Lesson 3

Representing Exponential Growth

3.1: Math Talk: Exponent Rules (5 minutes)

Warm-up

In grade 8, students studied how to multiply and divide numbers in exponential notation when the bases are the same. This warm-up 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 non-examples of a power of 2. Some examples are \(2^5\) and \(2^{100}\). Non-examples 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 whole-class discussion.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
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.

Speaking: MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, "I noticed that . . ." or "First, I _____ because . . . ." Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
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 non-zero 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 \(b^{\text-n} = \frac{1}{b^n}\) 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

Representation: Internalize Comprehension. Activate or supply background knowledge about exponents. Discuss the difference between repeated addition and repeated multiplication, while annotating visible examples of expressions to show the differences. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing; Language

Student Facing

  1. Complete the table. Take advantage of any patterns you notice.
    \(x\) 4 3 2 1 0
    \(3^x\) 81 27      
  2. Here are some equations. Find the solution to each equation using what you know about exponent rules. Be prepared to explain your reasoning.

    1. \(9^?\boldcdot 9^7 = 9^7\)
    2. \(\dfrac {9^{12}}{9^?}= 9^{12}\)
  3. 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 agreed-upon 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.

Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Before students share their explanations for the last question, present an incorrect answer and explanation. For example, “5 to the power of 0 is 0, because I multiplied to get it.” Ask students to identify the error, critique the reasoning, and write a correct explanation. As students discuss with a partner, listen for students who identify and clarify the ambiguous language in the statement. Invite students to share their critiques and corrected explanations with the class. Listen for and amplify the language students use to describe what happens when a number is raised to the power of 0 and highlight those who use patterns in the table. This helps students evaluate, and improve on, the written mathematical arguments of others, as they clarify their understanding of the power of 0.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

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).

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organization and problem solving. For example, students complete hours 0–3 and discuss the pattern they notice. Use annotations to show how the number of bacteria is changing from one hour to the next, then invite students to use what they notice to complete the table.
Supports accessibility for: Organization; Attention

Student Facing

  1. In a biology lab, 500 bacteria reproduce by splitting. Every hour, on the hour, each bacterium splits into two bacteria.

    1. Write an expression to show how to find the number of bacteria after each hour listed in the table.
    2. Write an equation relating \(n\), the number of bacteria, to \(t\), the number of hours.
    3. 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  
  2. In a different biology lab, a population of single-cell 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 \(80 \boldcdot 4^t\).)
Conversing: MLR2 Collect and Display. Before the whole-class discussion, invite students to discuss their thinking for the provided questions with a partner. Listen for and collect vocabulary and phrases students use to describe patterns in the table for the growth of bacteria. Display words and phrases such as “multiplier,” “exponentially,” and “doubling” for all to see. Remind students to suggest additions and to borrow language from the display as needed. This will help students read and use mathematical language during their partner and whole-group discussions.
Design Principle(s): Maximize meta-awareness

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).

Conversing: MLR8 Discussion Supports. Use this routine to help students describe the meaning of variables in exponential expressions. Arrange students in groups of 2. Invite Partner A to share their observations for the first graph and Partner B to share their observations for the second graph. Provide sentence frames such as “I can connect the equation to the graph by….” and “I notice that….” Invite the listener to press for additional details referring to features of the graphs, such as the vertical intercept, specific points, and the steepness of the curve. This will help students justify how features of the graph can be used to identify matching equations.
Design Principle(s): Support sense-making; Cultivate conversation

Student Facing

  1. 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. 
     

    Blank coordinate plane, origin O, with grid. Horizontal axis, labeled t, from 0 to 4, by 1’s. Vertical axis, labeled n, from 0 to 9,000, by 1,000’s.

    b. Graph \((t,p)\) when \(t\) is 0, 1, 2, 3, and 4. (If you get stuck, you can create a table.)

    Blank coordinate plane, origin O, with grid. Horizontal axis, labeled t, from 0 to 4, by 1’s. Vertical axis, labeled p, from 0 to 9,000, by 1,000’s.
  2. On the graph of \(n\), where can you see each number that appears in the equation?
  3. 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.

A discrete graph. Number of fish. Time in years.

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: Cool-down - Mice in the Forest (5 minutes)

Cool-Down

Cool-downs 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.

Graph of an exponential function, origin O.  number of days and cell population.

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 \((1,1500)\).