Lesson 3
Representing Exponential Growth
3.1: Math Talk: Exponent Rules (5 minutes)
Warmup
In grade 8, students studied how to multiply and divide numbers in exponential notation when the bases are the same. This warmup reviews these properties which students will use systematically as they work with exponential expressions in this unit.
Launch
Before starting the math talk, it may be helpful to take time to ensure that students understand the question. Ask students for examples and nonexamples of a power of 2. Some examples are \(2^5\) and \(2^{100}\). Nonexamples include \(100^2\) and \(5\boldcdot 2\). It may be useful to further remind students that, for example, \(2^5\) equals \(2 \boldcdot2 \boldcdot2 \boldcdot2 \boldcdot2\).
Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a wholeclass discussion.
Supports accessibility for: Memory; Organization
Student Facing
Rewrite each expression as a power of 2.
\(2^3 \boldcdot 2^4\)
\(2^5 \boldcdot 2\)
\(2^{10} \div 2^7\)
\(2^9 \div 2\)
Student Response
Student responses to this activity are available at one of our IM Certified Partners
Activity Synthesis
As students share their thinking as part of the math talk, be sure to demonstrate how to break each part of the expression into factors, i.e. \(2^5 \boldcdot 2 = (2 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2) \boldcdot 2=2^6\). Remind students that they can generalize the observations from examples such as this one. The exponents can be added when multiplying exponential expressions with the same base and subtracted when dividing those with the same base.
Design Principle(s): Optimize output (for explanation)
3.2: What Does $x^0$ Mean? (10 minutes)
Optional activity
This task reviews an important property of exponents which students have studied in grade 8, namely that if \(b\) is a nonzero number, then \(b^0 = 1\). This is a convention, one that allows the rule \(b^x \boldcdot b^y = b^{x+y}\) to remain true when \(x\) or \(y\) is allowed to be 0. An additional convention, which is not addressed in this task, states that \(b^{\textn} = \frac{1}{b^n}\) for a whole number \(n\). With these conventions, the equation \(b^x \boldcdot b^y = b^{x+y}\) is true for all integers \(x\) and \(y\).
Launch
Supports accessibility for: Memory; Conceptual processing; Language
Student Facing
 Complete the table. Take advantage of any patterns you notice.
\(x\) 4 3 2 1 0 \(3^x\) 81 27 
Here are some equations. Find the solution to each equation using what you know about exponent rules. Be prepared to explain your reasoning.
 \(9^?\boldcdot 9^7 = 9^7\)
 \(\dfrac {9^{12}}{9^?}= 9^{12}\)
 What is the value of \(5^0\)? What about \(2^0\)?
Student Response
Student responses to this activity are available at one of our IM Certified Partners
Student Facing
Are you ready for more?
We know, for example, that \((2+3)+5=2+(3+5)\) and \(2\boldcdot (3\boldcdot 5)=(2\boldcdot 3)\boldcdot 5\). The grouping with parentheses does not affect the value of the expression.
Is this true for exponents? That is, are the numbers \(2^{(3^5)}\) and \((2^3)^5\) equal? If not, which is bigger? Which of the two would you choose as the meaning of the expression \(2^{3^5}\) written without parentheses?
Student Response
Student responses to this activity are available at one of our IM Certified Partners
Anticipated Misconceptions
Students may think that \(a^0=0\). First, remind them that exponents are not the same as multiplication (for example \(4^3= 4\boldcdot 4\boldcdot 4\) and \(4 \boldcdot 4 \boldcdot 4\) is very different from \(4 \boldcdot 3\)). Next, ask them to use the patterns that they notice in the equations and tables to determine the correct value.
Activity Synthesis
Make sure students understand that \(5^0 = 1\) is an agreedupon definition. The reason for defining \(5^0\) this way is so that the property (\(5^a \boldcdot 5^b = 5^{a+b}\)) continues to hold when we allow 0 as an exponent.
Design Principle(s): Optimize output (for explanation); Maximize metaawareness
3.3: Multiplying Microbes (15 minutes)
Activity
This activity prompts students to build expressions of the form \(a \boldcdot b^x\) to encapsulate a type of pattern they have encountered several times so far, and to consider what \(a\) and \(b\) mean in the context of bacteria growth. They do so by writing numerical expressions that make explicit the key feature of exponential change, the repeated multiplication by the same factor, and then making a generalization of their repeated reasoning (MP8) using exponential notation. Since students are finally representing this pattern using an exponent, a quantity following this type of pattern is described as changing exponentially. The term growth factor is given to the multiplier or \(b\) in an expression of the form \(a \boldcdot b^x\).
Launch
Clarify that it is not necessary to compute the number of bacteria at the end of each hour; an expression would suffice. If needed, provide an example (e.g., write the expression for the first day as \(500 \boldcdot 2\) rather than as 1000).
Supports accessibility for: Organization; Attention
Student Facing

In a biology lab, 500 bacteria reproduce by splitting. Every hour, on the hour, each bacterium splits into two bacteria.
 Write an expression to show how to find the number of bacteria after each hour listed in the table.
 Write an equation relating \(n\), the number of bacteria, to \(t\), the number of hours.
 Use your equation to find \(n\) when \(t\) is 0. What does this value of \(n\) mean in this situation?
hour number of bacteria 0 500 1 2 3 6 t  In a different biology lab, a population of singlecell parasites also reproduces hourly. An equation which gives the number of parasites, \(p\), after \(t\) hours is \(p = 100 \boldcdot 3^t.\) Explain what the numbers 100 and 3 mean in this situation.
Student Response
Student responses to this activity are available at one of our IM Certified Partners
Anticipated Misconceptions
For the first question, some students may write either \(2 \boldcdot 500\) or \(500+500\) for the number of bacteria after one hour. Both are mathematically correct, but \(2 \boldcdot 500\) is more helpful for identifying a pattern, which will help generate an expression for the number of bacteria after \(t\) hours. If they struggle to complete the table, refocus their attention on the second row of the table and ask them if there is a different expression they could use for the number of bacteria after one hour.
Students may misread the directions and write the actual values in the table rather than expressions. Ask them to record the expression they used to determine the value in the table rather than the value itself.
Students may write something like \(500 \boldcdot 2 \boldcdot 2\boldcdot \text{. . .} \boldcdot 2\) with a note about there being \(t\) 2s. Encourage them to think how they might be able to write this expression more concisely.
Activity Synthesis
Invite students to share the expressions in their table and their generalized expression for the number of bacteria after \(t\) hours. Make connections between, for example, \(500 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2\), the more concise expression \(500 \boldcdot 2^5\), and the more general expression representing any number of hours \(500 \boldcdot 2^t\). Highlight that 500 is not only the initial number of bacteria, but the result of evaluating \(500 \boldcdot 2^0\).
Tell students that patterns like these, where a quantity is repeatedly multiplied by the same factor, the quantity is often described as changing exponentially. We can see why: an exponent is used to express the relationship. The term for the multiplier, 2, in the doubling relationship and 3 in the tripling relationship, is the growth factor.
Questions for discussion:
 “Is the growth of the bacteria characterized by common differences or common factors? How do you know?” (Common factors, since each time the hour increases by 1, the number of bacteria is multiplied by the same factor.)
 “In each row in the table, what does the value of 500 mean? Why doesn't it change?” (It is the initial bacteria population when they are first measured.)
 “What does \(2^0\) mean in this situation?” (\(2^0\) tells us no doubling has happened, so the original population of 500 is all we have.)
 “What do the 100 and 3 mean in the expression \(100 \boldcdot 3^t\)?” (100 is the initial population of the parasites when they are first measured and the number 3 is the growth factor, the number by which the population is multiplied each hour.)
 “If the starting parasites population is 80 but the population quadruples every hour, how will the expression change?” (It will be \(80 \boldcdot 4^t\).)
Design Principle(s): Maximize metaawareness
3.4: Graphing the Microbes (15 minutes)
Activity
Having just seen an example of the meaning of \(a\) and \(b\) in an exponential expression \(a \boldcdot b^x\), students now focus on interpreting these numbers using graphs. They graph the equations from the previous task, noticing that \(a\) is the vertical intercept of the graph while the number \(b\) determines how quickly the graph grows (since in these cases \(b>1\)). A larger value of \(b\) corresponds to a more rapid rate of growth for the bacteria population. The axes for the graphs have been labeled here, but in future activities, students will have to think strategically about how to label the axes to most effectively plot the points.
Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Display the equations from the previous activity for all to see (if they are not already visible).
Design Principle(s): Support sensemaking; Cultivate conversation
Student Facing

Refer back to your work in the table of the previous task. Use that information and the given coordinate planes to graph the following:
a. Graph \((t,n)\) when \(t\) is 0, 1, 2, 3, and 4.
b. Graph \((t,p)\) when \(t\) is 0, 1, 2, 3, and 4. (If you get stuck, you can create a table.)
 On the graph of \(n\), where can you see each number that appears in the equation?
 On the graph of \(p\), where can you see each number that appears in the equation?
Student Response
Student responses to this activity are available at one of our IM Certified Partners
Anticipated Misconceptions
Students may have trouble graphing the points, particularly finding the appropriate vertical (\(n\) or \(p\)) values. Ask them to find the coordinates of the grid points on the vertical axis and use that to estimate the vertical position of their points.
When calculating values by hand, many students may mistakenly write an expression like \(100 \boldcdot 3^2\) as \(300^2\). Remind them that the expression \(100 \boldcdot 3^2\) means \(100 \boldcdot 3 \boldcdot 3\).
Activity Synthesis
Make sure students recognize two key takeaways from this activity:
 The vertical intercept of the graph is the initial bacteria population size. It is the size of the population when first measured, or when \(t\), the number of hours since measurement, is 0.
 The growth factor of the populations is represented by how quickly the graph increases.
Lesson Synthesis
Lesson Synthesis
In this lesson we learned the term growth factor and used equations and graphs to represent situations with quantities that change exponentially. Ask students to identify the connections between the quantities in a situation, and the graph and expressions that represent it. Use an example from a classroom activity or a new example like this:
\(1000 \boldcdot 2^t\) represents a fish population after \(t\) years. Here is the graph of the yearly fish population. Display for all to see, and ask students where they can see the 1000 and the 2 in the graph.
Ask students:
 What was the fish population when the scientists first measured it? (1000)
 How can you tell from the graph? (It is the vertical intercept, the number of fish when \(t = 0\).)
 How is the fish population changing from year to year? (It's doubling.)
 How does the expression \(1000 \boldcdot 2^t\) represent the fish population after \(t\) years? (The 1000 is the starting population. Every year it doubles, so we multiply 1000 by 2, \(t\) times.)
3.5: Cooldown  Mice in the Forest (5 minutes)
CoolDown
Cooldowns for this lesson are available at one of our IM Certified Partners
Student Lesson Summary
Student Facing
In relationships where the change is exponential, a quantity is repeatedly multiplied by the same amount. The multiplier is called the growth factor.
Suppose a population of cells starts at 500 and triples every day. The number of cells each day can be calculated as follows:
number of days  number of cells 

0  500 
1  1,500 (or \(500 \boldcdot 3\)) 
2  4,500 (or \(500 \boldcdot 3\boldcdot 3\), or \(500 \boldcdot 3^2\)) 
3  13,500 (or \(500 \boldcdot 3\boldcdot 3 \boldcdot 3\), or \(500 \boldcdot 3^3\)) 
\(d\)  \(500 \boldcdot 3^d\) 
We can see that the number of cells (\(p\)) is changing exponentially, and that \(p\) can be found by multiplying 500 by 3 as many times as the number of days (\(d\)) since the 500 cells were observed. The growth factor is 3. To model this situation, we can write this equation: \(\displaystyle p = 500 \boldcdot 3^d\).
The equation can be used to find the population on any day, including day 0, when the population was first measured. On day 0, the population is \(500 \boldcdot 3^0\). Since \(3^0 = 1\), this is \(500 \boldcdot 1\) or 500.
Here is a graph of the daily cell population. The point \((0,500)\) on the graph means that on day 0, the population starts at 500.
Each point is 3 times higher on the graph than the previous point. \((1,1500)\) is 3 times higher than \((0,500)\), and \((2,4500)\) is 3 times higher than \((1,1500)\).