In this activity, students construct an exponential function to model population growth. Then they interpret the function when evaluated at certain rational numbers. They do so quantitatively, in terms of the population, and abstractly, in terms of an expression with a fractional exponent (MP2).

As students work, identify those who express the population of a given year as an exponential expression and those who use a calculator to estimate a numerical value. Both approaches are important: the first, because it gives an opportunity to interpret a fractional power, and the second, because it gives a concrete sense of the population growth.

If students do not yet correctly graph the function, consider asking:

• “How did you choose the horizontal and vertical dimensions for your graph?”

• “How could adjusting the horizontal and vertical dimensions help you see how the population grows between 2240 and 2270?”

Focus the discussion on the meaning of and in this context and on how to reason about these values. Display a table for all to see, like the one shown here, to fill in during the discussion:

Select previously identified students to share their responses for the population of Jada's base with the inputs 0.5 and 1.8, written as expressions ( and ) and as numerical values (about 68,938 and 127,003). Help students see that:

• We can think of the population in 2260, 2 decades after 2240, as because the 60% increase is applied twice. Although we can’t write the same kind of expanded expression for the population 0.5 decade after 2240, we can use properties of exponents to reason about its value.
• Here are three different ways to write the growth factor for the population 5 years after 2240: ,  , or .

Conclude by asking students to write three different ways to express the growth factor for the population 18 years after 2240 (,  , or ).

In this activity, students are given a graph of an exponential relationship to produce an equation and to reason about the context. Students then express the change in the function in terms of its decay rate, recalling the connection between decay factor and decay rate.

Begin the discussion by inviting students to share how they derived an expression representing the function and the meaning of 12 and (or 0.0625) in the expression (or ).

Next, focus the discussion on the interpretation of in this context and how its value could be estimated graphically and reasoned algebraically. Here are some questions to aid in the discussion.

• “How did you find the growth (or decay) factor for this situation?” (Because the amount of chemical decayed from 12 liters to 0.75 liters after 1 hour, the growth rate can be found using  or 0.0625.)
• “What does a point at mean in this situation?” (20 minutes after the cleaning begins, there are 4.8 liters of the chemical still in the lake.)
• “What percentage of the chemical is still in the lake after 20 minutes? What are some ways to calculate this value?” (About 40% is left after 20 minutes. This can be found using or .)
• “The fish living in the lake will be safe if the amount of chemical can be reduced to less than of a liter after 80 minutes of cleaning. Will the fish in this lake be safe? Explain your reasoning.” (Yes, because , or .)