Mathematics · Grade 7 · Geometric Relationships and Construction · Term 2

Geometric Transformations: Dilations

Understanding and performing dilations of 2D figures, including scale factor and center of dilation.

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

About This Topic

Dilations scale 2D figures uniformly from a fixed center point by a scale factor. In Ontario Grade 7 mathematics, within Geometric Relationships and Construction, students perform dilations on polygons and other shapes. They observe that distances from the center multiply by the scale factor, while angles, shape, and orientation stay the same. This builds skills to analyze transformations and construct accurate images.

Dilations introduce similarity between figures, linking to proportional reasoning across the curriculum. Students explain the center's role and scale factor's effect, preparing for advanced geometry like proofs and coordinate transformations. Practical applications appear in maps, art enlargements, and computer graphics, making the math relevant.

Active learning shines here because students construct dilations on graph paper or with rulers, measuring to verify properties firsthand. Collaborative comparisons reveal patterns in distances and angles, turning abstract scaling into concrete understanding that sticks through doing and discussing.

Key Questions

1. Analyze how a dilation changes the size of a figure while preserving its shape.
2. Explain the role of the scale factor and center of dilation in a transformation.
3. Construct a dilated image of a figure given a scale factor and center.

Learning Objectives

• Calculate the coordinates of the vertices of a dilated image given the original coordinates, a scale factor, and the center of dilation.
• Compare the side lengths and angle measures of a pre-image and its dilated image to identify invariant properties.
• Construct a dilated image of a 2D figure on graph paper using a specified scale factor and center of dilation.
• 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.
• Analyze the relationship between the distance of a vertex from the center of dilation and its corresponding vertex in the image.

Before You Start

Introduction to Geometric Shapes and Properties

Why: Students need to be familiar with basic 2D shapes (squares, triangles, etc.) and their properties, such as side lengths and angles.

Coordinate Plane Basics

Why: Understanding how to plot points and interpret coordinates is essential for performing dilations algebraically.

Introduction to Transformations (Translations, Reflections, Rotations)

Why: Prior exposure to other transformations helps students understand dilation as another type of geometric change.

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.

Watch Out for These Misconceptions

Common Misconception

Dilations change the shape or angles of the figure.

What to Teach Instead

Dilations preserve shape and angles because all distances scale proportionally from the center. Hands-on measuring during graph paper activities lets students check angles with protractors and compare side ratios, correcting this through direct evidence and group verification.

Common Misconception

Every point on the figure moves the same distance in a dilation.

What to Teach Instead

Distances from the center scale by the factor, so closer points move less. Active plotting in pairs helps students trace rays from the center and measure varying distances, building accurate mental models through repeated construction.

Common Misconception

The center of dilation must be inside or on the figure.

What to Teach Instead

The center can be anywhere, even outside. Exploration stations with varied centers show images on the same side or opposite, and peer teaching reinforces flexible positioning via shared drawings.

Active Learning Ideas

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.

35 min·Small Groups

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.

30 min·Pairs

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.

25 min·Whole Class

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.

40 min·Individual

Real-World Connections

• Architects and graphic designers use dilation to create scaled drawings and models. For example, a blueprint for a house is a dilation of the actual house, allowing builders to work with manageable dimensions.
• Cartographers use dilations when creating maps. A map is a scaled-down version of a large geographical area, where distances are proportionally reduced by a consistent scale factor to fit on paper or a screen.

Assessment Ideas

Quick Check

Provide students with a simple polygon (e.g., a triangle) on graph paper, a center of dilation, and a scale factor. Ask them to calculate the coordinates of the image's vertices and sketch the dilated figure. Check for accurate calculations and plotting.

Exit Ticket

Present students with two similar figures, one a dilation of the other, with the center of dilation marked. Ask them to determine the scale factor by measuring distances from the center and to explain in one sentence how the scale factor affects the size of the figure.

Discussion Prompt

Pose the question: 'What happens to the angles of a figure when it is dilated?' Have students discuss in pairs, referring to their constructions. Then, facilitate a whole-class discussion to solidify the understanding that angles remain congruent.

Frequently Asked Questions

What is a dilation in Grade 7 math?

A dilation scales a figure from a center point by a scale factor, enlarging or reducing size while keeping shape, angles, and orientation intact. Students construct these on coordinate grids, multiplying distances from the center. This matches Ontario curriculum expectations for geometric transformations and similarity.

How do you find the center of dilation?

The center is the fixed point where rays from original vertices pass through image vertices. Students identify it by drawing connecting lines or using trial points on grids. Practice with varied figures strengthens this skill, essential for accurate constructions in classwork.

How can active learning help students understand dilations?

Active approaches like grid plotting and partner verifications make scaling tangible. Students measure distances themselves, see proportional changes, and discuss discrepancies, which clears confusion faster than lectures. Collaborative builds boost confidence and retention, aligning with inquiry-based Ontario math practices for deeper geometric insight.

Why do dilations preserve shape?

Scaling from one center multiplies all distances proportionally, so relative side lengths and angles remain constant. Class demos with multiple factors illustrate this uniformity. Connecting to real enlargements, like photos, helps students grasp similarity without distortion.