At a glance. One lesson of about 60 minutes. It assumes enlargement from Year 9 and the surface area and volume formulae from lessons 1 to 3. Students use k, k² and k³ to find lengths, areas and volumes of similar solids, in either direction.
Calculator: GDC optional. The lesson is done by hand with fractions. The optional section shows GDC users how to cube a decimal scale factor, and reminds them to check the direction of the scaling.
What students will learn
Know
- Linear scale factor k gives area factor k² and volume factor k³.
- Corresponding lengths of similar solids are in the same ratio.
- Going from a ratio of areas or volumes to k needs a square root or a cube root.
Understand
- Area is length times length, so it scales by k twice; volume is length three times, so by k three times.
- A small change in size has a large effect on area and volume.
Do
- Find areas and volumes of similar solids from a scale factor.
- Find a scale factor from an area or volume ratio.
- Solve scale-model problems.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: scale factors for length, area and volume (questions 1 to 9).
- Criterion B. Investigating patterns: both figures. Students find the pattern k, k², k³ for several k and say why it holds.
- Criterion C. Communicating: writing the scale factor on its own line and naming whether it is for length, area or volume (worked examples, faded example).
- Criterion D. Applying mathematics in real-life contexts: the clay figures, the boat model and the cone slice.
The one thing to do with the figure
Settle the hook with the grid. Set k = 3 and ask students to count the unit squares in the grid. They see 9 without being told the rule. Then ask them to predict the volume factor for k = 3 (27) and check it against 162 / 6 on the brick.
In the second figure, ask students to find two pairs of lengths with the same area ratio, such as 2 : 3 and 6 : 9.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the clay figures. Collect votes and reasons. |
| 12 min | First figure; key idea; settle the hook. |
| 8 min | Second figure; worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A or D, return to the grid: 9 squares is a 3 by 3 grid, so k = 3. |
| 20 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 9. |
The answers
| Warm-up | 12 cm; 40 cm²; 16π cm². |
|---|---|
| Hook | 4 times the paint and 8 times the clay. |
| Worked examples | 300 cm²; 2400 litres; 81 cm². |
| Faded example | 810 ml. |
| 1. From a length ratio to a volume ratio | C. A keeps the area ratio; B squares it; D cubes it instead of the length ratio. |
| 2. Area of a similar solid | 180. 20 × 9. |
| 3. Volume of a similar solid | 96. 12 × 8. |
| 4. A decimal scale factor | 6.25 litres. 0.4 × 15.625. |
| 5. Length from an area ratio | 9. 18 ÷ 2. |
| 6. A scale model | 240 m². 0.6 × 400. |
| 7. Area from a volume ratio | 192. k = 4/3. |
| 8. A slice of a cone | 260. 270 − 10. |
| 9. Area and volume factors | B. A uses k for area; C multiplies by 2 and 3; D uses k² for volume. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Kept the area ratio, or squared or cubed it without finding k. | Draw 9 unit squares as a 3 by 3 grid and ask for k. |
| 60 or 540 in question 2 | Used k, or used k³. | Write the factor on its own line: k = 3, so k² = ____. |
| 24 or 48 in question 3 | Used k, or used k². | Ask: what is the unit of volume? cm³ has three lengths. |
| 1 or 2.5 in question 4 | Used k, or used k². | Cube the decimal on a calculator and write it down before multiplying. |
| 4.5 or 36 in question 5 | Used the area ratio as the length ratio, or went the wrong way. | Which statue is bigger, and does the answer have to be smaller? |
| 12 or 4800 in question 6 | Used k, or used k³. | Ask: is paint an area or a volume? |
| 256 or 144 in question 7 | Used the volume ratio, or used k. | Write the sequence: volume ratio, k, k², area. |
| 10 or 180 in question 8 | Gave the small cone, or took off one third. | Cube 1/3 and say what it means in words. |
| A, C or D in question 9 | Used k for area, multiplied by dimensions, or used k² for volume. | Use the grid: a 10 by 10 square for area; 10 layers of it for volume. |
Questions worth asking
- Why does doubling the lengths make the area four times as big? Listen for the 2 by 2 grid or “length times width, so two doublings”.
- Does a ratio of 1 : 4 for areas mean the lengths are in the ratio 1 : 4? No: take the square root, so 1 : 2.
- Are all circles similar? Are all rectangles? All circles are; rectangles are only if their side ratios match.
- Why can a large animal not be an enlarged mouse? Weight goes with k³ and bone strength with k².
Differentiation
A way in for anyone stuck
- Give a table of k from 1 to 5 with columns for k, k² and k³ already filled for k = 1 and 2.
- Build a 2 by 2 by 2 cube from 8 unit cubes and let students count faces and cubes.
- Write the three quantities in separate colours: length, area, volume.
Stretch
- A solid is enlarged so that its surface area doubles. By what factor do its lengths and its volume change?
- Two similar cones have volumes in the ratio 8 : 125. Find the ratio of their slant heights and of their curved surface areas.
Students learning in English as an additional language
“Similar” in maths means the same shape, not the same size. “Scale factor” is the number you multiply lengths by. Say “k squared” and “k cubed” aloud and write them next to the words “area” and “volume”.
| Word | Plain meaning | Example |
|---|---|---|
| similar | the same shape, with all lengths in the same ratio | a photo and its enlargement |
| scale factor | the number every length is multiplied by | k |
| corresponding | in the same position in each solid | the two heights |
| enlargement | a transformation that changes size and keeps shape | a copy on a photocopier at 150% |
| ratio | a comparison of two amounts by division | 1 : 2 |
| scale model | a small copy of a real object with every length reduced by the same factor | a toy train |
Sentence starters for explaining aloud or in writing:
- The length scale factor is … so the area scale factor is …
- I square the scale factor because …
- I cube the scale factor because …
Two things not to say
“Multiply by the scale factor.” Ask which quantity: length, area or volume.
“Squared for area, cubed for volume.” Tell students why: two lengths, three lengths.
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.
- AreaSimilar solids have a scale factor of 4. The small one has surface area 3 cm². Find the large one's.
48 cm². - Earlier work (lesson 3)A sphere has radius 3 cm. Find its surface area in terms of π.
36π cm².
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA model of a pagoda in Hue is made at 1 : 50. The model's volume is 0.8 litres. Find the volume of the pagoda in m³.
0.8 × 125 000 = 100 000 litres = 100 m³. - Working backTwo similar bottles hold 125 ml and 216 ml. The tall one is 12 cm. Find the other's height.
k = 5/6 for the small, so 10 cm.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA cube has its edge increased by 10%. By what percentage does the volume increase?
1.1³ = 1.331, so 33.1%. - OpenFind two similar solids where the large one has exactly 5 times the volume of the small one. Is the scale factor a whole number?
k = the cube root of 5, about 1.71, not a whole number.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
No device? Unit cubes and squared paper are enough. The With a GDC section is optional.