Proportion, rates, ratio and scale: a week of ready-to-teach maths

Five lessons for Year 9 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 unit-rate match cards, a town map to measure, 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: sheet 1 has the unit-rate match cards (Day 2), sheet 2 the town map (Day 4). Cut the cards before Day 2.
4. Calculators are needed every day, and rulers on Day 4.
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

Four tools describe a constant relationship between quantities. Direct proportion: one quantity is always the same multiple of the other, so its graph is a straight line through the origin. A rate compares quantities in different units, such as km/h or $/kg, and settles best-buy questions. A ratio compares quantities of the same kind, for sharing and mixing. Scale is a ratio between a drawing and the real thing. A model built from them goes round a cycle: formulate, solve, interpret in the situation, evaluate the assumptions, and report.

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 answers

Money is rounded to the cent and a unit price to the cent per unit. Rates carry both units, as in 72 km/h or 7.5 L/100 km. Ratios are written in simplest whole numbers once both parts are in the same unit. Every model answer ends with a sentence in the words of the situation, and the currency rate used all week is the unit’s 1 AUD = 0.65 USD, an example rate rather than today’s.

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 direct proportion and its constant; Day 2 turns it into rates and unit prices; Day 3 shares and mixes in ratios; Day 4 applies ratio to lengths on maps and plans; and Day 5 puts the tools together in a full modelling cycle with money. It builds on Year 8 rates and ratio and on percentages, and shares the modelling cycle with the Year 9 algebra modelling unit. It prepares Year 10 modelling of growth and decay in financial contexts, and of the scaling of objects.

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: 16 unit-rate match cards (Day 2).
• Cut-out sheet 2, one per pair: a town map at 1 : 10 000 with a scale bar (Day 4).
• If you have them: two or three real supermarket receipts or catalogue pages for best buys. No other equipment to buy.

Words for the word wall

direct proportion, constant of proportionality, unitary method, rate, unit rate, unit price, best buy, ratio, simplest form, share, scale, scale factor, scale bar, map, plan, exchange rate, model, assumption, evaluate, report

Dear families

This week in maths, Year 9 is modelling everyday problems with proportion, rates, ratio and scale: unit prices at the supermarket, fuel for a road trip, mixing and sharing, and distances on maps. Each model ends with a check: does the answer make sense, and what did we assume?

Try this at home

• Look for the unit price on a shelf label or online: which size of a product is the best buy?
• Ask: a car uses 8 L/100 km. How much fuel for a 250 km trip? (20 L.)
• Ask: share $90 between two people in the ratio 4 : 5. (Each part is $10: $40 and $50.)
• Ask: on a map at 1 : 50 000, how far is 3 cm? (150 000 cm, which is 1.5 km.)

My modelling week

Fill one row a day. Write a problem from class, the tool you used, and the answer in words.

Direct proportion

Two quantities are in direct proportion when one is always the same multiple of the other: y = kx. The constant k is the amount for one, so finding it once gives any amount, and the graph is a straight line through the origin.

We are learning to

• test a table for direct proportion by dividing each y by its x,
• find the constant of proportionality and use the unitary method,
• graph a direct proportion and read values from the line.

Success criteria

• I can tell whether a table shows direct proportion, and give the constant when it does.
• I can find the cost of any amount from the cost of one.

You need

The worksheet; calculators; rulers.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with the line on the board again.

Watch for these ideas

• Taking a constant difference for a constant multiple: 5, 8, 11 grows by 3 but is not proportional to 1, 2, 3.
• Dividing the wrong way: k is y ÷ x, the amount for one x.
• Drawing the line without (0, 0): a direct proportion always passes through the origin.

Answers

• Part A: A yes, k = 3.5; B yes, k = 1.5; C no (5, 4, 3.666667); D no (2, 4, 6).
• Part B: $59.50 ÷ 7 = $8.50 per m, so 12 m costs $102; 18 ÷ 4 = 4.5 L per min, so 81 L takes 18 min; $85.50 ÷ 45 = $1.90 per L, so 30 L costs $57.
• Part C: y = 2.5x passes through (0, 0), (1, 2.5), (2, 5), (3, 7.5), (4, 10), (5, 12.5), (6, 15); at x = 4.4, y = 11; y = 12.5 at x = 5.
• Part D: k = 18 ÷ 4 = 4.5, so y = 45 when x = 10, and x = 6 when y = 27.

Direct proportion

A. Proportional or not?

Divide each y by its x. If it is always the same, write the constant k. If not, say why.

B. The unitary method

Find the amount for one, then the amount asked for.

C. The graph of y = 2.5x

D. Stretch: find k first

y is directly proportional to x, and y = 18 when x = 4. Find y when x = 10, and x when y = 27.

Rates and best buys

A rate compares two quantities in different units and keeps both units: km/h, $/kg, L/100 km. A unit rate gives the amount for one unit, which makes fair comparisons possible: the best buy is the pack with the lowest price per unit.

We are learning to

• find a unit rate and write it with both units,
• use a rate to find a time, a distance or an amount,
• compare packs by unit price and decide the best buy.

Success criteria

• I can find the unit price of two packs and say which is the better buy.
• I can turn fuel used on a trip into litres per 100 km.

You need

The worksheet; calculators; the unit-rate match cards (cut-out sheet 1), one set per pair, cut out.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the match; start the second with Part A.

Watch for these ideas

• Dividing the wrong way round: $ ÷ kg gives a price per kilogram; kg ÷ $ gives kilograms per dollar.
• Comparing packs by total price, or by grams and kilograms mixed.
• Writing 40 minutes as 0.4 hours: it is 40 ÷ 60 = 2/3 of an hour.

Answers

• Part A: $1.30 per egg; 24 pages per minute; 72 beats per minute; 8 L per 100 km; 720 km/h.
• Part B: $2.20 and $2.10 per L: 2 L for $4.20 is better; $12.40 and $12.80 per kg: 750 g for $9.30 is better; $0.90 and $0.95 per roll: 24 rolls for $21.60 is better; $19 and $18 per kg: 200 g for $3.60 is better.
• Part C: the card key is with the mini-check answers.
• Part D: no. The last pair shows a smaller pack at a lower unit price; and a big pack is only a saving if it is used before it spoils.

Rates and best buys

A. Unit rates

Find each unit rate in the units given.

B. Best buys

Find the price per litre, per kilogram or per roll of each pack. Circle the better buy.

C. Rate match

Play rate match with the cards. Then write the unit rate of any two situations you matched, with the working.

D. Stretch: bigger is better?

Is the bigger pack always the better buy? Use Part B, and give one more reason a shopper might not choose the lowest unit price.

Ratio: sharing and mixing

A ratio compares quantities of the same kind. To share in a ratio, add the parts, find what one part is worth, then multiply; the shares add back to the whole. Mixtures and recipes scale the same way, and a ratio is simplified only once both parts are in one unit.

We are learning to

• share an amount in a two- or three-part ratio,
• simplify a ratio, changing both parts to the same unit first,
• scale a mixture or a recipe up and down.

Success criteria

• I can share an amount in a ratio and check that the shares add back to the whole.
• I can simplify 400 m : 2 km once both are in metres.

You need

The worksheet; calculators.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with Part B.

Watch for these ideas

• Dividing by one part of the ratio instead of the total number of parts.
• Simplifying across units: 400 m : 2 km is not 200 : 1.
• Scaling only one ingredient of a mixture, which changes the ratio.

Answers

• Part A: $150 in 2 : 3: $60, $90; 360 g in 5 : 4: 200 g, 160 g; 1.2 L in 1 : 5: 0.2 L, 1 L; $240 in 3 : 4 : 5: $60, $80, $100.
• Part B: 12 : 18 = 2 : 3; 2.5 : 1.5 = 5 : 3; 400 m : 2 km = 1 : 5; 45 min : 2 h = 3 : 8.
• Part C: concrete 3, 6, 12 buckets of cement, sand and gravel; 250 mL of cordial needs 1000 mL of water, 1.25 L of drink; pasta for 10 is 750 g.
• Part D: in 20 units of each tin, A gives 5 blue and B gives 8 blue: 13 blue to 27 white, 13 : 27. Not the same as adding the ratios, 3 : 6.

Ratio: sharing and mixing

A. Share it

Add the parts, find one part, then each share. Check that the shares add to the whole.

B. Simplest form

Write each ratio in simplest whole numbers. Change to the same unit first.

C. Mixtures and recipes

D. Stretch: mixing two mixes

Tin A holds blue and white paint in the ratio 1 : 3; tin B in the ratio 2 : 3. Equal amounts of the two tins are poured together. What is the ratio of blue to white now?

Scale: maps and plans

A scale is a ratio between a drawing and the real thing: at 1 : 200, every length on the plan is 200 times shorter than the real length, in the same unit. Multiply to go from the drawing to the real thing, divide to go back, and change units at the end.

We are learning to

• convert between drawn and real lengths at a given scale,
• measure distances on a map, with its scale or its scale bar,
• choose a scale that fits a drawing on a page.

Success criteria

• I can turn 8 cm on a 1 : 25 000 map into kilometres.
• I can measure a route on a map and give its real length in metres.

You need

The worksheet; calculators; rulers; the town map (cut-out sheet 2), one per pair.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the map; start the second with Part A.

Watch for these ideas

• Mixing units: the scale works in one unit, so 4 cm × 50 000 is 200 000 cm before it is 2 km.
• Multiplying when going from real to drawn.
• Measuring from the edge of a dot rather than its centre.

Answers

• Part A: 6 cm → 1.5 km; 13 m → 2.6 cm; 4.4 cm → 11 km; 3.2 cm → 1.6 m.
• Part B, at 100% (1 : 10 000, 1 cm = 100 m): School to Library 6.0 cm, about 600 m; School to Pool 4.5 cm, about 450 m; Library to Station 5.0 cm, about 500 m; Pool to Park 10.0 cm, about 1000 m; Station to Jetty 8.9 cm, about 890 m. Allow ± 1 mm of measuring (± 10 m).
• Part C: 1 : 50 gives 48 by 32 cm, too big; 1 : 100 gives 24 by 16 cm, too big; 1 : 200 gives 12 by 8 cm, fits. Use 1 : 200.
• Part D: the room is 4 m by 3 m = 12 m²; the plan is 12 cm². Lengths scale by 100, areas by 100 × 100 = 10 000.

Scale: maps and plans

A. Drawn and real

B. Routes on the town map

Measure each straight route on cut-out sheet 2 from dot to dot, to the nearest millimetre. Give its real length in metres.

C. Choose the scale

A school hall is 24 m by 16 m. Its plan must fit on a page 20 cm by 15 cm. Which of 1 : 50, 1 : 100, 1 : 200 will fit? Show the size of the drawing at each.

D. Stretch: what happens to area?

On a 1 : 100 plan a room is 4 cm by 3 cm. Find the real area of the room and the area on the plan. How many times bigger is the real area?

Modelling with money

A model turns a situation into a calculation and back: formulate it with named quantities and rates, solve, interpret the answer in the words of the situation, evaluate what was assumed, and report. Money problems are often chains of proportion: litres from kilometres, then dollars from litres.

We are learning to

• formulate a chain of rates for a cost and solve it,
• convert money at a constant exchange rate in both directions,
• evaluate a model’s assumptions and report the findings.

Success criteria

• I can work out the fuel cost of a trip and name one assumption the answer rests on.
• I can convert an amount to and from US dollars at a given rate.

You need

The worksheet; calculators; the unit’s self-check quiz for the game.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with Part B and the quiz.

Watch for these ideas

• Multiplying by the rate in both directions: going back to AUD divides.
• Using L/100 km as litres per kilometre: 840 km is 8.4 hundreds of kilometres.
• Rounding the litres before the cost, or giving an answer with no sentence in the situation’s words.

Answers

• Part A: 840 ÷ 100 × 7.5 = 63 L; 63 × $1.95 = $122.85, about $30.71 each for 4 people. It assumes the same fuel use and price all the way, and no detours.
• Part B: 500 AUD = 325 USD; 39 USD = 60 AUD; 1200 AUD = 780 USD.
• Part C: 2.4 kW × 3 h × 30 nights = 216 kWh; 216 × 32c = $69.12.
• Part D: a report gives the model, the answer $122.85, two assumptions, and the change: 10c a litre more adds 63 × $0.10 = $6.30, for $129.15.

Modelling with money

A. The road trip model

A family of 4 drives 840 km. Their car uses 7.5 L/100 km and petrol costs $1.95 a litre.

B. Exchange

Use 1 AUD = 0.65 USD. Round to the cent.

C. A heater’s running cost

A 2.4 kW heater runs for 3 hours a night for 30 nights. Electricity costs 32c per kWh. How much does the heating cost?

D. Stretch: report it

Write a short report on the road trip for the family: the model, the answer, two assumptions, and how the cost would change if petrol rose by 10c a litre.

Unit-rate match cards

For Day 2, one set per pair: 16 cards

Lay the cards face down. Take turns to turn two over. Keep them if one is a situation and the other is its unit rate.

Teacher note: the key is with the mini-check answers, so this sheet can go to pairs whole.

A town map at 1 : 10 000

For Day 4, one per pair

Measure each route in a straight line from the centre of one dot to the centre of the other. Use the scale, or lay a strip of paper along the route and read it against the scale bar.

Teacher note: printed at 100%, 1 cm on the map is 100 m; the scale bar works at any print size. The routes are in the Day 4 plan.

What we know: Proportion, rates, ratio and scale

Work on your own. Use a calculator. Show the rate or the one part you find first.

1. y is directly proportional to x, and y = 26 when x = 8. Find y when x = 12. ______________________
2. 2.5 kg of apples cost $11.25. What is the price per kilogram? ______________________
3. Which is the better buy: 375 mL for $2.40, or 1.25 L for $7.50? Show the unit prices. ______________________
4. A bus travels 207 km in 2 h 15 min. What is its average speed? ______________________
5. Share $96 in the ratio 5 : 3. ______________________
6. Write 1.5 kg : 600 g in simplest form. ______________________
7. On a 1 : 20 000 map a track is 7.5 cm long. How long is it in kilometres? ______________________
8. At 1 AUD = 0.65 USD, how many US dollars do 120 AUD buy? ______________________

Answers and marking guide

Answers with working

1. k = 26 ÷ 8 = 3.25, so y = 3.25 × 12 = 39.
2. $11.25 ÷ 2.5 = $4.50 per kg.
3. $6.40 and $6 per litre: the 1.25 L pack is the better buy.
4. 2 h 15 min = 2.25 h; 207 ÷ 2.25 = 92 km/h.
5. One part is $96 ÷ 8 = $12: $60 and $36.
6. 1500 g : 600 g = 5 : 2.
7. 7.5 × 20 000 = 150 000 cm = 1.5 km.
8. 120 × 0.65 = 78 USD.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working and the units of every rate.

Card key

Unit-rate match (Day 2): 126 km in 1.5 hours is 84 km/h; 48 km in 40 minutes is 72 km/h; 2.4 kg of nuts for $9.60 is $4/kg; 750 g of cheese for $3.60 is $4.80/kg; 45 L of water in 9 minutes is 5 L/min; 30 L of petrol for 400 km is 7.5 L/100 km; $63 for 3.5 hours of work is $18/h; 120 words in 2.5 minutes is 48 words/min.

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: direct proportion, rates and best buys, ratio, scale, and modelling with money.