Lesson 3

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.

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.

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.

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 \(y\).

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.

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.

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 \(a \boldcdot b^x\).

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).

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.

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 \(80 \boldcdot 4^t\).)

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).

Display the equations from the previous activity for all to see (if they are not already visible).

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 \(100 \boldcdot 3 \boldcdot 3\).

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.