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

Five lessons for Year 8 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 ratio bars, twelve situation cards, eight camp planning 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, one copy per pair or group: the situation cards are for Day 1, the camp cards for Day 4.
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

Real problems do not arrive as sums. Modelling turns a situation into mathematics in four steps: formulate (decide what matters, and whether the situation is a ratio or a rate), solve, interpret (say what the answer means in the situation, with units), and review (check that it makes sense and say what the model leaves out). A ratio compares parts of the same kind, so sharing $60 in 2 : 3 means 5 equal parts of $12. A rate compares different kinds, so $25 an hour scales to any number of hours. Money makes the review step matter: a model that ignores a booking fee or a budget can give the wrong decision.

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.

Prices in this pack

Prices, wages, fares and exchange rates are made up for the problems, close to typical Australian values, so that the working comes out cleanly. Students can bring today’s real prices into the review step: would the decision change?

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 decides which model a situation needs; Day 2 works ratio models; Day 3 works rate models with money; Day 4 runs the whole cycle on one open task; and Day 5 reviews models and their answers. It builds on ratios in Year 7 and rates earlier in Year 8, and prepares modelling with direct proportion, rates and scale, including financial contexts, in Year 9.

Materials for the week (one trip)

• From the classroom: pencils, rulers, calculators, scissors, this pack printed.
• Cut-out sheet 1, one per pair: 12 situation cards (Day 1).
• Cut-out sheet 2, one per group of three or four: 8 camp planning cards (Day 4).
• Optional: a catalogue, a phone plan brochure or a taxi fare table for real prices in the review steps.

Words for the word wall

model, formulate, solve, interpret, review, assumption, ratio, rate, part, share, ratio bar, wage, overtime, fixed cost, cost per unit, break-even, budget, exchange rate, reasonable

Dear families

This week in maths, Year 8 is using ratios and rates to solve real problems, many of them about money: sharing a prize fairly, working out pay with overtime, comparing phone plans and planning transport for a school camp. They learn to set up a problem, solve it, explain the answer and check whether it makes sense.

Try this at home

• Look at a phone, internet or electricity bill together. Which parts are fixed and which depend on how much you use?
• Ask: two people share a $90 prize in the ratio 1 : 2. How much each? ($30 and $60.)
• Ask: a job pays $24 an hour. What is the pay for 7.5 hours? ($180.)
• When you next buy something, ask: is the bigger pack really cheaper per kilogram?

My models this week

Fill one row a day. Write a real problem you met, whether it is a ratio or a rate, and its answer.

Printed from the free seegongsik Modelling with ratios and rates teaching pack · seegongsik.com/au/y8/measurement/AC9M8M07/pack

Ratio or rate? Formulating a model

Before solving, decide what kind of model a situation needs. Students sort situations into ratio and rate models, meet the four steps of modelling, and practise the first: formulating.

We are learning to

• decide whether a situation needs a ratio (parts of the same kind) or a rate (an amount per unit of something else),
• name the four steps of modelling: formulate, solve, interpret, review,
• write a model, and say what it assumes.

Success criteria

• I can say why sharing a prize is a ratio model but a wage is a rate model.
• I can write the model for a problem before I calculate anything.

You need

The worksheet and a calculator each; the situation cards from cut-out sheet 1, one set per pair, cut out.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Calling anything with two numbers a ratio.
• Jumping to a calculation before deciding what the numbers mean.
• Thinking a model is wrong because it leaves something out: every model does, so say what.

Answers

• Part A: rate 1, 3, 4, 7, 9, 10, 12; ratio 2, 5, 6, 8, 11. Answers: 1. $285; 2. $100 and $150; 3. 32.4 L; 4. 43.2 L; 5. 0.3 L; 6. 1.5 km; 7. $5.40; 8. 16 boys; 9. 22.5 minutes; 10. $80; 11. 4 bags; 12. 2.5 hours.
• Part B: 1. ratio 300 : 4, so flour = 300 ÷ 4 × 10 = 750 g; 2. rate $34/h, so pay = 34 × 5.5 = $187; 3. ratio 5 : 1, so cement = 40 ÷ 5 = 8 kg.
• Part C: 1. that every ingredient scales the same way; 2. a steady leak all day and night; 3. driving like the test, with no hills or traffic.
• Part D: answers vary; check that each situation names both quantities and whether they are the same kind.

Ratio or rate? Formulating a model

Formulate → solve → interpret → review

A. Situation cards

Solve any four cards here, with units:

B. Formulate, do not solve yet

For each problem, say whether it is a ratio or a rate, and write the calculation you would do.

C. What does the model assume?

D. Stretch: write your own

Write one ratio situation and one rate situation from your own life. For each, name the two quantities and solve it.

Sharing in a ratio

A ratio bar turns sharing into equal parts: add the ratio numbers, find one part, then build each share. Students share money two and three ways, work back from one share, and judge whether a ratio share is fair.

We are learning to

• share an amount in a ratio by finding the value of one part,
• work back from one share to the total and the other shares,
• interpret a share in the situation, and check that the shares add to the total.

Success criteria

• I can share $90 in the ratio 4 : 5 and check my answer.
• I can find the total when I know one share and the ratio.

You need

The worksheet and a calculator each.

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 of the ratio numbers instead of by their total.
• Giving shares that do not add up to the total.
• Mixing up the order: in 3 : 5, the first person gets 3 parts.

Answers

• Part A: $60 in 2 : 3: $24, $36; $840 in 3 : 4: $360, $480; $1500 in 2 : 3 : 5: $300, $450, $750.
• Part B: one part = $36 ÷ 3 = $12; the other share is 5 × $12 = $60, and the total is $96.
• Part C: 18 : 12 hours, so $450 is shared $270 and $180. Equal shares would give each class $225; which is fairer is a judgement: say why.
• Part D: answers vary; a fair rule should be agreed before the money is earned.

Sharing in a ratio

Add the ratio numbers to get the parts. Divide the total by the parts. Multiply to get each share. Check the shares add up.

A. Share it

B. Work back

Ali and Bo share some money in the ratio 3 : 5. Ali gets $36. How much does Bo get, and how much was there altogether?

C. Is it fair?

Two classes run a fundraising stall and make $450. Class A works 18 hours and Class B works 12 hours. Share the money in the ratio of the hours. Then say whether you think this is fair, and why.

D. Stretch: a fair rule

Three friends buy a raffle ticket together. Write a rule for sharing any prize fairly, and use it for a prize of $500.

Rates that cost money

Many prices are a rate, often with a fixed part added. Students model a week’s pay with overtime, compare two phone plans and find where they cost the same, build a taxi fare model, and say what each model leaves out.

We are learning to

• model pay as a rate times hours, with a higher rate for overtime,
• model a cost as a fixed part plus a rate, and compare two such models,
• review a money model: what does it leave out, and does that change the decision?

Success criteria

• I can work out a week’s pay when some hours are paid at time-and-a-half.
• I can find when two plans cost the same, and say which is cheaper either side.

You need

The worksheet and a calculator each.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Paying every hour at time-and-a-half, not just the hours over the normal week.
• Forgetting the fixed part of a plan or a fare.
• Choosing a plan from one usage only, without asking how much the person uses.

Answers

• Part A: 38 × $28 = $1064; 6 × $42 = $252; total $1316.
• Part B: 100 min: A $40, B $50; 200 min: A $50, B $50; 300 min: A $60, B $50. Same at 200 minutes; under that A is cheaper, over it B.
• Part C: fare = $4.20 + $2.10 × km; for 12 km, $4.20 + $25.20 = $29.40. It leaves out waiting time, tolls, booking fees and night rates.
• Part D: $4.20 + $2.10 × km = $50 gives km = ($50 − $4.20) ÷ $2.10 = 21.809524, so about 21 km.

Rates that cost money

A. A week’s pay

Sam earns $28 an hour for the first 38 hours of a week, and time-and-a-half (1.5 times the rate) for any hours after that. Last week Sam worked 44 hours. What was Sam’s pay?

B. Two phone plans

Plan A: $30 a month plus 10 cents a minute of calls. Plan B: $50 a month, calls included.

When do the plans cost the same? Which plan would you choose, and for whom? ______________________

C. A taxi model

A taxi charges $4.20 when you get in, then $2.10 for every kilometre. Write the model in words or symbols. What does a 12 km trip cost? Name two things the model leaves out.

D. Stretch: work backwards

You have $50 for the taxi. About how far can you go?

A modelling task: the school camp

Groups plan the transport for a school camp from eight information cards, two of which do not matter. They run the whole modelling cycle: choose what matters, find the options, interpret the cheapest, and review it against the rules and the budget.

We are learning to

• decide which information a model needs, and which it can leave out,
• list and cost the options that meet every condition,
• communicate a decision with its reasons, and review it.

Success criteria

• I can say which cards matter for the transport cost, and why the others do not.
• I can recommend the transport and explain why it is the best option within the rules.

You need

The worksheet and a calculator each; one set of camp cards (cut-out sheet 2) per group of three or four, cut out.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Using every card, including the distance and lunch cost.
• Forgetting the teachers when counting seats, or the teacher in each vehicle.
• Rounding a number of vehicles down: 1.84 coaches means 2.

Answers

• Part A: cards 1, 2, 4, 6, 7, 8 matter; cards 3 and 5 do not (prices include fuel, and lunch is not transport). People: 86 + 6 = 92.
• Part B: 2 coaches: 100 seats, $2400; 1 coach and 4 minibuses: 98 seats, $2560. Minibuses alone need 8, but there are only 6 teachers.
• Part C: book 2 coaches for $2400, within the $2500 budget; the next option costs $2560, over budget.
• Part D: answers vary: luggage space, a spare seat for a late helper, or splitting groups at the camp.

A modelling task: the school camp

Your group must book buses for a one-day school camp. Use the camp cards. Every person needs a seat.

A. Formulate

Which card numbers matter? Which do not, and why? How many people need seats?

B. Solve

C. Interpret and review

Write your recommendation to the principal in two or three sentences: what to book, what it costs, and why.

D. Stretch: what the model leaves out

Name two things a real camp organiser would also think about that your model does not include. Would they change your answer?

Review the model

The last step of modelling asks whether the answer makes sense and whether the model fits the situation. Students solve exchange-rate and mixing problems, judge whether models are appropriate, find a sharing mistake, and play the quiz as a class game.

We are learning to

• check an answer against the situation, not just the arithmetic,
• say when a model is appropriate and when it is not,
• explain a mistake and fix it.

Success criteria

• I can check a share by adding the parts back to the total.
• I can say one reason a model’s answer might not match real life.

You need

The worksheet and a calculator each; the self-check quiz at the bottom of the unit page for the game.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Treating a model’s answer as exact when it rests on assumptions.
• Checking the arithmetic but not whether the answer suits the situation.
• Dividing by the number of people instead of the number of parts.

Answers

• Part A: 1. 350 × 1.08 = 378 NZ dollars; 2. 3 + 1 = 4 parts, so 7.5 L of white and 2.5 L of black.
• Part B: 1. rough brick soaks up more paint, and walls usually need two coats; 2. people tire and take breaks; it is fine as an estimate of reading time, not of days; 3. the ratio ignores who paid; a share by contribution may be fairer.
• Part C: the shares $20 and $40 are 1 part and 2 parts of 3. In 2 : 3 there are 5 parts of $12: $24 and $36.
• Part D: answers vary; a good review names an assumption and says how the answer would change without it.

Review the model

A. Solve, then interpret

1. Suppose 1 Australian dollar buys 1.08 New Zealand dollars. How many New Zealand dollars do you get for 350 Australian dollars, before any fees?
2. Grey paint is mixed 3 parts white to 1 part black. How much of each is needed for 10 L of grey?

B. Is the model appropriate?

For each model, say whether its answer can be trusted, and why.

C. Find the mistake

Lee shared $60 in the ratio 2 : 3 and wrote $20 and $40. What went wrong? Give the right shares and a check.

D. Stretch: review your camp plan

Look back at your camp recommendation from Day 4. Write one assumption you made and how the answer would change if it were false.

Situation cards

For Day 1, one set per pair: twelve problems

Sort the cards by the model each problem needs: a rate (an amount per unit of something else) or a ratio (parts of the same kind).

Teacher note: rate: 1, 3, 4, 7, 9, 10, 12; ratio: 2, 5, 6, 8, 11. The answers are in the Day 1 plan.

Camp planning cards

For Day 4, one set per group: eight facts

You are booking the transport for a one-day school camp. Not every card matters for the transport. Decide which do.

Teacher note: cards 3 and 5 do not matter for the transport cost; the options and the answer are in the Day 4 plan.

What we know: Modelling with ratios and rates

Work on your own with a calculator. Show your model, then your answer with units.

1. Which needs a ratio model and which a rate model: “mix 2 parts blue to 5 parts yellow” or “petrol at $1.95 a litre”? ______________________
2. Share $72 in the ratio 4 : 5. Check your answer. ______________________
3. Two people share some money in the ratio 2 : 7. The larger share is $56. What was the total? ______________________
4. A job pays $22 an hour. What is the pay for 7.5 hours? ______________________
5. A taxi charges $3.80 plus $2.20 a kilometre. What does a 15 km trip cost? ______________________
6. Gym A charges $15 a visit. Gym B charges $40 a month plus $5 a visit. Which is cheaper for 6 visits a month? For how many visits do they cost the same? ______________________
7. Suppose 1 Australian dollar buys 0.65 US dollars. How many US dollars is $200 Australian? ______________________
8. A model says a 2 L tin of paint covers 32 m², so 10 L covers 160 m². Give one reason the real area painted could be less. ______________________

Answers and marking guide

Answers with working

1. The paint mix is a ratio (parts of paint to parts of paint); petrol is a rate (dollars per litre).
2. 4 + 5 = 9 parts of $8: $32 and $40; check 32 + 40 = 72.
3. One part = $56 ÷ 7 = $8; total = 9 × $8 = $72.
4. $22 × 7.5 = $165.
5. $3.80 + $2.20 × 15 = $36.80.
6. For 6 visits: A $90, B $70, so B. The same at 4 visits ($60).
7. 200 × 0.65 = 130 US dollars, before any fees.
8. Any sound reason, for example rough or porous walls, two coats, or paint left in the tin.

A quick three-level guide

Eight questions, four ideas. A student at standard writes a model before calculating and states each answer in the situation.

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: ratio or rate, sharing in a ratio, money rates, the camp task, and reviewing models.