Activity: Cool coffee
Purpose 
AOs 
Indicators 
Outcomes 
Snapshot
Learning experiences 
Cross curricular 
Assessment 
Spotlight
Purpose
Investigate the exponential graph associated with cooling coffee to develop a model and solve a problem.
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
 M76 Manipulate exponential expressions.
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Indicators
 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. Functions include:
 simple piecewise
 exponential y = e^{x} or y = b^{x}
 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.
 Understands effect of transformations in different representations.
 Understands the relationship between the graph of y = f(x) and the graphs of its transformations.
 Identifies and uses appropriate key features, that is, symmetry, period, amplitude, intercepts, maxima, minima, asymptotes, domain, and range.
 Makes links with solving equations (M77) and gradient functions (M79).
 Solves problems that involve manipulating rational, exponential, and logarithmic algebraic expressions. Methods that could be used in solving problems include:
 manipulating logarithms in order to solve exponential equations.
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Specific learning outcomes
Students will be able to:
 recognise types of nonlinear functions
 qualitatively sketch a graph for a given situation
 identify and describe key features of an exponential graph, asymptotes, domain, and range
 plot points to sketch a nonlinear curve, with or without technology
 understand effect of transformations in different representations
 understand the relationship between the graph of y = f(x) and the graphs of its transformations
 make connections between representations, such as f(x) notation, tables, mapping, equations, words, and graphs:
 write equations for exponential graphs and vice versa
 explain effect on graph/equation of changing parameters
 understand effect of transformations in different representations.
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Diagnostic snapshot(s)
Students need to have some prior understanding of linear functions and their graphs, rate of change, and exponential functions and their graphs.
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Planned learning experience
Investigation of the exponential graph associated with cooling coffee
When is coffee at an optimal temperature for drinking? Discuss how you might go about answering this question.
Setting the scene
View video clip
Joulies.
Pause at “Coffee isn’t always the right temperature” (22 sec).
Starter
Get students to draw a graph of coffee temperature against time
After a couple of minutes, ask:
 What temperature did the coffee start at?
 How long would it take to cool right down?
 Do you need to add anything to your graph?
Display some/all of the graphs for discussion. Discuss what the different graphs show.
 What is the same and what is different about these graphs?
 How might we group them? Explain.
 Linear, curve – which is the most sensible and why? (A linear that stops at 20^{o}C.)
Continue the video, stop again at 30 (or even 40 seconds) and discuss:
 What does “window of opportunity” mean?
 What is Dave’s problem?
 Discuss who likes their drinks scalding and who lukewarm?
 How could we get an idea what most coffee drinkers want?
Working with the data
Give out dataset 1.
Dataset 1: Temperature versus time after coffee is served
(Room temp = 20^{o}C)
time (min)
 temp (C)
 Difference between temperature and room temperature

 time (min)
 temp (C)
 Difference between temperature and room temperature

0
 95.0
 75.0

 32
 22.6
 2.6

1
 87.5
 67.5

 33
 22.3
 2.3

2
 80.8
 60.8

 34
 22.1
 2.1

3
 74.7
 54.7

 35
 21.9
 1.9

4
 69.2
 49.2

 36
 21.7
 1.7

5
 64.3
 44.3

 37
 21.5
 1.5

6
 59.9
 39.9

 38
 21.4
 1.4

7
 55.9
 35.9

 39
 21.2
 1.2

8
 52.3
 32.3

 40
 21.1
 1.1

9
 49.1
 29.1

 41
 21.0
 1.0

10
 46.2
 26.2

 42
 20.9
 0.9

11
 43.5
 23.5

 43
 20.8
 0.8

12
 41.2
 21.2

 44
 20.7
 0.7

13
 39.1
 19.1

 45
 20.7
 0.7

14
 37.2
 17.2

 46
 20.6
 0.6

15
 35.4
 15.4

 47
 20.5
 0.5

16
 33.9
 13.9

 48
 20.5
 0.5

17
 32.5
 12.5

 49
 20.4
 0.4

18
 31.3
 11.3

 50
 20.4
 0.4

19
 30.1
 10.1

 51
 20.3
 0.3

20
 29.1
 9.1

 52
 20.3
 0.3

21
 28.2
 8.2

 53
 20.3
 0.3

22
 27.4
 7.4

 54
 20.3
 0.3

23
 26.6
 6.6

 55
 20.2
 0.2

24
 26.0
 6.0

 56
 20.2
 0.2

25
 25.4
 5.4

 57
 20.2
 0.2

26
 24.8
 4.8

 58
 20.2
 0.2

27
 24.4
 4.4

 59
 20.1
 0.1

28
 23.9
 3.9

 60
 20.1
 0.1

Discuss in small groups and share with the class:
 Do you see any patterns in the data before plotting it?
Students plot data points (with or without technology):
 Do you see any patterns in the data after plotting points?
Ask the students to compare this graph with their starting graphs. Discuss and describe the features you notice (initial value, asymptotes, domain, and range).
Finding a model
Explore possible models with the students. Does this graph remind you of any graphs you have studied?
 The students could use guess and check or other methods, for example:
 Start by graphing T = 100 × 0.8^{t.}
 Investigate what happens if you change the 100 and/or 0.8.
 What happens in the long term?
 Will your model end up at zero?
 Should you adjust with plus something?
 The students repeat until they have a model that fits the data satisfactorily.
 Lead a class discussion with the students about:
 how changing the constants in the model changes the start and end temperatures
 what will happen in the long term.
Follow up with appropriate exercises to investigate y = Am^{x }+ C using graphics calculator, spreadsheets, or pen and paper. These could be ones that are created by the teacher or from an appropriate textbook or workbook.
Learn more:
The students make generalisations about how the constants in the model link to the curve and data collected for exponential cooling.
The students use their model to answer the original question: Between what times does the coffee temperature meet your requirement?
That is, they solve exponential equations with algebraic manipulation or appropriate technology.
Learn more:
Possible adaptations to the activity
 For a simpler model, plot time versus difference between coffee temp and room temp (third column in data set provided).
 Collect own data for cooling coffee.
 Option to go with the messy realstudentcollected data first (below) or the tidy dataset (see above).
“Messy” real studentcollected data Room temp = 21^{o}C

pretty cup

Insulated cup

Time: min
 Pretty cup temp
 Insulated cup temp

2.5
 75
 86

5
 70
 82

8.5
 65
 77

11
 63
 75

14
 60
 72

17
 56
 69

22.5
 51
 65

26
 49
 63

31.5
 46
 59

35
 43
 57

51
 37
 50

Explore the following: How do you get your coffee cool enough to drink in the shortest possible time? (Leave it black for a while, then add cream.)
Learn more:
Return to the
coffee cooling problem:
 Fit a model to this data, using technology.
 Discuss why it does not fit perfectly.
 Discuss the underlying variables.
 Discuss sources of error and variation.
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Crosscurricular links
 Science
 Hospitality (barista)
 Food technology
 Geography – demographics (exponential growth situations)
Extension/enrichment ideas
 Link y = am^{x} to y = Ae^{kx}
 Piecewise functions: explore 
 adding milk
 heating water to a boil, then letting it cool.
 Examine the effect of different types of cups.
 Given the graph of the cooling coffee, can you draw the graph of the gradient function? What does this mean in terms of the context?
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Planned assessment
This teaching and learning activity could lead towards assessment in the following achievement standard:
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Spotlight on
Pedagogy
 Facilitating shared learning by:
 having students working in groups
 students having learning conversations.
 Making connections to prior learning and experience by:
 providing reallife problems in which the context is relevant to students
 using problem solving strategies – including guess and check, looking for generalisations, using technology.
Key competencies
 Thinking:
 Using trial and improve, refining models, and making generalisations.
 Participating and contributing:
 Discussing ideas, contributing their graphs, and justifying their ideas.
 Relating to others:
 Discussing others’ ideas.
 Using text, language, and symbols:
 Students communicate mathematical exponential models and their representations.
Values
Students will be encouraged to value:
 problemsolving in everyday situations, including trial and improvement
 thinking reflectively, through rich classroom discussion opportunities
 the perception that mathematics can help them understand their world.
Māori/Pasifika
 Could use similar activity for hangi stones, hangi meal, etc, (same activity, different context, and data).
Planning for content and language learning
 Provide multiple opportunities for authentic language use with a focus on students using academic language.
 Is the language focus on key language?
 Do I make sure the students have many opportunities to notice and use new language?
Last updated September 9, 2018
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