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.

In this activity, students focus on situations in which quantities change exponentially with a growth factor that is between 0 and 1. The factor is still called the growth factor, though students should learn that sometimes people refer to it as a decay factor.

Students begin by 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).

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.

Make sure that students understand that a growth factor that is between 0 and 1 causes the value to decrease each time the factor is applied. Tell students that the quantity changes exponentially, though sometimes people use the more specific terms exponential growth and exponential decay to indicate whether the amount is increasing or decreasing. The multiplier is still called the growth factor, but when it is a positive number less than 1, the result decreases with every iteration. Sometimes people use decay factor to indicate that a quantity that decreases exponentially involves a positive factor that is less than 1, but it is still correct to use the term "growth factor."

Discuss questions such as:

• “What is the growth factor for this situation?” ()
• “When the growth factor is between 0 and 1, what do you expect to happen to the -values of points on the graph as you read from left to right? Explain your reasoning.” (I expect the values to decrease as they get closer and closer to the -axis. This happens because the -values are being multiplied by a fraction that is between 0 and 1, which decreases its value while still being positive.)

Time permitting, consider discussing:

• “What does a value of 2.5 mean?” (A value 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.)
• “Someone says that after seven weeks, the algae will disappear. Do you agree?” (No, if we keep multiplying by , the area will keep getting smaller, but never reach 0. No, after seven weeks there would be about 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 luminescence of a glow stick could change exponentially and use the graph to answer questions about the situation. The numbers, chosen explicitly to provide a realistic model of the luminescence 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 luminescence stays after hours.

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.

If students are unsure if or of the luminescence stays each hour, consider asking:

• “Tell me more about how much luminescence stays and how much goes away.”
• “Is the graph showing the luminescence that stays or the luminescence that goes away?”

The goal of this discussion is to 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 luminescence of the glow stick () after hours using an equation?” ( )
• “Where can we see the 9 lumens in the graph?” (the vertical intercept)
• “What about the ?” (The coordinates tell us that the brightness reduces by of the previous luminescence each hour, so of it, which is most of it, is left every hour.)

Also consider asking:

• “Will there be any luminescence remaining from the glow stick 12 hours after it starts glowing?”
• “If so, when will the glow stick stop glowing?”

These are difficult questions to answer. The mathematical model will never reach 0 because a positive quantity multiplied by is always positive. Practically speaking, however, the chemical reaction causing the glow stick to luminesce will end, unless more chemicals are added to the situation. This is a good opportunity to remind students that mathematical models are simplified descriptions of reality.