Geometric Transformations: Dilations

Active learning with hands-on tools helps students visualize how scale factors stretch or shrink figures from a center point. Using grid paper and measuring tools gives them direct evidence of how distances change while shapes stay the same. This approach builds intuition before moving to abstract reasoning, which is especially important for visual learners in Grade 7 geometry.

• Grade 7
• Mathematics
• 4 activities
• 25 min–40 min

Learning Objectives

1. Calculate the coordinates of the vertices of a dilated image given the original coordinates, a scale factor, and the center of dilation.
2. Compare the side lengths and angle measures of a pre-image and its dilated image to identify invariant properties.
3. Construct a dilated image of a 2D figure on graph paper using a specified scale factor and center of dilation.
4. Explain the effect of a scale factor greater than 1, between 0 and 1, and equal to 1 on the size of a dilated figure.
5. Analyze the relationship between the distance of a vertex from the center of dilation and its corresponding vertex in the image.

Project-Based Learning

⏱ 35 min·Small Groups

Small Groups: Grid Dilations

Provide grid paper with pre-drawn figures. Groups choose a center point and scale factor, plot corresponding image points by multiplying distances from the center, and connect to form the dilated shape. Discuss how sides and angles compare to the original.

Prepare & details

Analyze how a dilation changes the size of a figure while preserving its shape.

Facilitation Tip: During Grid Dilations, remind students to draw rays from the center through each vertex to locate image points precisely.

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·Pairs

Pairs Practice: Center Variations

Partners draw a triangle, then dilate it using three different centers: inside, on a vertex, and outside. They measure and record distance changes for each. Pairs swap papers to verify each other's work.

Prepare & details

Explain the role of the scale factor and center of dilation in a transformation.

Facilitation Tip: For Center Variations, circulate and ask guiding questions like, 'How does moving the center change the position of the image?' to prompt deeper observation.

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

⏱ 25 min·Whole Class

Whole Class Demo: Scale Factor Ladder

Project a figure. Teacher guides class to dilate it step-by-step with factors 0.5, 1, 2, and 3 from one center, plotting points together on a shared grid. Class notes size trends and measures key distances.

Prepare & details

Construct a dilated image of a figure given a scale factor and center.

Facilitation Tip: In Scale Factor Ladder, pause after each step to let students explain their predictions aloud 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

⏱ 40 min·Individual

Individual Challenge: Polygon Scaling

Students select a irregular polygon, pick a center, and apply a given scale factor to create the image. They label corresponding vertices and calculate side length ratios to confirm similarity.

Prepare & details

Analyze how a dilation changes the size of a figure while preserving its shape.

Facilitation Tip: For Polygon Scaling, provide rulers and protractors so students can verify congruence of angles and side ratios after dilation.

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 often start with physical tools like grid paper and rulers because geometric transformations are abstract until students can see and measure them. Avoid rushing to formulas; instead, let students discover patterns through repeated constructions. Research shows that students who plot points and measure distances themselves retain the concept longer. Use guided questions to steer correct thinking without giving away answers, and encourage students to compare results in small groups to catch their own errors.

What to Expect

Students will confidently identify the center of dilation, apply scale factors accurately, and explain why angles and shape remain unchanged. They will use measurements to verify their work and discuss transformations with peers to solidify understanding. By the end of these activities, they should articulate how scale factors affect size while preserving proportions.

Watch Out for These Misconceptions

Common Misconception

During Grid Dilations, watch for students who believe dilations distort shapes because they see the figure change size on the grid.

What to Teach Instead

Ask students to measure the angles with protractors and compare side lengths to the original, emphasizing that proportional scaling preserves shape. Have them present their findings to the group to reinforce the idea.

Common Misconception

During Center Variations, listen for students who assume all points move the same distance from their original positions.

What to Teach Instead

Guide them to measure distances from the center to each vertex in the original and image, then compare. Ask, 'Why does Point A move 2 cm but Point C moves 4 cm?' to highlight the role of the center.

Common Misconception

During Scale Factor Ladder, watch for students who assume the center must be inside or touching the figure.

What to Teach Instead

Provide examples with centers outside the figure and ask them to trace rays to locate image points. Have peers sketch centers in different positions to normalize this flexibility.

Assessment Ideas

Quick Check

After Grid Dilations, give students a polygon on graph paper, a center, and a scale factor. Ask them to calculate the new coordinates and sketch the image. Collect work to check for accurate plotting and proportional scaling.

Exit Ticket

During Center Variations, provide two similar figures with the center marked. Ask students to measure distances from the center, determine the scale factor, and write one sentence explaining how it affects the size.

Discussion Prompt

After Polygon Scaling, pose the question, 'What happens to the angles of a figure when it is dilated?' Have students discuss in pairs using their constructions, then facilitate a whole-class discussion to confirm that angles remain congruent.

Extensions & Scaffolding

• Challenge students to create a dilated image with a fractional scale factor (e.g., 0.5) and explain how it differs from whole-number scale factors.
• Scaffolding: Provide pre-labeled grid paper with the center marked and scale factor given to reduce setup time for struggling students.
• Deeper exploration: Have students explore how dilations relate to similar triangles by drawing a triangle, dilating it, and identifying corresponding parts in the original and image.

Key Vocabulary

Dilation	A transformation that changes the size of a figure but not its shape. It scales the figure from a central point.
Scale Factor	The ratio of the distance from the center of dilation to a point on the image to the distance from the center of dilation to the corresponding point on the original figure. It determines how much the figure is enlarged or reduced.
Center of Dilation	The fixed point from which all points on the original figure are scaled to create the dilated image.
Pre-image	The original figure before a transformation is applied.
Image	The figure that results after a transformation has been applied.