This task reviews an important property of exponents, which students have studied in grade 8, namely that if is a non-zero number, then . This is a convention, one that allows the rule to remain true when or is allowed to be 0. An additional convention, which is not addressed in this task, states that for a whole number . With these conventions, the equation is true for all integers and .

Students may think that . First, remind them that exponents are not the same as multiplication (for example , and is very different from ). 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 is an agreed-upon definition. The reason for defining this way is so that the property ( ) continues to hold when we allow 0 as an exponent.

For the first question, some students may write either or for the number of bacteria after one hour. Both are mathematically correct, but is more helpful for identifying a pattern, which will help generate an expression for the number of bacteria after 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 with a note about there being 2s. Encourage them to think how they might be able to write this expression more concisely.

Direct students’ attention to the reference created using Collect and Display. Ask students to share how they found the first expression. Invite students to borrow language from the display as needed and update the reference to include additional phrases as they respond. 

Invite students to share the expressions in their table and their generalized expression for the number of bacteria after  hours. Make connections between, for example,  , the more concise expression , and the more general expression representing any number of hours . Highlight that 500 is not only the initial number of bacteria, but the result of evaluating .

Tell students that in 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.

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 that 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 mean in this situation?” ( 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 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 parasite population is 80 but the population quadruples every hour, how will the expression change?” (It will be .)

Students may have trouble graphing the points, particularly finding the appropriate vertical ( or ) 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 as . Remind them that the expression means  .

Make sure that 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 , the number of hours since measurement, is 0.
• The growth factor of the populations is represented by how quickly the graph increases.