Mathematics · Grade 9

Active learning ideas

Dilations and Scale Factor

Active learning works for dilations because students need to physically see how distances from the center change and how rays guide movement. Working with hands-on tools lets them test scale factors in real time, making abstract concepts like proportional scaling concrete and memorable.

• Ontario Curriculum Expectations
• CCSS.MATH.CONTENT.8.G.A.3
• CCSS.MATH.CONTENT.8.G.A.4

• 30–45 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Project-Based Learning
• 35 min · Pairs

Graph Paper Scaling: Partner Dilations

Pairs plot a triangle on graph paper and select a center point. One partner applies a scale factor of 2, the other 0.5, then they verify distances from center to image points match the factor. Switch roles and compare results.

Explain how a dilation changes the size but not the shape of a figure.

Facilitation Tip

During Graph Paper Scaling, have students switch roles after each dilation to ensure both partners practice measuring and plotting.

What to look for

Provide students with a simple polygon on a coordinate grid, a center of dilation, and a scale factor. Ask them to calculate the coordinates of the image vertices and sketch the dilated figure. Check if their calculations are correct and if the resulting image is proportional to the original.

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

Activity 02

• Project-Based Learning
• 45 min · Small Groups

Digital Exploration: GeoGebra Dilations

In small groups, students open GeoGebra, create a polygon, choose a center, and adjust the scale factor slider. They record measurements of sides and distances before and after, noting patterns. Groups present one discovery to the class.

Predict the effect of a scale factor greater than one versus less than one on the image.

Facilitation Tip

In Digital Exploration, ask students to pause and predict the image before clicking the dilation tool, reinforcing visual estimation skills.

What to look for

Pose the question: 'If you dilate a square with a scale factor of 2, what happens to its area compared to the original square? What if the scale factor is 1/2?' Facilitate a discussion where students explain their reasoning, perhaps using coordinate examples or visual sketches to support their predictions.

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

Activity 03

• Project-Based Learning
• 30 min · Individual

Real-World Maps: City Planning Scale

Whole class views a city map projection. Individually, students dilate key landmarks from a central point using scale factors 1.5 and 0.75 on grid overlays. Discuss how scale affects planning decisions.

Analyze the relationship between the center of dilation and the corresponding points of the pre-image and image.

Facilitation Tip

For Real-World Maps, provide rulers and string so students can physically measure distances from the center to verify scale changes.

What to look for

Give students a pre-image and its dilated image on a coordinate plane, with the center of dilation marked. Ask them to determine the scale factor used for the dilation and write one sentence explaining how they found it.

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

Activity 04

• Project-Based Learning
• 40 min · Small Groups

Prediction Relay: Scale Factor Challenges

Teams line up; first student predicts image coordinates for a dilation, passes to next for verification on grid. Correct predictions score points; rotate roles. Debrief misconceptions as a class.

Explain how a dilation changes the size but not the shape of a figure.

What to look for

Provide students with a simple polygon on a coordinate grid, a center of dilation, and a scale factor. Ask them to calculate the coordinates of the image vertices and sketch the dilated figure. Check if their calculations are correct and if the resulting image is proportional to the original.

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

A few notes on teaching this unit

Teach dilations by starting with physical objects on grids before moving to coordinates, as hands-on work builds intuition. Avoid rushing to formulas; instead, let students notice patterns in how scale factors stretch or shrink distances from the center. Research shows that discussing negative scale factors early prevents later confusion with flips or reflections.

Students will correctly perform dilations on coordinate grids, explain how scale factors affect distances from the center, and predict image positions with confidence. They will use precise measurements and clear reasoning to justify their transformations.

---

Watch Out for These Misconceptions

• During Graph Paper Scaling, watch for students who assume the shape changes during dilation.

  Have partners use protractors to measure angles in both pre-image and image, confirming they remain equal, and ask them to mark parallel sides to reinforce preservation of shape.

• During Real-World Maps, watch for students who think scale factors apply uniformly across the entire figure.

  Use string to measure distances from the center to each vertex before and after dilation, guiding students to see that only distances along rays scale by the factor.

• During Digital Exploration, watch for students who confuse negative scale factors with reflections.

  Have students use the GeoGebra slider to observe the 180-degree rotation and record how points move along rays, then discuss why this is different from a flip.

---

Methods used in this brief

• Project-Based Learning