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

Math Models: Planning with Math in Real Life

C4 A1About 45 minutes
Learning goals
  • I can say what question a real-life problem is asking and what information I need.
  • I can make assumptions and explain why I made them.
  • I can make a model with a drawing, a table of values, a pattern or an equation, and use it to make a prediction.
  • I can check my model, revise it when something changes, and share what I found.

Words to know

mathematical modelling
Using math to understand and solve a real-life problem, step by step.
model
A drawing, table, pattern or equation that shows a real-life situation in math.
assumption
Something we decide to treat as true so we can solve the problem.
prediction
What we think will happen, based on our model.
revise
To change your model or answer to make it better.

Remember from Grade 2

In Grade 2 you used math to solve real-life problems, like a mitten drive and planning picnic tables. You followed the modelling steps. In Grade 3 you have more math tools: multiplication, division, numbers to 1000, tables of values and variables. Let's put them to work!

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.

The four modelling steps

  1. Understand the problem. What is the question? What do we know? What else do we need to find out?
  2. Make assumptions. What will we treat as true? Why?
  3. Make a model and solve. Use a drawing, a table of values, a pattern or an equation. Use it to make a prediction.
  4. Check, revise and share. Does the answer make sense in real life? If something changes, revise the model. Explain your answer and your assumptions.
Worked example

Example 1: Hot chocolate for the skating party

Room 8 is having a skating party at the outdoor rink. There are 24 students and 3 adults. One package of hot chocolate mix makes 8 cups. How many packages should they buy?

  1. Understand: The question is How many packages do we need? We know: 24 + 3 = 27 people, and 1 package makes 8 cups.
  2. Assume: Every person comes and drinks 1 cup. (We need this, because the problem does not tell us how many cups each person drinks.)
  3. Model: Make a table of values. 1 package → 8 cups, 2 → 16, 3 → 24, 4 → 32.
  4. 3 packages make only 24 cups. 24 is less than 27, so 3 people would have no hot chocolate.
  5. 4 packages make 32 cups. 32 is more than 27. Everyone gets a cup, with 32 − 27 = 5 cups left over.
  6. Check: 4 × 8 = 32, and 32 is more than 27. ✔

They should buy 4 packages, if each person drinks 1 cup.

Cups of hot chocolate on a park bench beside an outdoor skating rink with skaters in the distance.
A skating party is a real-life problem: how much hot chocolate do we need? A math model helps the class plan before they buy. AI-generated image
PackagesCups of hot chocolate
18
216
324
432
540
The model: a table of values that adds 8 cups for each package.

Revising the model

The teacher says: "It will be very cold that day. Let's plan for 2 cups each." Our assumption changed, so we revise the model.

Now we need 27 + 27 = 54 cups. Extend the table: 5 → 40, 6 → 48, 7 → 56. 6 packages make only 48 cups, which is not enough. 7 packages make 56 cups, which is enough, with 2 cups left over.

A new assumption can change the answer. That's okay! Good modellers explain which assumption they used.

Worked example

Example 2: Saving for a field trip

A trip to the science centre costs $20. Maya has $5. She plans to save $3 each week from doing chores. In which week will she have enough?

  1. Understand: The question is When will Maya have at least $20?
  2. Assume: She saves $3 every week and does not spend any of it.
  3. Model: A growing pattern. Start at $5, add $3 each week. Week 1: $8, week 2: $11, week 3: $14, week 4: $17, week 5: $20.
  4. We can also write it as an equation with a variable: after w weeks, Maya has 5 + (3 × w) dollars. For w = 5: 5 + 15 = 20.
  5. Check: 5 jumps of $3 is $15, and $5 + $15 = $20. ✔

Maya will have $20 in week 5, if she saves $3 every week.

WeekMoney saved
Start$5
1$8
2$11
3$14
4$17
5$20
Maya's savings model. The operation add $3 repeats.
$0$5$10$15$20$5Start$8Week 1$11Week 2$14Week 3$17Week 4$20Week 5Goal: $20
Maya's savings model as a bar graph. Each week the bar grows by $3. In week 5 it reaches the dashed goal line at $20.
Worked example

Example 3: How many buses?

The Grade 3 classes are going to an apple orchard in the fall. 62 students and 8 adults are going. Each school bus holds 48 people. How many buses are needed?

  1. Understand: We need seats for 62 + 8 = 70 people. Each bus holds 48.
  2. Assume: Everyone rides the bus, and a bus cannot hold more than 48 people.
  3. Model: 1 bus holds 48. 2 buses hold 48 + 48 = 96.
  4. 1 bus is not enough: 48 is less than 70. 2 buses are enough: 96 is more than 70.
  5. Check: 70 − 48 = 22 people would be left without a seat on 1 bus. So we really need a 2nd bus.

They need 2 buses. There will be 96 − 70 = 26 empty seats.

Just for fun

Why did the math model bring a warm jacket to the skating party?

Because it assumed it would be cold!

Work together

Modelling is great with a partner or a group. Listen to other people's ideas. Someone might make a different assumption and get a different answer, and both answers can make sense! Ask, "What did you assume? Why?"

Different assumptions, different answers

How many books will Room 8 read in October? Jaden assumes each of the 24 students reads 2 books: 24 + 24 = 48 books. Priya assumes each student reads 3 books: 24 + 24 + 24 = 72 books.

Both models make sense. The way to find out is to collect real data: keep a class reading chart for October, then compare it with the predictions.

Your turn

1

What is an assumption in a math model?

2

A box of granola bars has 6 bars. Your class has 25 students, and each student gets 1 bar. How many boxes do you need?

3

In the hot chocolate example, what would change if only 20 people came and each had 1 cup?

4

Ben has $10 and saves $5 each week. Make a table for weeks 1 to 4.

5

A van holds 7 children. How many vans are needed for 20 children? What do you assume?

6

Why is it important to check your answer against real life?

Hands-on activity

Plan a family snack

You need: paper, a pencil and a grown-up.

The problem: Plan a snack for everyone in your home (or for a family visit). How much should you prepare?

  1. Understand: Who is eating? What snack? (Apple slices, crackers or popcorn.)
  2. Assume: Decide how much each person eats. For example, each person eats 6 crackers. Write your assumption.
  3. Model: Make a table of values: 1 person → 6 crackers, 2 people → 12, …
  4. Solve: How many crackers in all? How many packages, if a sleeve has 30?
  5. Check and revise: Serve the snack. Was it enough? Too much? Change your assumption for next time.
  6. Share: Tell your family what you found and what you assumed.

Grown-up note: please do any cutting (like slicing apples) yourself. Popcorn is a choking risk for children under 4, so keep it away from little ones.

More practice: warm-up

1

What are the four modelling steps?

2

Name two kinds of math models.

3

1 bag holds 10 apples. How many apples in 4 bags?

4

Each table seats 4 people. How many people can sit at 5 tables?

5

What does revise mean?

More practice: on your own

1

Each pack has 8 juice boxes. A soccer team has 15 players. How many packs are needed if each player gets 1 juice box? How many are left over?

2

Ava reads 20 pages each day. Her book has 150 pages. Make a model. In how many days will she finish?

3

A class is collecting 300 cans for the food bank. They collect 50 cans each week. In which week will they reach their goal?

4

In question 3, the class collects 100 cans in week 2 (not 50). Revise the model.

5

Two students predict how many glasses of water the class drinks in a day. One assumes each of 24 students drinks 2 glasses, and one assumes 3 glasses. Find both predictions. How could you find out which is closer?

Challenge

1

Your class (25 students) is making a paper chain for a winter celebration. Each student makes 10 links a day. Your goal is 1000 links. Will you reach the goal in 3 days? Explain your model and assumptions.

2

A family is planting a vegetable garden. They want 30 tomato plants. Each row fits 6 plants. They have room for 4 rows. Is that enough? What could they change?

3

Make up your own real-life problem that could be solved with the modelling steps. Write one assumption you would make.

Check your understanding

Quiz

1.

What is the first step in mathematical modelling?

2.

Which is an assumption?

3.

Each package makes 8 cups. How many packages do you need for 30 cups?

4.

Sam has $4 and saves $4 each week. How much does he have after 3 weeks?

5.

Why might two students get different answers to the same modelling problem?

6.

A bus holds 48 people. How many buses are needed for 50 people?

For parents and teachers

Big idea: Mathematical modelling (C4) means using math to make sense of a real situation: understand the problem, make assumptions, build a model (table, pattern, equation, drawing), and check and revise. In Grade 3, models can use multiplication, division, tables of values and variables.

Common difficulties:

  • Forgetting real life when checking. 70 people ÷ 48 seats does not mean "1 and a bit" buses: you need 2. Ask, "Does this make sense?"
  • Not stating assumptions. Encourage your child to say "I assumed…" out loud.
  • Thinking there is only one right answer. With different assumptions, different answers can both be reasonable.

How to help: Involve your child in real planning: how many hot dogs for a barbecue, how many weeks to save for a toy, how many bags of leaves to rake. These everyday problems build modelling skills and social-emotional skills like cooperation and perseverance (strand A).