Similarity Lab: Length and Area

Overview

In this second lesson of the eight-lesson unit, students use proportional reasoning to find missing lengths in similar figures. They then connect a linear scale factor to changes in perimeter and area by enlarging shapes on grid paper and explaining why area changes by the square of the scale factor.

Learning intentions

Students will:

• identify and use a scale factor between similar figures;
• calculate missing side lengths using equivalent ratios;
• connect scale factor to perimeter and area;
• explain why area changes by the square of the linear scale factor.

Success criteria

• I can determine the scale factor between two similar figures.
• I can use the scale factor to find an unknown length.
• I can predict how perimeter and area change when a figure is enlarged.
• I can justify why area is multiplied by the scale factor squared.

Curriculum links

• Similarity and proportional reasoning: solve problems involving corresponding lengths in similar figures.
• Measurement: investigate and calculate perimeter and area of two-dimensional shapes.
• Spatial sense: represent and analyse enlarged figures on a grid.
• Mathematical processes: communicate reasoning, make connections, and verify solutions.

Lesson structure (30 minutes)

1. 0–4 min · Retrieval hook. Display the opening questions on the retrieval and hook slides: “A rectangle is enlarged by a scale factor of 3. What happens to one side, the perimeter, and the area?” Students answer independently, then compare with a partner. Invite two contrasting predictions, without confirming the area answer yet.

2. 4–10 min · Guided modelling. Use the scale-factor modelling slides to model two similar rectangles: the smaller has dimensions 3 cm by 5 cm and the larger has dimensions 6 cm by 10 cm. Establish that corresponding lengths are multiplied by 2, so the scale factor is 2. Model a missing-length example, such as (4:x=6:9), and solve (x=6) by multiplying by the scale factor (1.5). Students annotate the matching sides and state the ratio they used.

3. 10–16 min · Grid enlargement investigation. Give each student grid paper and display the instructions on the grid investigation instructions. Students draw or use a 2-by-3 rectangle, enlarge it by a scale factor of 2, and record the original and enlarged perimeter and area. They calculate: original perimeter 10 units, enlarged perimeter 20 units; original area 6 square units, enlarged area 24 square units. Students discuss what changed by 2 and what changed by 4.

4. 16–23 min · Independent practice. Distribute the similar figures and area practice worksheet. Students complete questions involving missing corresponding lengths, perimeter after enlargement, and area after enlargement. Include a prompt asking students to complete the sentence: “If the scale factor is (k), area is multiplied by ___ because ___.” Circulate and check that students use corresponding sides rather than adding or subtracting lengths.

5. 23–27 min · Peer checking and explanation. Display the checking prompts on the peer-checking discussion slides. In pairs, students compare one missing-length solution and one area solution. They must identify the scale factor, check the units, and ask, “Where is the square of the scale factor shown?” Partners correct errors in a different colour and explain one solution aloud.

6. 27–30 min · Exit calculation. Students complete the final question shown on the exit question slide: “A similar shape is enlarged by a scale factor of 3. Its original area is 8 cm² and one original side is 5 cm. What is the new area and the corresponding side length?” Students show calculations: new side (=15) cm and new area (=8\times3^2=72) cm². Collect responses as students leave.

Resources

• the complete similarity lesson slide deck
• the similar figures and area practice worksheet
• Grid paper
• Rulers and pencils
• Coloured pens or pencils for peer checking
• Whiteboard and markers
• Calculators, if routinely permitted

Assessment

• Listen during retrieval and modelling for correct identification of corresponding sides and the difference between linear and area scale factors.
• During grid work and worksheet practice, check calculations, units, and students’ written explanations of (k^2).
• Use the exit calculation to identify whether students can apply a scale factor to both a length and an area; reteach with a visual grid model if needed.

Differentiation

• Provide a scale-factor prompt card on the worksheet: “Find a pair of corresponding sides; divide larger by smaller; multiply matching lengths by (k).”
• Support students who need it with pre-drawn rectangles, colour-coded corresponding sides, a ratio table, and partner talk before written explanations.
• For EAL learners, provide the sentence frames “The scale factor is ___ because ___” and “Area is multiplied by ___ because ___”; explicitly model length units versus square units.
• Extend early finishers by asking them to design a shape with an area of 12 square units, enlarge it by a non-integer scale factor, and predict the new area before calculating.