Grade by Grade
C. Algebra · Patterns, Variables, Coding and Modelling

Math Models: Solving Real-Life Problems

C4 A1.4 A1.6About 45 minutes
Learning goals
  • I can tell what question a real-life problem is asking.
  • I can decide what information I need and what I will assume (guess is true).
  • I can make a model with a drawing, a table, a pattern or a number sentence.
  • I can check my answer, change my model if I need to, and share my thinking.

Words to know

model
A drawing, table, pattern or number sentence that shows a real-life problem in math.
mathematical modelling
Using math to understand and solve a real-life problem, step by step.
assumption
Something we decide to treat as true so we can solve the problem.
information
The facts and numbers we know or need to find out.
revise
To change your work to make it better.

Math helps us in real life

Math is not only on worksheets. People use math to plan parties, build bridges, share food and get ready for winter.

When we use math to solve a real-life problem, we make a model. A model can be a drawing, a table, a pattern or a number sentence. Making and using a model is called mathematical modelling.

A cardboard box overflowing with colourful knitted mittens, with more mittens piled beside it in a school hallway.
A mitten drive is a real-life problem that math can help with. A model helps you plan how many mittens to collect each week to reach a goal. AI-generated image

The four modelling steps

  1. Understand the problem. What is the question? What do we know? What do we need to find out?
  2. Make assumptions. What will we treat as true? (For example: Every child will come to the party.)
  3. Make a model and solve. Use a drawing, table, pattern or number sentence.
  4. Check, revise and share. Does the answer make sense? Do we need to change our model? Tell others what you found and how.
1Understandthe problem2Makeassumptions3Modeland solve4Check,revise, shareIf it does not make sense, go back and try again
The modelling steps. The dashed line shows that we can go back and revise.
Worked example

The mitten drive

Room 12 is collecting mittens for families in the community. Their goal is 100 mittens. In week 1 they collected 20 mittens. In week 2 they had 40 mittens in all. When will they reach their goal?

  1. Understand: The question is In which week will we have 100 mittens? We know: 20 after week 1, 40 after week 2.
  2. Assume: The class keeps collecting 20 mittens each week.
  3. Model: Make a table that grows by 20: week 1: 20, week 2: 40, week 3: 60, week 4: 80, week 5: 100.
  4. Check: 20 + 20 + 20 + 20 + 20 = 100. ✔ Five jumps of 20 make 100.
  5. Share: If we keep collecting 20 mittens each week, we will reach 100 in week 5.

They will reach 100 mittens in week 5, if they keep collecting 20 each week.

Week12345
Mittens in all (prediction)20406080100
The model: a growing pattern that adds 20 each week.

Revising the model

In week 3, it was very cold and many families donated. The class had 70 mittens, not 60! Our assumption was not true this week.

Good modellers revise. Now: 70 after week 3. If they collect 20 more each week: week 4: 90, week 5: 110. They still reach the goal in week 5, and they go past it! If they collected 30 more in week 4, they would reach 100 in week 4.

Worked example

Picnic tables for a class trip

Room 12 has 22 people going to the park for a picnic: 19 children and 3 adults. Each picnic table seats 5 people. How many tables do they need?

  1. Understand: We need enough tables for 22 people, 5 people per table.
  2. Assume: Everyone comes. Each table really holds 5 people, no more.
  3. Model: Skip count by 5s, one table at a time: 1 table: 5, 2 tables: 10, 3 tables: 15, 4 tables: 20, 5 tables: 25.
  4. 4 tables seat only 20, and 20 is less than 22. 2 people would have no seat.
  5. 5 tables seat 25, and 25 is more than 22. Everyone fits, with 3 empty seats.
  6. Check and share: We need 5 tables. 4 tables are not enough because they only seat 20.

They need 5 tables. There will be 3 empty seats.

🙂🙂🙂🙂🙂🙂🙂🙂🙂🙂🙂🙂🙂🙂🙂🙂🙂🙂🙂🙂🙂🙂
A model of 22 people, 5 at each table. The 5th table has only 2 people, but it is still needed.

Talk about your thinking

Modelling is better with a partner. Listen to each other's ideas. You might make different assumptions and get different answers, and both can make sense! Explain why you chose your assumption.

Just for fun

What if your model assumed every person at the picnic eats 100 sandwiches? 🥪 You would need a mountain of bread! Checking your assumptions saves the day (and the bread).

Worked example

Different assumptions, different answers

Jonah wants to know how many glasses of juice his family drinks in a week. Last Monday they drank 4 glasses.

  1. Jonah's assumption: They drink 4 glasses every day. Model: count by 4s for 7 days: 4, 8, 12, 16, 20, 24, 28 glasses.
  2. His sister's assumption: They drink 4 glasses on school days (5 days) and 6 glasses on Saturday and Sunday. Model: 4, 8, 12, 16, 20 for the school days, then 20 + 6 + 6 = 32 glasses.
  3. Both models make sense. The best way to know is to collect data: keep track for a real week, then compare.

Jonah's model says 28, his sister's says 32. The answers are different because the assumptions are different.

Your turn

1

What is a model in math?

2

Lucas reads 10 pages each night. He wants to read a 50-page book. What is the question? What will you assume?

3

Solve Lucas's problem with a model.

4

A class of 18 children is going skating. Each car holds 4 children. How many cars are needed?

5

Why might two people get different answers to the same real-life problem?

6

After you get an answer, what should you do?

Hands-on activity

Plan a family movie night

Work with your family to plan a movie night snack. Use the four steps.

  1. Understand: How many people will come? What snack will you have (popcorn, apple slices, crackers)?
  2. Assume: How much will each person eat? (Example: Each person eats 6 crackers.)
  3. Model: Draw it, make a table or skip count. (Example: 4 people, 6 crackers each: 6, 12, 18, 24 crackers.)
  4. Check and share: Count out the real snack. Was your model right? Did anyone want more? How would you revise your model next time?

What to look for: your child can say their assumption and explain how the model shows the problem.

More practice: warm-up

1

What does assumption mean?

2

Name one kind of model.

3

What does revise mean?

4

A plant grows 2 cm each week. It is 2 cm tall after week 1. Assume it keeps growing the same way. How tall is it after week 3?

5

What is the first step of modelling?

More practice: on your own

1

Your class wants to save $50 for a class plant. Each week you save $10. Make a table. In which week do you reach $50?

2

There are 27 children going on a field trip. Each van holds 10 children. How many vans? Explain.

3

Mila says, Our garden had 5 ripe tomatoes on Monday and 10 on Tuesday. If it keeps going like this, it will have 25 on Friday. What assumption is she making? Is it always true?

4

You want to give 2 cookies to each of 9 friends. Make a model. How many cookies do you need?

5

A model said your family would need 12 granola bars for a camping trip, but you ran out after 10. What should you do next time?

Challenge

1

A school has 45 children in the snow fort club. Each fort team has 10 children. How many teams can they make? Will anyone be left over? What could they do?

2

Two classes collect cans for a food bank. Class A collects 10 cans each day. Class B collects 15 cans each day. Both start at 0. Which class reaches 60 cans first? On which day? What did you assume?

3

Plan a model: How many minutes do you spend brushing your teeth in one week? Tell your assumption and solve.

Check your understanding

Quiz

1.

What is mathematical modelling?

2.

Which is an assumption?

3.

23 people need seats. Each table holds 5. How many tables are needed?

4.

A model predicted 60, but the real answer was 70. What should you do?

5.

Why can two people get different answers to the same real-life problem?

6.

You read 5 pages each day. Assume you keep going. After 6 days, how many pages have you read?

For parents and teachers

Mathematical modelling (overall expectation C4) is about using math to understand real-life situations. It has no single right method. Children understand the problem, make assumptions, build a model (drawing, table, pattern, number sentence), check, revise and share. This lesson also practises Strand A skills: communicating and listening to others' ideas (A1.4) and thinking critically and creatively (A1.6).

Common mistakes

  • Jumping to a number without saying what the question is. Ask: What are we trying to find out?
  • Not noticing leftovers (22 people need 5 tables of 5, not 4). Ask: Does everyone fit?
  • Thinking there is only one right answer. Celebrate different reasonable assumptions.

How to help

  • Turn everyday planning into modelling: snacks for guests, how many days until a holiday, how many weeks to save for a toy.
  • Ask What are you assuming? and How could we check?
  • Afterwards, compare the model to what really happened and talk about how to revise it.