Mathematical Modelling: a week of ready-to-teach maths

Five days of lessons for Year 6 Number. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet, plus story and calculation cards, price cards from two shops, a mini-check and every answer.

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. Cut out the story and calculation cards (Days 2 and 3) and the price cards (Day 5) once, one set per pair or group.
4. Open the free interactive unit on your board. Every plan tells you which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and answers are all on the one page.

No maths background needed

This pack is written for the busy generalist teacher. Each plan explains the idea in plain words, lists the misconceptions children bring, and gives the answers. The big idea of the week: a real question becomes mathematics and then becomes a real answer again. Modelling runs in four stages. Formulate the question as calculations; choose the operations and a sensible method, mental, written or calculator; solve; then interpret the answer in the situation and justify the choices. The last stage matters most: 2.7 buses means 3 buses.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 50 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point if your timetable runs two shorter sessions.

Words to watch

A model here is the set of calculations that stands for the real situation. To formulate is to turn the words into calculations; to interpret is to say what the answer means in the situation; to justify is to give the reason for each choice. A unit price is the price of one item, or of 100 g; profit is the money taken minus the costs.

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 meets the four stages of the cycle; Days 2, 3 and 4 each look closely at one or two of them; and Day 5 puts them together in one larger model, a sausage sizzle fundraiser. The week draws on everything in Year 6 Number: decimals, fractions and percentages of amounts, and estimating to check. It prepares Year 7, where modelling problems are solved with rates, ratios and percentage increase and decrease.

Materials for the week (one trip)

• From the classroom: scissors, pencils, this pack printed.
• A calculator for each pair (a tablet or laptop calculator is fine). A spreadsheet is a good extra on Day 5 if you have one.
• Cut out once: the story and calculation cards, one set per pair, and the price cards, one set per group of three or four.

Words for the word wall

model, formulate, operation, strategy, solve, interpret, justify, unit price, budget, cost, profit, round up, round down, reasonable

Dear families

This week in maths, Year 6 uses maths to answer real questions: can we afford it, which is the better buy, how many buses do we need? We turn each question into calculations, choose a quick way to work them out, and then say what the answer means.

Try this at home

• At the shops, compare two sizes of the same product. Which is the better buy, and how do you know?
• Plan a family meal on a budget. What would it cost, and is there money left over?
• Ask: 7 children need a lift and each car has room for 4 of them. Is the answer 1.75 cars?
• When you calculate something together, say what the answer means in a full sentence.

My real questions this week

Fill one row a day. Write a real question, the calculation you used and what the answer means.

Printed from the free seegongsik Mathematical Modelling teaching pack · seegongsik.com/au/y6/number/AC9M6N09/pack

The modelling cycle

Mathematical modelling uses maths to answer a real question in four stages: formulate the question as calculations, choose the operations and a method, solve, then interpret the answer in the situation and justify it. Children follow the cycle through budget problems and name the stage of each step in solved ones.

We are learning to

• name the four stages of the modelling cycle and say what happens in each,
• follow the cycle to answer a budget question,
• answer the real question in a sentence, not just with a number.

Success criteria

• I can say which stage a step of a solution belongs to.
• I can use the four stages to decide whether $70 buys 3 soccer balls and a pump.

You need

The worksheet, one per child, and a calculator for each pair.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the party budget on the board.

Watch for these ideas

• Calculating straight away, before deciding what the question asks.
• Stopping at a number: $5 on its own does not say whether the party fits the budget.
• Thinking the cycle runs once: a model that does not fit the situation is checked and changed.

Answers

• Part A, the steps in printed order: problem 1: Solve, Interpret, Formulate, Choose; problem 2: Choose, Interpret, Solve, Formulate.
• Part B: formulate 3 × $18.95 + $12.50 and compare it with $70; choose 3 × $19 = $57, take off 3 × 5c; solve $56.85 + $12.50 = $69.35; interpret: yes, with 65c to spare.
• Part C: $36 ÷ $2.50 = 14.4. 14 cupcakes bring in $35, not enough, so the stall must sell 15 ($37.50).

The modelling cycle

A. Which stage is it?

Write Formulate, Choose, Solve or Interpret beside each step.

B. Follow the cycle

The class wants 3 soccer balls at $18.95 each and a $12.50 pump. The budget is $70. Can the class buy them?

C. Your turn

A fete stall sells cupcakes at $2.50 each. The ingredients cost $36. How many cupcakes must the stall sell to cover the cost?

Formulate

Formulating turns the words of a real question into calculations: which numbers matter, which operations the situation describes, and in what order. It also means noticing when a question cannot be answered yet because something is missing. Children sort needed from extra numbers, name what is missing and match stories to calculations.

We are learning to

• choose the operations a story describes, and the order to do them in,
• find the numbers a question needs and leave out the ones it does not,
• say what is missing when a question cannot yet be answered.

Success criteria

• I can write a calculation for 5 tickets at $12.50 each and a $6 program.
• I can say why 4 + 3 + 8 does not model drinks for 4 friends at $3 each and an $8 cake.

You need

The worksheet, one per child, a calculator for each pair, and the story and calculation cards (cut-outs), one set per pair.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the last two questions on the board.

Watch for these ideas

• Using every number in the story, even one the question does not need.
• Adding all the numbers because the question asks for a total, when some must be multiplied first.
• Doing the steps in the order the numbers appear, not the order the situation needs.

Answers

• Part A: (1) not needed 2 km; $4.20 ÷ 6, 70c; (2) not needed 9 am; $50 − ($18.95 + $3.40), $27.65; (3) not needed $2.50; 28 × 3 = 84, then 84 ÷ 10, 9 packs (8.4 rounded up).
• Part B: (1) how much the class saves each week; (2) the cost of the bus and the number of students; (3) how many books, and the price of each.
• Part C (other correct forms are fine): (1) 5 × $12.50 + $6 = $68.50; (2) ($45 + $5) ÷ 4 = $12.50; (3) $80 − 15% of $80 = $68; (4) $100 ÷ $12.50 = 8 weeks; (5) $36 − 3/4 of $36 = $9.

Formulate

A. Which numbers matter?

Circle the numbers you need and cross out the one you do not. Then write the calculation and work it out.

B. What is missing?

Each question cannot be answered yet. Write what you would need to know.

1. The class is saving for a $240 excursion. How many weeks will it take?
I need to know: ______________________________________________

2. Every student pays an equal share of the bus. How much does each pay?
I need to know: ______________________________________________

3. Is the school’s $500 enough for new library books?
I need to know: ______________________________________________

C. Write the calculation

Choose and solve

Once a question is formulated, choose operations and a method that suit the numbers: mental for friendly numbers, written or calculator for awkward ones. Then solve step by step. Children choose methods, compare unit prices to find the better buy, and solve problems in several steps, with a calculator where it helps.

We are learning to

• choose a mental, written or calculator method to suit the numbers,
• compare two offers by finding the unit price of each,
• solve a problem in several steps and keep track of each step.

Success criteria

• I can explain why 25% of $48 needs no calculator but 13% of $286 does.
• I can show which cereal box is the better buy.

You need

The worksheet, one per child, a calculator for each pair, and the story and calculation cards from Day 2.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the solving vote.

Watch for these ideas

• Reaching for a calculator for 25% of $48, or doing 13% of $286 in your head and slipping.
• Thinking the bigger pack is always the better buy.
• Losing a step: finding the discount and forgetting to take it off.

Answers

• Part A (other methods with a good reason are fine): 25% of $48 = $12, mental: a quarter of $48; 37 × $4.95 = $183.15, calculator, or 37 × $5 take away 37 × 5c; $19.99 + $5.01 = $25, mental: the cents make a dollar; 13% of $286 = $37.18, calculator; 1/3 of $96 = $32, mental: $90 and $6, each shared by 3; 8 × $2.25 = $18, mental: 8 × $2 and 8 × 25c.
• Part B: cereal 90c and 84c per 100 g, so 750 g box for $6.30; juice 90c and 85c a can, so 10 cans for $8.50; pencils 30c and 31c a pencil, so 12 pencils for $3.60.
• Part C: (1) 2 × $45 + 3 × $22.50 = $157.50; 10% off is $15.75, so they pay $141.75. Taking 10% off each ticket first gives the same. (2) 3 lots of three and 1 single: $17, against $20 for 10 singles.

Choose and solve

A. Choose a method

For each calculation, choose mental, written or calculator. Then work it out.

B. The better buy

Find the unit price of each offer with a calculator, then circle the better buy.

C. Solve in steps

1. A family of 2 adults and 3 children visits the zoo. Adult tickets are $45 and child tickets $22.50. A family pass takes 10% off the total. What does the family pay?

2. Raffle tickets are $2 each or 3 for $5. Kai wants 10 tickets. What is the cheapest way to buy them?

Interpret and justify

The last stage reads the answer back into the situation. A calculator answer often needs work before it answers the question: 2.7083333 buses means 3 buses, but 6.25 bows means 6. Children decide how each answer fits its situation, judge whether answers are sensible, and write full answers that say what the result means and why the method was right.

We are learning to

• read a calculator answer back into the situation, rounding up, down or not at all,
• check that an answer is sensible against an estimate,
• write a full answer that interprets the result and justifies the method.

Success criteria

• I can explain why 130 ÷ 48 means 3 buses, not 2.
• I can write an answer sentence that someone else could check.

You need

The worksheet, one per child, and a calculator for each pair.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the claims in Part B.

Watch for these ideas

• Writing the calculator answer as the answer: 2.7083333 buses.
• Always rounding to the nearest whole number: 8.3333333 packs of sausages still needs 9.
• Giving a number with no words: $10.25 without saying who pays it, or why.

Answers

• Part A: (1) 2.7083333, round up: 3 buses; (2) 8.3333333, round up: 9 packs; (3) 6.25, round down: 6 bows; (4) 14.285714, round down: 14 packs; (5) 18.75, keep it: $18.75 each.
• Part B: Ari, not sensible: it is $117; Bea, sensible ($34); Cy, sensible (108 sausages); Dev, not sensible: it is $15.
• Part C: (1) Each student pays $10.25: the entry for all 28 is $126, the bus makes it $287, and sharing that equally among 28 gives $10.25. (2) Shop A is cheaper: 20% off is $18, so Shop A charges $72 and Shop B $75, a difference of $3.

Interpret and justify

A. Round up, round down or keep it?

The calculator has done the division. Decide what the answer means in the situation.

B. Is it sensible?

Estimate first. Write sensible or not sensible, and why.

1. Ari says: “26 pool tickets at $4.50 cost $1170.”
____________________________________________________________

2. Bea says: “15% off a $40 shirt makes it $34.”
____________________________________________________________

3. Cy says: “9 packs of 12 sausages are enough for 100 people.”
____________________________________________________________

4. Dev says: “Sharing $90 among 6 friends gives each $540.”
____________________________________________________________

C. Write a full answer

1. The class of 28 is going to the museum. Entry is $4.50 each and the bus costs $161 in all. Here is the working: 28 × $4.50 = $126; $126 + $161 = $287; $287 ÷ 28 = $10.25. Write the answer in a full sentence and say why the working is right.

2. A $90 tent is 20% off at Shop A, or $15 off at Shop B. Which shop is cheaper, and by how much? Justify your answer.

Model a sausage sizzle

Today groups run the whole cycle on one larger problem: the shopping for a sausage sizzle fundraiser. They decide what to find out, choose a shop for each item from the price cards, work out the packs and the cost, set a price, and justify the plan, including what happens if fewer people come.

We are learning to

• formulate a larger problem as a set of smaller questions,
• choose between two offers and solve with a calculator or a spreadsheet,
• interpret the totals as a price and a profit, and justify the plan.

Success criteria

• I can show that our shopping list feeds 120 people at the lowest cost.
• I can explain our selling price and our profit, and what changes if only 100 people come.

You need

The worksheet, one per child, a calculator for each pair, and the price cards (cut-outs), one set per group.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after the shopping list; start Session B by setting the price.

Watch for these ideas

• Buying 120 ÷ 22 = 5.45 loaves: loaves come whole, so 6 (round up).
• Choosing the cheaper pack instead of the cheaper total for 120 people.
• Forgetting that the shopping is paid for first: if fewer people come, the costs stay the same.

Answers

• Shopping, the cheaper shop for each item: sausages Shop B, 4 packs, $78 (Shop A $84); bread Shop B, 7 loaves, $16.80 (Shop A $19.80); onions Shop A, 3 bags, $8.70 (Shop B $11.80); sauce Shop B, 1 bottle, $6.90 (Shop A $7.60); napkins Shop B, 1 pack, $5.50 (Shop A $5.70).
• Total $115.90, about 97c a sausage. Shop B sausages are 65c each, Shop A 70c.
• At $2.50 each, 120 sales take $300, a profit of $184.10; if 100 come, $250 and $134.10. 47 sales cover the costs. Other prices with a good reason are fine. Going further: $1.50 (100 sales take $150; one step less takes $100).

Model a sausage sizzle

1. Formulate

What do you need to find out? List your questions.

2. Choose

Which shop for each item? Will you work it out mentally, on paper, with a calculator or with a spreadsheet?

3. Solve: the shopping list

4. Interpret and justify

Why is this a good plan? ______________________________________________________________

Story cards and calculation cards

Cut out the twelve story cards and the twelve calculation cards once, one set for each pair.

Story match (Days 2 and 3)

1. Shuffle all twenty-four cards and lay them face down in rows.
2. Take turns to turn over two cards. A story and the calculation that models it make a pair: keep them and say why the calculation fits. If not, turn them back.
3. On Day 3, a pair counts only when you also work out the answer and say what it means.
4. When all the cards are gone, the player with more pairs wins.

Teacher note: the pairs are drinks for 4 friends at $3 each, and an $8 cake, 4 × 3 + 8 = $20; a $4 drink, a $3 packet of chips and an $8 cake, 4 + 3 + 8 = $15; 4 boxes each hold 3 bags of 8 lollies, 4 × 3 × 8 = 96; 6 friends share a $30 bill equally, 30 ÷ 6 = $5; 6 concert tickets cost $30 each, 6 × 30 = $180; you have $30 and spend $6, 30 − 6 = $24; 20% of the $45 raised goes to the library, 0.2 × 45 = $9; a $45 game has 20% off, 45 − 0.2 × 45 = $36; a $45 game and a $20 controller, 45 + 20 = $65; a quarter of $60 is saved, 60 ÷ 4 = $15; 4 weekly payments of $60, 4 × 60 = $240; a $60 jacket has a quarter off, 60 − 60 ÷ 4 = $45.

Price cards: the sausage sizzle

Cut out the ten price cards once, one set for each group.

Sausage sizzle shopping (Day 5)

1. Sort the cards into five pairs: the same item from each shop.
2. For each pair, work out how many packs feed 120 people, and what they cost.
3. Keep the card with the lower cost for each item, and lay the five cards out as your shopping list.
4. Copy the list onto the worksheet and add up the total.

Teacher note: the cheaper shop for each item is sausages Shop B (4 packs, $78), bread Shop B (7 loaves, $16.80), onions Shop A (3 bags, $8.70), sauce Shop B (1 bottle, $6.90), napkins Shop B (1 pack, $5.50), a total of $115.90. No mix of the two shops costs less for any item.

What we know: Mathematical Modelling

Work on your own. Show your working and say what each answer means.

1. Name the four stages of the modelling cycle, in order. ______________________________
2. Write a calculation for 3 tickets at $16.50 each and a $7 program, and work it out. ______________________________
3. “Can the class afford a fish tank for the classroom?” What do you need to know first? ______________________________
4. Which is the better buy: 400 g for $3.20 or 1 kg for $7.50? Show how you know. ______________________________
5. 150 students are going to the zoo. Each bus holds 45. How many buses are needed? ______________
6. A $120 scooter is 25% off. Is $85 enough to buy it? ______________________________
7. Mia saves $7.50 a week. How many weeks until Mia has $60? ______________
8. Ben says: “4 friends share a $70 bill, so each pays $280.” Is that sensible? Explain. ______________________________

Answers and marking guide

Answers

1. Formulate, choose, solve, interpret.
2. 3 × $16.50 + $7 = $56.50.
3. The price of the tank (and anything it needs) and how much money the class has to spend.
4. 80c and 75c per 100 g, so 1 kg for $7.50 is the better buy.
5. 150 ÷ 45 = 3.3333333, so 4 buses (round up).
6. The sale price is $90, so no, $5 short.
7. $60 ÷ $7.50 = 8 weeks.
8. No: sharing makes each part smaller than the bill. $70 ÷ 4 = $17.50 each; $280 is 4 × $70.

A quick three-level guide

Eight questions, four ideas. A child at standard answers most questions and can say what each answer means 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: the cycle, formulating, choosing and solving, interpreting and justifying, and the sausage sizzle model.