Mathematical Modelling: a week of ready-to-teach maths
Five days of lessons for Year 4 Number. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet, plus cut-out story cards, number sentence cards and problem cards, a mini-check and every answer.
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.
- Cut out the two card sheets once; the cards are reused across the week.
- Open the free interactive unit on your board. Every plan tells you which picture to show and which buttons to press.
- 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 is a cycle of four steps: read the situation, write it as a number sentence, calculate, and say what the answer means back in the situation. Most of the work is in the first and last steps.
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.
Every answer has a unit
In this pack 25 is a number, but $25 or 25 pencils is an answer, so children finish every problem with a sentence that names the unit. A calculator is welcome for numbers like $348 + $276, as long as its answer is checked against the situation.
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 | Children learn and do | On screen |
|---|---|---|---|
| 1 | From a story to a number sentence | Decide whether a story joins, takes away, makes equal groups or shares, and write the number sentence | From real life to maths; Formulate a number sentence |
| 2 | Additive situations | Model joining, taking away and comparing with a bar, including money and change | Additive situations |
| 3 | Multiplicative situations | Model equal groups, arrays, sharing and grouping, and tell adding from multiplying | Multiplicative situations |
| 4 | Interpret the answer | Write the answer as a sentence about the situation, and check that it makes sense | Interpret the answer |
| 5 | The modelling cycle | Run the whole cycle on practical problems, some with two steps, using a calculator where it helps | The modelling cycle |
How the week builds
Day 1 is the first step of the cycle on its own: turning a story into a number sentence. Days 2 and 3 split the stories into the two kinds the descriptor names, additive and multiplicative, with money in both. Day 4 is the last step: answering in words. Day 5 runs the whole cycle. It builds on the mental strategies from Efficient Strategies (AC9M4N06) and on estimating to check (AC9M4N07).
Materials for the week (one trip)
- From the classroom: scissors, pencils, this pack printed.
- For Day 5: a few calculators, one per group (a tablet calculator is fine), and play money if you have it.
- Cut out once, use all week: the story cards, number sentence cards and problem cards in this pack.
Words for the word wall
situation, model, number sentence, add, subtract, multiply, divide, equal groups, share, total, difference, how many more, each, change, interpret
Dear families
This week in maths, Year 4 learns to solve real problems: turning a situation into a number sentence, working it out, and saying what the answer means. Many of our problems are about money.
Try this at home
- Share a real problem, such as: we need 3 packets of 8 bread rolls. How many rolls is that? What would you write?
- At the shops, ask: two drinks at $3 each and a sandwich at $7. Is that adding, multiplying, or both?
- Ask your child to finish with a sentence, not just a number: “We have 24 rolls.”
- Make up a story for a number sentence together, such as 20 ÷ 4.
My real problems this week
Fill one row a day. Write a problem from home, its number sentence, and the answer as a sentence.
| Day | The problem | Number sentence | The answer, as a sentence |
|---|---|---|---|
| Monday | |||
| Tuesday | |||
| Wednesday | |||
| Thursday | |||
| Friday |
Printed from the free seegongsik Mathematical Modelling teaching pack · seegongsik.com/au/y4/number/AC9M4N08/pack
From a story to a number sentence
Children read a short story and decide what is happening in it: joining, taking away or comparing, making equal groups, or sharing. That decision gives the operation, and the numbers in the story give the number sentence: 4 packs of 6 pencils is 4 × 6. They solve it and say the answer with its unit, 24 pencils.
We are learning to
- decide whether a story joins, takes away or compares, makes equal groups, or shares,
- choose the operation that matches what is happening,
- write the story as a number sentence and solve it.
Success criteria
- I can write a number sentence for a story.
- I can say what is happening in the story that tells me the operation.
You need
The story cards and number sentence cards (cut-out sheet 1), one set per pair. The worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Four kinds of story Tell four quick stories, one of each kind, while children act them out with counters. Build a chart: joining is adding, taking away or comparing is subtracting, equal groups is multiplying, sharing or grouping is dividing. Ask: “In this story, is anything joined, taken away, put in equal groups or shared?” |
| 30 min | Story to sentence Show the board (see the box). Pairs then match the story cards to the number sentence cards, and each child does Part A. Ask: “Which words in the story told you to multiply?” |
| 10 min | Write your own Children do Part B, then swap with a partner, who writes the number sentence for each story. Ask: “Your partner wrote 6 + 4 for your 6 × 4 story. What could you change so the story says multiply?” |
Two half-sessions instead? Split after the card match; start Session B with Part A.
Watch for these ideas
- Hunting for key words: “more” does not always mean add. How many more? is a subtraction.
- Writing the numbers in story order: Mum is 41 and Gran is 68 needs 68 − 41, not 41 − 68.
- Adding the numbers in an equal-groups story: 4 packs of 6 is 4 × 6, not 4 + 6.
Answers
- Part A: add, 27 + 25 = 52, 52 children; multiply, 8 × 6 = 48, 48 eggs; subtract, 45 − 19 = 26, $26; divide, 32 ÷ 4 = 8, 8 stickers each; subtract, 68 − 41 = 27, 27 years older; multiply, 5 × 9 = 45, $45.
- Part B: stories vary. 6 × 4 needs equal groups (6 bags of 4 apples); 24 − 6 needs taking away or comparing (24 birds and 6 fly away). Card match: the two numbers on each story card appear together on one sentence card only.
From a story to a number sentence
A. What is happening? Write the number sentence and the answer
| Story | Add, subtract, multiply or divide? | Number sentence | Answer |
|---|---|---|---|
| There are 27 boys and 25 girls in the hall. How many children are there? | |||
| A box holds 6 eggs. How many eggs are in 8 boxes? | |||
| Tom had $45 and spent $19. How much does he have left? | |||
| 32 stickers are shared equally by 4 friends. How many does each friend get? | |||
| Mum is 41 and Gran is 68. How much older is Gran? | |||
| 5 friends each pay $9 for the movies. How much do they pay altogether? |
B. Write a story for each number sentence
6 × 4
24 − 6
Additive situations
Additive stories join amounts, take an amount away, or compare two amounts. A bar model shows which: one long bar cut into parts, or two bars side by side. If the whole is unknown, add the parts. If a part or the gap between two bars is unknown, subtract. Change is a part too: the part of the note that is left after paying.
We are learning to
- draw a bar model for a joining, taking-away or comparing story,
- use the bar to decide whether to add or subtract,
- solve money problems, including change, and answer with the unit.
Success criteria
- I can draw a bar model for a story and write the number sentence from it.
- I can explain why I added or subtracted by pointing to the unknown on my bar.
You need
The add and subtract story cards from Day 1. The worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Draw the story Read the shirt and cap problem and draw one long bar on the board, cut into $24 and $18, with a question mark over the whole. Ask: “Do we know the whole or the parts? So do we add or subtract?” |
| 30 min | Bars and sentences Show the board (see the box). Children do Part A: draw the bar, write the number sentence, and answer with the unit. Ask: “Ben’s tower problem says ‘more’. Why is it a subtraction?” |
| 10 min | Two steps Children do Part B, then sort the Day 1 add and subtract story cards by bar: whole unknown, part unknown, or compare. Ask: “Can you solve it with one number sentence? With two?” |
Two half-sessions instead? Split after the first two problems of Part A; start Session B with the fete.
Watch for these ideas
- Adding in a comparing story because it says “more”: 63 + 48. The gap between the bars is 63 − 48.
- Adding the note and the price to find change. Change is less than the note, so subtract.
- Stopping at the number. 15 is not finished; 15 more blocks is.
Answers
- Part A: the whole is unknown, so add: 24 + 18 = 42, $42; a part is unknown, so subtract: 50 − 27 = 23, $23; the whole is unknown, so add: 136 + 85 = 221, $221; compare two bars, so subtract: 63 − 48 = 15, 15 more blocks.
- Part B: 40 − 15 − 8 = 17, or 15 + 8 = 23 and 40 − 23 = 17. She has $17 left.
Additive situations
A whole cut into parts: $18 and $7 more
Two bars to compare: 61 pages and 45 pages
A. Draw a bar model, write the number sentence, and answer
Answer: ____________
Answer: ____________
Answer: ____________
Answer: ____________
B. Lily has $40. She buys a hat for $15 and socks for $8. How much does she have left?
Multiplicative situations
Multiplicative stories have equal groups. When the number of groups and the number in each group are known, multiply to find the total: 7 packs of 6 pencils is 7 × 6. When the total is known, divide. Sharing finds how many are in each group; grouping finds how many groups. Money works the same way: 5 tickets at $8 each is 5 × $8.
We are learning to
- recognise equal groups, arrays, sharing and grouping in a story,
- multiply when the total is unknown and divide when it is known,
- tell an adding story from a multiplying one.
Success criteria
- I can say whether a story is additive or multiplicative, and why.
- I can write and solve a number sentence for an equal-groups, sharing or grouping story.
You need
Counters, a handful per pair. The multiply and divide story cards from Day 1. The worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Equal or not? Show the board (see the box). For each situation, pairs make it with counters before the number sentence is shown. Ask: “Are the groups equal? Which words in the story tell you?” |
| 30 min | Groups, rows and shares Children do Part A: a quick picture of groups or rows, the number sentence, and the answer with its unit. Ask: “36 oranges in bags of 4: do we know how many bags, or how many in each bag?” |
| 10 min | Add or multiply? Children do Parts B and C, then share their two stories for 24 ÷ 3. Ask: “Ella’s answer is 7. Seven what? Does that fit 4 drinks at $3 each?” |
Two half-sessions instead? Split after the first three problems of Part A; start Session B with the $45.
Watch for these ideas
- Adding the two numbers in an equal-groups story: 4 drinks at $3 is 4 × 3, not 4 + 3.
- Dividing the wrong way round: 36 oranges in bags of 4 is 36 ÷ 4, not 4 ÷ 36.
- Treating sharing and grouping as different operations. Both are division; only the question changes.
Answers
- Part A: equal groups, 7 × 6 = 42, 42 pencils; equal groups (money), 5 × 8 = 40, $40; grouping, 36 ÷ 4 = 9, 9 bags; sharing, 45 ÷ 5 = 9, $9 each; array, 6 × 9 = 54, 54 plants.
- Part B: stories vary. Sharing: 24 apples shared by 3 children, 8 each. Grouping: 24 apples in bags of 3, 8 bags. Part C: no. Equal groups mean multiply: 4 × 3 = 12, so the drinks cost $12.
Multiplicative situations
A. Draw the groups or rows, write the number sentence, and answer
Answer: ____________
Answer: ____________
Answer: ____________
Answer: ____________
Answer: ____________
B. Write two stories for 24 ÷ 3
A sharing story (how many in each group):
A grouping story (how many groups):
C. Ella buys 4 drinks at $3 each. She writes 4 + 3 = 7. Is she right? Explain.
Interpret the answer
The last step of modelling goes back to the situation. 30 ÷ 5 = 6 is finished maths but not a finished answer: in the story it means each friend gets $6. Children write every answer as a sentence that names the unit, and ask whether it makes sense. An answer bigger than the amount being shared, or 80 cars for 20 children, means the model went wrong.
We are learning to
- say what a number means back in the situation,
- write the answer as a sentence that answers the question,
- check that an answer makes sense, and fix the model when it does not.
Success criteria
- I can write my answer as a sentence with the unit.
- I can explain why an answer does or does not make sense in the situation.
You need
Mini whiteboards for the opening. The worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Six what? Write 30 ÷ 5 = 6 for $30 shared by 5 friends, and ask what the 6 is. Then show the board (see the box) and choose a meaning for each model. Ask: “Is the 6 friends, dollars or something else? How do you know?” |
| 30 min | Answer in a sentence Children do Part A: number sentence, calculation, then a full sentence that answers the question. Partners read each other’s sentences aloud. Ask: “Does your sentence answer the question that was asked, or a different one?” |
| 10 min | Does it make sense? Children do Parts B and C, then explain one fix to the class. Ask: “How could Mr Lee get 80 cars? What should he have done instead?” |
Two half-sessions instead? Split after Part A; start Session B with the meanings in Part B.
Watch for these ideas
- Stopping at the bare number. 140 is not an answer; the tickets cost $140 is.
- Answering a different question: the canteen sold 145 pies on Monday, but the question asks how many more.
- Accepting an answer that cannot be: sharing $30 can never give each friend $90.
Answers
- Part A: 28 × 5 = 140. The tickets cost $140. 96 ÷ 4 = 24. Each table gets 24 pencils. 145 − 98 = 47. Monday sold 47 more pies.
- Part B: $9 for each friend; 56 apples altogether. Part C: no, the children are shared into cars of 4, so 20 ÷ 4 = 5 cars; no, each friend gets less than the $30 shared: 30 ÷ 3 = $10.
Interpret the answer
A. Write the number sentence, work it out, and answer in a sentence
B. Ring the meaning that fits the situation
C. Does it make sense? If not, fix it.
The modelling cycle
Children run the whole cycle on practical problems: read the situation, formulate a number sentence (or two, for a two-step problem), calculate with a strategy that suits the numbers or a calculator when they are large, and interpret the answer in a sentence. The class picnic ends with a decision rather than a number: is $120 enough?
We are learning to
- run the four steps of the modelling cycle on a practical problem,
- solve two-step problems, including money,
- choose a mental strategy or a calculator, and check that the answer makes sense.
Success criteria
- I can show all four steps of the cycle for a problem.
- I can explain my answer in terms of the situation.
You need
The problem cards and the four step cards (cut-out sheet 2), one set per group. A few calculators. The worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Four steps Show the board (see the box), then have groups lay out the four step cards in a circle as a mat. Ask: “Which step do you think takes the most thinking? Why?” |
| 30 min | Round the mat Groups move each problem card round the mat, writing the number sentences and a sentence answer. Then each child does Part A, the class picnic. Ask: “This problem has two steps. What do you need to find first?” |
| 10 min | Share and check Each group presents one problem and the class checks that the answer makes sense. Finish with the self-check quiz as a class game. Ask: “Is $104 sensible for 26 bottles at $4? How could you check it quickly?” |
Two half-sessions instead? Split after the problem cards; start Session B with the picnic.
Watch for these ideas
- Stopping after one step of a two-step problem: 54 is the child tickets only.
- Trusting a calculator without a rough idea first. $348 + $276 is about $600.
- Giving the cost when the question asks for a decision: $104 does not say whether $120 is enough.
Answers
- Problem cards: A: 26 × 4 = 104, $104; B: 348 + 276 = 624, $624 (a calculator suits it); C: 84 ÷ 7 = 12, 12 rows; D: 165 − 98 = 67, $67; E: 6 × 9 = 54, then 54 + 15 = 69, $69; F: 3 × 24 = 72, then 72 − 50 = 22, 22 balloons.
- Part A: 3 + 1 = $4 a child, and 26 × 4 = 104 (or 26 × 3 = 78 and 26 × 1 = 26). Yes, $120 is enough, with $16 left over.
The modelling cycle
A. The class picnic
B. My problem card
Copy the problem from your card, then go round the four steps.
Story cards and number sentence cards
Cut out the twelve story cards and the twelve number sentence cards, one set per pair. Partners match each story to its number sentence (Day 1), then sort the stories by bar model (Day 2) and by equal groups (Day 3).
Story cards
Number sentence cards
Teacher note: there are three stories for each operation. The sentence cards leave the answer off, so pairs solve each one once it is matched.
Modelling step cards and problem cards
Cut out the four step cards and the six problem cards (Day 5), one set per group. Lay the step cards in a circle as a mat, and move each problem card round it.
Step cards
Problem cards
Teacher note: cards E and F need two steps, and card B suits a calculator. The answers are on the Day 5 plan.
What we know: Mathematical Modelling
For questions 1 to 5, write the number sentence and the answer with its unit.
- There are 9 bags with 5 apples in each bag. How many apples are there? ______________________
- Raj had $60 and spent $35. How much does he have left? ______________________
- 48 children make 6 equal teams. How many children are in each team? ______________________
- Ella has 17 marbles and wins 8 more. Additive or multiplicative? ____________ ______________________
- 3 books cost $7 each. What do they cost altogether? ______________________
- 36 ÷ 4 = 9 for “$36 shared equally by 4 friends”. Write the answer as a sentence: ______________________
- Sam says 5 pencils at $2 each cost $52. Does that make sense? ______ What should it be? ______
- Tickets cost $12 for an adult and $8 for a child. What do 2 adult tickets and 3 child tickets cost? ______________________
How did this week feel? Circle one: I can teach it · I am getting there · I need more help
Answers and marking guide
Answers
- 9 × 5 = 45: there are 45 apples.
- 60 − 35 = 25: he has $25 left.
- 48 ÷ 6 = 8: there are 8 children in each team.
- Additive: 17 + 8 = 25, so she has 25 marbles.
- 3 × 7 = 21: the books cost $21 (multiplicative: equal groups).
- Each friend gets $9.
- No: 5 × 2 = 10, so the pencils cost $10. ($52 looks like 5 and 2 written side by side.)
- 2 × 12 = 24 and 3 × 8 = 24, then 24 + 24 = 48: the tickets cost $48.
A quick three-level guide
| Idea | Working towards | At standard | Beyond |
|---|---|---|---|
| Formulating (Q1 to Q5) | picks the operation from a key word, or adds the numbers in an equal-groups story | writes a correct number sentence for joining, taking away, comparing, equal groups and sharing | writes two different stories for the same number sentence |
| Interpreting (Q6, Q7) | gives a bare number | answers in a sentence with the unit, and spots an answer that cannot be right | explains what went wrong in a model and fixes it |
| Two steps with money (Q8) | stops after the first step | plans both steps and combines them | checks the total with an estimate before or after calculating |
Eight questions, three ideas. A child at standard writes the right number sentence for each kind of story and finishes with a sentence that says what the answer means.
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 | Writes number sentences | Additive models | Multiplicative models | Answers in a sentence | Runs the cycle |
|---|---|---|---|---|---|
The five columns are the five days: number sentences, additive, multiplicative, answering in a sentence, and the whole cycle.