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

Math Modelling: Planning a Winter Carnival Hot Chocolate Stand

C4 A1About 45 minutes

Learning goals

I can use the modelling process to plan a real event.
I can state my assumptions and explain why they are reasonable.
I can build a model with expressions, a table, an inequality or code, and use it to make predictions.
I can check and revise my model when something changes, and share a clear recommendation.

Words to know

mathematical model
Math (expressions, equations, inequalities, tables, graphs or code) that represents a real situation so we can make predictions and decisions.

assumption
Something we decide to treat as true so we can build the model, like half the students will buy a cup.

variable (in a model)
A quantity that can change, like the number of cups sold.

profit
The money left over after paying the costs: money in − costs.

revise
Change the model when new information arrives or an assumption turns out to be wrong.

Remember from Grade 5

In Grade 5 you used the modelling steps to plan a class trip. You made assumptions, used expressions, equations, inequalities and code, and revised your model when the facts changed.

In Grade 6 your models can use everything from this unit: linear patterns and their equations, expressions with decimal tenths, equations with multiple terms, inequalities with two operations, and efficient code.

The modelling process: 1 Understand, 2 Assume, 3 Model and solve, 4 Check and revise, 5 Share. An arrow goes from Check and revise back to Model and solve.

The modelling steps

Understand the problem. What is the question? What do we know? What do we still need?
Make assumptions. What will we treat as true? Why is that reasonable?
Model and solve. Use an expression, equation, inequality, table, graph or code. Make a prediction.
Check and revise. Does the answer make sense? What if something changes?
Share. Give your recommendation, your assumptions, your model, and what you learned.

Let's start with a plan

The Grade 6 class is running a hot chocolate stand at the school's winter carnival. All the profit goes to the local food bank. They have two choices (these are pretend prices for our model):

Option A: rent a hot-water urn for $30, then pay $0.50 per cup for the mix and the paper cup.
Option B: buy ready-made hot chocolate for $1.50 per cup.

They will sell each cup for $2. Which option should they choose, and how many cups must they sell to raise $150?

A table with a large hot-water urn, stacks of paper cups and cups of hot chocolate in a snowy schoolyard during a winter carnival. — A hot chocolate stand needs an urn, mix and cups. Every cup sold adds to both the cost and the money raised, which is what the model has to compare. AI-generated image

Worked example

Example 1: Which option costs less?

Write an expression for the cost of each option for c cups. Use a table to decide when each option is cheaper.

1. Understand. We need to compare the costs of A and B for different numbers of cups.
2. Assume. The prices stay the same all day, and the class needs one cup of mix for every cup sold.
3. Model. Option A: 30 + 0.5c. Option B: 1.5c. Both are linear: A grows by $0.50 per cup and B grows by $1.50 per cup.
Make a table (see below). At 30 cups both cost $45 (30 + 15 = 45, and 1.5 × 30 = 45).
4. Check. Below 30 cups, B is cheaper (no rental fee). Above 30 cups, A is cheaper, because each extra cup costs only $0.50.

If they sell more than 30 cups, Option A costs less. At exactly 30 cups, the options cost the same.

Cups (c)	Option A: 30 + 0.5c	Option B: 1.5c	Cheaper
10	$35	$15	B
20	$40	$30	B
30	$45	$45	same
40	$50	$60	A
60	$60	$90	A
120	$90	$180	A

Comparing the two options. The costs are equal at 30 cups.

The table as a graph. Option B (dashed) starts at $0 but climbs faster; Option A (solid) starts at $30 for the urn. The lines cross at 30 cups, where both cost $45.

Worked example

Example 2: Predict the sales and the profit

There are 240 students at the school. About how many cups will they sell, and what profit will Option A make?

Assume about 1 in 2 students buys one cup (it's a cold day, and families might buy some too, so this may be low). 240 ÷ 2 = 120 cups.
Money in: 120 × $2 = $240.
Costs (Option A): 30 + 0.5 × 120 = 30 + 60 = $90.
Profit: 240 − 90 = $150.

About 120 cups and a profit of $150 for the food bank, if our assumptions are right.

Worked example

Example 3: An inequality and code for the goal

With Option A, how many cups must they sell to make a profit of at least $150?

Each cup brings in $2 and costs $0.50, so each cup adds $1.50 of profit. Then subtract the $30 urn.
Profit = 1.5c − 30. The goal: 1.5c − 30 ≥ 150.
Boundary: 1.5c − 30 = 150, so 1.5c = 180. Count: 1.5 × 100 = 150 and 1.5 × 20 = 30, so 1.5 × 120 = 180. c = 120, and it is included (≥).
Check with code: SET c = 0, then REPEAT UNTIL 1.5 × c − 30 ≥ 150 with ADD 10 to c inside, then SAY c. Profit at 110 cups is 165 − 30 = 135 (not enough). At 120 cups it is 180 − 30 = 150 ✔. It says 120.

c ≥ 120: they must sell at least 120 cups. That matches the prediction from Example 2, so the goal is reachable but tight.

Worked example

Example 4: Check and revise

The forecast changes: the carnival day will be mild, and the class now expects only 1 in 4 students to buy a cup. Revise the model.

New prediction: 240 ÷ 4 = 60 cups.
Option A still costs less at 60 cups ($60 versus $90). Profit: 60 × 2 − 60 = 120 − 60 = $60. That's far below $150.
Idea: raise the price to $2.50. Now each cup adds 2.50 − 0.50 = $2 of profit. Profit = 2c − 30.
Goal: 2c − 30 ≥ 150, so 2c ≥ 180, so c ≥ 90. Still more than 60 cups.
Another idea: also sell to families at the carnival. If 30 family members buy too, that's 60 + 30 = 90 cups ✔.

Revised plan: charge $2.50 and sell to families as well as students, aiming for at least 90 cups. Revising showed that the first plan would fall short on a mild day.

A complete model answer has

the question in your own words
your assumptions, and why they are reasonable
your model (expression, table, inequality or code) with units like $ or cups
your answer and how you checked it
what would make you revise it
a short recommendation someone could act on

Modelling is teamwork: listen to other people's assumptions. There can be more than one good model.

Just for fun

Modelling tip (and a warning): Always state your assumptions. Ours quietly assumed nobody would ask for 47 marshmallows in one cup. One customer like that and the whole model melts!

Hands-on activity

Model a family plan

Choose a real plan with your family. For example:

Rent or buy? Renting skates or snowshoes costs a fee each time; buying costs more once. After how many outings is buying cheaper?
Snack budget: for a movie night or a party, with a fixed cost (like a pizza) plus a cost per guest. How many guests fit the budget?

Use real prices from a flyer, a website or a store sign with your grown-up, or make up reasonable pretend prices.

Understand: write the question.
Assume: list at least 2 assumptions.
Model: write an expression for each option, and make a table. Use an inequality if there is a budget.
Check and revise: change one assumption. Does your answer change?
Share: explain your recommendation to your family in 3 or 4 sentences.

What to look for: an expression with a fixed part and a per-item part (like 30 + 0.5c), a table that shows where one option becomes cheaper, stated assumptions, and a clear recommendation.

Your turn

What do Option A and Option B cost for 50 cups?

Which option should the class choose if they expect to sell only 20 cups? Why?

Name two assumptions used in Example 2.

With Option A and a price of $2, what is the profit for 80 cups?

Why would Option B be a poor choice if they sell each cup for $1.50?

What does this code say?

SET c = 0

REPEAT UNTIL 2 × c − 30 ≥ 150

ADD 10 to c

SAY c

Write a two-sentence recommendation for the class, using Examples 1 to 3.

More practice: warm-up

What is an assumption?

Evaluate 30 + 0.5c when c = 10.

Evaluate 1.5c when c = 10.

Name the five modelling steps in order.

Why do we check and revise a model?

More practice: on your own

The urn rental goes up to $45. Now Option A costs 45 + 0.5c. At how many cups do the options cost the same?

At a price of $2.50, using Option A ($30 urn), what is the profit for 100 cups?

At a price of $2.50, how many cups must they sell for a profit of at least $150? (Profit = 2c − 30.)

A bigger school has 300 students. If 1 in 2 buys a cup at $2 (Option A), what is the profit?

Give one weakness of the class's model, and how you could improve it.

Challenge

One volunteer can serve about 40 cups per hour. The stand is open for 2 hours. How many volunteers are needed to serve 120 cups, if each volunteer works the whole time?

If you graphed the cost of each option (cups across, dollars up), which option's points would rise more steeply? Where would the two sets of points meet?

Option C: the community centre lends an urn for free, but its mix costs $0.75 per cup. Compare C (0.75c) with A (30 + 0.5c) for 120 cups. When is each cheaper?

Check your understanding

Quiz

What is the first step in the modelling process?

Which is an assumption?

Option A costs 30 + 0.5c and Option B costs 1.5c. Which is cheaper for 60 cups?

For how many cups do Options A and B cost the same?

Why should you revise a model?

They sell 120 cups at $2 each. Their costs are $90. What is the profit?

For parents and teachers

Big idea: Grade 6 students apply the process of mathematical modelling to represent, analyse, make predictions and provide insight into real-life situations (C4). This lesson pulls the whole unit together: linear expressions (30 + 0.5c), decimal tenths, inequalities (1.5c − 30 ≥ 150), tables and code.

Common difficulties:

Giving a number with no assumptions. Ask: "What did you assume? What if that changes?"
Forgetting the fixed cost (the $30 rental) or the units ($, cups).
Thinking there is one right answer. Different assumptions lead to different, equally reasonable models.

How to help: Model real family decisions together: renting versus buying sports equipment, phone or streaming plans, or a party budget. Have your child write the expressions and a table, then explain their recommendation to you. Social-emotional learning and mathematical processes (A1): modelling builds reasoning, communication and the confidence to revise a plan when facts change. The prices in this lesson are pretend; real prices vary.