Lesson 7

Using Negative Exponents

7.1: Exponent Rules (5 minutes)


This warm-up prepares students to work with expressions involving negative exponents. Students have studied exponents and their properties in grade 8 and encountered negative exponents at that time. The goal here is to review the fact that for integer exponents \(m\) and \(n\) and a non-zero base \(b\), this property holds: \(b^m \boldcdot b^n = b^{m+n}\).

Student Facing

How would you rewrite each of the following as an equivalent expression with a single exponent?

  • \(2^4 \boldcdot 2^0\)
  • \(2^4 \boldcdot 2^{\text-1}\)
  • \(2^4 \boldcdot 2^{\text-3}\)
  • \(2^4 \boldcdot 2^{\text-4}\)

Student Response

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Activity Synthesis

Review, if needed, the conventions that \(2^0 = 1\) and \(2^{-1} = \frac{1}{2}\), emphasizing that these conventions guarantee that \(2^a \boldcdot 2^b = 2^{a+b}\) for any two integers \(a\) and \(b\).

7.2: Coral in the Sea (15 minutes)


Up to this point in the unit, students have written and interpreted equations representing exponential change which were meaningful for non-negative input values. In this lesson, they interpret the meaning of a negative value in a context. The independent variable is time, \(t\), so negative values of \(t\) refer to times before \(t\) is 0.

For the last question, to find the time when the coral had the given volume, students may:

  • Calculate the values of \(y\) when \(t\) is -3, -4, -5, etc. (i.e., extend the table of values, if they previously created one) until they reach 37.5 or a value close to it.
  • Repeatedly divide by 2 (or multiply by \(\frac12\)) the value of \(y\) when \(t\) is -2 until it approaches or hits 37.5.
  • Use the equation \(1,\!200 \boldcdot 2^t = 37.5\) to find the value of \(2^t\) and then reason about \(t\) from there.

Look for students using these or other strategies so they can share later.

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


Arrange students in groups of 2. Encourage them to think quietly about the questions before discussing with their partner. Creating a table or spreadsheet may help students organize the work in the second question. If needed, encourage students to do so.

Reading, Conversing, Writing: MLR5 Co-Craft Questions. To help students make sense of the language of exponential situations, start by displaying the task statement “A marine biologist estimates that a structure of coral has a volume of 1,200 cubic centimeters and that its volume doubles each year.” Give students 1–2 minutes to write their own mathematical questions about the situation. Invite students to share their questions with the class, then reveal the activity’s questions. Listen for and amplify any questions referring to the value of \(t\), especially those wondering about time in the past. This will build student understanding and interpretation of the language used to represent negative exponents.
Design Principle(s): Maximize meta-awareness; Cultivate conversation
Representation: Internalize Comprehension. Begin the activity with concrete or familiar contexts. Encourage students to create a timeline next to their tables, selecting a starting point year for zero and then labeling the years correlated with the positive exponents prior to labeling years associated with negative exponents.
Supports accessibility for: Conceptual processing; Memory

Student Facing

A marine biologist estimates that a structure of coral has a volume of 1,200 cubic centimeters and that its volume doubles each year.

Coral underwater with fish.
  1. Write an equation of the form \(y=a \boldcdot b^t\) representing the relationship, where \(t\) is time in years since the coral was measured and \(y\) is volume of coral in cubic centimeters. (You need to figure out what numbers \(a\) and \(b\) are in this situation.)
  2. Find the volume of the coral when \(t\) is 5, 1, 0, -1, and -2.
  3. What does it mean, in this situation, when \(t\) is -2?
  4. In a certain year, the volume of the coral is 37.5 cubic centimeters. Which year is this? Explain your reasoning.

Student Response

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Anticipated Misconceptions

Students may struggle to think of how to start finding the values for \(a\) and \(b\). Suggest that they make a table of corresponding values of \(t\) and \(y\), and think about starting with \(t\) and calculating \(y\).

Activity Synthesis

Review how students found an equation, \(y = 1,200 \boldcdot 2^t\), to represent the volume of coral. Make sure that they recall that the 1,200 represents the 1,200 cubic centimeters of coral when it was first measured and the 2 indicates the doubling of its volume each year.

Select students to share their interpretations of negative values of \(t\) and how they found the value of \(t\) when given a volume. Ask questions such as:

  • “In this situation, what does it mean to say that when \(t\) is -3, \(y\) is \(1,200 \boldcdot 2^{\text-3}\) or 150?” (3 years before the coral was measured to be 1,200 cubic centimeters, its volume was 150 cubic centimeters).
  • “How did you estimate or find when the volume of the coral was 37.5 cubic centimeters?”

It may be helpful to display a graph of \(y = 1,200 \boldcdot 2^t\), such as the one embedded in the digital version of these materials. Display the graph (either with or without the points from the table plotted) and ask students to identify the volume of coral when \(t\) is -3 and the time when the volume of coral is 37.5 cubic centimeters.

If using the digital applet, consider initially turning off the function and the point, and hiding the expression list. Ask students to gesture at points on the graph corresponding to questions in the activity. Then, turn on the function and the point for demonstration. Remember that if you click on a plotted point, its coordinates are revealed.

7.3: Windows of Graphs (15 minutes)

Optional activity

The goal of this activity is to address a skill important to successfully using graphing technology: choosing an appropriate graphing window. Students are given three graphs representing a relationship from an earlier task. They comment on their effectiveness in representing the relationship and consider ways to adjust the graphing window to improve the information that the graph shows.

Choosing an appropriate window for a graph representing a situation characterized by exponential change can be challenging. Because exponential functions eventually grow very quickly, if the window is too small or too large, then the function may not be visible or may not show features that are interesting. Students could use an interactive graphing tool to experiment and decide on an appropriate graphing window. It is also helpful, however, to consider the context, the initial amount, and the growth factor.


After analyzing three attempts at graphing, students are instructed to use graphing technology to create a version that is better. Provide access to graphing technology (Desmos is available under Math Tools). Sudents may need instructions or a refresher on how to change the graphing window in the technology they are using before they can be successful with this task.

Engagement: Develop Effort and Persistence. Break the class into small discussion groups and then invite a representative from each group to report back to the whole class. Each group can report back on all three graphs, or, to allow for increased student agency, each group can be responsible for reporting back on a single graph of their choosing.
Supports accessibility for: Language; Social-emotional skills; Attention

Student Facing

The volume, \(y\), of coral in cubic centimeters is modeled by the equation \(y = 1,\!200 \boldcdot 2^x\) where \(x\) is the number of years since the coral was measured. Three students used graphing technology to graph the equation that represents the volume of coral as a function of time.


Two points on coordinate plane, origin O.


Graph of function on a grid.


graph of function on grid.

For each graph:

  1. Describe how well each graphing window does, or does not, show the behavior of the function.
  2. For each graphing window you think does not show the behavior of the function well, describe how you would change it.
  3. Make the change(s) you suggested, and sketch the revised graph using graphing technology.

Student Response

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Anticipated Misconceptions

Some students might not know how to begin to gauge the fitness of a graphing window. Encourage them to consider whether the quantity of interest is increasing over time or decreasing over time, and how this information might be conveyed by the graph. For example, this can help students decide whether the vertical intercept should be near the top or bottom of the graph. 

Students might also make a table of values to find the \(y\)-values when \(x\) is 1, 2, and 3. Prompt them to discuss whether those points show up on each of the three graphs and whether those points are needed to show how the quantities are changing and, if so, how to adjust the graphing window to show the points.

Activity Synthesis

Invite students to share their observations of the three graphs and their suggestions for improving the readability or meaningfulness of a graph. Discuss questions such as:

  • “To graph the coral structure for 5 years, what would be a reasonable graphing window?”
  • “If we want to graph the coral structure for 10 years, what would be a reasonable graphing window, assuming that the equation is still valid?”
  • “What happens if the range we choose is too small, for example, 10,000?”
  • “What happens if the range we choose is too big, for example, 100,000,000?”

Consider showing students, using your graphing technology of choice, how the graph changes when the horizontal dimension is 10 years and the vertical dimension is adjusted from 10,000, to 100,000, to 1,000,000 (as in Graph B), and then to 10,000,000 cubic centimeters.

Representing, Conversing: MLR7 Compare and Connect. As students share their observations of the three graphs with the class, call students’ attention to the different ways the scale on the vertical axis is represented graphically. Take a close look to determine where the same point is represented on each graph. Wherever possible, amplify student words and actions that describe the connections between a specific feature of one mathematical representation and a specific feature of another representation.
Design Principle(s): Maximize meta-awareness

7.4: Measuring Meds (15 minutes)


In this activity, students use a description and table of values to write an equation that represents them. Then, they interpret the equation for negative values of the exponent and produce a graph. They also interpret the numbers in the equation in terms of the context.

For the final question, only an estimate is possible. Expect different answers that range between 2 and 3 hours before the medicine was measured, as a more precise answer cannot be given right now. Look for students who use the graph and try to extend it to values in between integer hours. Invite them to share during the discussion.


If students continue to use graphing technology while working on this task, they are likely to enter their equation and see a smooth curve. It is not the intention that they try and sketch a curve at this time. Draw their attention to the instructions that say to plot the points.

Action and Expression: Provide Access for Physical Action. Provide access to graph paper, or tools and assistive technologies, such as graphing calculators or graphing software.
Supports accessibility for: Visual-spatial processing; Conceptual processing; Organization

Student Facing

A person took some medicine but does not remember how much. Concerned that she took too much, she has a blood test every hour for several hours.

    1. Time \(t\) is measured in hours since the first blood test and amount of medicine in her body, \(m\), is measured in milligrams. What is the growth factor? That is, what is \(b\) in an equation of the form \(m=a \boldcdot b^t\) ? What is \(a\)?
    2. Find the amounts of medicine in the patient’s body when \(t\) is -1 and -3. Record them in the table.
    \(t\), time (hours) \(m\), medicine (mg)
    0 100
    1 50
    2 25
  1. What do \(t=0\) and \(t=\text-3\) mean in this context?
  2. The medicine was taken when \(t\) is -5. Assuming the person did not have any of the medication in her body beforehand, how much medicine did the patient take?
  3. Plot the points whose coordinates are shown in the table. Make sure to draw and label tick marks on the axes.
    Blank coordinate plane, no grid, Origin O. Horizontal axis labeled “t”. Vertical axis, labeled “m”. Vertical axis intersects horizontal axis at origin.
  4. Based on your graph, when do you think the patient will have:

    1. 500 mg of medicine remaining in the body
    2. no medicine remaining in the body

Student Response

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Student Facing

Are you ready for more?

Without evaluating them, describe each of the following quantities as close to 0, close to 1, or much larger than 1.

\(\displaystyle \frac{1}{1-2^{\text-10}}\qquad\qquad \frac{2^{10}}{2^{10}+1}\qquad\qquad \frac{2^{\text-10}}{2^{10}+1}\qquad\qquad \frac{1-2^{\text-10}}{2^{10}} \qquad\qquad \frac{1+2^{10}}{2^{\text-10}}\)


Student Response

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Anticipated Misconceptions

Students may struggle with understanding the negative time values. It may help to give clock values to the numbers. For example, the first blood test was done at 5:00 (\(t = 0\)). Ask students what \(t\) would be when it is 6:00 and 4:00.

Activity Synthesis

Ask students how they labeled the axes on the graph. For the domain, it needs to include times from 3 hours before the medicine was measured to 2 hours after. (Students may also choose to include 5 hours before the first measurement.) The vertical axis should include 0 up to at least 800. (Higher, if students include the value for \(t=\text-5\) in the table.)

Use the last question about when there was 500 mg of medicine to begin a discussion about values between integer hours. Elicit from students how the graph might look if we consider non-integer time values. If students still have access to graphing technology for this activity, they can enter their equation and see the curve for themselves.

Consider displaying the following graphs for all to see. Explain that if the amount of medicine in the bloodstream is plotted at time intervals smaller than 1 hour, the graph may look like the first graph. We sometimes graph a relationship where a quantity changes continuously using a curve, as in the second graph. Students will have more opportunities to consider non-integer domain values in upcoming lessons.

Points plotted on coordinate plane, showing exponential decay. Horizontal axis from negative 2 to 3, labeled t. Vertical axis from 0 to 500, labeled m.
Graph of a continuous curve showing exponential decay. Horizontal axis from negative 2 to 3, labeled t. Vertical axis from 0 to 500, labeled m.

To follow up on the last question, consider discussing: How long could the decrease by half go on? Is there a point at which it is no longer practical or reasonable to use the mathematical model?

Lesson Synthesis

Lesson Synthesis

We looked at some situations where negative values representing times are meaningful. Review how we can use equations to understand a quantity before and after a certain time.

Suppose \(p= 90,\!000 \boldcdot 3^t\) represents the population, \(p\), of a colony of bacteria, \(t\) days since Monday. Ask students to respond to these prompts quietly.

  • Think of a possible value for \(t\). What does it represent in this situation?
  • For your value of \(t\), determine the corresponding value of \(p\). What does it represent in this situation?
  • Describe in words what the model says about this bacteria colony.

Invite students who chose different values to share their responses. Be sure to select at least one student who chose a negative value for \(t\). For the last prompt, highlight relevant vocabulary like initial amount and growth factor. If time permits, draw a set of axes for all to see and plot points corresponding to the values shared by students.

7.5: Cool-down - Invasive Fish (5 minutes)


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Student Lesson Summary

Student Facing

Equations are useful not only for representing relationships that change exponentially, but also for answering questions about these situations.

Suppose a bacteria population of 1,000,000 has been increasing by a factor of 2 every hour. What was the size of the population 5 hours ago? How many hours ago was the population less than 1,000?

We could go backwards and calculate the population of bacteria 1 hour ago, 2 hours ago, and so on. For example, if the population doubled each hour and was 1,000,000 when first observed, an hour before then it must have been 500,000, and two hours before then it must have been 250,000, and so on.

Another way to reason through these questions is by representing the situation with an equation. If \(t\) measures time in hours since the population was 1,000,000, then the bacteria population can be described by the equation:

\(\displaystyle p = 1,\!000,\!000 \boldcdot 2^t\)

The population is 1,000,000 when \(t\) is 0, so 5 hours earlier, \(t\) would be -5 and here is a way to calculate the population:

\(\displaystyle \begin{align} 1,\!000,\!000 \boldcdot 2^{\text-5} &= 1,\!000,\!000 \boldcdot \frac{1}{2^5} \\ &= 1,\!000,\!000 \boldcdot \frac{1}{32} \\ &= 31,\!250 \end{align}\)

Likewise, substituting -10 for \(t\) gives us \(1,\!000,\!000 \boldcdot 2^{\text-10}\) or \(1,\!000,\!000 \boldcdot \frac{1}{2^{10}}\), which is a little less than 1,000. This means that 10 hours before the initial measurement the bacteria population was less than 1,000.

Video Summary

Student Facing