around the origin does!

A″ (−1, −1), B″ (−4, −1), C″ (−1, −3). Two reflections, one in each axis, give the same result as a 180° rotation around the origin.

Worked example

Example 2: Does the order matter?

Start at P (2, 3). Plan 1: translate 4 units right, then reflect in the y-axis. Plan 2: reflect in the y-axis first, then translate 4 units right. Do they end in the same place?

Plan 1: (2, 3) → translate 4 right → (6, 3) → reflect in the y-axis → (−6, 3).
Plan 2: (2, 3) → reflect in the y-axis → (−2, 3) → translate 4 right → (2, 3). It's back where it started!
(−6, 3) and (2, 3) are different points.

No. The order changed the result. Always do the steps in the order given.

The two plans from Example 2 on a grid. Same two steps, different order, different endings: Plan 1 lands at (−6, 3), but Plan 2 lands back at (2, 3).

Plan	After step 1	After step 2
Translate 4 right, then reflect in the y-axis	(6, 3)	(−6, 3)
Reflect in the y-axis, then translate 4 right	(−2, 3)	(2, 3)

Two plans for P (2, 3) with the same steps in a different order.

Worked example

Example 3: Find the combination

Triangle ABC has A (1, 1), B (4, 1), C (1, 3). Its final image is A″ (−1, −4), B″ (−4, −4), C″ (−1, −2). Describe a combination of transformations that does this.

Compare the shapes. In ABC, B is to the right of A, and C is above A. In the image, B″ is to the left of A″, but C″ is still above A″. Left and right swapped, but up and down did not. That is what a reflection in the y-axis does.
Reflect ABC in the y-axis: (−1, 1), (−4, 1), (−1, 3).
Compare with the final image: every x matches, and every y is 5 less. So slide 5 units down.

A reflection in the y-axis, then a translation 5 units down. (Other answers can also be right, such as translating 5 units down first and then reflecting in the y-axis.)

Just for fun

Getting dressed is a combination where order matters: socks then shoes works much better than shoes then socks!

Track it in a table

For combinations, make a table with one column for each step. Write the coordinates after every step. If you make a mistake, the table shows you exactly where. Staying organized like this helps you stay calm with long problems, and you can check each step on grid paper.

Hands-on activity

Design a transformation pattern

You need: grid paper, a ruler, coloured pencils and tracing paper (or baking parchment).

Draw axes from −8 to 8. Draw a small shape in quadrant I, like a flag or the letter L made of grid squares.
Choose a combination of two transformations, for example "reflect in the y-axis, then translate 2 units down". Write it down.
Before drawing, predict the final coordinates of each vertex in a table.
Draw the image after step 1 in pencil and after step 2 in colour. Was your prediction right?
Repeat with other combinations to fill all four quadrants and make a pattern.
Give your finished pattern to your grown-up. Can they figure out which combinations you used?

Math talk: "When did the order of your steps matter?"

Your turn

Start at (3, 1). Translate 2 units left, then reflect in the x-axis. Where does it end?

Start at (3, 1). Reflect in the x-axis first, then translate 2 units left. Where does it end? Is it the same as the last answer?

Start at (2, 4). Reflect in the y-axis, then rotate 180° around the origin. Where does it end? Which single reflection gives the same result?

Start at (−1, 2). Rotate 90° clockwise around the origin, then translate 3 units right.

Start at (4, −2). Translate 3 units up, then reflect in the y-axis.

Start at (1, 2). Plan A: translate 3 units down, then rotate 180° around the origin. Plan B: rotate 180°, then translate 3 units down. Compare.

More practice: warm-up

In a combination, what shape do you start with for step 2?

Start at (2, 2). Translate 1 unit right, then 1 unit up.

Reflect (3, 5) in the y-axis, and then reflect the image in the y-axis again. Where does it end?

A point is translated 2 units right, then 5 units right. Describe one translation that does the same thing.

Two rotations of 90° clockwise around the same point make one rotation of how many degrees?

More practice: on your own

Start at (−3, 1). Reflect in the x-axis, then translate 4 units right and 2 units up.

Start at (5, 2). Rotate 90° counterclockwise around the origin, then reflect in the y-axis.

Triangle (1, 1), (3, 1), (1, 2) is translated 4 units left, then reflected in the x-axis. Give the final vertices and the quadrant.

Use (1, 2), "translate 3 units right" and "reflect in the y-axis" to show that order can matter.

A point ends at (2, −3) after a reflection in the x-axis. Where did it start?

Challenge

A point is reflected in the y-axis and then translated 6 units right. It ends at (4, 1). Where did it start?

Square ABCD has A (1, 1), B (3, 1), C (3, 3), D (1, 3). Its final image is A″ (−1, −2), B″ (−3, −2), C″ (−3, −4), D″ (−1, −4). Describe a combination of two transformations that does this.

Predict: where does (3, −2) end up after six rotations of 90° clockwise around the origin?

Check your understanding

Quiz

Start at (2, 5). Reflect in the x-axis, then reflect in the y-axis. Where does it end?

Reflecting in the x-axis and then in the y-axis gives the same result as which single transformation?

Start at (1, 1). Translate 3 units left, then 2 units down. Where does it end?

Start at (2, 0). Rotate 90° counterclockwise around the origin, then translate 4 units right.

What does the double prime in A″ tell you?

Which statement is true?

For parents and teachers

Big idea: E1.4 asks students to describe and perform combinations of translations, reflections and rotations, and to predict the results. The key habit is that the image from one step becomes the starting shape for the next step.

Common difficulties:

Applying both steps to the original shape. Insist on writing the coordinates after each step in a table.
Assuming order never matters. Try Example 2 together so your child sees a case where it does.
Working backwards. To find a starting point, undo the steps in reverse order.

Social-emotional learning (A1): Long, multi-step problems can feel overwhelming. Encourage your child to break them into steps, use a table, and check each step. Praise persistence.

How to help: Make patterns together on grid paper using one shape and different combinations. Ask "Can you predict where it will land before you draw it?"