Students will be able to interpret an exponential graph in the context of a "bunny problem" and write an exponential given a graph and two points on the function. Students will use negative exponents during this process and interpret a negative exponent as "division" in order to make meaning of this.

Students are first exposed to exponential graphs in the context of a realistic example of exponential growth that they have explored extensively through data tables.

30 minutes

Some students will immediately see how to interpret the graph and to find the equation using the same tools as the word problems. Other students will need some coaching about how to do this. The scaffold that seems to help students most is asking them to organize the information into a data table. Then I ask them, “What is the missing information here?” They then are able to see that they need to find the growth factor or multiplier. If the only method that they can come up with to do this is guess and check, that is totally fine. They will be motivated later on to figure out a new method, and many students will write their own equations and figure out how to solve those on their own.

When it comes to the data tables at the bottom of the page, it is very important that students develop a method that does not involve only guess and check. The idea is for them to develop shortcuts. I found that many of my students used their calculators to guess multipliers over and over again, but this shortchanges them from the learning associated with dealing with the fractions. The ideal strategy is to break down the numbers into their prime factors, which makes it easier to identify the multiplier.

Obviously by this point in the unit, all students have figured out that the data tables of exponential functions have constant multipliers, but this warm-up offers the chance to discuss this idea more abstractly. Also, when you ask students to describe their process, they often use the word *difference* when they mean *quotient*. It is a great opportunity to discuss the difference between these words, and it also is a chance to highlight the fact that these functions are not linear.

30 minutes

10 minutes