Modelling with ratios: a week of ready-to-teach maths

Five lessons for Year 7 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 a house plan with furniture, modelling cards, 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 at 100%: the house plan and furniture (Day 1), one per pair, and modelling cards (Day 4), one set per group.
4. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.

The idea in plain words

A ratio compares quantities that change together. A scale of 1 : 100 means every length in the real thing is 100 times its length on the plan, so multiply to go from the plan to the real thing and divide to go back. Sharing in a ratio means finding one part first: add the parts, divide the total by them, then multiply for each share. Modelling wraps the maths in its situation: decide what the ratio is, choose a way to show it (a bar model, a table or a double number line), solve, and then answer in the situation’s own terms, rounding the way the situation needs.

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 ratios

Write a scale as 1 : 100 and a ratio as 3 : 5, in the order the question names the parts. Put lengths in one unit before multiplying, and give every answer with its unit and a sentence about the situation: “The kitchen is 4 m wide.” Rulers are needed on Days 1 and 2, and calculators from Day 3.

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

Days 1 and 2 use a ratio as a scale, on plans and maps; Day 3 shares quantities in a ratio; Day 4 scales recipes and mixtures and starts the modelling cards; and Day 5 solves practical problems from start to finish. It builds on ratios (AC9M7N08) and on converting metric units, and it leads into rates in Year 8 (AC9M8M05).

Materials for the week (one trip)

• From the classroom: pencils, rulers, scissors, calculators, this pack printed at 100%.
• Cut-out sheet 1, one per pair: the house plan and six pieces of furniture (Day 1).
• Cut-out sheet 2, one set per group: 8 modelling cards (Day 4).
• A local street map or a road atlas page with its scale, if you have one (Day 2).

Words for the word wall

ratio, part, share, scale, scale factor, scale bar, plan, model, map, enlarge, reduce, bar model, double number line, unitary method, mixture, model (to model a problem), interpret, justify

Dear families

This week in maths, Year 7 is using ratios to solve real problems: reading plans and maps, sharing amounts fairly, and scaling recipes and mixtures.

Try this at home

• Find the scale on a map, a model kit or a house plan, and work out a real length from it.
• Mix a drink or a cleaning product by its label’s ratio, and work out how much for a bigger batch.
• Halve or double a recipe together, and decide what to do with an egg that does not halve neatly.
• Ask: if a road is 3 cm on a map at 1 : 25 000, how long is it? Is that a walk or a drive?

Ratios this week

Fill one row a day with a place you met a ratio, the ratio, and what it told you.

Printed from the free seegongsik Modelling with ratios teaching pack · seegongsik.com/au/y7/measurement/AC9M7M06/pack

Plans

A scale is a ratio between a plan and the real thing. At 1 : 100, every real length is 100 times its length on the plan, so 1 cm on the plan is 1 m in the house. Students measure a house plan, try furniture in the rooms, and convert both ways at other scales.

We are learning to

• read a scale as a ratio of plan length to real length,
• multiply by the scale to go from the plan to the real thing, and divide to go back,
• change units after scaling, from centimetres to metres.

Success criteria

• I can find a real length from a plan, and a plan length from a real one.
• I can decide whether a piece of furniture fits a room, and say why.

You need

Cut-out sheet 1 printed at 100% (one per pair), rulers, scissors and the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the house; start the second with Part C.

Watch for these ideas

• Multiplying by the scale and keeping the wrong unit: 300 cm, not 300 m.
• Dividing when going from the plan to the real thing. The real thing is bigger.
• Measuring from the end of the ruler instead of from 0.

Answers

• Part A, printed at 100%: Bedroom 1 4 cm by 3.5 cm, 4 m by 3.5 m; Bedroom 2 3.5 cm by 3.5 cm, 3.5 m by 3.5 m; Living 6 cm by 4.5 cm, 6 m by 4.5 m; Kitchen 4 cm by 4.5 cm, 4 m by 4.5 m; the house 10 m by 9 m. Accept 0.2 m either way.
• Part B: (a) 2.2 m; (b) yes: 4 − 1.8 = 2.2 m to spare across (1.1 m each side) and 4.5 − 1 = 3.5 m the other way; (c) 6 × 4.5 = 27 m², and 27 × $45 = $1215.
• Part C: (a) 6.2 × 50 = 310 cm = 3.1 m; (b) 1800 cm ÷ 200 = 9 cm; (c) 200 cm ÷ 25 = 8 cm; (d) 2.5 m = 250 cm, and 250 ÷ 5 = 50, so 1 : 50.
• Part D: at 1 : 100, 1 cm² on the plan is 1 m by 1 m, so 12 m². At 1 : 200, 1 cm is 2 m, so 1 cm² is 4 m²: 48 m².

Plans

A. The house

Use the house plan on cut-out sheet 1. Measure each room in centimetres, then give its real size in metres.

B. Furniture

1. The couch is 2.2 cm long on the plan. How long is the real couch?   ____________
2. Will the table fit in the kitchen with at least 1 m clear on every side? Show why.   ________________________
3. Carpet costs $45 per square metre. How much does carpet for the living room cost?   ____________

C. Other scales

1. A plan is drawn at 1 : 50. A wall measures 6.2 cm on it. How long is the real wall?   ____________
2. A shed 18 m long is drawn at 1 : 200. How long is it on the plan?   ____________
3. A model kitchen is built at 1 : 25. The real bench is 2 m long. How long is the model bench?   ____________
4. On a plan, 5 cm stands for 2.5 m. What is the scale?   ____________

D. Stretch

A room measures 12 cm² on a 1 : 100 plan. What is its real area? What would it be on a 1 : 200 plan with the same 12 cm²?

Maps

A map scale is a ratio too, and a much bigger one: at 1 : 25 000, 1 cm on the map is 25 000 cm on the ground, which is 250 m. Students turn map lengths into real distances and back, measure a route with a scale bar, and choose a scale.

We are learning to

• find a real distance from a map length and its scale, and the other way,
• change centimetres to metres and kilometres: 100 000 cm is 1 km,
• use a scale bar, and choose a scale for a map.

Success criteria

• I can find a real distance from a map and give it in sensible units.
• I can explain how I chose a scale for a map.

You need

The worksheet printed at 100%, rulers, calculators, and a local map with its scale if you have one.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Dividing by 100 to get kilometres. There are 100 000 cm in a kilometre.
• Reading 1 : 25 000 as 1 cm to 25 000 m. Both numbers are in the same unit.
• Multiplying a real distance by the scale to get a map length. The map is smaller: divide.

Answers

• Part A: (a) 3 × 25 000 = 75 000 cm = 0.75 km; (b) 7.2 × 25 000 = 180 000 cm = 1.8 km; (c) 2.5 km = 250 000 cm, ÷ 25 000 = 10 cm.
• Part B: (a) 6.5 km; (b) 3.6 km; (c) 30 km.
• Part C, printed at 100%: A to B 4 cm, B to C 6.5 cm, and 2.5 cm is 1 km, so through B is 4.2 km; straight is 8 cm, 3.2 km; the straight road is 1 km shorter. Accept 0.1 km either way.
• Part D: 4000 km = 400 000 000 cm, and 400 000 000 ÷ 25 = 16 000 000, so the scale must be at least 1 : 16 000 000. A rounder 1 : 20 000 000 makes the map 20 cm wide, with room for a margin.

Maps

100 cm = 1 m · 1000 m = 1 km · 100 000 cm = 1 km

A. At 1 : 25 000

1. A road is 3 cm long on the map. How long is the real road?   ____________
2. A road is 7.2 cm long on the map. How long is the real road?   ____________
3. Two towns are 2.5 km apart. How far apart are they on the map?   ____________

B. Other maps

1. On a 1 : 100 000 map, a distance measures 6.5 cm. What is the real distance?   ____________
2. On a 1 : 40 000 map, a distance measures 9 cm. What is the real distance?   ____________
3. On a 1 : 250 000 map, a distance measures 12 cm. What is the real distance?   ____________

C. A route

D. Stretch

A map of Australia must fit across a page about 25 cm wide. Australia is about 4000 km wide. Choose a sensible scale for the map, and explain your choice.

Sharing in a ratio

To share in a ratio, find one part first: add the parts, divide the total by that number, then multiply for each share. A bar model shows every step. Students share in two- and three-part ratios and work back from one share to the whole.

We are learning to

• share an amount in a two- or three-part ratio,
• draw a bar model to show the parts and one part’s value,
• work back from one share to the other share and the total.

Success criteria

• I can share an amount in a ratio and check that the shares add to the total.
• I can find the total when I know one share.

You need

The worksheet and 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 the total by one number of the ratio, such as $84 ÷ 3, instead of by all the parts.
• Giving the shares in the wrong order. The first share goes with the first number.
• Leaving out a part in a three-part ratio. Add all three.

Answers

• Part A: (a) 7 parts of $12: $36 and $48; (b) 9 parts of 50 g: 100 g and 350 g; (c) 3 parts of 0.5 L: 0.5 L and 1 L.
• Part B: (a) 6 parts of 120 g: 120 g, 240 g, 360 g; (b) 15 parts of $100: $400, $500, $600.
• Part C: (a) one part is 9, so the other share is 45 and the total 63; (b) one part is 3, so there are 9 red and 33 altogether.
• Part D: by hours, 3 : 5 gives $36 and $60; equally, $48 each. Either can be fair with a reason: hours pay for work done, and equal shares suit a stall that needed both people to run.

Sharing in a ratio

Label each bar with one part’s value, then group the parts.

A. Two parts

B. Three parts

C. Work back

1. Two shares are in the ratio 2 : 5. The smaller share is 18. Find the larger share and the total.   ____________
2. A bunch of flowers is red and white in the ratio 3 : 8. There are 24 white flowers. How many red, and how many altogether?   ____________

D. Stretch

Two students ran a cake stall that made $96. One worked 3 hours and the other 5 hours. Should they share the money in the ratio of their hours, or equally? Work out both, then choose and justify.

Mixtures and recipes

A recipe or a mixture keeps its ratio when the batch changes, so every part is multiplied by the same factor. Students scale a recipe, solve the modelling cards, and compare two mixtures.

We are learning to

• scale a recipe by finding the amount for one serve first,
• find each part of a mixture from its ratio and the total,
• compare two mixtures by the fraction that is one part.

Success criteria

• I can scale a recipe up or down and check that the ratio stays the same.
• I can say which of two mixtures is stronger and show why.

You need

Modelling cards from cut-out sheet 2 (one set per group, cut out before the lesson), calculators and the worksheet.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Adding the same amount to each ingredient instead of multiplying, which changes the ratio.
• Comparing mixtures by one part alone. Compare the fraction of the whole that part makes.
• Mixing millilitres and litres in one calculation.

Answers

• Part A: for 1, eggs 0.25, milk 30 mL, flour 20 g; for 12, 3, 360 mL, 240 g; for 20, 5, 600 mL, 400 g.
• Part B (cards): 1 3 L of cordial (and 15 L of water); 2 50.3 cm; 3 1.5 L red and 4.5 L white; 4 2 L orange, 4 L pineapple, 8 L lemonade; 5 2.7 km; 6 750 g; 7 $1200, $720, $480; 8 36 mL.
• Part C: (a) A is 2 of 11 = 0.182, B is 3 of 16 = 0.188, so B has more cordial; (b) A is 2 of 5 = 0.400, B is 3 of 8 = 0.375, so A has more blue.
• Part D: 2 L in 3 : 1 is 1.5 L orange and 0.5 L mango; with 1 L more mango it is 1.5 : 1.5, which is 1 : 1.

Mixtures and recipes

A. Scale a recipe

Pancakes for 8. Find the amounts for 1 pancake first.

B. Modelling cards

C. Which is stronger?

1. Mix A is 2 : 9 and mix B is 3 : 13, cordial to water, in cups. Which has more cordial in it? Show why.   ______________
2. Mix A is 2 : 3 and mix B is 3 : 5, blue to yellow paint, in litres. Which has more blue in it? Show why.   ______________

D. Stretch

Juice is 3 parts orange to 1 part mango. 1 L of mango is poured into 2 L of this juice. What is the new ratio of orange to mango?

Modelling

Modelling puts the ratio inside a situation: decide what the ratio is, choose a picture, solve, then answer in the situation’s own terms, rounding the way it needs. Students solve six practical problems, then play the quiz.

We are learning to

• set up a ratio from a practical situation,
• interpret an answer, rounding up or down as the situation needs,
• justify the method and the picture we chose.

Success criteria

• I can solve a practical ratio problem and give the answer in a sentence.
• I can explain why I rounded my answer the way I did.

You need

The worksheet, calculators, and the unit’s self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after problem 3; start the second with problem 4.

Watch for these ideas

• Rounding by the usual rule when the situation says otherwise: 2.7 packets means buying 3.
• Sharing the raffle prize equally, or by adding the amounts paid, instead of in their ratio.
• Reading 1.625 hours as 1 hour 62 minutes. The 0.625 of an hour is 0.625 × 60 = 37.5 minutes.

Answers with working

1. 20 m = 2000 cm, and 2000 ÷ 87 = 22.99, so about 23 cm.
2. 5 L ÷ 25 = 0.2 L of oil, which is 200 mL.
3. The ratio is 5 : 6 : 9, 20 parts of $12.50: $62.50, $75.00, $112.50.
4. 26 × 25 000 = 650 000 cm = 6.5 km; 6.5 ÷ 4 = 1.625 hours, about 1 hour 38 minutes.
5. 300 ÷ 4 = 75 g each, × 18 = 1350 g, which is 2.7 packets: buy 3, because 2 packets is not enough.
6. Cement is 1 part in 6: 360 ÷ 6 = 60 kg, which is 3 bags.
7. 160 m = 16 000 cm, and 16 000 ÷ 40 = 400, so 1 : 400 just fits (the 110 m side is then 27.5 cm); 1 : 500 leaves a margin.
8. Jug A has 0.4 L of cordial and jug B 0.2 L, so 0.6 L of cordial to 3.4 L of water: 0.6 : 3.4 = 3 : 17.

Modelling

For each problem: write the ratio, solve it, and answer in a sentence about the situation.

Problems (7 and 8 are the stretch)

1. A model railway is built at a scale of 1 : 87. A real carriage is 20 m long. How long is the model carriage, to the nearest centimetre?
2. A mower runs on 25 parts petrol to 1 part oil. How much oil goes with 5 L of petrol? Give the answer in millilitres.
3. Three friends put in $5, $6 and $9 for raffle tickets and win $250. They share the prize in the ratio of what they paid. How much does each get?
4. On a 1 : 25 000 map, a bush walk measures 26 cm. How long is the walk, and about how long does it take at 4 km an hour?
5. A pasta recipe for 4 uses 300 g of pasta. Pasta comes in 500 g packets. How many packets are needed for 18 people?
6. A concrete mix is cement, sand and gravel in the ratio 1 : 2 : 3 by mass. Cement comes in 20 kg bags. How many bags make 360 kg of mix?
7. Stretch. A poster of the school oval, 160 m by 110 m, must fit on paper 40 cm by 28 cm. Choose a scale and justify it.
8. Stretch. Jug A holds 2 L of cordial and water mixed 1 : 4. Jug B holds 2 L mixed 1 : 9. Both are poured into a bowl. What is the ratio of cordial to water in the bowl?

A house plan at 1 : 100

Day 1. Print at 100% if you can: then 1 cm on the plan is 1 m in the house. If the print size is different, use the scale bar instead. Cut out the furniture, drawn at the same scale, to try in the rooms.

Modelling cards

Day 4. Cut out the 8 cards. For each card: write the ratio, choose a bar model, a table or a double number line and say why, solve it, then answer in a sentence about the situation.

What we know: Modelling with ratios

Work on your own. Show your working and give each answer with its unit.

1. A plan is drawn at 1 : 50. A wall is 7.4 cm long on the plan. How long is the real wall?   ____________
2. On a 1 : 30 000 map, a road measures 5 cm. How long is the real road, in kilometres?   ____________
3. A model boat is built at 1 : 40. The real boat is 4.8 m long. How long is the model?   ____________
4. Share $210 in the ratio 2 : 5.   ____________

5. Share 900 g of fruit in the ratio 2 : 3 : 4.   ____________
6. A recipe for 15 biscuits uses 180 g of butter. How much butter for 40 biscuits?   ____________
7. Green paint is 2 parts blue to 3 parts yellow. A room needs 6.5 L of green. Blue comes in 1 L tins. How many tins of blue must be bought? Explain the rounding.
8. 48 students travel on two buses in the ratio 5 : 3. Draw a bar model to find how many are on each bus, and say why a bar model suits.

Answers and marking guide

Answers with working

1. 7.4 × 50 = 370 cm = 3.7 m.
2. 5 × 30 000 = 150 000 cm = 1.5 km.
3. 4.8 m = 480 cm, and 480 ÷ 40 = 12 cm.
4. 7 parts of $30: $60 and $150.
5. 9 parts of 100 g: 200 g, 300 g, 400 g.
6. 180 ÷ 15 = 12 g a biscuit, × 40 = 480 g.
7. Blue is 2 of 5 parts: 6.5 ÷ 5 × 2 = 2.6 L, so buy 3 tins. 2 tins would leave the paint short, so round up.
8. A bar of 8 equal parts, each 6 students: 30 and 18 students. A bar model suits because it shows the whole and the equal parts at once.

A quick three-level guide

Eight questions, four ideas. A student at standard scales both ways with the right units, shares by finding one part, and answers a practical problem in a sentence, rounding as the situation needs.

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: plans, maps, sharing in a ratio, mixtures and recipes, and modelling.