Activity: Coffee costs
Purpose 
AOs 
Indicators 
Outcomes 
Snapshot
Learning experiences 
Cross curricular 
Assessment 
Spotlight
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:
 M72 Display the graphs of linear and nonlinear functions and connect the structure of the functions with their graphs
 M77 Form and use linear, quadratic, and simple trigonometric equations.
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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(xB)+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 M77, manipulating expressions M76, and gradient functions M79.
 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…
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Specific learning outcomes
Students will be able to:
 form algebraic expressions
 form quadratic equations
 graph quadratic functions (with or without technology)
 solve quadratic equations
 interpret features of quadratic functions
 use concepts of income, price, cost, and profit to solve problems
 use coordinate geometry techniques to find the equation of a line
 interpret gradient of a line.
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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 coordinate geometry to find equation of a straight line.
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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 do you notice about your solutions to these questions?
 What advice would you give the coffee shop owner?
Problem 4: Brownies and coordinate 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 coordinate 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 lessable 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.
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Crosscurricular links
 Business studies
 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.
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Planned assessment
This teaching and learning activity could lead towards assessment in the following achievement standards:
 AS91257 Mathematics and statistics 2.2 Apply graphical methods in solving problems.
 AS 91261 Mathematics and statistics 2.6 Apply algebraic methods in solving problems.
 AS 91269 Mathematics and statistics 2.14 Apply systems of equations in solving problems.
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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:
 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 July 30, 2015
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