At a glance. One lesson of about 60 minutes. Similar triangles, the scale factor, the AA, SSS and SAS conditions, shadows and parallel lines in a triangle.
Calculator: no calculator. Most scale factors are friendly, and the reasoning is the lesson. Question 6 needs one short division, 60 ÷ 7, to 1 decimal place, so allow paper working.
What students will learn
Know
- Similar triangles have equal angles and sides in the same ratio.
- The scale factor is new length ÷ old length.
- AA, SSS and SAS are conditions that show triangles are similar.
Understand
- Similarity is multiplicative: every side is multiplied by the same number.
- Equal angles force proportional sides.
Do
- Decide whether two triangles are similar, naming the condition.
- Find missing sides using a scale factor.
- Solve problems with shadows and parallel lines.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: finding missing sides in similar triangles (questions 2 to 5).
- Criterion B. Investigating patterns: students predict ratios before moving the extra-length slider and state when triangles are not similar. With the second figure they find that a line parallel to a side always gives equal ratios.
- Criterion C. Communicating: the matching sides and the condition (AA, SSS, SAS) are named in the working.
- Criterion D. Applying mathematics in real-life contexts: indirect measurement with shadows, and the photograph enlargement (questions 4 and 7).
The one thing to do with the figure
Set scale factor 2 with no extra, and ask what the ratios will be. Then raise the extra slider by 1 before the class answers: “What will happen to the ratios? What about the angles?”
Ask: if one ratio differs, is the triangle similar? Do not accept “almost”. The additive trap (4, 5, 6 to 6, 7, 8) is worth drawing on the board next to the figure.
A possible order
| 5 min | Warm-up, then answers. |
|---|---|
| 8 min | Hook: the tree; guesses and reasons. |
| 12 min | The figure; the three conditions in words. |
| 12 min | Worked examples; parallel lines in pairs. |
| 5 min | Question 1, the check. |
| 13 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 2 : 3; 180 degrees; 10. |
|---|---|
| Worked examples | Tree: 9 m. Triangle Q: 30 cm and 35 cm. |
| Faded example | BC = 15 cm. |
| 1. AA condition | B. C confuses similar with congruent. |
| 2. Scale factor | 12 cm. |
| 3. Matching sides | 39 cm. |
| 4. Shadows | 10 m. |
| 5. Missing side | 12. |
| 6. Ratios, 1 d.p. | 8.6 cm. |
| 7. Enlargement | 24 cm. |
| 8. Parallel lines | 12 cm. |
| 9. Which pair must be similar | C. B has only one matching angle; D has no angle condition at all. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| 7, 23, 11, 9 or 31 in questions 2, 3, 5, 6, 7 | Adds the difference instead of multiplying. | Ask: if every side doubled, would you add the same number to each? |
| 1.33, 13.5, 4.2 or 54 | Uses the scale factor upside down. | Ask whether the answer should be bigger or smaller than the starting side. |
| 12 or 22.5 in question 4 | Subtracts the shadows, or uses the ratio upside down. | Draw both triangles and compare ratios of height to shadow. |
| 11 or 16 in question 8 | Uses DB or DE alone. | Ask which big triangle's side matches AD. |
| A, B or D in question 9 | Accepts a condition that is not enough. | Ask for two triangles that fit the condition but are not similar. |
Questions worth asking
- If all three angles are equal, why must the sides be in proportion? Listen for “the shape cannot change, only the size”.
- Does a scale factor of 1/2 make a similar triangle? Yes, a reduction. Ask what the area of the new triangle does.
- Can two triangles have the same side lengths and not be similar? No: if all three sides are equal they are congruent, which is similar with scale factor 1.
Differentiation
A way in for anyone stuck
- Colour matching sides and angles on the same pair of triangles before any calculation.
- Give a table: small side, large side, scale factor, so that students fill the ratio first.
- Use whole-number scale factors for the first three questions.
Stretch
- A triangle is enlarged by scale factor 3. By what factor does its area grow? (9.)
- Find the height of a building using a mirror on the ground and your own height.
Students learning in English as an additional language
Use the pairs “bigger / smaller” and “matching / corresponding” with arrows on a diagram.
| Word | Plain meaning | Example |
|---|---|---|
| similar | the same shape, not necessarily the same size | a photo and its enlargement |
| scale factor | the number you multiply a length by to get the matching length | 12 ÷ 2 = 6 |
| corresponding | in the same position in each shape; matching | the two shortest sides |
| ratio | a comparison of two numbers by dividing | 12 : 6 = 2 : 1, a scale factor of 2 |
| congruent | exactly the same shape and size | scale factor 1 |
Sentence starters for explaining aloud or in writing:
- The scale factor is … ÷ … = …
- The triangles are similar because …
- The side that matches … is …
Two things not to say
“Add the difference to every side.” The additive trap is the main error in the unit.
“Cross-multiply.” Students then cannot say what the numbers mean. Name the scale factor first.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- Scale factorTriangle A has a side of 5 cm; the similar triangle B has the matching side 20 cm. What is the scale factor?
4. - Missing sideSides 3, 4, 5 and a similar triangle with shortest side 9. Find the longest side.
15.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ShadowA 1.8 m person casts a 1.2 m shadow. A building casts a 20 m shadow. How tall is the building?
30 m. - MapTwo similar triangles on a map have matching sides of 4 cm and 10 cm. If the first has a side of 6 cm, how long is the second's matching side?
15 cm.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningAre 3, 4, 5 and 6, 8, 11 similar?
No: 6 ÷ 3 = 2, 8 ÷ 4 = 2 but 11 ÷ 5 = 2.2. - From Lesson 4Find the distance between (0, 0) and (6, 8), then say what scale factor takes the 3, 4, 5 triangle to it.
10; scale factor 2.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard and by touch; “Show me the steps” gives four cases in words.
On paper: cut-out triangles and a ruler do the same job; for the shadows, a metre stick outdoors works well.