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

Five days of lessons for Year 5 Number. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet, plus twelve situation cards to sort, eight party price cards for a modelling task, 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 situation cards and the party price cards once, one set per 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: modelling turns a real situation into maths and back again. Find what the story asks and which numbers matter, choose the operation (combining or comparing amounts is additive; equal groups, rates and sharing are multiplicative), calculate, then say what the answer means in the story: 26 children with juice in packs of 6 need 5 packs, not 4 and a bit.

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.

Calculators

Once a number sentence is written, a calculator is fine: the modelling is in choosing the operations and reading the answer back into the story. On Days 4 and 5 one calculator per group helps with the money.

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 to 3 walk the modelling cycle one step at a time: understand the story, choose the operation, write and solve the number sentence. Day 4 models money problems in several steps, and Day 5 puts the whole cycle to work on a real plan. It builds on Year 4’s modelling with number sentences (AC9M4N08) and this year’s multiplication, division and estimation (AC9M5N06, AC9M5N07, AC9M5N08), and it leads to Year 6, where children model problems with fractions, decimals and percentages and justify their choices (AC9M6N09).

Materials for the week (one trip)

• From the classroom: scissors, envelopes for the cards, mini whiteboards, this pack printed.
• For Days 4 and 5: one calculator per group.
• Cut out once: the twelve situation cards (Day 2) and the eight party price cards (Day 5). No maths equipment to buy.

Words for the word wall

model, situation, formulate, number sentence, operation, additive, multiplicative, unknown, interpret, communicate, budget

Dear families

This week in maths, Year 5 uses maths to solve real problems. We read the problem, decide what it asks, choose the operation, work it out, and say what the answer means. For example, 26 children need juice boxes that come in packs of 6: 26 ÷ 6 = 4 r 2, so we buy 5 packs.

Try this at home

• Plan a picnic together: how many of each thing do you need, and what will it cost?
• At the shops, compare a multipack with buying the same items one at a time.
• Ask your child which numbers in a word problem matter, and which do not.
• Make up a story for 24 × 6, then one for 24 ÷ 6.

My problems this week

Fill one row a day. Find a real problem at home and model it.

Printed from the free seegongsik Mathematical Modelling teaching pack · seegongsik.com/au/y5/number/AC9M5N09/pack

From the story to the maths

Children meet the modelling cycle: formulate, choose operations, calculate, interpret, communicate. Today is the first step: find what a story asks and which numbers matter, and draw a bar model of the situation before calculating.

We are learning to

• name the five steps of the modelling cycle,
• find the question in a story and the numbers that matter,
• draw a bar model of a story before calculating.

Success criteria

• I can underline the question and cross out a number the story does not need.
• I can draw a bar model for an ‘altogether’ story and for an ‘equal groups’ story.

You need

Mini whiteboards. Rulers. The worksheet, one per child.

Lesson flow (about 50 minutes)

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

Watch for these ideas

• Using every number in the story. Read the question first, then choose the numbers it needs.
• Calculating before knowing what is asked. Underline the question before anything else.
• A bar model that does not match the story: equal groups need equal parts.

Answers

• Part A: 1. not needed 11 and 30; 4 × 12 = 48, so 48 stickers; 2. not needed 9; 46 − 13 = 33, 33 + 8 = 41, so 41 people; 3. not needed 3; 6 × 9 = 54, 54 − 25 = 29, so $29.
• Part B: 238 + 175 = 413 cans; 7 × 24 = 168 pencils (one bar in two parts; 7 equal parts). Part C: Formulate, Choose operations, Calculate, Interpret, Communicate.

From the story to the maths

A. Underline the question. Circle the numbers you need. Cross out the ones you do not.

B. Draw a bar model, then solve

C. Number the five steps in order, 1 to 5

Which operation?

Children decide which operation models a story. Additive situations put amounts together, take some away or compare them; multiplicative situations have equal groups, a rate such as $3 each, or a sharing. The words help, but the situation decides.

We are learning to

• tell additive situations from multiplicative ones,
• choose the operation that models a story,
• explain the choice with the words and the situation.

Success criteria

• I can sort the situation cards and give a reason for each pile.
• I can explain why ‘how much more’ is a subtraction.

You need

The situation cards (cut-out sheet 1), one set per group. Mini whiteboards. The worksheet, one per child.

Lesson flow (about 50 minutes)

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

Watch for these ideas

• Choosing by one key word: ‘more’ does not always mean add. ‘How much more’ compares, so subtract.
• Thinking only sharing is division. Packing eggs in cartons of 12 is division too.
• Dividing the wrong way round: 4 ÷ 96. Divide the total by the number of groups, or by the size of each group.

Answers

• Part A: add A, G, J; subtract C, H, I; multiply B, E, L; divide D, F, K.
• Part B (any one card from each pile): A 312 + 297 = 609 children; B 8 × 15 = 120 chairs; C 186 − 149 = 37 seats; D 144 ÷ 12 = 12 cartons; E 24 × 3 = $72; F 96 ÷ 4 = $24; G 1250 + 875 = 2125 m; H 52 − 38 = $14; I 900 − 275 = 625 litres; J 2460 + 385 = 2845 books; K 250 ÷ 5 = 50 cm; L 35 × 8 = 280 legs. Part C: any stories that give 24 + 6 = 30, 24 × 6 = 144, 24 ÷ 6 = 4.

Which operation?

A. Sort the situation cards. Write the card letters and the clue.

B. Solve one card from each pile

C. Same numbers, three stories

Write the number sentence

Children write a number sentence for a story, with a box for the unknown wherever it falls: 14 × 25 = □, or □ × 8 = 112 when the number of boxes is unknown. They solve it with an efficient strategy, or a calculator for big numbers, and give the answer with its unit.

We are learning to

• write a number sentence with a box for the unknown,
• solve it with an efficient strategy, or a calculator for big numbers,
• write the answer with its unit, in a sentence.

Success criteria

• I can write □ × 8 = 112 for ‘some boxes of 8 crayons hold 112 crayons’.
• I can answer with the unit: 14 boxes, not just 14.

You need

Mini whiteboards. The worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the baker story again.

Watch for these ideas

• Using the numbers in the order they appear: 48 − 37 for Leo. The unknown is the start: □ − 37 = 48.
• Answering the sentence, not the story: 14 instead of 14 boxes. Read the question again.
• Reaching for the calculator first. It cannot choose the operation; the number sentence does.

Answers

• Part A: 1. 14 × 25 = □, 350 trees; 2. □ × 8 = 112, 14 boxes; 3. □ − 37 = 48, $85; 4. 300 ÷ 25 = □, 12 pieces; 5. 1265 + 1480 = □, 2745 pages; 6. 420 − □ = 64, 356 people.
• Part B: each answer put in the box makes the sentence true. Part C: any stories; 15 × □ = 90 gives □ = 6, □ ÷ 4 = 23 gives □ = 92.

Write the number sentence

A. Write a number sentence with a box for the unknown, then solve it

B. Check two answers

Put two of your answers back into their number sentences. Is each sentence true?

______________________________     ______________________________

C. A story for a sentence

Money in steps

Children model money problems that need more than one step. Each step is one operation and has a meaning, such as ‘the cost of the entry’, so each step can be checked on its own. They also compare a pack price with buying the same items one at a time.

We are learning to

• break a money problem into steps, one operation each,
• say what each step’s answer means,
• compare a pack price with single items, and explain which is better.

Success criteria

• I can solve a two-step money problem and say what each step means.
• I can decide which is the better buy and explain why.

You need

One calculator per group. The worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after problem 1 of Part A; start Session B with the movie problem again.

Watch for these ideas

• Stopping after the first step: the cost of the entry is not the change.
• Taking the wrong number into the next step. Write what each number means beside it.
• Comparing a pack price with one single price. Compare the same number of items.

Answers

• Part A: 1. $18 + $12 = $30 (the total cost); $30 ÷ 5 = $6 (what each friend pays). 2. 4 × $7.50 = $30 (the entry); $30 + $2 = $32 (the total); $50 − $32 = $18 (the change). 3. $210 − $66 = $144 (what she still needs); $144 ÷ $12 = 12 (the lawns to mow).
• Part B: Juice boxes: 6 singles cost $4.50, so the pack is cheaper by 30c; Pencils: 10 singles cost $4, so the pack is cheaper by 50c; Muesli bars: 8 singles cost $6, so singles are cheaper by 40c. Part C: any problem whose steps each say what they mean.

Money in steps

A. Solve in steps. Write what each step means.

B. The better buy

C. Your turn

Write a two-step money problem for a partner. Solve it first, step by step.

Plan the party

Children put the whole modelling cycle to work on a real plan: party food for 26 children on a $120 budget. They work out how many of each thing, how many packs (whole packs only, so they round up), the cost, the total and the money left, then explain the plan to someone else.

We are learning to

• read each answer back into the situation, with its unit,
• round up when only whole packs can be bought,
• communicate a solution so someone else can follow it.

Success criteria

• I can explain why 26 juice boxes in packs of 6 means buying 5 packs.
• I can write a plan that says what we buy, what it costs, and whether it fits the budget.

You need

The party price cards (cut-out sheet 2), one set per group. One calculator per group. The worksheet, one per child.

Lesson flow (about 50 minutes)

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

Watch for these ideas

• Buying 4 packs of juice because 26 ÷ 6 is 4 r 2: 2 children would miss out. Whole packs mean rounding up.
• Adding the pack prices without multiplying by the number of packs.
• A number with no words: $114 needs a sentence. The party costs $114, and $6 is left.

Answers

• Part A: juice boxes 26 needed, 5 packs of 6, $21; mini pizzas 26 needed, 4 packs of 8, $38.40; cupcakes 26 needed, 3 packs of 12, $43.20; paper plates 26 needed, 2 packs of 25, $6.20; cups 26 needed, 2 packs of 20, $5.20. Total $114, so it fits, with $6 left.
• Part B (one each): balloons 2 packs for $9, so no; napkins 1 pack for $2.40, so yes. Part C: any note that names the items, the total and the money left.

Plan the party

A. Plan food for 26 children. Each child gets one of each. The budget is $120.

Total cost: __________    Does it fit the budget? yes / no    Money left: __________

B. Something extra?

With the money left, can you also buy balloons or napkins, one for each child? Show your working.

C. Tell the principal

Write a note: what you will buy, what it costs, and whether it fits the budget.

Situation cards

Cut out the twelve cards (Day 2), one set per group. Read each story and sort it by the operation that models it: add, subtract, multiply or divide. Be ready to say which words told you.

Teacher note: three cards for each operation. The sort and the answers are on the Day 2 plan.

Party price cards

Cut out the eight cards (Day 5), one set per group. Each card shows how many come in a pack and what one pack costs.

Teacher note: the shop sells whole packs only, so children round up the number of packs. The plan for 26 children is on the Day 5 plan.

What we know: Mathematical Modelling

Work on your own. Show your thinking if you can.

1. Each of 9 teams has 14 players. How many players are there? Number sentence and answer: __________________
2. A shop had 503 apples and sold 268. How many apples are left? Number sentence and answer: __________________
3. Some packs of 6 cans hold 84 cans. How many packs are there? Write a number sentence with a box, then solve it: __________________
4. Tickets cost $8 each. How much do 13 tickets cost?   __________
5. 3 sandwiches cost $6.50 each. You pay with $20. What is your change?   __________
6. 58 children go on an excursion in minibuses that take 12. How many minibuses are needed?   __________
7. Sam packs 40 pencils into boxes of 10: 40 ÷ 10 = 4. What does the 4 mean?
8. Which is the better buy: 4 pens for $3.60, or pens at 95c each? Explain.

Answers and marking guide

Answers

1. 9 × 14 = 126 players
2. 503 − 268 = 235 apples
3. □ × 6 = 84, so □ = 14 packs
4. 13 × $8 = $104
5. 3 × $6.50 = $19.50; $20 − $19.50 = $0.50 change
6. 58 ÷ 12 = 4 r 10, so 5 minibuses (round up: everyone goes)
7. 4 boxes of 10 pencils
8. 4 singles cost $3.80, so the pack of 4 is cheaper by 20c

A quick three-level guide

Eight questions, four ideas. A child at standard answers most questions and can say why, by naming the operation the situation needs and what the answer means in it.

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: finding the question, choosing the operation, writing the number sentence, money in steps, and explaining the answer.