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# Activity: Coffee costs

## Purpose

Investigate price, income, and profit functions associated with cafe coffee sales.

## Achievement objectives

In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:

• M7-2 Display the graphs of linear and non-linear functions and connect the structure of the functions with their graphs
• M7-7 Form and use linear, quadratic, and simple trigonometric equations.

## Indicators

• Uses algebra and geometry to link points and lines on the Cartesian plane.
• Demonstrates understanding of functions and the relationships between the graph of functions, y=f(x), and the graphs of their transformations of the form, y=Af(x-B)+C.
• Makes connections between representations, such as f(x) notation, tables, mapping, equations, words and graphs:
• Writes equations for graphs and vice versa.
• Explains effect on graph / equation of changing parameters.
• Identifies and uses appropriate key features, that is, symmetry, intercepts, maxima, minima, domain, and range.
• Makes links with solving equations M7-7, manipulating expressions M7-6, and gradient functions M7-9.
• Solves problems that can be modelled by linear [or] quadratic … equations and interprets solutions in context.
• Uses completing the square and quadratic formula for solving quadratic equations.
• Demonstrates understanding of the relationship between an equation and its solutions…

## Specific learning outcomes

Students will be able to:

• form algebraic expressions
• graph quadratic functions (with or without technology)
• interpret features of quadratic functions
• use concepts of income, price, cost, and profit to solve problems
• use co-ordinate geometry techniques to find the equation of a line
• interpret gradient of a line.

## Diagnostic snapshot(s)

Students need:

• an understanding of linear and quadratic functions and their graphs
• to be able to solve linear and quadratic equations
• to be able to use co-ordinate geometry to find equation of a straight line.

## Planned learning experiences

### Setting the scene

Discuss with students ideas they have about how the price of a cup of coffee affects the income, customer numbers, and profits at a cafe or coffee shop.

My local cafe charges \$4.00 per coffee and sells, on average, 500 coffees each day. Costs have recently increased, so the coffee shop owner wants to increase the price he charges for a coffee. However, he estimates that, for every 10c increase in price, he will sell 5 fewer coffees each day.

#### Problem 1: What is income at \$4.00 per cup of coffee?

If the owner wants to increase total income by \$200, how much would he have to charge for a cup of coffee?

Discussion: We want to express the income as a function of the increase in what a customer pays for a coffee:

• Discuss what x could be (the price increase in dollars).
• Find an expression for the price of a coffee in terms of x.
• Find an expression for the number of coffees sold in terms of x.
• Find an expression for income.
• How can we use this expression to determine how much the cafe should charge for a cup of coffee to increase their income by \$200?

Encourage students to use a range of methods including graphing, calculus, guess and check, etc.

Explore the solution(s):

• Why are there two solutions to the problem?
• Which one would be better for the coffee shop?
• Why?

#### Problem 2: How much should the cafe charge to maximise their income?

Encourage students to use a range of methods including graphing, calculus, guess and check, etc.

Discussion:

• How many customers has the coffee shop lost?
• Would you pay this price for a coffee?
• What other things should the owner of the coffee shop consider in making decisions about raising prices?

#### Problem 3: Profit = price - cost

The cost of making one coffee is \$2.00.

• Express profit as a function of selling price.
• What price maximises the profit on coffee sales?
• What advice would you give the coffee shop owner?

#### Problem 4: Brownies and co-ordinate geometry (modelling)

The cafe manager believes that the number of brownies sold is related to their price. The following table shows the sales data for the last two seasons.

 Price (x) Number of Brownies sold (y) Winter (June, July, Aug) \$3.50 932 Spring (Sept, Oct, Nov) \$4.20 722

The manager assumes that the relationship between price and the number of brownies sold might be linear. Investigate this, for example, by sketching a graph of the number of brownies sold versus price

Assuming the manager is correct, and there is a linear relationship:

• Use co-ordinate geometry methods to find a model to link the number of brownies sold per season and the price.
• Discuss what your model tells you about this relationship. You should refer to features of your linear model and explain what happens when the price is very low or very high.
• Estimate the number of brownies sold over 3 months if the price was set to \$4.00. (The brownie supplier can supply 400 brownies per month.) What price should the manager charge so that every brownie could be expected to be sold?

Discuss the manager’s assumptions:

• What other models could sensibly link brownie sales and price?
• What other factors should the manager consider?

### Possible adaptations to the activity

Modify the initial price, numbers sold, and costs, or adapt to different products.

Suggest a variety of strategies to answer the problem: graphing software, graphics calculators, calculus, quadratic equation solving strategies, etc.

The students can work in small groups.

For an able class, present the problem with no scaffolding. Be more directive with less-able groups.

Overlaying income curve and profit curves in different colours on one graph provides good talking points.

Investigate other possible models for brownie prices and sales.

• Food technology
• Economics

### Extension/enrichment ideas

• Modify profit scenarios with sales of other products.
• Modify scenario with physical constraints, for example, maximum number of customers and/or coffees sold, etc.

## Planned assessment

This teaching and learning activity could lead towards assessment in the following achievement standards:

## Spotlight on

### Pedagogy

• Enhancing the relevance of new learning by:
• providing appropriate levels of challenge
• encouraging students to explain their thinking.
• Facilitating shared learning through:
• appropriate groupings of students
• students working in groups
• making connections.
• Providing sufficient opportunities to learn by:
• allowing sufficient time to solve tasks
• providing activities with differentiated entry and exit points, including extension and enrichment
• problem solving.

### Key competencies

• Thinking:
• Relating solutions to the context.
• Participating and contributing:
• Working collaboratively.
• Using language, symbols, and text:
• Solving algebraic problems.

### Values

Students will be encouraged to value:

• perseverance – with an initially complex problem
• curiosity – thinking about assumptions
• seeing that mathematical processes can be useful in the real world.

### Māori/Pasifika

Coffee could be transferred to any other consumable product. For example, hangi meals, handicrafts.

### Planning for content and language learning

Last updated September 9, 2018