Mathematical modelling with rational numbers: a week of ready-to-teach maths

Five lessons for Year 7 Number. 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 catalogue cards, GST pair 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: GST pair cards (Day 3) and catalogue cards (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

Mathematical modelling turns a real question into mathematics and back again. You decide which numbers matter and which operations capture the situation, you calculate, and then you say what the answer means: the jacket costs $60, so it fits a $70 budget. Most money problems this week take a percentage of a base and then take it off or add it on: 25% off $80 removes $20, and 10% GST on $60 adds $6. Comparing value means putting options on the same footing, such as the price per 100 g.

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.

Money, rounding and calculators

Write money with a dollar sign, and with two decimal places when there are cents: $13.60, $4.05. Round to the nearest cent once, at the end of a calculation. Every answer is a sentence too: the price falls by $12, from $48 to $36. Days 1 and 2 build mental and written strategies without calculators; on Days 3 to 5 calculators are welcome for checking, and on Day 5 a spreadsheet suits the task well.

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 modelling cycle that every later day uses; Day 2 models discounts and markups; Day 3 adds GST to bills; Day 4 compares value with unit prices; and Day 5 puts the whole cycle to work on one open task. It draws on the fractions, decimals, percentages and ratios of the last units, and it prepares the percentage change, rates and financial modelling of Year 8.

Materials for the week (one trip)

• From the classroom: pencils, scissors, small envelopes for the cards, this pack printed.
• Cut-out sheet 1, one per pair: 12 GST pair cards (Day 3).
• Cut-out sheet 2, one per pair: 12 catalogue cards (Day 4).
• Calculators for Days 3 to 5; a spreadsheet on Day 5 if devices are handy.
• Optional: real receipts, catalogues or shelf labels students bring from home.

Words for the word wall

model, real problem, translate, solve, interpret, check, percentage, discount, sale price, markup, selling price, multiplier, GST, before GST, including GST, subtotal, total, unit price, per kilogram, per 100 g, better buy, cost, takings, profit, budget

Dear families

This week in maths, Year 7 is using mathematics to solve money problems: sale prices, price rises, GST, splitting bills and finding the better buy. We practise turning a real question into a calculation and then saying what the answer means.

Try this at home

• At the shops, find the unit price on a shelf label, such as the price per 100 g, for two sizes of one product. Which is better value?
• A $40 item is 25% off. What is the sale price? ($30.)
• Four people share a $90 bill equally. How much does each pay? ($22.50.)
• Look at a receipt together. Does it show how much GST is included?

Money maths this week

Fill one row a day with a money question you met: the maths you did, and what the answer meant.

Printed from the free seegongsik Mathematical modelling with rational numbers teaching pack · seegongsik.com/au/y7/number/AC9M7N09/pack

The modelling cycle

Modelling turns a real question into maths and back again: real problem, translate, solve, interpret. Students practise the two steps most often skipped, writing the maths before calculating and saying what the answer means.

We are learning to

• describe the four stages of the modelling cycle,
• write the calculation a situation needs before working it out,
• interpret an answer in the situation and check that it makes sense.

Success criteria

• I can write the calculation for “a $64 game is 25% off” before working it out.
• I can say whether an answer makes sense, and why.

You need

The worksheet. No calculators today.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Stopping at the percentage: 25% of $80 is the discount, $20, not the sale price.
• Leaving a bare number. The answer is a sentence: the jacket costs $60.
• Missing an answer that cannot be right, such as a sale price above the old price.

Answers

• Part A: 1 solve, 2 interpret, 3 real problem, 4 translate.
• Part B: 1. $72 ÷ 4; each friend pays $18. 2. 20% of $35 = $7, then $35 − $7; the cap costs $28. 3. 4% of $250 = $10, then add it on; after a year the balance is $260. 4. $3.20 × 2 × 5; the week of bus trips costs $32.
• Part C: 1. no, a discount lowers the price: $60 − $9 = $51; 2. yes, 3 × $15 = $45.
• Part D: any story that takes 10% off $40, such as a sale, giving $36. A year of months costs 25 × 12 = $300, so paying $270 up front saves $30.

The modelling cycle

A. Name the stage

Each line is one stage of the same problem. Write real problem, translate, solve or interpret.

B. Translate, then solve

Write the calculation first. Then work it out and write the answer as a sentence.

1. Four friends share a $72 pizza order equally.
   Calculation: ____________________________________
   Answer: ________________________________________
2. A $35 cap is 20% off.
   Calculation: ____________________________________
   Answer: ________________________________________
3. A savings balance of $250 earns 4% interest in a year.
   Calculation: ____________________________________
   Answer: ________________________________________
4. The bus fare is $3.20 each way, for 5 school days.
   Calculation: ____________________________________
   Answer: ________________________________________

C. Does it make sense?

1. Kim works out 15% off a $60 shirt and gets $69. Does it make sense? If not, fix it.
2. Jo splits a $45 bill between 3 people and says each pays $15. Does it make sense? If not, fix it.

D. Stretch

Write a real problem that the calculation 40 × 0.9 could model, then solve and interpret it. ________________________________

A phone plan costs $25 a month, or $270 for a year paid up front. Which is better value, and by how much? ________________

Discounts and markups

A discount finds a percentage of the price and takes it off; a markup finds a percentage of the cost and adds it on. One multiplication does both steps at once: 25% off is × 0.75, and a 25% markup is × 1.25.

We are learning to

• calculate a discount and a sale price,
• calculate a markup and a selling price,
• choose an efficient strategy, such as one multiplication, and compare offers.

Success criteria

• I can find the sale price of a $56 item with 25% off in two ways.
• I can say why 25% off is the same as multiplying by 0.75.

You need

The worksheet. No calculators today.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Subtracting the percentage itself: 25% off $56 is not $31.
• Multiplying by 0.25 for 25% off. That gives the discount, not the sale price.
• Thinking 20% off and then 10% off is 30% off. The second discount comes off a smaller price.

Answers

• Part A: $12 off, $36; $9 off, $81; $18 off, $102; $14 off, $21; $12.50 off, $237.50.
• Part B: $15 on, $75; $9 on, $27; $60 on, $210; $1.70 on, $10.20.
• Part C: 1. $15 off (20% is only $14); 2. 25% off ($30 off); 3. the same, $9 off either way.
• Part D: $100 → $80 → $72, which is 28% off, not 30%. $56 is 80% of the old price, so $56 ÷ 0.8 = $70.

Discounts and markups

A. Sale prices

B. Markups

C. Which offer is better?

1. A $70 item: $15 off, or 20% off? ________________________

2. A $120 item: $25 off, or 25% off? ________________________

3. A $45 item: $9 off, or 20% off? ________________________

D. Stretch

A $100 jacket is 20% off, then a further 10% off the sale price. Is that the same as 30% off? Show why. ________________________

After a 20% discount, a pair of jeans costs $56. What was the price before the sale? ________________

GST and bills

The goods and services tax (GST) is 10% on most goods and services in Australia. Shop prices already include it, but a quote or a bill often lists prices before GST, and the total adds a tenth: the price × 1.1.

We are learning to

• find 10% GST on a price before GST and the total,
• complete a bill with a subtotal, GST and a total,
• find the GST already inside a shop price.

Success criteria

• I can add GST to $46: $4.60 GST, $50.60 in all.
• I can explain why the GST inside a $77 price is $7, not $7.70.

You need

The GST pair cards from cut-out sheet 1 (one set per pair), the worksheet, calculators for checking.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Taking 10% of a price that already includes GST. The GST inside $77 is a tenth of $70.
• Moving the decimal point the wrong way: 10% of $12.40 is $1.24, not $124.
• Rounding along the way. Work in cents and round once, at the end.

Answers

• Part A: $30: $3.00 GST, $33.00; $85: $8.50 GST, $93.50; $12.40: $1.24 GST, $13.64; $250: $25.00 GST, $275.00; $7.95: $0.80 GST, $8.75.
• Part B: subtotal $306.50, GST $30.65, total $337.15.
• Part C: $55 ÷ 11 = $5.00; $132 ÷ 11 = $12.00; $8.80 ÷ 11 = $0.80.
• Part D: $100 + 10% = $110, and 10% off $110 is $99: no, the second 10% is of a bigger amount. $1200 plus GST is $1320, so the $1300 quote is cheaper by $20.

GST and bills

A. Add GST

B. Complete the plumber’s bill

C. The GST inside a shop price

A shop price is the price before GST plus a tenth of it: 11 tenths. So the GST is one eleventh of the shop price.

D. Stretch

Add 10% to $100, then take 10% off the result. Do you get back to $100? Explain. ________________________

One builder quotes $1200 plus GST. Another quotes $1300 including GST. Which is cheaper, and by how much? ________________

Best buys

To compare packs fairly, put them on the same footing: divide the price by the quantity to get a unit price, such as the price per item, per kilogram or per 100 g. Shelf labels in Australian supermarkets show a unit price for this reason.

We are learning to

• find a unit price per item, per kilogram, per litre or per 100 g,
• compare two packs and choose the better buy,
• explain when the bigger pack is not the better choice.

Success criteria

• I can show that 4 for $5.60 ($1.40 each) beats 3 for $4.50 ($1.50 each).
• I can give one reason to buy the smaller pack.

You need

The catalogue cards from cut-out sheet 2 (one set per pair), the worksheet, calculators for checking.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Choosing the lower sticker price. Compare the price per unit.
• Comparing a price per kilogram with a price per 100 g. Use one unit for both.
• Thinking bigger is always cheaper. Some pairs on the cards go the other way.

Answers

• Part A: popcorn $0.75 per bag; rice $2.70 per kg; milk $1.55 per L; cheese $1.50 per 100 g.
• Part B: 1. 100 bags ($6.50 per 100 bags, not $7.00); 2. 2 L ($2.10 per L, not $2.20); 3. 1 kg ($1.80 per kg, not $1.90); 4. 200 g ($1.20 per 100 g, not $1.25).
• Part C: for example, it may go off before it is used, there may be nowhere to store it, or the money is not there today.
• Part D: 750 g $3.20 per kg, 2 kg $3.15 per kg, so the 2 kg bag, by $0.05 a kilogram. 400 g $7.00, 1 kg $6.50, 1.5 kg $6.80 per kg, so 1 kg.
• Cards, better buy first: rice 5 kg bag ($2.20 per kg); yoghurt 1 kg tub ($0.55 per 100 g); pasta 500 g packet ($0.30 per 100 g); apples bag of 10 ($0.65 per apple); orange juice 2 L bottle ($1.80 per L); muesli bars box of 12 ($0.50 per bar).

Best buys

A. Find the unit price

B. Which is the better buy?

Find both unit prices, then circle the better buy.

1. Tea bags: 50 bags for $3.50, or 100 bags for $6.50. Compare per 100 bags. ________________________
2. Juice: 1.5 L for $3.30, or 2 L for $4.20. Compare per L. ________________________
3. Flour: 1 kg for $1.80, or 2 kg for $3.80. Compare per kg. ________________________
4. Biscuits: 200 g for $2.40, or 300 g for $3.75. Compare per 100 g. ________________________

C. Not always the bigger pack

Give two reasons a shopper might buy the smaller pack even when the bigger pack is better value.

D. Stretch

Pasta: 750 g for $2.40, or 2 kg for $6.30. Which is better value, and by how much per kilogram? ________________

Mixed nuts: 400 g for $2.80, 1 kg for $6.50, 1.5 kg for $10.20. Which is the best buy? ________________

A modelling task

One open task uses the whole week: students plan a sausage sizzle for the school fete, work out costs, takings and profit, and justify a price. Calculators or a spreadsheet are welcome today.

We are learning to

• turn a practical problem into a set of calculations,
• choose a representation, such as a table or a spreadsheet,
• interpret results and justify a decision in the situation.

Success criteria

• I can set out the costs in a table and find the profit.
• I can justify a price with numbers and a reason.

You need

The worksheet, calculators (a spreadsheet if devices are handy), the self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after question 4; start the second with question 5.

Watch for these ideas

• Buying part of a pack: 2.4 bottles of sauce means buying 3.
• Mixing up takings and profit. Profit is takings minus costs.
• Rounding the cost of a serve before multiplying. Keep the full value until the end.

Answers with working

1. sausages 10 × $9.60 = $96.00; bread 6 × $2.80 = $16.80.
2. onions 2 × $3.50 = $7.00; sauce 120 ÷ 50 = 2.4, so 3 × $4.00 = $12.00.
3. $96.00 + $16.80 + $7.00 + $12.00 = $131.80.
4. takings 120 × $3.00 = $360.00; profit $360.00 − $131.80 = $228.20.
5. $131.80 ÷ 120 is about $1.10 a serve, so each sale makes about $1.90.
6. 15% of $96.00 = $14.40; the sausages cost $81.60 and the profit rises to $242.60.
7. $131.80 ÷ $3.00 = 43.93, so 44 serves (43 would fall short).
8. food for 100 costs $115.40, takings $350.00, profit $234.60: $6.40 more than at $3.00; either choice with a reason, such as more money raised or more people fed.

A modelling task: the sausage sizzle

Your class is running the sausage sizzle at the school fete. Each serve is one sausage on a slice of bread, with onions and sauce. The plan is to sell 120 serves at $3.00 each.

Questions (7 and 8 are the stretch)

1. How many packets of sausages and loaves of bread are needed for 120 serves, and what do they cost?
2. How many bags of onions and bottles of sauce are needed? You cannot buy part of a bottle.
3. What is the total cost of the food?
4. If all 120 serves sell at $3.00 each, what are the takings? What is the profit?
5. What does one serve cost to make, to the nearest cent? How much does each sale make?
6. The butcher offers the school 15% off the sausages. How much does that save, and what is the new profit?
7. Stretch. How many serves must sell at $3.00 just to cover the cost of the food in question 3?
8. Stretch. At $3.50 a serve the class expects to sell only 100. Work out the food to buy and the profit for that plan. Which price would you choose, and why?

GST pair cards

Cut out the 12 cards. Day 3: lay them face down and play memory. A pair is a price before GST and the same price with 10% GST added.

Catalogue cards

Cut out the 12 cards. Day 4: match the two packs of each product, find each unit price, and mark the better buy. The bigger pack is not always it.

What we know: Modelling with money

Work on your own. Show your working, and write each answer as a sentence.

1. Name the four stages of the modelling cycle, in order. ________________________________________
2. A $52 pair of shoes is 25% off. What is the sale price? ________________
3. A shop buys a toaster for $30 and marks it up 40%. What is the selling price? ________________
4. A bill is $64 before GST. Find the GST and the total. ________________
5. Rice: 500 g for $3.40, or 1.5 kg for $9.60. Which is the better buy? Show the unit prices. ________________
6. Five friends share a $64.50 bill equally. How much does each pay? ________________
7. A $90 jacket is 20% off, and a voucher takes a further $10 off the sale price. What is the final price? ________________
8. Sam works out that a $40 item with 15% off costs $46. Explain the mistake and give the right price. ________________________________

Answers and marking guide

Answers with working

1. Real problem, translate to maths, solve, interpret the result (and check it makes sense).
2. 25% of $52 is $13, so $39; or 52 × 0.75.
3. 40% of $30 is $12, so it sells for $42.
4. GST $6.40; total $70.40.
5. 500 g is $6.80 a kilogram; 1.5 kg is $6.40 a kilogram; the 1.5 kg bag is the better buy.
6. $64.50 ÷ 5 = $12.90 each.
7. $90 → $72 after 20% off, then $72 − $10 = $62.
8. Sam added the $6 instead of taking it off; a discount lowers the price, so $34.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working and states each answer as a sentence about 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: the modelling cycle, discounts and markups, GST, best buys, and the modelling task.