The shape of the lesson
By the end of the lesson, learners can
- What makes shapes similar: Use the lab to enlarge a shape by different scale factors.
- Similar triangles: In the similar-triangles tab, slide the parallel line up and down inside the big triangle.
- Similarity in context: Shadows, photos, models and ramps all hide similar triangles or similar shapes.
- Match: pair up corresponding sides (opposite equal angles). Redraw triangles separately if they overlap.
- Divide: \(k = \text{new} \div \text{old}\) from one known matching pair.
- Multiply: new length = old length \(\times k\); old length = new length \(\div k\).
- Nested triangles: use whole sides from the shared vertex.
- Reasons (Extended): two pairs of equal angles, each with its geometric reason.
One picture every idea returns to
In similar shapes, corresponding angles are equal and corresponding lengths are in the same ratio. Scale factor \(k = \dfrac{\text{length on new shape}}{\text{matching length on old shape}}\); multiply any old length by \(k\) to get the new one.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Same shape, different size.. Similar shapes are enlargements of each other: every angle matches and every length is multiplied by the same scale factor. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: What makes shapes similar | Use the lab to enlarge a shape by different scale factors. Project tab 1 of the lab (Similarity lab, similar shapes) and let a student drive it while the class predicts. Model the worked example Testing for similarity on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Similar triangles | In the similar-triangles tab, slide the parallel line up and down inside the big triangle. Project tab 2 of the lab (Similarity lab, similar triangles) and let a student drive it while the class predicts. Model the worked example Nested triangles on the board, one line at a time. Your turn: In the crossed triangles above, BC = 5 cm. Find CD. | Pairs |
| 21 to 30 min | Teach and check: Similarity in context | Shadows, photos, models and ramps all hide similar triangles or similar shapes. Model the worked example Shadows on the board, one line at a time. Your turn: A photo 10 cm wide and 15 cm high is enlarged to be 24 cm wide. How high is the enlargement? | Round Table |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 20 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Similarity in context, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Round Table
One sheet travels round the team. Each student adds one line of working or one idea and passes it on, so the solution is built four hands at a time and every contribution is visible.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| Adding the same amount does not make shapes similar. | A 4 by 6 rectangle with 3 cm added to each side becomes 7 by 9, and \(7 \div 4 \ne 9 \div 6\). Similarity is about MULTIPLYING. |
| In nested triangles the scale factor uses the WHOLE side AB, not the extra piece DB. | Using \(6 \div 4\) here gives 9 cm for BC, which is wrong. |
| Keep each ratio the same way round. | Height over shadow for the girl equals height over shadow for the tree: \(\dfrac{h}{14} = \dfrac{1.8}{2.4}\). Mixing a height with a shadow on one side gives nonsense. |
Similar shapes: worksheet
6 questions, 20 marks. Written in the style of the papers; no past paper question is reproduced.
State the two conditions for two polygons to be similar.
A triangle has sides 4 cm, 6 cm and 7 cm. A similar triangle has longest side 17.5 cm. Find its other two sides.
Rectangle A is 8 cm by 12 cm, rectangle B is 10 cm by 15 cm and rectangle C is 12 cm by 16 cm. Which rectangles are similar to A? Show your working.
In triangle ABC, D lies on AB and E lies on AC, with DE parallel to BC. AD = 6 cm, DB = 4 cm, AE = 7.5 cm and DE = 9 cm.
Find (a) BC (b) EC.
(Extended) PQ is parallel to RS. Lines PS and QR cross at T. PT = 3 cm, TS = 7.5 cm and PQ = 4 cm.
(a) Show that triangles PQT and SRT are similar, giving reasons. (b) Find RS.
A photo is 10 cm wide and 15 cm high. Mia wants a similar enlargement that fits inside a frame 30 cm wide and 40 cm high, as large as possible.
(a) Show that the photo cannot fill the frame exactly. (b) Find the width and height of the largest similar enlargement that fits.
Answer key and marking notes
- Corresponding angles are equal.
- Corresponding sides are in the same ratio (all multiplied by the same scale factor).
- The longest sides match: \(k = 17.5 \div 7 = 2.5\).
- \(4 \times 2.5 = 10\) cm and \(6 \times 2.5 = 15\) cm.
- A to B: \(10 \div 8 = 1.25\) and \(15 \div 12 = 1.25\). Equal, so B is similar to A.
- A to C: \(12 \div 8 = 1.5\) but \(16 \div 12 = 1.33\). Not equal, so C is not similar to A.
- AB = 6 + 4 = 10 cm, so \(k = 10 \div 6 = \dfrac{5}{3}\).
- (a) BC \(= 9 \times \dfrac{5}{3} = 15\) cm.
- (b) AC \(= 7.5 \times \dfrac{5}{3} = 12.5\) cm, so EC \(= 12.5 - 7.5 = 5\) cm.
- (a) Angle PTQ = angle STR (vertically opposite angles).
- Angle QPT = angle RST (alternate angles, PQ parallel to RS).
- Two pairs of equal angles, so the triangles are similar (P matches S, Q matches R).
- (b) \(k = TS \div TP = 7.5 \div 3 = 2.5\), so RS \(= 4 \times 2.5 = 10\) cm.
- (a) Width factor \(30 \div 10 = 3\); height factor \(40 \div 15 = 2.67\). Different, so no similar enlargement fills both.
- (b) The smaller factor, 2.666..., is the largest that fits both ways.
- Width \(= 10 \times \dfrac{40}{15} = 26.7\) cm (3 s.f.); height \(= 15 \times \dfrac{40}{15} = 40\) cm.
6 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
A rectangle measures 6 cm by 9 cm. A similar rectangle has a shorter side of 10 cm. Find the length of its longer side.
- A15 cm
- B13 cm
- C6.67 cm
- D5.4 cm
In the diagram, DE is parallel to BC. AD = 4 cm, DB = 6 cm and DE = 6 cm. Find BC, marked \(x\) cm.
- A15
- B9
- C12
- D2.4
At the same time of day, a metre stick standing upright casts a shadow 0.7 m long and a tree casts a shadow 12.6 m long. How tall is the tree?
- A18 m
- B8.82 m
- C12.9 m
- D13.3 m
Which pair of shapes is ALWAYS similar?
- AAny two squares
- BAny two rectangles
- CAny two isosceles triangles
- DAny two rhombuses
In the diagram, AB is parallel to CD, and AD and BC cross at E. AE = 6 cm, ED = 9 cm and BE = 5 cm. Find EC, marked \(x\) cm.
- A7.5
- B3.33
- C8
- D10.8
A triangle has sides 5 cm, 7 cm and 9 cm. A similar triangle has a perimeter of 52.5 cm. Find the length of its longest side.
- A22.5 cm
- B19.5 cm
- C17.5 cm
- D40.5 cm
Quiz key
Scale factor \(k = 10 \div 6 = \tfrac{5}{3}\), so the longer side is \(9 \times \tfrac{5}{3} = 15\) cm. The tempting 13 cm adds 4 cm to the 9 cm because the short side grew by 4 cm; similar shapes are MULTIPLIED, not added to. 6.67 cm uses the factor upside down (\(10 \times \tfrac{6}{9}\)), and 5.4 cm is \(6 \times 9 \div 10\).
AB \(= 4 + 6 = 10\) cm, so the scale factor from ADE to ABC is \(10 \div 4 = 2.5\) and \(x = 6 \times 2.5 = 15\). The tempting 9 uses DB instead of AB (\(6 \div 4 \times 6\)), but DB is not a side of either triangle. 12 adds 6 cm as if the extra piece of AB were added to DE, and 2.4 uses the factor upside down.
\(\dfrac{h}{12.6} = \dfrac{1}{0.7}\), so \(h = 12.6 \div 0.7 = 18\) m. The tempting 8.82 m multiplies \(12.6 \times 0.7\), which mixes a height with a shadow (the ratio is the wrong way up). 12.9 m adds the 0.3 m difference between the stick and its shadow, and 13.3 m adds 0.7 m; similar triangles need a multiplier, not a difference.
Every square has four \(90^\circ\) angles and four equal sides, so any square is an enlargement of any other. The tempting rectangles always have equal angles, but their sides need not be in the same ratio (a 2 by 3 rectangle is not similar to a 2 by 5 one). Isosceles triangles can have different angles, and rhombuses have equal sides but can have different angles.
Angle BAE = angle CDE and angle ABE = angle DCE (alternate angles, AB parallel to CD), and the angles at E are vertically opposite, so the triangles are similar with A matching D and B matching C. Scale factor \(= ED \div AE = 9 \div 6 = 1.5\), so \(x = EC = 5 \times 1.5 = 7.5\). The tempting 3.33 matches the sides the wrong way (\(5 \times 6 \div 9\)). 8 adds the 3 cm difference, and 10.8 is \(6 \times 9 \div 5\).
The first perimeter is \(5 + 7 + 9 = 21\) cm, so the scale factor is \(52.5 \div 21 = 2.5\) and the longest side is \(9 \times 2.5 = 22.5\) cm. The tempting 19.5 cm shares the extra \(52.5 - 21 = 31.5\) cm equally, adding 10.5 cm to each side; that changes the shape. 17.5 cm is a third of the perimeter, and 40.5 cm adds all of the extra 31.5 cm to the longest side.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.