Modelling with percentages: a week of ready-to-teach maths
Five lessons for Year 8 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 situation cards, multiplier cards, a mini-check and every answer with its working.
Start here: five minutes to Monday
- Skim the week at a glance on the next page.
- Print the five days. Each day is two A4 sheets: a plan and a worksheet.
- Print the two cut-out sheets, one copy per pair, and cut them before Day 1: the situation cards are for Day 1, the multiplier cards for Day 2.
- Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
- Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.
The idea in plain words
A mathematical model turns a real situation into numbers, works on the numbers, and turns the answer back into the situation. It runs in a cycle: formulate (what are we finding, and what are we assuming?), compute (choose an efficient way), interpret (what does the number mean here?) and review (is it reasonable, and do the assumptions still hold?). Percentages are the everyday language of money: 15% off, 10% GST, a 3% rise. Three moves cover most of them: a percentage of an amount, a change by a multiplier such as × 0.85 for 15% off, and one amount as a percentage of another. A good answer is a sentence with units, rounded 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.
Money and calculators
Money answers are given to the cent ($21.60, not $21.6) and rounded only at the end. Calculators are welcome from Day 2 once a student can say roughly what the answer should be; spreadsheets suit Days 4 and 5 if your class has them. Prices in this pack are made-up but realistic; GST is the Australian 10%.
The week at a glance
One lesson a day for a week. Each day stands on the day before, so run them in order.
| Day | Lesson | Students learn and do | On screen |
|---|---|---|---|
| 1 | From a situation to a number | Turn short stories into calculations and worded answers, match situation, calculation and answer cards, and judge whether an answer makes sense | From a situation to a number and back |
| 2 | Three percentage moves | Find a percentage of an amount, change an amount with a multiplier, and write one amount as a percentage of another | Three percentage moves; multiplier cards |
| 3 | Discounts and GST | Work sale prices two ways, add or find the 10% GST, and test whether two discounts make one | A 15% discount, worked through |
| 4 | The modelling cycle | Plan a sausage sizzle fundraiser: formulate with assumptions, compute costs and profit, interpret and review | The modelling cycle |
| 5 | Interpret, then review | Compare two phone plans, change an assumption and watch the answer move, write a recommendation, and play the quiz | Interpret, then review; self-check quiz |
How the week builds
Day 1 practises the two ends of modelling, getting into the numbers and back out again; Day 2 gives three percentage moves to work with; Day 3 uses them on real prices with discounts and GST; Day 4 runs the whole cycle on one larger problem; and Day 5 asks how an answer changes when an assumption does. It builds on Year 7 percentages and GST, and prepares the linear models of Year 8 algebra and the financial mathematics of Years 9 and 10.
Materials for the week (one trip)
- From the classroom: scissors, pencils, calculators (from Day 2), this pack printed.
- Cut-out sheet 1, one per pair: 24 situation, calculation and answer cards (Day 1).
- Cut-out sheet 2, one per pair: 24 change and multiplier cards (Day 2).
- Optional: a spreadsheet for Days 4 and 5, and a real supermarket catalogue for Day 3. No other equipment to buy.
Words for the word wall
model, formulate, assumption, compute, interpret, review, reasonable, percentage, percentage of, increase, decrease, multiplier, discount, sale price, GST, profit, cost, income, per unit, best buy, round to the cent
Dear families
This week in maths, Year 8 is using percentages to solve real money problems: discounts, GST, tips, splitting bills and comparing phone plans. Students learn to set up a problem, work it out efficiently, explain the answer in a sentence, and check whether it makes sense, including what they had to assume.
Try this at home
- Look at a receipt together. Find the GST: in Australia it is one eleventh of a GST-inclusive total.
- Ask: a $60 pair of shoes is 25% off. What do they cost? ($45: a quarter of $60 is $15.)
- Ask: is 20% off and then another 10% off the same as 30% off? (No: on $100 it gives $72, not $70.)
- Compare two phone or streaming plans you know. Which is cheaper for your family, and what are you assuming?
My money maths this week
Fill one row a day. Find a percentage in real life (a sale, a receipt, a news story) and do one calculation with it.
| Day | Where I found it | My calculation and answer |
|---|---|---|
| Monday | ||
| Tuesday | ||
| Wednesday | ||
| Thursday | ||
| Friday |
Printed from the free seegongsik Modelling with percentages teaching pack · seegongsik.com/au/y8/number/AC9M8N05/pack
From a situation to a number and back
Modelling starts and ends in words. Students turn short stories into calculations, write worded answers with units, match situation, calculation and answer cards, and judge whether an answer makes sense.
We are learning to
- pick out the question, the numbers and the operation in a short money or measurement story,
- write the calculation and a worded answer with units, rounded the way the story needs,
- decide whether an answer is reasonable, and fix it when it is not.
Success criteria
- I can turn “Six friends split a $92.70 bill.” into a calculation and a sentence.
- I can explain why “3.5 buses” is not a sensible answer.
You need
The worksheet; the situation, calculation and answer cards (cut-out sheet 1), one set per pair. No calculators today.
Lesson flow (about 60 minutes)
| 10 min | Into the numbers, and out again Show the board (see the box). Then model “Six friends split a $92.70 bill.”: 92.70 ÷ 6 = 15.45, so “Each pays $15.45.” Ask: “The calculator would say 15.45. Why is the sentence better than the number?” |
| 20 min | Match the triads Pairs lay out the 8 situation cards and find each one’s calculation card and answer card. Each pair explains one match aloud. Ask: “Two cards both multiply by 1.1. How did you tell which story each fits?” |
| 30 min | Write, judge, invent Parts A to C; early finishers do Part D. Share Part B, asking what the unreasonable answers forgot. Ask: “Which answers in Part B were right as numbers but wrong for the situation?” |
Two shorter sessions instead? Split after the triads; start the second with Part A.
Watch for these ideas
- Stopping at the number: an answer needs its unit and its meaning.
- Rounding the wrong way: buses, boxes and tins round up; shared money rounds to the cent.
- Choosing the operation from a key word (“share” always divides) rather than from the situation.
Answers
- Part A: 1. 86.40 ÷ 4 = 21.60; Each pays $21.60. 2. 4.80 × 2.5 = 12; They cost $12. 3. 345 ÷ 3 = 115; Each instalment is $115. 4. 240 ÷ 4 = 60; There are 60 km to go. 5. 12 ÷ 10 = 1.20 < 1.35; The box, at $1.20 a pen. 6. 200 ÷ 57 ≈ 3.5; Four buses (three are not enough).
- Part B: 1. reasonable (10% of $24 is $2.40); 2. not reasonable (money stops at cents: $16.67 each (one gets $16.66)); 3. not reasonable (more people need more flour: 450 g); 4. reasonable (3.5 × 7 = 24.5); 5. not reasonable (you cannot book half a bus: 4).
- Part C: any story whose numbers and question fit the calculation; check that the answer sentence has a unit.
- Part D: estimates such as 85 ÷ 4 ≈ 21 and 5 × 2.5 ≈ 12 catch slips like a misplaced decimal point.
From a situation to a number and back
A. Calculation and sentence
| Situation | Calculation | Answer sentence |
|---|---|---|
| 1. Four friends share a $86.40 pizza order equally. | ||
| 2. Bananas cost $4.80 a kilogram. How much for 2.5 kg? | ||
| 3. A $345 camp fee is paid in three equal instalments. | ||
| 4. A 240 km trip is three-quarters done. How far is left? | ||
| 5. Pens are $1.35 each or $12 for a box of 10. Which is cheaper per pen? | ||
| 6. A bus seats 57. How many buses for 200 students? |
B. Is it reasonable?
Circle yes or no. If no, write a better answer and why.
| 1. A $24 book has 10% off. It now costs $21.60. | yes · no | |
| 2. Three people share $50 equally. Each gets $16.666... | yes · no | |
| 3. A recipe for 4 uses 300 g of flour. For 6 people: Use 200 g. | yes · no | |
| 4. A car uses 7 L per 100 km. On a 350 km trip: It uses 24.5 L. | yes · no | |
| 5. 200 students need buses that seat 57. Book 3.5 buses. | yes · no |
C. Write the situation
Write a short money story for each calculation, then answer it in a sentence: 150 × 0.8 · 18.60 ÷ 3
D. Stretch: estimate first
For Part A questions 1 and 2, write a quick estimate before the exact answer. How does an estimate catch a misplaced decimal point?
Three percentage moves
Most percentage problems use one of three moves: a percentage of an amount, a change by a multiplier, or one amount as a percentage of another. Students practise each with an efficient strategy and match changes to multipliers.
We are learning to
- find a percentage of an amount using 10%, 5% and 25% as stepping stones,
- increase or decrease an amount in one step with a multiplier such as × 0.85,
- write one amount as a percentage of another, and undo a percentage change.
Success criteria
- I can find 20% off $75 in one step with a multiplier: $75 × 0.8 = $60.
- I can explain why adding 20% back to the sale price does not give the original price.
You need
The worksheet; the change and multiplier cards (cut-out sheet 2), one set per pair; calculators.
Lesson flow (about 60 minutes)
| 10 min | Three moves on a bar Show the board (see the box). For each move, ask what the efficient first step is: 10% is a tenth, 25% a quarter. Ask: “Why is 12 out of 48 the same percentage as 1 out of 4?” |
| 15 min | Match the multipliers Pairs match the 12 change cards to the multiplier cards. Settle “up 100%” (× 2) and “down 1%” (× 0.99) as a class. Ask: “Which multipliers are less than 1, and what does that tell you?” |
| 35 min | Of, change, as a percentage Parts A to C; early finishers do Part D. Share Part D: $60 after 20% off came from $75, not from $72. Ask: “Why does 20% of the sale price differ from 20% of the original?” |
Two shorter sessions instead? Split after the multiplier cards; start the second with Part A.
Watch for these ideas
- Using × 0.15 for 15% off: that finds the discount, not the price left.
- Writing 3% as × 1.3: it is × 1.03.
- Undoing a discount by adding the same percentage back.
Answers
- Part A: 10% of $65 = $6.50; 25% of $48 = $12; 15% of $60 = $9; 5% of $34 = $1.70; 12.5% of $80 = $10; 30% of $45 = $13.50.
- Part B: up 10% on $40: × 1.1 = $44; down 15% on $60: × 0.85 = $51; up 25% on $36: × 1.25 = $45; down 40% on $125: × 0.6 = $75; up 3% on $200: × 1.03 = $206.
- Part C: 18 of 24 = 75%; 7 of 20 = 35%; 450 of 1500 = 30%; 3 of 8 = 37.5%.
- Part D: $64 ÷ 0.8 = $80; $45 ÷ 0.75 = $60.
Three percentage moves
A. A percentage of an amount
Use a stepping stone (10%, 5%, 25%...) and show it.
B. Change it in one step
| Change | Multiplier | New amount |
|---|---|---|
| $40, up 10% | ||
| $60, down 15% | ||
| $36, up 25% | ||
| $125, down 40% | ||
| $200, up 3% |
C. As a percentage
| 1. What percentage: 18 of the 24 students handed in the form? | |
| 2. What percentage: a team won 7 of its 20 games? | |
| 3. What percentage: $450 saved towards a $1500 bike? | |
| 4. What percentage: 3 of the 8 slices are left? |
D. Stretch: undo the discount
a price after 20% off is $64. And a price after 25% off is $45. Find each original price. Then explain why adding the percentage back to the sale price gives the wrong answer.
Discounts and GST
Prices meet percentages in two places: discounts and the 10% GST. Students work sale prices two ways, add GST or find it inside a price, and test whether two discounts in a row make one.
We are learning to
- find a sale price by subtracting the discount, or in one step with a multiplier,
- add 10% GST with × 1.1, and find the GST inside a price by dividing by 11,
- explain why two discounts in a row are not the same as one discount of their sum.
Success criteria
- I can find 15% off $80 two ways and get $68 both times.
- I can find the GST in a $33 price and explain why it is not $3.30.
You need
The worksheet, one per student; calculators; a supermarket catalogue or receipts if you have them.
Lesson flow (about 60 minutes)
| 10 min | Two ways to a sale price Show the board (see the box). Then 15% off $80: $80 − $12 = $68, or $80 × 0.85 in one step. Ask: “Which way is faster on a calculator, and why does it give the same answer?” |
| 15 min | GST in and out A $30 item plus 10% GST costs $33. So GST is one eleventh of the $33, not a tenth. Pairs check with three receipt totals. Ask: “The GST was 10% of $30. Why is it not 10% of $33?” |
| 35 min | Sales, GST, double discounts Parts A to C; early finishers do Part D. Share Part C with the prices worked step by step. Ask: “Two 50% discounts in a row: is the item free? What is left, and why?” |
Two shorter sessions instead? Split after GST in and out; start the second with Part A.
Watch for these ideas
- Finding the GST in a GST-inclusive price as 10% of it.
- Adding two discounts: 20% then 10% is 28% off, not 30%.
- Rounding in the middle of a calculation instead of at the end.
Answers
- Part A: 15% off $40: $40 − $6 = $34 (× 0.85); 30% off $89.90: $89.90 − $26.97 = $62.93 (× 0.7); 12.5% off $64: $64 − $8 = $56 (× 0.875); 40% off $150: $150 − $60 = $90 (× 0.6).
- Part B: $50 + GST = $55; $123 + GST = $135.30; $8.99 + GST = $9.89; GST inside $55 is $5, $22 is $2, $143 is $13.
- Part C: 20% then 10% off $100 = $72, but 30% off = $70; 50% then 50% off $60 = $15, but 100% off = $0. The second discount comes off a smaller price.
- Part D: $60: $51 or $48; $80: $68 or $68; $110: $93.50 or $98. They match at $80: below that the $12 off is better, above it the 15%.
Discounts and GST
A. Sale prices, two ways
| Price and discount | Subtract the discount | One step with a multiplier |
|---|---|---|
| $40, 15% off | ||
| $89.90, 30% off | ||
| $64, 12.5% off | ||
| $150, 40% off |
B. GST
Add 10% GST to each price before GST. Then find the GST already inside each GST-inclusive price.
| $50 before GST: price with GST = | $55 includes GST: GST = | ||
| $123 before GST: price with GST = | $22 includes GST: GST = | ||
| $8.99 before GST: price with GST = | $143 includes GST: GST = |
C. Two discounts or one?
| $100: 20% off, then 10% off the new price. Is that the same as 30% off? | |
| $60: 50% off, then 50% off the new price. Is that the same as 100% off? |
D. Stretch: 15% off or $12 off?
A shop offers either 15% off or $12 off. Work out both prices for items costing $60, $80, $110. For which prices is each offer better? Find the price where they are equal.
The modelling cycle
One larger problem, run through all four stages. Students plan a sausage sizzle fundraiser: formulate with stated assumptions, compute costs and profit, interpret the answer for the class, and review what would change it.
We are learning to
- formulate a problem by naming what to find and listing the assumptions,
- compute costs, takings and profit efficiently, rounding packs up,
- interpret and review the answer, saying which assumption matters most.
Success criteria
- I can list at least three assumptions behind my profit and say why each matters.
- I can explain why the number of sausage packs is rounded up, not to the nearest.
You need
The worksheet, one per student; calculators; a spreadsheet if your class has one.
Lesson flow (about 60 minutes)
| 10 min | Four stages Show the board (see the box), then read the fundraiser brief at the top of the worksheet aloud. Ask: “Before any numbers: what are we trying to find out, and what will we have to assume?” |
| 35 min | Run the cycle Pairs work Parts A to C in order, writing each stage’s heading. Stop at 20 minutes to compare profits and assumptions. Ask: “Two pairs have different profits. Is one wrong, or did they assume different things?” |
| 15 min | Review together Each pair reports one assumption that would change the answer most. Early finishers do Part D. Ask: “If it rains and only 120 sell, are we still better off than doing nothing?” |
Two shorter sessions instead? Split after Part B; start the second with the interpretation.
Watch for these ideas
- Buying 11.7 packs: shops sell whole packs, so round up.
- Forgetting the fixed costs (gas, onions) that do not grow with sales.
- Treating takings as profit.
Answers
- Part A: find the profit from selling 140 sausages at $3; assumptions as listed (plus any sensible others).
- Part B: 12 packs of sausages ($138), 18 loaves ($63), 3 bottles of sauce ($18), onions $9, gas $25: cost $253. Takings 140 × $3 = $420; profit $167, about 39.8% of the takings.
- Part C: e.g. “Selling 140 at $3, the sizzle raises about $167.” The biggest risk is selling fewer: at 120sold (buying for 120), cost $213.50, takings $360, profit $146.50.
- Part D: profit rises by $70 at $3.50 if the same number sell; whether they would is the assumption to review.
The modelling cycle: a sausage sizzle
Your class runs a sausage sizzle to raise money. You expect to sell 140 sausages in bread at $3 each. Sausages come in packs of 12 for $11.50; bread in loaves of 8 slices for $3.50; a bottle of sauce ($6) serves 60. Onions cost $9 and gas hire $25.
A. Formulate
We are finding: ____________________________________________________________
Tick the assumptions you agree with and add one of your own.
| every sausage bought is sold | |
| each sausage gets one slice of bread | |
| one bottle of sauce serves 60 | |
| the gas bottle is hired for the day | |
B. Compute
| Item | How many, and why | Cost |
|---|---|---|
| Sausages | ||
| Bread | ||
| Sauce | ||
| Onions and gas | ||
| Total cost | ||
| Takings | ||
| Profit |
C. Interpret and review
Write one sentence telling the class what the sizzle will raise. Then: which assumption, if it were wrong, would change the answer most?
D. Stretch: change one thing
What if the price went up to $3.50? Recompute the profit, then say what new assumption you have made.
Interpret, then review
An answer is only as good as its assumptions. Students compare two phone plans for different users, change an assumption and watch the recommendation move, write the advice in a sentence, then play the quiz as a class game.
We are learning to
- compare two options by working out each one for several situations,
- say how an answer changes when an assumption changes,
- write a recommendation that names the assumption it depends on.
Success criteria
- I can say which phone plan suits someone who uses 15 GB a month, with the numbers.
- I can write advice that says when it would change.
You need
The worksheet, one per student; calculators; the self-check quiz on the unit page for the game.
Lesson flow (about 60 minutes)
| 10 min | One answer, three stories Show the board (see the box). Each button changes one assumption, and the answer and its review change with it. Ask: “The maths was right every time. So what made the answer change?” |
| 30 min | Which plan? Parts A to C; early finishers do Part D. Share Part C advice: the best ones give a usage at which the advice flips. Ask: “Below what usage is Plan A cheaper? How sure are you of next month’s usage?” |
| 20 min | Self-check quiz as a class game Teams of four hold up one, two or three fingers for their option, then reveal and read the explanation aloud. A team that was wrong says which stage of the cycle tripped it. Ask: “Before we reveal it: which percentage move does this question need?” |
Two shorter sessions instead? Split after Part B; start the second with Part C and the quiz.
Watch for these ideas
- Comparing only the monthly fees and ignoring extra data charges.
- Giving advice without the assumption it depends on (usage).
- Charging part of a gigabyte: extra data here is charged per whole GB started.
Answers
- Part A: 6 GB: Plan A $25, Plan B $39; 12 GB: Plan A $35, Plan B $39; 14 GB: Plan A $45, Plan B $39; 20 GB: Plan A $75, Plan B $39.
- Part B: Plan A is cheaper up to 12 GB ($35); from 13 GB Plan B is cheaper ($40 against $39).
- Part C: e.g. “Use Plan A if you stay under 13 GB a month; otherwise Plan B.” Accept any advice that names the usage it depends on.
- Part D: a 1.5% card surcharge on $68 is $1.02, so $69.02; the $2 flat fee on the board is bigger for any price under $133.33.
Interpret, then review
Plan A: $25 a month with 10 GB of data, then $5 for each extra GB (or part of a GB). Plan B: $39 a month with 30 GB, no extra charges up to that.
A. Cost for different users
| Data used in a month | Plan A cost (working) | Plan B cost | Cheaper |
|---|---|---|---|
| 6 GB | |||
| 12 GB | |||
| 14 GB | |||
| 20 GB |
B. Where does it flip?
Find the largest whole number of GB for which Plan A is still the cheaper (or equal) choice. Show how you know.
C. Write the advice
Write two sentences for a friend choosing a plan. Name the assumption your advice depends on and say when it would change.
D. Stretch: a percentage fee or a flat fee?
A shop adds a 1.5% card surcharge instead of the board’s $2 card fee. Work out the total for the $68 jacket. For which prices is the $2 flat fee the bigger charge?
Situation, calculation and answer cards
For Day 1, one set per pair: eight situations
Eight calculations
Eight answers
Teacher note: the triads are A $60 shirt is 25% off. 60 × 0.75 → It costs $45. / A $40 meal plus a 10% tip. 40 × 1.1 → You pay $44. / $72 shared by 3 people. 72 ÷ 3 → Each pays $24. / 9 of 12 plants flowered. 9 ÷ 12 × 100 → 75% flowered. / 2.5 kg of mince at $12 a kilogram. 12 × 2.5 → It costs $30. / A $120 bill with 10% GST added. 120 × 1.1 → The total is $132. / Saving $15 a week for a $105 game. 105 ÷ 15 → It takes 7 weeks. / A $200 phone drops 30% in value. 200 × 0.7 → It is worth $140.
Change and multiplier cards
For Day 2, one set per pair: twelve changes
Match each change to the number you multiply by to make it in one step.
Twelve multipliers
Teacher note: up 10% → × 1.1, down 15% → × 0.85, up 2.5% → × 1.025, down 40% → × 0.6, up 100% → × 2, down 5% → × 0.95, up 25% → × 1.25, down 12.5% → × 0.875, up 3% → × 1.03, down 50% → × 0.5, up 60% → × 1.6, down 1% → × 0.99.
What we know: Modelling with percentages
Work on your own. You may use a calculator from Question 4. Show your working.
- Find 25% of $56 without a calculator. ______________________
- A $45 game is 20% off. Find the sale price in one step. ______________________
- 27 of the 36 students came to the excursion. What percentage is that? ______________________
- A receipt total of $88 includes 10% GST. How much of it is GST? ______________________
- A $50 jacket is 10% off, then another 10% off the new price. Is that 20% off? Show why. ______________________
- After 10% off, a book costs $36. What was the original price? ______________________
- 5 friends share a $132.40 bill equally. Write the answer as a sentence. ______________________
- Pencils come in packs of 10. A school needs 87. How many packs, and what did you assume? ______________________
How did this week feel? Circle one: I can teach it · I am getting there · I need more help
Answers and marking guide
Answers with working
- 25% is a quarter: $56 ÷ 4 = $14.
- $45 × 0.8 = $36.
- 27/36 = 75%.
- $88 ÷ 11 = $8 (the price before GST is $80).
- $50 × 0.9 × 0.9 = $40.50, but 20% off is $40: not the same.
- $36 ÷ 0.9 = $40.
- $132.40 ÷ 5 = $26.48: “Each friend pays $26.48.”
- 87 ÷ 10 = 8.7, so 9 packs (rounding up), assuming packs cannot be split.
A quick three-level guide
| Idea | Working towards | At standard | Beyond |
|---|---|---|---|
| Percentage moves (Q1, Q3) | confuses the percentage with the amount | finds a percentage of an amount and writes an amount as a percentage | chooses an efficient stepping stone each time |
| Changes and money (Q2, Q4, Q5) | uses × 0.2 for 20% off, or 10% of a GST-inclusive price | uses multipliers, finds GST as one eleventh, and explains double discounts | undoes a change with division |
| Undoing and modelling (Q6, Q8) | adds the percentage back, or rounds packs to the nearest | undoes a discount and rounds up where the situation needs it | states the assumption behind each answer |
| Communicating (Q7) | gives a bare number | answers in a sentence with units, rounded to the cent | explains how the answer would change |
Eight questions, four ideas. A student at standard answers most questions with working and writes money answers to the cent, in a sentence.
Weekly class record
Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.
| Name | Situations | Percentage moves | Discounts, GST | Modelling cycle | Review |
|---|---|---|---|---|---|
The five columns are the five days: situations, percentage moves, discounts and GST, the modelling cycle, and review.