Proportion and scaling: a week of ready-to-teach maths

Five lessons for Year 10 Measurement. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus twelve scale factor cards to match, twelve scaling cards to sort, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets, one copy per pair. Cut the 12 scale factor cards before Day 1 and the 12 scaling cards before Day 2, and keep each set in an envelope.
4. Calculators and rulers are used every day. Small cubes for Days 2 and 4 and a street map for Day 3 help, but every task also works from the sheets.
5. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.

The idea in plain words

Two shapes are similar when one is a scaled copy of the other: every length is multiplied by the same scale factor k. Areas do not follow the lengths: they are multiplied by k², and volumes by k³. So a model at half size needs a quarter of the paint and holds an eighth of the water. Maps use the same idea, written as a ratio such as 1 : 25 000. Because volume outgrows surface area, a big object has less surface for its size, which is why giants and ice blocks behave as they do. Put together, these let students model a practical problem, say what they assumed, check the answer, improve the model and report it.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

How we write these

The scale factor k is always the new length divided by the original, so an enlargement has k above 1 and a reduction k below 1. Map scales are written 1 : n with the same unit on both sides. Answers are rounded only at the last step, to a sensible number of figures for the situation.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 sets up the scale factor; Day 2 follows it into area and volume; Day 3 uses it on maps and plans; Day 4 shows what k² against k³ means for real objects; and Day 5 puts it all into one modelling task with a report. It builds on Year 9, where students enlarged shapes and used similar triangles, and it prepares the modelling and reporting of senior mathematics and science.

Materials for the week (one trip)

• From the classroom: calculators, rulers, scissors, envelopes for the cards, this pack printed.
• Cut-out sheet 1, one per pair: 12 scale factor cards (Day 1).
• Cut-out sheet 2, one per pair: 12 scaling cards (Day 2).
• If you have them: small cubes (Days 2 and 4), a local street map (Day 3), a banana and a tape measure (Day 5). No other equipment to buy.

Words for the word wall

similar, scale factor, enlargement, reduction, corresponding sides, ratio, scale, map scale, area factor, volume factor, surface area, square-cube law, model, assumption, evaluate, report

Dear families

This week in maths, Year 10 is learning what happens to lengths, areas and volumes when an object is scaled up or down, and how to use scale to plan real things, from maps to models.

Try this at home

• Ask: a 30 cm pizza and a 40 cm pizza: how many times as much pizza is the bigger one? (About 1.8 times: area grows with the square.)
• Ask: a recipe fills a round cake tin 20 cm across. How much mixture for a tin 30 cm across, just as deep? (2.25 times.)
• Ask: a road map says 1 : 250 000. How far is 4 cm on it? (10 km.)
• Ask: why does a bag of small ice cubes melt sooner than one big block of the same ice? (More surface for the same ice.)

My scaling week

Fill one row a day. Write a map, model, plan or copy you saw, its scale, and one real length it gives.

Similar shapes and the scale factor

Two shapes are similar when one is a scaled copy of the other: matching angles are equal and every pair of matching sides is in the same ratio, the scale factor k. Students find k, test pairs of triangles, find missing lengths and match six scale changes to their factors.

We are learning to

• find a scale factor as the new length divided by the original,
• decide whether two shapes are similar by checking every pair of matching sides,
• use k to find a missing length, and a shadow to find a height.

Success criteria

• I can show that triangles of 5, 12, 13 cm and 7.5, 18, 19.5 cm are similar, with k = 1.5.
• I can find the height of a tree from its shadow and a stick’s.

You need

Calculators; a metre ruler and a sunny spot for a real shadow if you can; the scale factor cards (cut-out sheet 1), one set per pair; the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Adding instead of multiplying: 10 by 15 to 20 by 25 adds 10 to each side and is not similar.
• Dividing the wrong way: k is new over original, so a reduction has k below 1.
• Mixing units: a 15 cm model and a 4.5 m statue need one unit before dividing.

Answers

• Part A: 2, 2, 2: similar, k = 2; 1.5, 1.5, 1.5: similar, k = 1.5; 1.5, ≈ 1.33, ≈ 1.29: not similar; 0.75, 0.75, 0.75: similar, k = 0.75; 1.5, 1.5, ≈ 1.67: not similar.
• Part B: k = 8 ÷ 5 = 1.6, so the sides are 6.4, 8, 9.6 cm; k = 15 ÷ 18 ≈ 0.833, so the short side is 12 × 15 ÷ 18 = 10 cm.
• Part C: the card key is with the mini-check answers.
• Part D: 1.5 ÷ 2 = h ÷ 14, so h = 10.5 m. It assumes flat ground, an upright stick and tree, and parallel rays measured at the same moment.

Similar shapes and the scale factor

A. Similar or not?

The sides are listed in matching order. Divide each side of the second triangle by the matching side of the first. The first row is done for you.

B. Find the missing lengths

C. Match the scale factors

Match each scale change card to its scale factor card. Write the letters.

D. Stretch: how tall is the tree?

Length, area and volume

When every length of an object is multiplied by k, its areas are multiplied by k² and its volumes by k³. Students use the three factors forwards and backwards, judge two pizzas by area, and sort twelve quantities by how they scale.

We are learning to

• multiply lengths by k, areas by k² and volumes by k³,
• work back to k from an area or a volume factor,
• decide whether a real quantity behaves like a length, an area or a volume.

Success criteria

• I can explain why a 40 cm pizza is not just one third more than a 30 cm pizza.
• I can find k when a model has one thousandth of the volume.

You need

Calculators; eight small cubes per group to build a 2 by 2 by 2 block, if you have them; the scaling cards (cut-out sheet 2), one set per pair; the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Scaling an area by k: doubling every length gives four times the area.
• Square-rooting a volume factor: k is the cube root of a volume factor.
• Thinking mass follows length: the mass of a solid follows its volume.

Answers

• Part A: k = 2: 2, 4, 8; k = 3: 3, 9, 27; k = 5: 5, 25, 125; k = 0.5: 0.5, 0.25, 0.125; k = 10: 10, 100, 1000; k = 1.5: 1.5, 2.25, 3.375.
• Part B: k = √49 = 7; k = 1 ÷ ³√1 000 = 0.1, so the deck area is 0.1² = 0.01, one hundredth of the real one. The 30 cm pizza has 706.9 cm², or 47.1 cm² a dollar; the 40 cm has 1256.6 cm², or 52.4 cm² a dollar, the better buy.
• Part C: the card key is with the mini-check answers.
• Part D: 297 ÷ 210 ≈ 1.414 and 420 ÷ 297 ≈ 1.414, so they are similar with k ≈ 1.414. Two A4 sheets make one A3, so k² = 2 and k = √2.

Length, area and volume

A. Three factors

Every length of a solid is multiplied by k. Fill in how many times the lengths, the areas and the volume are multiplied. The first row is done for you.

B. Work back, then buy a pizza

C. Sort the cards: by k, k² or k³?

Every length of each object is multiplied by k. Write each card’s letter under the factor its quantity changes by.

D. Stretch: why A3 is A4 scaled

Maps and plans

A scale such as 1 : 25 000 says one unit on the map stands for 25 000 of the same unit on the ground: the scale factor is 25 000. Students convert map and ground lengths both ways, find a scale from one known distance, and see that an area on a map is scaled by the square of the scale.

We are learning to

• change a map length into a ground length at a ratio scale, and back,
• find the scale of a map or plan from one known distance,
• scale an area on a map by the square of the scale factor.

Success criteria

• I can find how far 4.6 cm is on a 1 : 50 000 map.
• I can explain why a 2 cm by 3 cm park on a 1 : 25 000 map covers 6 × 25 000² cm², not 6 × 25 000.

You need

Calculators; rulers; a local street map or a printed topographic map if you have one; the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Losing a unit: 25 000 cm is 250 m, not 25 km.
• Dividing on the way out: a ground length is the map length times the scale.
• Scaling an area by the scale, not by its square.

Answers

• Part A: 1 : 100: 7.5 cm stands for 7.5 m; 1 : 25 000: 4 cm stands for 1 km; 1 : 50 000: 4.6 cm stands for 2.3 km; 1 : 1 000 000: 12 cm stands for 120 km; 1 : 50 000: 7 cm stands for 3.5 km; 1 : 250 000: 16 cm stands for 40 km.
• Part B: 320 000 ÷ 8 = 40 000, so 1 : 40 000; 600 ÷ 12 = 50, so 1 : 50.
• Part C: 500 m by 750 m, so 375 000 m² or 37.5 hectares. The area is multiplied by 25 000² = 625 000 000.
• Part D: 23 × 87 = 2 001 cm, about 20 m; the roof is 80.5 cm² × 7 569 ≈ 60.9 m².

Maps and plans

A. Map to ground, and ground to map

Fill the gaps. Give ground lengths in metres or kilometres. The first row is done for you.

B. Find the scale

C. An area on a map

D. Stretch: a model train

The square-cube law

Scale a solid by k and its surface area grows by k² while its volume grows by k³, so a bigger object has less surface for each unit of volume. Students follow a growing cube, explain why crushed ice melts sooner, and test whether a giant could be a scaled-up person.

We are learning to

• find the surface area to volume ratio of a cube and watch it fall as the cube grows,
• explain that surface area grows by k² and volume by k³,
• use the law to explain an everyday effect.

Success criteria

• I can show that a 4 cm cube has less surface for its volume than a 1 cm cube.
• I can explain why a giant 10 times as tall as a person would break its legs.

You need

Calculators; 27 small cubes per group (interlocking, wooden or sugar cubes) if you have them; the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Comparing surface areas alone: the big cube has more surface, but less for its volume.
• Thinking the ratio stays fixed: it is divided by k as the solid grows.
• Expecting strength to grow with weight: a bone’s strength follows its cross-section area.

Answers

• Part A: 1 cm: 6, 1, 6; 2 cm: 24, 8, 3; 3 cm: 54, 27, 2; 4 cm: 96, 64, 1.5; 5 cm: 150, 125, 1.2; 10 cm: 600, 1 000, 0.6.
• Part B: 64 small cubes with 384 cm² of surface against 96 cm² for the block, 4 times as much, so the crushed ice takes in heat faster and melts first.
• Part C: mass × 1 000; bone cross-section × 100; so each square centimetre of bone carries 10 times the load, and the legs would break.
• Part D: 6s² = s³ when s = 6 cm (216 and 216); doubling the side halves the ratio, since 6 ÷ 2s = (6 ÷ s) ÷ 2.

The square-cube law

A. A cube that grows

Fill in the surface area, the volume, and the surface area divided by the volume. The first row is done for you.

B. Crushed ice

C. Could a giant be a scaled-up person?

D. Stretch: where the numbers meet

Model and report

A model turns a practical question into mathematics: say what we assume, solve, check the answer against the situation, improve the model and report. Students plan a giant fibreglass banana for a main street: its size, the paint it needs, its mass, and a better model.

We are learning to

• state the assumptions a model makes,
• use k, k² and k³ to solve a practical problem,
• evaluate a model, change it, and report what we found.

Success criteria

• I can work out the paint for a 12 m banana and say what I assumed.
• I can explain why a hollow shell’s mass grows by k², not k³.

You need

Calculators; a real banana and a tape measure if you have them; the worksheet; the self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part B; start the next with Part C.

Watch for these ideas

• Scaling the paint by k: it grows by k² = 3 600, not by 60.
• Dividing by 100 for square metres: one square metre is 10 000 cm².
• Stopping at the first answer: a model is checked against the situation and changed.

Answers

• Part A: k = 1 200 ÷ 20 = 60, so the giant is 3.5 × 60 = 210 cm, or 2.1 m, across. It assumes the giant has the shape of a real banana, and that a banana is a cylinder.
• Part B: π × 3.5 × 20 ≈ 219.9 cm²; × 3 600 ≈ 791 681 cm² ≈ 79.2 m². Two coats: 158.3 ÷ 12 ≈ 13.2 L, so 4 tins.
• Part C: π × 1.75² × 20 ≈ 192.4 cm³; × 216 000 ≈ 41.6 m³, about 42 t if solid. Shell: 79.2 × 0.01 ≈ 0.79 m³, about 1.4 t. The shell’s material follows its surface, so k².
• Part D: a report names the assumptions (cylinder, no ends, a 1 cm shell, the paint’s cover), the method (k, k², k³), the findings, and an improvement, such as adding the two ends (about 6.9 m² more) or allowing for the taper.

Model and report: a giant banana

A town plans a giant fibreglass banana, 12 m long, for its main street. A real banana is about 20 cm long and 3.5 cm across. Model the real banana as a cylinder.

A. Set up the model

B. The paint

C. The mass, and a better model

D. Stretch: report to the town

Scale changes and their scale factors

For Day 1, one set per pair. Six cards describe a scale change; six give a scale factor k, the new length divided by the original. Match them.

Teacher note: the key is with the mini-check answers.

By k, k² or k³?

For Day 2, one set per pair. Every length of each object is multiplied by k. Sort the cards into three piles: the quantities that change by k, by k² and by k³.

Teacher note: the key is with the mini-check answers.

What we know: Proportion and scaling

Work on your own. A calculator is allowed. Show your working.

1. Are triangles of 6, 8, 10 cm and 9, 12, 15 cm similar? If so, give the scale factor. ______________________
2. A solid is enlarged with k = 4. How many times are its surface area and its volume multiplied? ______________________
3. A 1 : 50 model house has a door 4.2 cm tall. How tall is the real door, in metres? ______________________
4. On a 1 : 25 000 map, two towns are 6.4 cm apart. How far apart are they? ______________________
5. A big rug is similar to a small rug and has 9 times its area. The small rug is 80 cm long. How long is the big rug? ______________________
6. When every length of a solid is doubled, its surface area divided by its volume, circle one:   doubles  ·  stays the same  ·  halves
7. A 1 : 20 model house takes 30 mL of paint. About how much paint would the real house take for the same coat? ______________________
8. At the same moment, a 1.2 m post casts a 1.5 m shadow and a building casts a 20 m shadow. How tall is the building? Write one assumption. ______________________

Answers and marking guide

Answers with working

1. The ratios are 1.5, 1.5, 1.5: similar, with k = 1.5.
2. Surface area × 4² = 16; volume × 4³ = 64.
3. 4.2 × 50 = 210 cm = 2.1 m.
4. 6.4 × 25 000 = 160 000 cm = 1.6 km.
5. k = √9 = 3, so 80 × 3 = 240 cm, or 2.4 m.
6. It halves: the surface is multiplied by 4 and the volume by 8.
7. Paint covers area: 30 × 20² = 12 000 mL = 12 L.
8. 1.2 ÷ 1.5 × 20 = 16 m. Assumptions: flat ground, upright post and building, parallel rays.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working and uses k² and k³ without prompting.

Card keys

Scale factor cards (Day 1): k = 2.5: G and A; k = 0.5: D and J; k = 30: E and L; k = 1.5: B and H; k = 0.25: K and F; k = 15: I and C. Scaling cards (Day 2): by k: O, T, V; by k²: N, Q, S, W; by k³: M, P, R, U, X.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: similar shapes, length, area and volume, maps and plans, the square-cube law, and the modelling task.