Enlargements (Positive Scale Factors)

Active learning helps students grasp enlargements because the visual and tactile nature of these activities makes abstract scaling rules concrete. Hands-on work with rays, grids, and shapes lets students see how scale factors change distances, areas, and positions in real time, building lasting understanding beyond diagrams alone.

• Year 9
• Mathematics
• 4 activities
• 25 min–40 min

Learning Objectives

1. Calculate the coordinates of an enlarged image given an object, a center of enlargement, and a positive integer or fractional scale factor.
2. Describe the effect of a given scale factor on the lengths of sides and the area of a 2D shape.
3. Construct an accurate enlargement of a given 2D shape using a specified center of enlargement and scale factor.
4. Analyze the relationship between the center of enlargement, the object's vertices, and the corresponding image's vertices.

Problem-Based Learning

⏱ 35 min·Pairs

Ray Drawing: Integer Scales

Mark the centre on grid paper. Draw rays from the centre through each vertex of the object shape. Use a ruler to mark points at twice the distance along each ray for scale factor 2, then connect to form the image. Pairs measure and compare side lengths and areas.

Prepare & details

How does doubling the side lengths of a shape affect its area?

Facilitation Tip: During Ray Drawing: Integer Scales, have students use different colored pencils for object and image rays to avoid confusion.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

Problem-Based Learning

⏱ 40 min·Small Groups

Geoboard Construction: Fractional Scales

Stretch elastic bands on geoboards to form simple shapes. Identify a centre off-board. Direct rays to scale vertices by 0.5 towards the centre, plotting new positions. Groups record coordinates and verify enlargement properties.

Prepare & details

Analyze the relationship between the center of enlargement, the object, and the image.

Facilitation Tip: For Geoboard Construction: Fractional Scales, encourage students to stretch rubber bands slowly to see proportional movement toward the center.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

Problem-Based Learning

⏱ 30 min·Small Groups

Area Hunt: Scale Patterns

Provide pre-drawn shapes and centres. Students enlarge with factors 1.5, 2, 3 on squared paper, calculate areas before and after. In small groups, tabulate results to identify the square-of-scale rule through patterns.

Prepare & details

Construct an enlargement of a shape with a given positive scale factor and center.

Facilitation Tip: In Area Hunt: Scale Patterns, ask students to label all side lengths and areas on their cut-out shapes to make comparisons explicit.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

Problem-Based Learning

⏱ 25 min·Whole Class

Centre Hunt: Reverse Engineering

Give object-image pairs. Students trial centres by drawing rays, finding intersections. Whole class shares successful centres and scale factors, discussing verification methods.

Prepare & details

How does doubling the side lengths of a shape affect its area?

Facilitation Tip: During Centre Hunt: Reverse Engineering, provide a variety of centers—inside, on the edge, and outside the shape—to challenge assumptions.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

Teaching This Topic

Teachers should model precise ray drawing and labeling, emphasizing that rays must pass through corresponding vertices of the object and image. Avoid rushing through fractional scale factors, as these often reveal hidden gaps in proportional reasoning. Research suggests frequent quick-checks during hands-on work help students internalize the difference between linear and area scaling before misconceptions take root.

What to Expect

Successful learning looks like students confidently drawing rays, calculating new coordinates, and explaining how scale factors affect lengths and areas. They should also justify their enlargement choices using measurements and peer feedback, showing clear links between linear and area scaling.

Watch Out for These Misconceptions

Common Misconception

During Ray Drawing: Integer Scales, watch for students who assume area scales by the same factor as length.

What to Teach Instead

Ask these students to cut out their enlarged shape and the original, then physically compare areas by overlaying or tiling with unit squares to reveal the true fourfold increase.

Common Misconception

During Geoboard Construction: Fractional Scales, watch for students who place image points further from the center than the object.

What to Teach Instead

Have them trace rays on transparencies, overlaying object and image to visually confirm points move proportionally closer to the center for factors less than 1.

Common Misconception

During Centre Hunt: Reverse Engineering, watch for students who assume the center must lie inside the shape.

What to Teach Instead

Prompt them to plot multiple centers on the same diagram, then use rays to confirm that centers outside still produce valid enlargements with consistent proportional distances.

Assessment Ideas

Quick Check

After Ray Drawing: Integer Scales, provide a simple shape on a coordinate grid with center (2,3) and scale factor 2. Ask students to calculate the coordinates of the enlarged vertices and sketch the image, focusing on correct ray construction.

Exit Ticket

After Geoboard Construction: Fractional Scales, ask students to draw an original triangle, mark a center of enlargement, and enlarge it by 0.5. They must write one sentence explaining how the area of the new triangle relates to the original, using their measurements.

Discussion Prompt

During Area Hunt: Scale Patterns, display two similar triangles with marked centers. Ask students to verify the enlargement by measuring one pair of corresponding sides, calculating the scale factor, and explaining why the area ratio is the square of the scale factor.

Extensions & Scaffolding

• Challenge: Ask students to enlarge a shape by a scale factor of 1.5, then calculate the exact area ratio using algebra.
• Scaffolding: Provide pre-drawn rays with measurements for students to complete the enlargement step-by-step.
• Deeper exploration: Have students create an enlargement puzzle where peers must identify the scale factor and center from a partial diagram.

Key Vocabulary

Center of Enlargement	The fixed point from which all distances are measured when creating an enlargement. Lines are drawn from this point through the vertices of the original shape.
Scale Factor	The ratio of the length of a side on the image to the corresponding side on the object. A scale factor greater than 1 increases the size, while a scale factor between 0 and 1 decreases the size.
Object	The original 2D shape before it has been enlarged or reduced.
Image	The resulting shape after an enlargement or reduction has been applied to the object.
Enlargement	A transformation that changes the size of a shape but not its angles or proportions, based on a center of enlargement and a scale factor.