C. Algebra · Patterns, Equations, Inequalities, Coding and Modelling

Math Models: Predicting and Planning in Real Life

C4 A1About 45 minutes

Learning goals

I can follow the modelling steps to solve a real-life problem.
I can make and explain assumptions.
I can use a pattern, table of values, graph or equation as a model to make predictions.
I can check my model and revise it when something changes.

Words to know

mathematical model
Math (like a table, a graph, a pattern or an equation) that shows a real-life situation so we can make predictions.

assumption
Something we decide to treat as true so we can solve the problem, like the birds eat the same amount every day.

prediction
What we think will happen, based on our model.

revise
Change the model to make it better.

Remember from Grade 3

In Grade 3 you used the modelling steps to plan real events, like a skating party. In Grade 4 you have even more tools for a model: growing patterns, tables of values, graphs, equations and decimal tenths. A good model helps you predict what will happen.

The modelling steps. The dashed line shows that we can go back and revise.

The modelling steps

Understand the problem. What is the question? What do we know? What else do we need?
Make assumptions. What will we treat as true? Why?
Make a model and solve. Use a pattern, table, graph or equation. Make a prediction.
Check, revise and share. Does the answer make sense? If something changes, revise the model. Explain your thinking and your assumptions.

Worked example

Example 1: The class reading challenge

Room 12 has read 8 books so far this month. Their goal is 50 books. Last week they read 12 books. In which week will they reach the goal?

Understand: The question is when will the class reach 50 books? We know they start at 8 and read 12 last week.
Assume: The class reads 12 books every week. (We need this, because we don't know the future. Last week is our best clue.)
Model: This is a growing pattern: start at 8, add 12 each week. Make a table: after week 1: 20, week 2: 32, week 3: 44, week 4: 56.
After week 3 they have 44, which is less than 50. After week 4 they have 56, which is more than 50.
Check: 8 + 4 × 12 = 8 + 48 = 56 ✔. That makes sense: the class needs 42 more books, and 3 weeks give only 36, but 4 weeks give 48.

With our assumption, the class reaches the goal during week 4.

Week	Books read in total
0	8
1	20
2	32
3	44
4	56

The model: start at 8 and add 12 each week. Week 0 is today.

A graph of the model. The points climb by 12 each week. The goal of 50 is passed between week 3 and week 4.

Revise the model when something changes

The teacher says week 2 is the winter break, and the class will only read about 6 books that week. Our assumption is no longer true, so we revise.

New table: week 1: 8 + 12 = 20. Week 2: 20 + 6 = 26. Week 3: 26 + 12 = 38. Week 4: 38 + 12 = 50.

Now the class reaches exactly 50 at the end of week 4. A good modeller checks the model again whenever the facts change.

Worked example

Example 2: How long will the birdseed last?

The class has a bird feeder. A bag holds 5 kg of seed. Each day the class puts out 0.5 kg. How many days will the bag last?

Understand: The question is how many days? We know the bag is 5 kg and they use 0.5 kg a day.
Assume: They use 0.5 kg every day, including weekends (a family volunteer fills it), and no seed is spilled.
Model: A shrinking pattern in tenths: 5.0, 4.5, 4.0, 3.5, 3.0, 2.5, 2.0, 1.5, 1.0, 0.5, 0. Count the jumps: 10.
Another way: 0.5 kg is 5 tenths, and 5 kg is 50 tenths. Write an equation: 5 × d = 50 (in tenths). So d = 10.
Check: 10 days × 0.5 kg = 5.0 kg ✔

The bag will last about 10 days, if our assumptions are true.

Thinking about assumptions

What if a cold snap brings more birds, and the class has to put out 1 kg a day? Revise: 5 kg ÷ 1 kg = 5 days. When we change an assumption, the prediction changes too. That is why we always share our assumptions with our answer.

A good answer to a modelling problem includes...

The answer to the question (with units: days, books, kg).
The assumptions you made.
Your model (a table, a graph, a pattern or an equation).
A check: does the answer make sense in real life?

Just for fun

Weather forecasters use math models every day. That's why they say "60% chance of rain" instead of "Maybe? Bring an umbrella just in case, I guess?"

Hands-on activity

Model it at home: how long will it last?

You need: something your family uses up a little at a time, like a box of cereal, a bag of apples, a roll of paper towels or a jar of peanut butter; paper and a pencil.

Understand: Ask how many days will it last? Find out how much there is (count the apples, or read the mass on the box).
Assume: Decide how much your family uses each day. Ask your grown-up. Write it down as an assumption.
Model: Make a table of values or a shrinking pattern. Predict the day it will run out.
Check: Put the prediction on the calendar. Each day, record how much is left. Was your model close?
Revise and share: If guests came or someone skipped breakfast, revise your model. Tell your family what you found and what your assumptions were.

What to look for: your child names at least one assumption, uses a pattern or table, and explains why the real answer might be different from the prediction.

Your turn

Leo has saved $14. He saves $6 each week. He wants a $50 toque-and-mitten set. Make a model. In which week will he have enough? State your assumption.

A snowman is 1.5 m tall. On warm days it melts 0.3 m each day. Make a model. After how many warm days will it be gone?

In question 2, what is one reason the real answer might be different?

A team needs 40 juice boxes for a tournament. Juice boxes come in packs of 8. Write an equation and solve it.

Name the four modelling steps.

Why do we write down our assumptions?

More practice: warm-up

What is an assumption?

A class plants 3 trees each spring. How many trees after 4 springs (starting with 0)?

Name two things you can use as a math model.

After making a prediction, what step comes next?

A plant is 10 cm tall and grows 2 cm a week. Predict its height after 3 weeks.

More practice: on your own

The school's recycling bin gets 4 kg of paper each day. It holds 30 kg. It is empty on Monday morning. Will it be full by the end of Friday? Make a model.

A family floods a backyard rink. Each cold night they add about 0.5 cm of ice. They want the ice to be 4.0 cm thick. It starts with no ice. How many nights will it take? State your assumption.

A graph shows money saved: week 1 $5, week 2 $10, week 3 $15. Use the model to predict week 8.

Your class wants enough chairs for a family math night: 45 people. Chairs come stacked in piles of 9. How many piles? What assumption are you making?

In Example 1, what if the class read 15 books each week instead of 12? When would they reach 50?

Challenge

A class is planning a winter walk. They walk 0.6 km the first week and plan to walk 0.2 km more each week after that (0.6, 0.8, 1.0, ...). In which week will they first walk 2.0 km? Make a table.

Two classes compete in a book challenge. Class A has 20 books and reads 5 a week. Class B has 8 books and reads 9 a week. Use a model to find when Class B catches up.

Make up your own modelling problem about your family or community. Write the question, one assumption, your model and the answer.

Check your understanding

Quiz

Which is the first modelling step?

Why do we make an assumption?

A bag has 3 kg of birdseed. The class uses 0.5 kg a day. About how many days will it last?

A class has read 10 books and reads 10 more each week. When will they reach 40 books?

Your model says the bag lasts 10 days, but more birds come in a cold snap. What should you do?

Which is the best way to share a modelling answer?

For parents and teachers

Big idea: The Ontario curriculum (C4) asks Grade 4 students to apply the process of mathematical modelling to represent, analyse, make predictions, and provide insight into real-life situations. Modelling is open-ended: different assumptions can give different, equally good answers.

Common difficulties:

Skipping assumptions. Ask "What are you pretending is true?" every time.
Not checking against real life. Ask "Does that make sense? Could it really happen?"
Thinking a revision means the first model was wrong. Revising is part of the process when new information arrives.
Decimals in models (0.5 kg a day). Counting in tenths helps.

How to help: Model real family questions: how long the milk will last, when you'll have saved enough for something, how long to shovel the driveway. Let your child make assumptions, test the prediction, and talk about why it was or wasn't right (Strand A: connecting, reflecting and communicating).