Scale Factors in Action

Overview

In this second lesson of the eight-lesson unit Similarity to Solid Design, students use scale factors to enlarge and reduce two-dimensional shapes. Building on their understanding of equivalent ratios and measurement, they calculate corresponding lengths, compare perimeters and areas, and justify whether two shapes are similar.

Learning intentions

Students will:

• Use a scale factor to enlarge or reduce every length in a two-dimensional shape.
• Calculate how perimeter and area change under enlargement or reduction.
• Identify similar and non-similar pairs using corresponding side ratios.
• Explain their reasoning using mathematical language, diagrams and units.

Success criteria

• I can use the formula: new length = original length × scale factor.
• I can label corresponding sides on an original shape and its image.
• I can explain that perimeter scales by (k), while area scales by (k^2).
• I can justify whether two shapes are similar by showing that all corresponding sides use the same scale factor.

Saskatchewan Mathematics 9 Outcomes and Indicators

• Geometry/transformations: Aligns with the Saskatchewan Mathematics 9 outcome and relevant indicators on the official curriculum page that support scale factors, corresponding sides, and similarity of shapes.
• Official source: Saskatchewan Ministry of Education curriculum page: https://curriculum.gov.sk.ca/CurriculumOutcomeContent?id=153

Lesson structure (30 minutes)

1. 0–3 min · Warm-up. Teacher displays the question on the hook and warm-up slides: “A side changes from 2 cm to 5 cm. What is the scale factor?” Students solve independently, then compare with a partner. Confirm (k=\frac{5}{2}=2.5), and ask whether this is an enlargement or reduction.

2. 3–10 min · Mini-lesson. Using the scale factor teaching slides, teacher models that every corresponding length must be multiplied by the same factor. For (k=\frac{3}{2}), an original length of 4 cm becomes (4\times\frac32=6) cm. Emphasise:

• new length = original length × (k)
• perimeter scales by (k)
• area scales by (k^2)

Teacher works through a (4\text{ cm}\times6\text{ cm}) rectangle enlarged by 1.5: its new dimensions are (6\text{ cm}\times9\text{ cm}); the area changes from (24\text{ cm}^2) to (54\text{ cm}^2). Students annotate the formula and draw arrows from each original side to its image in their notes. Address the common mistakes of using different scale factors for different sides and omitting square units for area.

3. 10–18 min · Guided practice. Teacher distributes grid paper and models the first step from Scale Factor Lab worksheet: identify the original polygon, choose the stated scale factor, and multiply each side length. Students enlarge the irregular polygon on grid paper, using a ruler and coloured pencils to show corresponding sides. Teacher circulates, checking that students label original and image sides clearly and use one consistent scale factor.

4. 18–26 min · Collaborative lab. In pairs, students complete the remaining tables and comparison questions on Scale Factor Lab worksheet. They calculate missing lengths, decide whether shape pairs are similar, and write one explanation in words. Prompt pairs to test all corresponding sides rather than relying only on appearance. Students should encounter and explain examples such as:

• 2 to 6 gives (k=3)
• 5 to 4 gives (k=0.8)
• ratios 3:5 and 6:10 describe similar shapes
• ratios 4:7 and 8:15 do not describe similar shapes

Teacher uses the collaborative task and discussion slides to display time reminders and the prompt: “How do you know one scale factor works for every corresponding side?”

5. 26–28 min · Formative check. Teacher checks one labelled original/image pair and one written justification from each pair, asking students to explain their choice. Students correct labels, calculations or units in a different colour. Quickly identify students needing a follow-up using whole-number factors and pre-drawn grids.

6. 28–30 min · Closure. Teacher displays the partner-share prompt on the plenary slide: “I know shapes are similar because…” Partners complete the sentence using evidence about corresponding sides and a constant scale factor. Invite two students to share, then collect the success criteria exit ticket slips if an individual final check is needed.

Resources

• the Scale Factors in Action slide deck
• Scale Factor Lab worksheet
• Grid paper
• Rulers
• Coloured pencils
• Student mathematics notes or notebooks
• Board or document camera

Assessment

• During the warm-up and mini-lesson, check whether students distinguish the scale factor from the new length and apply it consistently.
• During the grid task and lab, inspect one labelled pair, one completed table and one written similarity justification per pair.
• Use the closure response or exit slip to identify whether students can justify similarity and remember that area uses square units and (k^2).

Differentiation

• Support students with whole-number scale factors, pre-drawn grids, a multiplication template and the sentence starter: “The shapes are similar because each corresponding side is multiplied by ___.”
• Provide a small reference box showing (k=\frac{\text{image length{\text{original length), new length = original × (k), perimeter × (k), and area × (k^2).
• For EAL learners and students requiring additional support, pair verbal explanation with arrows, colour coding, labelled examples and explicit vocabulary such as original, image, corresponding, enlarge and reduce.
• Extend confident students by asking them to investigate and explain why an area changes by (k^2), then create a similar shape with a fractional scale factor.