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

Math Modelling: Planning a Class Trip

C4 A1About 45 minutes
Learning goals
  • I can use the modelling process to plan for a real-life situation.
  • I can say what assumptions I am making and why.
  • I can build a model with expressions, equations, inequalities, tables, graphs or code.
  • I can check my model, revise it when something changes, and share my thinking.

Words to know

mathematical model
A way to use math (an expression, equation, table, graph or code) to describe a real situation and make predictions.
assumption
Something we decide to treat as true so we can make the model, like every student will come.
prediction
What the model says will happen.
revise
Change the model when new information comes in or an assumption turns out to be wrong.
insight
Something new you understand about the situation because of the model.

Remember from Grade 4

In Grade 4 you used the modelling steps to plan things like a reading challenge and to figure out how long birdseed would last. You made assumptions, used a table or equation, and revised your model when something changed.

In Grade 5 you have new tools for your models: expressions with variables, equations up to 100 and inequalities up to 50, decimal hundredths for money, and even code.

The modelling process1Understand2Assume3Model,solve4Check,revise5Sharego back and revise if needed
The modelling process: 1 Understand the problem, 2 Make assumptions, 3 Model and solve, 4 Check and revise, 5 Share. A dashed arrow goes from Check and revise back to Assume, because you may need to go around again.

The 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 is that reasonable?
  3. Model and solve. Use an expression, equation, inequality, table, graph or code. Make a prediction.
  4. Check and revise. Does the answer make sense? If something changes, revise the model.
  5. Share. Explain the answer, your assumptions and your model, and any insight you gained.

Let's start with a story

Ms. Tremblay's Grade 5 class wants to go on a field trip to a maple sugar bush in March. The class will raise money with a bake sale. How many cookies do they need to sell?

Metal sap buckets hang on maple trees in a snowy forest, with a steaming sugar shack behind them.
A maple sugar bush in March. Sap drips into the buckets, and steam rises from the sugar shack where the sap is boiled into syrup. AI-generated image
Worked example

Example 1: Planning the bake sale

The bus costs $180. Entry to the sugar bush is $5 per student. There are 24 students. Cookies sell for $0.75 each. How many cookies must the class sell?

  1. 1. Understand. We need the total cost of the trip, then the number of cookies that makes that much money.
  2. 2. Assume. All 24 students go. The class keeps all the money from the cookies (a family donates the ingredients). Every cookie baked is sold.
  3. 3. Model. Cost for s students: 180 + 5 × s. With s = 24: 180 + 120 = $300.
  4. Cookie money: 4 cookies sell for 4 × $0.75 = $3.00. So every $3 needs 4 cookies. $300 is 100 groups of $3, so the class needs 100 × 4 = 400 cookies.
  5. 4. Check. 400 cookies × $0.75: 100 groups of 4 cookies, each group making $3, is $300 ✔. Is 400 cookies reasonable? That's about 17 cookies per student to bake and sell. It's a lot, but possible with family help.

The class needs to sell 400 cookies. We assumed all 24 students go and that the ingredients are donated.

Cookies soldMoney raised
4$3.00
40$30.00
100$75.00
200$150.00
400$300.00
A table helps the class track their progress toward $300. Each row is found from the pattern: 4 cookies make $3.

Revise the model when something changes

The sugar bush calls: groups of 20 or more students get a lower price of $4 per student! Our model changes:

180 + 4 × 24 = 180 + 96 = $276.

How many cookies now? $276 ÷ $3 = 92 groups of 4 cookies, so 92 × 4 = 368 cookies. That's 32 fewer cookies to bake!

Another change: 2 students can't go. Now s = 22, which is still 20 or more, so the price is still $4. Cost: 180 + 4 × 22 = 180 + 88 = $268. Notice that the bus costs the same no matter how many students go, so the cost per student goes up when fewer students go. That's an insight!

first planbus $180$120= $300group pricebus $180$96= $27622 studentsbus $180$88= $268$0$50$100$150$200$250$300bus (same every time)entry for students
How the trip cost changes as the model is revised. The bus stays $180; only the entry part changes: $300, then $276, then $268.
Worked example

Example 2: An inequality model for the bus

The bus has 48 seats. All 24 students are going, and some parents want to come as helpers. How many adults, a, can come?

  1. Understand: students + adults must fit in 48 seats. That means the total must be less than 49 (48 or fewer).
  2. Assume: the teacher also needs a seat, so 25 seats are already taken.
  3. Model: 25 + a < 49. Boundary point: a = 24.
  4. Check: a = 23: 25 + 23 = 48 < 49 ✔ (every seat full). a = 24: 49 < 49 ✘. So a can be 0 to 23.

Up to 23 adults can come. If our assumption changes (for example, the bus driver says one seat must stay empty), we would revise: 25 + a < 48, so up to 22 adults.

Worked example

Example 3: A model in code

Write code that finds how many weeks of bake sales the class needs if they sell $45 of cookies each week and need $276.

  1. SET money = 0, SET weeks = 0.
  2. REPEAT UNTIL money > 275: ADD 45 to money, ADD 1 to weeks. Then SAY weeks.
  3. Trace: 45, 90, 135, 180, 225, 270, 315. After 6 weeks they have $270, which is not enough. After 7 weeks they have $315.

7 weeks. (We used money > 275 because, with whole-dollar amounts, that means $276 or more.) Insight: they will have $315 − $276 = $39 left over, which could pay for hot chocolate!

A complete model answer has

  • The answer to the question, with units (cookies, dollars, weeks).
  • Your assumptions and why they are reasonable.
  • Your model: an expression, equation, inequality, table, graph or code.
  • A check: does it make sense in real life?
  • Any insight you gained, like the bus costs the same no matter how many go.

Real problems are messy. Different people can make different assumptions and still have good models. Listen to other ideas and explain yours kindly.

Just for fun

Funny but true: it takes about 40 litres of maple sap boiled down to make just 1 litre of maple syrup. That's a lot of sap for one stack of pancakes, which is why a sugar bush in March smells so sweet!

Hands-on activity

Model a family plan

You need: paper, a pencil and a family plan to model, for example:

  • a family movie night: how much will snacks cost?
  • a birthday party: how many pizzas or cupcakes do you need?
  • saving up for something: how many weeks will it take?

Steps:

  1. Understand: write the question. Ask your grown-up for real numbers (prices, how many people).
  2. Assume: write at least two assumptions. (Example: each person eats 2 slices of pizza.)
  3. Model: write an expression or equation. Make a table if it helps.
  4. Check: does your answer make sense? Ask your grown-up.
  5. Revise: your grown-up changes one thing ("Grandma is coming too!"). Update your model.
  6. Share: explain your plan to your family in 1 minute.

What to look for: written assumptions, a model with a variable, an answer with units, and a sensible revision. For example, for pizza: slices = 2 × p; with 7 people that's 14 slices; with 8 slices per pizza, 2 pizzas (16 slices) are enough.

Your turn

1

The bus costs $180 and entry is $5 per student. Write an expression for the cost for s students.

2

Use your expression to find the cost for 30 students.

3

Cookies sell for $0.75 each. How much money do 8 cookies make?

4

The class needs $150. How many cookies at $0.75 must they sell?

5

Name one assumption in Example 1, and say what would happen if it were wrong.

6

A bus has 52 seats. 27 are taken. Write and solve an inequality for how many more people, p, can get on.

More practice: warm-up

1

What is an assumption?

2

What is the first step of the modelling process?

3

Evaluate 180 + 5 × s for s = 10.

4

Why do we revise a model?

5

How many quarters ($0.25) make $3.00?

More practice: on your own

1

Each student needs 2 bottles of water for the trip. Water comes in packs of 12. How many packs for 24 students and 2 teachers? State your assumption.

2

The class sells $45 of cookies each week. Make a table for weeks 1 to 4.

3

The sugar bush tour is 90 minutes. The bus ride is 55 minutes each way. Lunch is 30 minutes. If the bus leaves school at 9:00, when will it get back? State your assumptions.

4

Change one assumption in the previous question: a 15-minute washroom stop on the way back. When will the bus return?

5

Why is it good to share your assumptions along with your answer?

Challenge

1

The bus costs $180 no matter how many students go. Entry is $4 per student. Find the cost per student for 20 students and for 30 students. What insight do you get?

2

A second plan: sell hot chocolate for $1.50 a cup instead of cookies. How many cups make $276?

3

Write code that finds how many cookies (sold in groups of 4 for $3) the class needs to sell to have more than $275. What does it SAY?

Check your understanding

Quiz

1.

What is an assumption in a math model?

2.

The bus costs $180 and entry is $5 per student. What is the cost for 20 students?

3.

Cookies sell for $0.75. How many must be sold to make $30?

4.

The price drops from $5 to $4 per student. What should you do with your model?

5.

Which step comes after making assumptions?

6.

Why does each student pay less when more students go on the trip?

For parents and teachers

Big idea: Grade 5 students apply the process of mathematical modelling to represent, analyse, make predictions and provide insight into real-life situations (C4). Modelling is open-ended: different assumptions lead to different (but still good) answers. What matters is that your child states assumptions, uses a clear model, checks it and revises it when things change.

Common difficulties:

  • Skipping assumptions. Ask, "What are you pretending is true?"
  • Stopping at a number. Ask, "Does that make sense in real life? Can you buy 4.3 packs of water?"
  • Not revising. Change one fact on purpose and have your child update the plan.

How to help: Real family plans make the best models: a party, a trip, a savings goal. Let your child do the math and present the plan. Social-emotional learning (A1): modelling involves teamwork. Encourage your child to listen to other ideas and explain their own respectfully.