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Transformations

Students classify and perform rigid transformations—translations, rotations, and reflections—and the non-rigid transformation of dilation on the coordinate plane.

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Beginner 15–20 minutes Triangles & Transformations

Triangles & Transformations Page 2 of 8

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At a Glance

Transformations

Binder SectionTriangles & Transformations

DifficultyBeginner

Estimated Time15–20 minutes

PrerequisitesNone

Essential Question

How do transformation rules predict the location and size of an image?

Build Understanding

A transformation maps every point of a preimage to a corresponding point in an image. Label the image with prime marks and apply the rule to every vertex.

RIGID TRANSFORMATIONS
Rigid transformations preserve both size and shape. Corresponding lengths and angle measures remain equal.

Translations slide a figure without turning or flipping it. The rule (x,y) -> (x+a,y+b) moves every point a units horizontally and b units vertically.

Rotations turn a figure around a fixed center through a stated angle and direction. For example, a 90-degree counterclockwise rotation about the origin follows (x,y) -> (-y,x).

Reflections flip a figure across a line of reflection. Reflection over the x-axis follows (x,y) -> (x,-y), and reflection over the y-axis follows (x,y) -> (-x,y).

NON-RIGID TRANSFORMATIONS
A dilation is non-rigid because it changes a figure's size unless the scale factor is 1. A dilation preserves shape and angle measures while multiplying every length by the absolute value of scale factor k. About the origin, its rule is (x,y) -> (kx,ky). If k > 1, the image is an enlargement; if 0 < k < 1, it is a reduction.

A sequence may combine transformations. Because order can change the final image, perform and record each transformation in the order given.

Visual Model

4Thousands 3Hundreds 2Tens 7Ones

4,327 = 4,000 + 300 + 20 + 7

Goals

Learning Targets

Classify translations, rotations, and reflections as rigid transformations.
Classify dilation as a non-rigid transformation.
Explain what each transformation preserves and changes.
Apply coordinate rules to every vertex.
Describe a transformation using its direction, distance, center, angle, line, or scale factor.
Compose more than one transformation in the stated order.

Click each word

Interactive Vocabulary

Reveal one step at a time

Worked Examples

Example 1

Translate (−2,3) right 5 and down 1

Add (5,−1).

Example 2

Rotate (4,−1) 90° counterclockwise about the origin

Use (x,y)→(−y,x).

Ask and explain

Common Questions

What does a prime mark mean? It labels the image after a transformation.

Which transformations are rigid? Translations, rotations, and reflections.

Which transformation is non-rigid? Dilation, because it generally changes size.

Does order matter in a sequence? Usually yes.

Does a dilation move points? Yes, unless a point is the center.

What does a dilation preserve? Shape and angle measures, but not usually length or area.

Notice and correct

Common Misconceptions

Changing coordinate signs without using the correct rule

State the transformation and rule first.

Confusing the preimage and image

Use unprimed and primed labels.

Assuming every transformation preserves size

Dilations change size unless the scale factor is 1.

Learn Through Video

Watch, pause, and explain

Use the chapter markers, key vocabulary, and reflection prompts to stay actively engaged while you watch.

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Now Playing

Translate a Shape on the Coordinate Plane - Midnight Math Tutor

1 Translate a Shape on the Coordinate Plane - Midnight Math Tutor 2 Rotate a Shape 90 Degrees Counterclockwise - Midnight Math Tutor 3 Thinking About Dilations - Khan Academy

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Warm-Up Guided Independent Challenge

Quick Start

Warm-Up

Warm-up 1

Translate (1,−2) by ⟨3,4⟩.

We Do

Guided Practice

Guided 1

Translate (2,-3) right 4 and up 1. Answer: (6,-2).

Guided 2

Reflect (-5,2) over the y-axis. Answer: (5,2).

Guided 3

Rotate (3,1) 90 degrees counterclockwise. Answer: (-1,3).

Guided 4

Dilate (-2,4) by k = 3 about the origin. Answer: (-6,12).

Guided 5

Name a transformation that changes orientation. Answer: reflection.

You Do

Independent Practice

Independent 1

Reflect (−6,2) across y=x.

Independent 2

Dilate (4,−2) by factor 1/2 from the origin.

Independent 3

Describe the change from (1,2) to (−2,1).

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Success Criteria

Before you begin, I can…

I can name the transformation from its coordinate changes.
I apply the rule to every vertex.
I check preserved lengths, angles, orientation, or scale as appropriate.

Question 1

Which statement shows the key idea in this lesson?

Question 2

Which habit best supports accuracy?

Question 3

What should a student do after solving?

Question 4

Which explanation is strongest?

Question 5

How can an error be found?