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Grade 8 · Mathematics · 30 minutes

Unit 3 · Geometry and bivariate evidence · Lesson 1 of 3

Transformations, Congruence and Similarity

Track vertices through rigid motions and a dilation.

Mr. S Teaches · Original starter mini-lesson · MS8-MATH-04

Learning objective

Describe coordinate transformations and justify congruence or similarity from a transformation sequence.

Before you begin

  • Coordinate plane
  • Scale factors

Gather your materials

  • Graph paper and ruler
  • Pencil
  • Optional tracing paper

Language to know

rigid motion
A movement that preserves lengths and angles.
dilation
A scaling from a center by one factor.
congruent
Same size and shape.
similar
Same shape, possibly with a different size.

Understand the concept

Translations, rotations and reflections preserve lengths and angles, so a sequence of them produces a congruent figure. A dilation changes lengths by its scale factor but preserves angle measures, producing a similar figure. Coordinate rules communicate how each point changes; they should be checked on every vertex, not guessed from one.

Teach it step by step

  1. Plot triangle A(1,1), B(3,1), C(1,2). Translate it by (+2,−1) and record all three new vertices.
  2. Reflect the original triangle across the y-axis and compare corresponding side lengths using the grid.
  3. Dilate the original by scale factor 2 centered at the origin; measure horizontal and vertical sides.
  4. Explain which pair is congruent and which is similar but not congruent.
  5. Ask the learner to describe a sequence from the original triangle to a translated reflection.

Make the thinking visible

Worked example or model response

Translation gives A′(3,0), B′(5,0), C′(3,1), congruent to ABC. Dilation gives (2,2),(6,2),(2,4): side lengths double, so the image is similar but not congruent to ABC.

Guided practice

Reflect P(−2,3) and Q(1,1) across the x-axis. Then dilate those original points by 1/2 from the origin.

Independent practice

1. Translate (−3,2) by (+4,+1). 2. A triangle is reflected and then rotated. Explain whether the image is congruent. 3. Is a scale-factor-3 image of a nondegenerate triangle congruent to its original?

Quick assessment

Does a dilation centered at the origin with factor 2 follow (x,y)→(x+2,y+2)? Correct the rule.

Answer key & assessment criteria

Answers and teaching guidance

  1. Guided: Reflections: (−2,−3),(1,−1). Half-size dilation: (−1,1.5),(0.5,0.5).
  2. Independent 1: (1,3). 2: Yes, each operation preserves distances and angles. 3: No; lengths triple although corresponding angles stay equal.
  3. Quick check: The dilation rule is (x,y)→(2x,2y); adding 2 is a translation.

Look for: Look for an explanation of the choices, not only a final answer. Require units and a check against the context.

Make it work for your learner

More support

Use tracing paper and colored corresponding vertices to separate movement from scaling.

Extension challenge

Compare reflecting then translating with translating then reflecting and find when their results differ.

Selected contributions · Official sources

Standards connections

Contributes to Common Core standard

8.G.A.2

Common Core · Grade 8

Use rigid-motion sequences to justify congruence.

Read the official standard ↗

This lesson contributes selected skills; it does not fully address the standard.

Source and mapping checked 2026-11-04. This mini-lesson contributes selected skills; it does not claim full attainment.

Contributes to Common Core standard

8.G.A.3

Common Core · Grade 8

Describe transformations with coordinate rules.

Read the official standard ↗

This lesson contributes selected skills; it does not fully address the standard.

Source and mapping checked 2026-11-04. This mini-lesson contributes selected skills; it does not claim full attainment.

Contributes to Common Core standard

8.G.A.4

Common Core · Grade 8

Use a dilation and rigid motions to explain similarity.

Read the official standard ↗

This lesson contributes selected skills; it does not fully address the standard.

Source and mapping checked 2026-11-04. This mini-lesson contributes selected skills; it does not claim full attainment.