At a glance. One lesson of about 60 minutes. It assumes enlargement and ratio. Students use k, k² and k³ in both directions, find k from an area or volume ratio, apply percentage changes, and handle a cone cut parallel to its base.
Calculator: GDC optional. Cube roots are awkward by hand unless the ratio is a perfect cube. A GDC keeps k unrounded; every example can also be done by choosing the numbers so that k is a fraction.
What students will learn
Know
- Lengths multiply by k, areas by k² and volumes by k³.
- Similar means every corresponding length has the same ratio.
- A volume ratio needs a cube root and an area ratio a square root to give k.
Understand
- The powers come from the number of lengths multiplied together.
- A chain k → k² → k³ lets one ratio give the others.
- A percentage increase is a factor minus one.
Do
- Find unknown lengths, areas and volumes of similar solids.
- Find k from a ratio of areas or volumes.
- Find the volume of a frustum by removing a similar cone.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: scale factors on similar solids (questions 2 to 8).
- Criterion B. Investigating patterns: the what-if, why larger animals need thicker legs.
- Criterion C. Communicating: the explanation of why the powers differ, in the key card and the exit ticket.
- Criterion D. Applying mathematics in real-life contexts: models, vases, lorries and tins, and the question of whether a pair is similar.
The one thing to do with the figure
Ask for the two numbers before moving the slider. Say: “If every edge is doubled, what will the surface area be?” Hold up fingers for the answer, then check. Repeat for the volume.
Then set the scale factor to 0.5 and ask which way the numbers move and why the volume falls the most.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the lantern. Collect predictions for the paper and the oil. |
| 12 min | The figure with predictions; the key idea; settle the hook. |
| 10 min | Warning card and worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A or B, return to the number of lengths in a volume. |
| 18 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 | 144 cm²; 20 cm³; 3 ÷ 2. |
|---|---|
| Hook | The paper is multiplied by 16 and the oil by 64. |
| Worked examples | 180 cm² and 405 mL; 9 : 25 and 27 : 125; about 58.7%. |
| Faded example | 2 000 cm³. |
| 1. Volume scale factor | D. A uses k, B uses k², C works out k³ as 3k. |
| 2. Area scale factor | 600. 96 × 6.25. |
| 3. Model to real volume | 96. 12 000 × 8000 ÷ 1 000 000. |
| 4. Cost that follows area and cost that follows volume | 494. 468.75 + 25. |
| 5. From areas to volumes | 320. 135 × 64 ÷ 27. |
| 6. A slice between two cuts of a cone | 152. 216 − 64. |
| 7. Capacity ratio to area factor | 1.84. 2.5 to the power 2 ÷ 3. |
| 8. Percentage increases | 72.8. 1.2³ = 1.728. |
| 9. From volume ratio to area ratio | B. k = 2 : 5, so areas are 4 : 25. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or C in question 1 | Used k or k² for a volume, or worked out k³ as 3 × k. | Count the lengths multiplied: three for a volume. |
| 240 or 1500 in question 2 | Used k, or k³, for an area. | An area is length times length, so k². |
| 4.8 or 0.24 in question 3 | Used k² or k for a volume. | Write the factor as 20 × 20 × 20. |
| 250 or 531 in question 4 | Swapped the powers for clay and glaze, or put both costs on k³. | Label each cost: follows volume (k³) or follows area (k²). |
| 240 or 758.5 in question 5 | Used the area ratio for a volume, or cubed it. | Write the chain: area ratio, k, volume ratio. |
| 288 or 240 in question 6 | Used k or k² for each small cone. | Write each k, cube it, and subtract the two cones. |
| 2.5, 1.58 or 6.25 in question 7 | Used the volume factor, a square root, or its square. | Find k first with a cube root. |
| 44, 20 or 198.6 in question 8 | Repeated the area change, gave the length change, or cubed the area factor. | Write the factor 1.44 and the chain. |
| A, C or D in question 9 | Gave the length ratio, squared the volume ratio, or gave the volume ratio itself. | Cube-root first, then square. |
Questions worth asking
- Why does the volume factor have three copies of k? A volume is length times width times height, and each is multiplied by k.
- Does the rule work if only one dimension changes? No: the solids must be similar, with all lengths in the same ratio.
- What is the ratio of heights if the volume ratio is 1 : 2? 1 : ³√2, about 1 : 1.26.
- How can a larger tin hold more but need less label per mL? Capacity grows with k³ and area only with k², so area per mL falls.
Differentiation
A way in for anyone stuck
- Build a cube from 8 unit cubes and ask how many unit squares cover it and how many unit cubes fill it.
- Give a table with columns for length, area and volume factors and the powers of k.
- Use perfect cubes (8, 27, 64) for the first questions.
Stretch
- Show that if the volume of a solid is multiplied by V, then its surface area is multiplied by V to the power 2 ÷ 3.
- A cone of height 30 cm is cut parallel to its base so that the two parts have equal volume. Find the height of the small cone to one decimal place.
Students learning in English as an additional language
“Similar” means the same shape and a different size. A “scale factor” is the number every length is multiplied by. A “ratio” compares two quantities with a colon. “Corresponding” lengths match in the two solids. Say “k squared” and “k cubed” aloud.
| Word | Plain meaning | Example |
|---|---|---|
| similar solids | solids with the same shape, one an enlargement of the other | two cans in the ratio 3 : 5 |
| scale factor | the number every length is multiplied by | k = 1.4 for an enlargement |
| corresponding | matching in the two solids | the two lids of two cans |
| frustum | a cone with its top cut off by a plane parallel to the base | a plant pot |
| ratio | a comparison of two quantities by division | 1 : 20 for a model |
Sentence starters for explaining aloud or in writing:
- The scale factor of the lengths is … because …
- Volumes are multiplied by … because …
- These solids are / are not similar because …
Two things not to say
“Double the size, double the volume.” Ask: which size?
“Just use k squared.” Count the number of lengths in what you are finding.
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.
- AreaTwo similar boxes have heights 5 cm and 15 cm. The smaller has surface area 70 cm². Find the larger's.
630 cm². - VolumeA model is 1 : 10. The real object's volume is 5000 cm³. Find the model's volume.
5 cm³.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA model of a Moai statue is 1 : 25 and holds 64 cm³ of clay. How many litres of clay does the full-size statue need?
1000 L, because 64 × 15 625 = 1 000 000 cm³. - Earlier work (Year 9)A rectangle is enlarged by scale factor 2.5. The original area is 8 cm². Find the new area.
50 cm².
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA scale model of a ship has a surface area 1 ÷ 400 of the real one. Find the length scale and the volume ratio.
1 : 20 and 1 : 8000. - ProofShow that for a cone cut at the midpoint of its height, the frustum has 7 times the volume of the small cone.
k = 1 ÷ 2, so the small cone is 1 ÷ 8 and the frustum 7 ÷ 8 of the whole.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to a slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
A calculator is optional. Choose perfect cubes to keep the work exact.