Mathematics • Year 7 • Unit 3 • Lesson 17
Missing Sides — Mixed Challenge
Combine scale-factor and proportion methods, maps, models and shadows. Spot a plausible mistake about going from BIG to small, then design your own scaled shape.
1. Mixed problems
Each question mixes ideas. Show your working. 2 marks each
1.1 Solve: x / 7 = 3 / 4 .
1.2 Two similar triangles. SF (small → big) = 2.5. Small has sides 4, 6, 8. Find ALL three sides of the big triangle.
1.3 Map scale 1 : 200. A house is 3 cm wide on the map. Find the real width, in metres.
1.4 Big triangle sides 30, 24, x. Small similar triangle has corresponding sides 10, 8, 6. Find x.
1.5 A 1.8 m student casts a 1.2 m shadow. At the same moment a tree casts a 9 m shadow. Find the tree's height.
1.6 △LMN ∼ △STU. LM = 14, MN = 21, ST = 6. Find TU and the scale factor from △LMN to △STU.
2. Find the mistake
Another Year 7 student tried to find a missing side when going from a BIG triangle to a small one. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Problem: A large triangle has sides 24 and 18. A similar SMALL triangle has corresponding side 8 and unknown side x. Find x.
Line 1: Pick a known pair: 24 ↔ 8.
Line 2: Scale factor = 24 ÷ 8 = 3.
Line 3: Apply SF to the unknown: x = 18 × 3 = 54.
Line 4: Answer: x = 54.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full, including the corrected final answer.
Stuck? Sanity-check: we're going from BIG (18) to small (x). Should x be bigger or smaller than 18? The student got x = 54 — bigger! That can't be right.3. Open-ended challenge — design a billboard
This question has more than one correct answer. Show your work clearly. 4 marks
3.1 You design a small mock-up advertisement that is a 12 cm × 18 cm rectangle. The real billboard will be similar to your mock-up (same shape, scaled up). The real billboard's longest side must be between 3 m and 5 m (300 cm and 500 cm).
Design three different billboards that all satisfy this constraint, each with a different scale factor. For each one:
(i) Choose a valid scale factor.
(ii) Compute both dimensions of the real billboard, in cm AND in m.
(iii) Confirm the longest side is between 300 cm and 500 cm.
Bonus: What is the LARGEST possible scale factor? What is the SMALLEST? (Give answers as decimals, to 2 d.p.)
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Solve x/7 = 3/4
Cross-multiply: 4x = 21, so x = 5.25 (or 21/4).
1.2 — Multiply each side by 2.5
4 × 2.5 = 10, 6 × 2.5 = 15, 8 × 2.5 = 20. Big triangle sides: 10, 15, 20.
1.3 — Map scale 1 : 200
Real = 3 × 200 = 600 cm = 6 m.
1.4 — Big triangle missing side
SF (small → big) = 30 ÷ 10 = 3 (or 24 ÷ 8 = 3 — consistent). x = 6 × 3 = 18.
1.5 — Tree from student + shadow
tree / 1.8 = 9 / 1.2. So tree = 1.8 × (9 ÷ 1.2) = 1.8 × 7.5 = 13.5 m.
1.6 — △LMN ∼ △STU
SF (LMN → STU) = ST ÷ LM = 6 ÷ 14 = 3/7.
TU = MN × 3/7 = 21 × 3/7 = 9.
2 — Find the mistake
(a) The mistake is on Line 3 (and the direction taken in Line 2 was already misleading).
(b) The student went from BIG (24) to small (8), so the scale factor IN THAT DIRECTION is 8 ÷ 24 = 1/3 (a reduction), NOT 3. They then multiplied by 3 instead of dividing — making the small triangle's side BIGGER than the large triangle's side, which is impossible.
(c) Corrected working:
Line 1: Pick a known pair: 24 ↔ 8. (unchanged)
Line 2 (fixed): Going BIG → small, SF = 8 ÷ 24 = 1/3 (or use the BIG → small ratio of 1/3).
Line 3 (fixed): x = 18 × 1/3 = 6 (or equivalently 18 ÷ 3 = 6).
Line 4 (fixed): Answer: x = 6.
The fix: when going BIG → small, DIVIDE by the scale factor (or multiply by its reciprocal). Sanity-check: x should be SMALLER than 18 because the small triangle is smaller.
3 — Open-ended challenge (sample solutions)
Small mock-up: 12 cm × 18 cm. Longest side of real billboard must satisfy 300 ≤ 18 × SF ≤ 500, so 16.67 ≤ SF ≤ 27.78.
SF = 20: Real = 240 cm × 360 cm = 2.4 m × 3.6 m. Longest side 360 cm ✓.
SF = 25: Real = 300 cm × 450 cm = 3.0 m × 4.5 m. Longest side 450 cm ✓.
SF = 27: Real = 324 cm × 486 cm = 3.24 m × 4.86 m. Longest side 486 cm ✓.
Bonus: Largest SF = 500 ÷ 18 ≈ 27.78 (giving longest side exactly 500 cm). Smallest SF = 300 ÷ 18 ≈ 16.67 (giving longest side exactly 300 cm).
Marking: 1 mark per valid SF + dimension pair (3 marks), 1 mark for correct bonus (max SF and min SF, both to 2 d.p.).