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# Activity: Cool coffee

## 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:

• M7-2 Display the graphs of linear and non-linear functions and connect the structure of the functions with their graphs
• M7-6 Manipulate exponential expressions.

## 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(x-B)+C. Functions include:
• simple piecewise
• exponential y = ex or y = bx
• 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 (M7-7) and gradient functions (M7-9).
• 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.

## Specific learning outcomes

Students will be able to:

• recognise types of non-linear 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 non-linear 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.

## 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.

## 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 20oC.)

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 = 20oC)

 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 × 0.8t.
• 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 = Amx + 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.

### 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 real-student-collected data first (below) or the tidy dataset (see above).

#### “Messy” real student-collected data Room temp = 21oC

 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.)

• 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.

• Science
• Hospitality (barista)
• Food technology
• Geography – demographics (exponential growth situations)

### Extension/enrichment ideas

• Link y = amx to y = Aekx
• Piecewise functions: explore -
• 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?

## Planned assessment

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

## 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 real-life 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:

• problem-solving 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 August 16, 2019