Dilations and Similarity

Active learning builds spatial reasoning when students manipulate shapes rather than watch demonstrations. For dilations and similarity, hands-on graphing and physical models turn abstract scale factors into visible patterns in side lengths, perimeters, and areas.

• 10th Grade
• Mathematics
• 4 activities
• 30 min–45 min

Learning Objectives

1. Calculate the scale factor of a dilation given corresponding side lengths of two similar polygons.
2. Compare the perimeters of two similar polygons by applying the scale factor to the original perimeter.
3. Analyze how the area of a polygon changes when its dimensions are scaled by a factor of k, resulting in an area scaling factor of k squared.
4. Explain the conditions required for two polygons to be classified as similar, focusing on angle congruence and side proportionality.
5. Demonstrate how to use similar triangles and a known height to indirectly measure the height of an inaccessible object, such as a tall tree or building.

Project-Based Learning

⏱ 35 min·Pairs

Pairs: Graph Paper Dilations

Partners draw a triangle on grid paper, then create dilations from a chosen center using scale factors 2 and 0.5. They measure sides, perimeters, and areas before and after, recording ratios in a table. Pairs compare results and check angle equality to confirm similarity.

Prepare & details

Analyze how a change in scale factor affects the perimeter versus the area of a polygon.

Facilitation Tip: During Graph Paper Dilations, ask pairs to label original and image points with matching colors to make corresponding sides and angles obvious before measuring.

Setup: Flexible workspace with access to materials and technology

Materials: Project brief with driving question, Planning template and timeline, Rubric with milestones, Presentation materials

• Apply
• Analyze
• Evaluate
• Create
• Self-Management
• Relationship Skills
• Decision-Making

Project-Based Learning

⏱ 45 min·Small Groups

Small Groups: Shadow Proportions

Groups measure shadows and heights of classmates and yard objects at noon. They form proportions assuming similar triangles from the sun's rays, solving for unknown heights. Groups graph their data and discuss how scale factors relate real-world sizes.

Prepare & details

Differentiate what defines two figures as similar in terms of transformations.

Facilitation Tip: In Shadow Proportions, give each small group a fixed light source and objects of different heights so they repeatedly align edges and record data together.

Setup: Flexible workspace with access to materials and technology

Materials: Project brief with driving question, Planning template and timeline, Rubric with milestones, Presentation materials

• Apply
• Analyze
• Evaluate
• Create
• Self-Management
• Relationship Skills
• Decision-Making

Project-Based Learning

⏱ 40 min·Whole Class

Whole Class: Transparency Transformations

Project figure pairs; class identifies potential similarities. Students overlay transparencies to test dilations and other transformations, noting scale factors. Facilitate a vote and explain criteria, with volunteers demonstrating at the board.

Prepare & details

Explain how similarity can be used to measure objects that are physically inaccessible.

Facilitation Tip: For Transparency Transformations, circulate with a ruler to check that students slide—not rotate—the transparency when verifying congruence of rotated or reflected shapes.

Setup: Flexible workspace with access to materials and technology

Materials: Project brief with driving question, Planning template and timeline, Rubric with milestones, Presentation materials

• Apply
• Analyze
• Evaluate
• Create
• Self-Management
• Relationship Skills
• Decision-Making

Project-Based Learning

⏱ 30 min·Individual

Individual: Scale Model Calculations

Each student chooses a room object, measures it, and designs a scale model with k=0.25. They compute required material lengths and areas, then sketch and label. Share one insight in a quick class roundup.

Prepare & details

Analyze how a change in scale factor affects the perimeter versus the area of a polygon.

Facilitation Tip: While students work on Scale Model Calculations, provide calculators but require hand-drawn diagrams so they connect numeric outputs to visual proportions.

Setup: Flexible workspace with access to materials and technology

Materials: Project brief with driving question, Planning template and timeline, Rubric with milestones, Presentation materials

• Apply
• Analyze
• Evaluate
• Create
• Self-Management
• Relationship Skills
• Decision-Making

Teaching This Topic

Teachers anchor similarity in transformations first, not just ratios, because students need to see that only dilations preserve shape while changing size. Avoid rushing to the formula; instead, insist on visual and numeric evidence side by side. Research shows that students who physically dilate shapes on grids develop stronger proportional reasoning than those who only compute scale factors from given coordinates.

What to Expect

Students will confidently verify similarity by showing equal angles and proportional sides, distinguish linear from quadratic scaling, and articulate how center and scale factor control a dilation. They will use grid measurements and transformations to justify each step.

Watch Out for These Misconceptions

Common Misconception

During Graph Paper Dilations, watch for students who assume the area scales by the same factor as the sides.

What to Teach Instead

Have students outline each dilated shape in a different color, count grid squares to find area, and plot ordered pairs (scale factor, area) on a class chart to reveal the quadratic pattern.

Common Misconception

During Transparency Transformations, watch for students who equate any size change with similarity.

What to Teach Instead

Give each group a transparency with a pre-marked polygon and a non-dilation tool (e.g., scissors) to show how cutting alters angles and ruins proportionality.

Common Misconception

During Shadow Proportions, watch for students who think any two triangles formed by shadows are automatically similar.

What to Teach Instead

Require students to measure both shadows and one height, then use a ratio table to test proportionality before concluding similarity.

Assessment Ideas

Exit Ticket

After Scale Model Calculations, give each student a rectangle with sides 5 cm by 7 cm and ask them to dilate it by 1.5 centered at the origin on graph paper, then compute the perimeter and area of both shapes and explain how each scales.

Quick Check

During Graph Paper Dilations, collect one pair of corresponding sides from each pair and ask students to state the scale factor and write the ratio of the longer side to the shorter side to verify proportionality.

Discussion Prompt

After Shadow Proportions, pose the prompt and facilitate the class discussion: ‘Using the shadow method you just practiced, how would you determine the height of a tree if your height is 1.7 m, your shadow is 0.85 m, and the tree’s shadow is 6 m? Students must sketch the triangles and explain the proportional reasoning step by step.

Extensions & Scaffolding

• Challenge students to design a composite figure (two non-overlapping polygons) and dilate it by two different scale factors, then prove the final image is similar to the original.
• Scaffolding: Provide a partially labeled table for Shadow Proportions so students fill in only one missing entry before predicting the next.
• Deeper exploration: Introduce negative scale factors and ask students to analyze what a reflection combined with a dilation looks like on the coordinate plane using Transparency Transformations.

Key Vocabulary

Dilation	A transformation that changes the size of a figure but not its shape. It is defined by a center point and a scale factor.
Scale Factor (k)	The ratio of the lengths of corresponding sides of two similar figures. A scale factor greater than 1 indicates enlargement, while a factor between 0 and 1 indicates reduction.
Similar Figures	Two figures are similar if they have the same shape but not necessarily the same size. Their corresponding angles are congruent, and the ratio of their corresponding side lengths is constant (the scale factor).
Corresponding Sides	Sides in similar figures that are in the same relative position and are proportional to each other.