Dilations and Non-Rigid Transformations

Active learning works well for dilations because students need to see the dynamic relationship between the original figure, the center point, and the scaled image. Moving beyond static textbook diagrams helps learners internalize that scale factors greater than 1 expand figures, while scale factors between 0 and 1 shrink them, all while angles remain constant.

• 10th Grade
• Mathematics
• 3 activities
• 20 min–35 min

Learning Objectives

1. Calculate the coordinates of an image after a dilation centered at the origin with a given scale factor.
2. Compare the perimeters and areas of a polygon and its image after a dilation, analyzing the effect of the scale factor.
3. Justify the effect of a dilation on angle measures and side lengths, explaining why shape is preserved but size changes.
4. Construct a dilation of a polygon on the coordinate plane using a given center and scale factor.
5. Differentiate between rigid transformations and dilations based on their impact on size and shape.

Experiential Learning

⏱ 35 min·Pairs

Desmos Exploration: Scale Factor Slider

Students create a polygon in Desmos and apply a dilation with a variable scale factor using a slider. They record perimeter and area at five different scale factor values, plot the results on a table, and write a conjecture about the relationship between scale factor and area change.

Prepare & details

Compare the effects of rigid and non-rigid transformations on geometric figures.

Facilitation Tip: During the Desmos Exploration, circulate and ask each group to explain how changing the scale factor affects the distance from the center to a specific vertex.

Setup: Varies; may include outdoor space, lab, or community setting

Materials: Experience setup materials, Reflection journal with prompts, Observation worksheet, Connection-to-content framework

• Apply
• Analyze
• Evaluate
• Self-Awareness
• Self-Management
• Social Awareness

Gallery Walk

⏱ 30 min·Small Groups

Gallery Walk: Dilation or Distortion?

Post eight to ten before-and-after figure pairs around the room. Some are true dilations; others are distortions where horizontal and vertical scales differ. Student groups determine which are true dilations and, for valid ones, identify the center of dilation and scale factor using the ratio of corresponding side lengths.

Prepare & details

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

Facilitation Tip: For the Gallery Walk, require each student to measure two corresponding angles in a displayed pair of figures and post their measurements next to the image.

Setup: Wall space or tables arranged around room perimeter

Materials: Large paper/poster boards, Markers, Sticky notes for feedback

• Understand
• Apply
• Analyze
• Create
• Relationship Skills
• Social Awareness

Think-Pair-Share

⏱ 20 min·Pairs

Think-Pair-Share: Reverse Engineer the Dilation

Present a pre-image and image on the coordinate plane without labeling the center or scale factor. Students work individually to determine the center and scale factor, compare their method with a partner, and identify any differences before a class discussion of multiple valid approaches.

Prepare & details

Construct a dilation of a figure on the coordinate plane and justify the coordinates of the image.

Facilitation Tip: During the Think-Pair-Share, listen for students who reference the center point when explaining why a dilation is not a translation.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Teaching This Topic

Teachers should introduce dilations by first focusing on the center point rather than the scale factor, because many students default to the origin without conceptual understanding. Use physical tools like rubber bands or grid paper to model dilations before moving to digital platforms. Research shows that students grasp non-rigid transformations better when they physically manipulate the figure and measure changes themselves, so avoid over-reliance on pre-made applets until students have built intuition.

What to Expect

Successful learning looks like students confidently identifying the center of dilation, predicting the effect of a given scale factor, and distinguishing dilations from other transformations. They should articulate why angle measures stay the same even though side lengths change, and justify their answers with measurements and sketches.

Watch Out for These Misconceptions

Common Misconception

During the Desmos Exploration: Scale Factor Slider, watch for students who believe the shape changes when the scale factor is altered.

What to Teach Instead

Interrupt their exploration and ask them to measure two corresponding angles in the original and image triangles using the built-in measurement tools, then compare the values side by side.

Common Misconception

During the Gallery Walk: Dilation or Distortion?, watch for students who assume the center of dilation is always inside the figure.

What to Teach Instead

Point to an image where the center is outside the polygon and ask students to trace the rays from the center to each vertex to confirm the direction of scaling.

Assessment Ideas

Quick Check

After the Desmos Exploration, ask students to calculate the coordinates of the image triangle after a dilation with scale factor 2 centered at (1, 1). They should write the new coordinates and explain why the perimeter ratio is equal to the scale factor.

Exit Ticket

During the Gallery Walk, collect student notes that include measured corresponding angles and a written statement explaining why the figures are similar despite changes in size.

Discussion Prompt

After the Think-Pair-Share, facilitate a whole-class discussion where students compare a dilation to a translation, using examples from their activity sheets to defend whether the transformation preserves size, shape, and angle measures.

Extensions & Scaffolding

• Challenge: Provide a complex polygon and ask students to perform two successive dilations (e.g., scale factor 2 followed by scale factor 1/2) and justify why the final image is congruent to the original even though it appears smaller in between.
• Scaffolding: Give students a partially completed table with three known coordinates of the original triangle and the center of dilation; ask them to find the remaining coordinates after a dilation with scale factor 3.
• Deeper exploration: Have students design a dilation puzzle where a peer must determine the center and scale factor given only the original and image polygons.

Key Vocabulary

Dilation	A transformation that changes the size of a figure but not its shape. It enlarges or shrinks a figure by a scale factor from a fixed center point.
Scale Factor	The ratio of the length of an image segment to the length of its corresponding pre-image segment in a dilation. It determines how much the figure is enlarged or shrunk.
Center of Dilation	A fixed point from which all dilations are measured. Distances from the center to corresponding points on the pre-image and image are proportional.
Non-Rigid Transformation	A transformation that changes the size or shape of a figure. Dilations are the primary example in this unit.