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.

• Grade 9
• Mathematics
• 4 activities
• 30 min–45 min

Learning Objectives

1. Calculate the coordinates of image points after a dilation centered at the origin and at an arbitrary point.
2. Compare the scale factor's effect on the lengths of corresponding sides between a pre-image and its dilation.
3. Analyze the relationship between the scale factor and the area of a figure and its dilation.
4. Explain how the center of dilation influences the position of the image relative to the pre-image.
5. Construct the image of a polygon under a dilation given the pre-image, center, and scale factor.

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.

Prepare & details

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.

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

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.

Prepare & details

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.

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

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.

Prepare & details

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.

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·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.

Prepare & details

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

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

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.

What to Expect

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

Common Misconception

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

What to Teach Instead

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.

Common Misconception

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

What to Teach Instead

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.

Common Misconception

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

What to Teach Instead

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.

Assessment Ideas

Quick Check

After Graph Paper Scaling, provide a quick-check sheet with a triangle on a grid and a center point. Ask students to calculate the image vertices for a scale factor of 3 and sketch the result, then check their work against a provided answer key.

Discussion Prompt

During Digital Exploration, pause the activity and ask students to discuss in pairs how dilating a square with a scale factor of 2 changes its area. Circulate to listen for explanations that connect scale factor to area growth (scale factor squared) and correct any misconceptions immediately.

Exit Ticket

After Real-World Maps, give students an exit ticket with a pre-image and its dilated image on a coordinate plane. Ask them to determine the scale factor and write one sentence explaining how they found it, using the center marked on the grid to justify their answer.

Extensions & Scaffolding

• Challenge students to dilate a figure with a fractional center coordinate during the Prediction Relay, adding complexity to their calculations.
• For students who struggle, provide pre-labeled rays on graph paper during Graph Paper Scaling to highlight the path of movement.
• Deeper exploration: Ask students to compare dilations with different centers but the same scale factor, analyzing how the center’s location changes the final image position.

Key Vocabulary

Dilation	A transformation that changes the size of a figure but not its shape. It produces an image that is similar to the original figure.
Scale Factor	The ratio of the length of a side of the image to the length of the corresponding side of the pre-image. It determines whether the dilation is an enlargement or a reduction.
Center of Dilation	The fixed point from which all points of the pre-image are scaled to create the image. All corresponding points lie on lines passing through this center.
Image	The resulting figure after a geometric transformation, such as a dilation, has been applied to the pre-image.
Pre-image	The original figure before a geometric transformation is applied.