E. Spatial Sense · Spatial Sense: 3D Objects, Transformations, Circles, Surface Area and Volume

Transformations on the Cartesian Plane

E1.4 A1About 50 minutes

Learning goals

I can describe translations, reflections and rotations on the Cartesian plane using coordinate rules.
I can reflect shapes in the axes and in other vertical or horizontal lines.
I can predict the coordinates and the quadrant of an image before I draw it.
I can identify which transformation was performed by comparing coordinates.

Words to know

Cartesian plane
A grid made by a horizontal x-axis and a vertical y-axis crossing at the origin (0, 0). It has four quadrants.

image
The shape you get after a transformation. Its vertices are labelled with primes, like A′.

coordinate rule
A rule that shows what happens to every point, like (x, y) → (x + 3, y − 2).

line of reflection
The mirror line in a reflection.

centre of rotation
The fixed point a shape turns around.

orientation
Which way a shape faces. A reflection reverses the orientation; translations and rotations do not.

congruent
Same size and same shape. Translations, reflections and rotations always give congruent images.

Remember from Grade 6

In Grade 6 you plotted points in all four quadrants, reflected shapes in the x-axis and the y-axis, rotated shapes around the origin by 90°, 180°, 270° and 360°, and combined transformations. You did most of it by counting squares.

In Grade 7 you will describe transformations with coordinate rules, reflect in lines other than the axes, and use rules to predict where a shape will land.

Let's start with a game designer

In a video game, the computer moves characters by changing their coordinates with rules. "Move 3 right and 2 down" becomes (x, y) → (x + 3, y − 2). A rule is fast, and it works for every point at once. Today you'll think like a game designer.

Transformation	Coordinate rule	Example: (2, 1) goes to
translation a right, b up (negative a = left, negative b = down)	(x, y) → (x + a, y + b)	3 right, 2 down: (5, −1)
reflection in the x-axis	(x, y) → (x, −y)	(2, −1)
reflection in the y-axis	(x, y) → (−x, y)	(−2, 1)
rotation 90° clockwise about the origin	(x, y) → (y, −x)	(1, −2)
rotation 90° counterclockwise about the origin	(x, y) → (−y, x)	(−1, 2)
rotation 180° about the origin	(x, y) → (−x, −y)	(−2, −1)

Coordinate rules for transformations. For rotations, you can still check with the Grade 6 counting method: for a 90° clockwise turn, right → down, up → right, left → up, down → left.

Triangle A(2, 1), B(5, 1), C(2, 4) is rotated 90° clockwise about the origin. Using (x, y) → (y, −x), the image is A′(1, −2), B′(1, −5), C′(4, −2). The image is congruent to the original and moves from quadrant I to quadrant IV.

Worked example

Example 1: Write a translation rule

A translation moves A(−3, 2) to A′(1, −1). Write the rule, then use it to find the image of B(0, 5).

Change in x: from −3 to 1 is 4 right (+4).
Change in y: from 2 to −1 is 3 down (−3).
Rule: (x, y) → (x + 4, y − 3).
B(0, 5) → (0 + 4, 5 − 3) = (4, 2).

Rule (x, y) → (x + 4, y − 3); B′(4, 2).

Worked example

Example 2: Reflect in a line that isn't an axis

Reflect P(5, 1) and Q(0, 3) in the vertical line x = 2 (the line through all points whose x-coordinate is 2).

P(5, 1) is 5 − 2 = 3 units to the right of the line. Its image is 3 units to the left: x = 2 − 3 = −1. The y-coordinate stays the same.
Q(0, 3) is 2 units to the left of the line. Its image is 2 units to the right: x = 2 + 2 = 4.
Check: the line x = 2 should be exactly halfway between each point and its image. Halfway between 5 and −1 is 2 ✓. Halfway between 0 and 4 is 2 ✓.

P′(−1, 1) and Q′(4, 3).

Reflecting in the line x = 2: each image is the same distance from the line as the original, on the other side. The y-coordinates do not change.

Worked example

Example 3: Which transformation was it?

Triangle A(2, 1), B(5, 1), C(2, 4) has image A′(1, −2), B′(1, −5), C′(4, −2). Identify the transformation.

The image is the same size, so it is a translation, reflection or rotation.
Look at A(2, 1) → A′(1, −2): the numbers switched places and one sign changed. That is a sign of a 90° rotation.
Test (x, y) → (y, −x): A → (1, −2) ✓, B → (1, −5) ✓, C → (4, −2) ✓.

A rotation of 90° clockwise about the origin (the picture above).

Worked example

Example 4: Predict, then check

A shape is in quadrant II. It is rotated 180° about the origin and then translated 2 units up. Which quadrant will the image be in? Check with the point (−4, 3).

Predict: a 180° turn sends quadrant II to the opposite quadrant, IV. A small move up might keep it in IV.
Check: (−4, 3) → 180°: (4, −3) → 2 up: (4, −1).
(4, −1) has positive x and negative y: quadrant IV ✓

Quadrant IV (for this point; a point closer to the x-axis could cross into quadrant I, so always check the vertices).

Congruent or not?

Translations, reflections and rotations slide, flip or turn a shape. They never change its size or angles, so the image is always congruent. A dilation (last lesson) changes the size, so its image is similar but not congruent (unless the scale factor is 1). A reflection also reverses the orientation: going around the vertices A → B → C, the image goes the other way around.

Hands-on activity

Floor-grid game

You need: masking tape (or chalk outside on a driveway, with a grown-up's OK), a small toy, paper and a pencil.

Make big x- and y-axes on the floor with tape. Mark 5 steps each way from the origin.
Put the toy on a point, like (2, 3).
A partner calls a rule, for example "(x, y) → (−x, y)" or "rotate 90° counterclockwise about the origin".
Before moving the toy, predict its new coordinates and quadrant. Write them down.
Move the toy and check. Switch roles.

Math talk: "Which rules swap the numbers? Which rules only change signs? Which keep the toy in the same quadrant?"

Just for fun

Why did the triangle feel dizzy?

It rotated 90°, then 90° again, then again, then again... and ended up right back where it started. (4 × 90° = 360°!)

Your turn

Use the rule (x, y) → (x − 4, y + 5) on (3, −2).

Reflect (−4, 6) in the x-axis.

Rotate (2, 5) 90° counterclockwise about the origin.

Rotate (−3, 1) 180° about the origin.

Write the rule for the translation that moves (1, 1) to (−2, 4).

Reflect (6, 2) in the line x = 3.

Which transformations always give an image that is congruent to the original?

More practice: warm-up

Use (x, y) → (x + 1, y + 1) on (2, 3).

Reflect (5, −1) in the y-axis.

Rotate (4, 4) 180° about the origin.

Rotate (0, 3) 90° clockwise about the origin.

Does a reflection change a shape's orientation?

More practice: on your own

Rotate triangle P(1, 2), Q(3, 2), R(1, 5) 90° clockwise about the origin.

Reflect (−2, 4) in the horizontal line y = 1.

(2, 3) → (−2, 3) and (5, 1) → (−5, 1). Name the transformation.

(1, 4) → (−4, 1) and (3, 2) → (−2, 3). Name the transformation.

A triangle with vertices (2, 3), (4, 3) and (3, 5) is reflected in the x-axis and then translated 10 units left. Which quadrant is the image in?

Challenge

Translate a shape with (x, y) → (x + 3, y − 1), then with (x, y) → (x − 5, y + 4). Write one rule that does both.

Reflect A(2, 5) in the vertical line x = −1.

Sam says a rotation of 270° counterclockwise about the origin has the rule (x, y) → (y, −x). Is he right? Explain.

Check your understanding

Quiz

Which rule is a reflection in the x-axis?

Use the rule (x, y) → (x − 2, y + 3) on (4, −1).

Rotate (3, 1) 90° clockwise about the origin.

Which transformation always keeps a shape the same size?

Reflect (7, 2) in the line x = 4.

The rule (x, y) → (−x, −y) is ...

For parents and teachers

Big idea: E1.4 asks students to describe and perform translations, reflections and rotations on a Cartesian plane and to predict the results. Grade 6 did this by counting; Grade 7 adds coordinate rules, reflections in lines like x = 2, and identifying a transformation from coordinates.

How to help at home: Play the floor-grid game, or plot points on grid paper. Always ask for a prediction first ("Which quadrant will it land in?"), then check.

Common difficulties:

Mixing up the x-axis and y-axis rules. Reflecting in the x-axis changes the y-coordinate. Think: the x-axis is the mirror, so points flip up and down.
Mixing up the 90° clockwise and counterclockwise rules. Check with the counting method, or with tracing paper.
Reflecting in x = 2 as if it were the y-axis. Measure the distance from the line, not from the origin.

This lesson builds A1 skills: making and testing predictions, and using clear math language and notation.