Lesson 5

This warm-up asks students not only to identify that a quantity is changing linearly or exponentially, but also to identify a term when the preceding value is not given.

For the last entry in each table, students may struggle to find the actual value to write in the table. Explain that an expression representing the value is all that’s needed.

Focus the discussion on how students used the patterns in the tables to generate a \(y\)-value for a non-consecutive value of \(x\). Make sure students are able to express the pattern of repeated addition in the first table, using an expression like \(2.5 + 7.5x\), and that they can express the pattern of repeated multiplication in the second table using an exponential expression like \((2.5) \boldcdot 4^x\). Highlight the words linear and exponential to describe how the quantities are changing suggested by the tables.

In this activity, students focus on producing and interpreting a graph representing a quantity that decays exponentially. They can then use the graph to make some estimates about the context, which is about algae control. After students create their graph, use the activity synthesis to encourage students to interpret the meaning of input values that are not whole numbers.

Note that the axes on the given graph aren’t scaled. Students will need to make sure that the vertical intercept can be plotted and that it is not too close to 0. If students do the work to decide how to scale the axes, they are creating a coherent representation of the relationship (MP2).

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

To introduce the context and keep students thinking about exponential growth and decay, consider presenting a riddle:

A certain area of the pond is covered in algae. Each day, the algae-covered area doubles. If it takes 24 days for the algae to completely cover the pond, how many days did it take to cover half of the pond? Given the word "half," students may be inclined to immediately say that it takes 12 days. But if they ponder the situation a bit, they should recognize that it takes 23 days to cover half the pond, because it only takes one more day or one more doubling to cover the whole pond.

Students may find it challenging to choose a scale for the axes in a way that helps them plot the points and see a pattern. If they are still struggling to choose a scale for the axes after a few minutes, ask students to think about the greatest and least vertical coordinates they need to show on the graph and what the height of each rectangle on the grid should be to show these values. In addition, if they get stuck plotting points, suggest that they first make a table of values.

Once students have created reasonable graphs, focus the discussion on the non-integer values of \(t\). Discuss questions such as:

• “What does a \(t\) value of 2.5 mean?” (a value \(t\) of 2.5 means 2.5 weeks or a little more than 17 days after treatment begins.)
• “How did you estimate the area covered by algae after a certain non-whole-number of weeks?” (We can use the graph to estimate the area of algae, which is between the area in week 2 and that in week 3, though not too much precision can be given. It is definitely less than 27 square yards and more than 9 square yards. Following the general trend of the points decreasing less and less rapidly as time goes on, an estimate of 15 is reasonable.)

Time permitting, consider discussing:

• “Someone says that after seven weeks, the algae will disappear. Do you agree?” (Sample responses: No, if we keep multiplying by \(\frac13\), the area will keep getting smaller, but never reach 0. No, after seven weeks there would be about \(\frac{1}{9}\) of a square yard or 1 square foot covered by algae. Yes, rounded to the nearest square yard, there are 0 square yards covered by the algae after seven weeks.)

Here students make sense of a graph representing a situation characterized by exponential decay. They justify why the amount of insulin in a patient's body could change exponentially and use the graph to answer questions about the situation. The numbers, chosen explicitly to provide a realistic model of the insulin context, are more complex than what students have seen so far. They will need to apply their understanding both to identify the growth factor and to write an expression showing how much insulin remains after \(m\) minutes.

In this activity, students are building skills that will help them in mathematical modeling (MP4). They don't decide which model to use, but students have an opportunity to figure out how to justify that an exponential model is a good model for the data. Also, they are prompted to construct an equation to represent the model in a scaffolded way.

Explain that insulin is a hormone that is needed to process glucose (sugar) in the body. For people with diabetes, a disorder in which the body cannot process glucose properly or does not produce enough insulin, injections of insulin are sometimes needed. Once in the bloodstream, insulin breaks down fairly rapidly, so additional injections are needed over the course of a day to control the person’s glucose.

Display the graph for all to see. Ask students to observe the graph and be prepared to share one thing they notice and one thing they wonder. Give students a moment to share their observations and questions with a partner.

Students may notice that the insulin values decrease by \(\frac{1}{10}\) each minute and say, in response to the first two questions, that \(\frac{1}{10}\) of the original insulin remains after each minute. Ask them what \(\frac{1}{10}\) of 10 mg is. Then ask them if that is the amount of insulin that remains after 1 minute. If needed, ask them what fraction of 10 mg is the amount of insulin that remains after 1 minute. Is that the same as the fraction of 9 mg of insulin that remains after 2 minutes?

Invite students to share their responses with the class. If not already mentioned in students’ explanations, highlight the connections between the equation, the graph, and the quantities in the situation. Ask students:

• “How can we use an equation to express the milligrams of insulin in the body (\(I\)) after \(m\) minutes using an equation?” (\(I = 100 \boldcdot \left(\frac{9}{10}\right)^{m}\))
• “Where can we see the 100 mg in the graph?” (the vertical intercept)
• “What about the \(\frac{9}{10}\)?” (The coordinates tell us that \(\frac{1}{10}\) of insulin gets removed each minute, so \(\frac {9}{10}\) of it, which is most of it, is left in the body every minute. The amount of insulin decays relatively slowly, which shows in the points dropping gradually.)

Also consider asking:

• “Will there be any insulin remaining in the bloodstream an hour after the injection?”
• “If so, when will the body completely run out of insulin?”

These are difficult questions to answer. The mathematical model will never reach 0 because a positive quantity multiplied by \(\frac{9}{10}\) is always positive. Practically speaking, however, the insulin given to the patient will eventually all disappear, unless a new dose is injected. The model also doesn't account for any insulin that the body produces naturally. This is a good opportunity to remind students that mathematical models are simplified descriptions of reality.