Mathematics · Grade 8

Active learning ideas

Dilations and Scale Factor

Active learning helps students see how scale factors stretch or shrink figures proportionally, correcting common visual misconceptions. Hands-on work with grids, geoboards, and projections builds spatial reasoning that static diagrams cannot provide.

• Ontario Curriculum Expectations
• 8.G.A.3
• 8.G.A.4

• 25–40 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Project-Based Learning
• 30 min · Pairs

Pairs: Coordinate Grid Dilations

Partners plot a simple polygon on graph paper and choose a center point outside it. They apply a scale factor of 2 or 0.5 by multiplying distances from the center to each vertex, then connect new points. Pairs measure sides and angles to confirm similarity.

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

Facilitation Tip

During Coordinate Grid Dilations, circulate to ensure pairs measure distances from the center before and after dilation to reinforce proportional reasoning.

What to look for

Provide students with a simple triangle plotted on a coordinate grid and a center of dilation. Ask them to calculate the coordinates of the image vertices after a dilation with a scale factor of 2. Check their calculations for accuracy in applying the scale factor to each coordinate.

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

Activity 02

• Project-Based Learning
• 40 min · Small Groups

Small Groups: Geoboard Scale Challenges

Equip groups with geoboards and rubber bands to create initial figures. Select a center peg and scale factor, then build the dilated image nearby. Groups use rulers to check proportional distances and share findings on chart paper.

Analyze the effect of a scale factor greater than one versus less than one on a figure's dimensions.

Facilitation Tip

For Geoboard Scale Challenges, ask groups to record scale factors and side lengths in a shared table to make patterns visible for discussion.

What to look for

Present students with an image of a square and its dilated version. Ask them to determine the scale factor used for the dilation and explain how they know. Include a prompt asking them to describe the relationship between the side lengths of the original and the image.

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

Activity 03

• Project-Based Learning
• 25 min · Whole Class

Whole Class: Shadow Projection Demo

Project light on objects like blocks to cast shadows on a wall. Measure object-to-light and shadow distances to calculate scale factors. Class discusses how changing light position shifts the center of dilation.

Construct the image of a figure after a given dilation from a center point.

Facilitation Tip

In the Shadow Projection Demo, pause after each center change to let students sketch rays and predict image positions before measuring.

What to look for

Pose the question: 'If you dilate a figure with a scale factor of 0.5, what happens to its area compared to the original?' Facilitate a class discussion where students share their predictions and reasoning, connecting the scale factor to the change in area.

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

Activity 04

• Project-Based Learning
• 35 min · Individual

Individual: GeoGebra Experiments

Students access a dilation tool applet, input figures, centers, and factors. They record changes in side lengths and areas, then test predictions for composite dilations. Submit screenshots with observations.

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

Facilitation Tip

During GeoGebra Experiments, have students save at least three trials with different centers to compare how position affects the image.

What to look for

Provide students with a simple triangle plotted on a coordinate grid and a center of dilation. Ask them to calculate the coordinates of the image vertices after a dilation with a scale factor of 2. Check their calculations for accuracy in applying the scale factor to each coordinate.

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

A few notes on teaching this unit

Teachers should model dilation steps slowly, emphasizing the role of the center and scale factor before students work independently. Avoid rushing to abstract rules; let students discover that scale factors apply to all points uniformly. Research shows that students grasp proportionality better when they connect it to real-world examples, like shadows or maps, so anchor explanations in concrete contexts.

By the end of these activities, students will accurately plot dilated figures, explain how scale factors affect dimensions, and verify proportional relationships. They will also recognize that dilations preserve shape regardless of center position.

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Watch Out for These Misconceptions

• During Coordinate Grid Dilations, watch for students who assume the shape changes because the coordinates look different.

  Have pairs measure angles with protractors and compare side lengths before and after dilation to show angles stay the same and sides scale uniformly.

• During Geoboard Scale Challenges, watch for students who reverse the effect of scale factors less than 1.

  Ask groups to measure original and dilated shapes, then record the scale factor as a fraction (e.g., 1/2) to clarify that smaller factors shrink the figure.

• During Shadow Projection Demo, watch for students who think the center must be inside the figure to dilate it.

  After changing the light source position, have students trace rays from the new center to vertices to see how the image shifts while shape is preserved.

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Methods used in this brief

• Project-Based Learning