If they try to write an expression using multiplication, students may have trouble determining the value to multiply with the original cost in each question because the percent changes (8%, 30%, and 35%) are not associated with familiar benchmark fractions. Remind students that “percent” means “per 100” and can be written as a fraction with a denominator of 100 or as a decimal in hundredths. Also, for this task they can use other expressions— for example, expressions that use addition or subtraction—to make the calculations. But the expressions using multiplication will be very important moving forward.

The purpose of this discussion is to use methods for understanding exponential change and apply them to percentage increase and decrease situations.

Invite previously selected students to share their expressions. Sequence the discussion of the strategies by the order listed in the Activity Narrative. If possible, record and display their work for all to see.

Connect the different responses to the learning goals by asking questions such as:

• “How can we see that all of the representations are equivalent?” (Using the distributive property and evaluating the sum inside the parentheses.)

• “Does this remind you of other equivalent expressions found in this unit?” (Yes, we did something similar when looking at items gaining or losing a fraction of their value.)

• “If the same percentage increase (or decrease) were to be applied again, which expression would be the easiest to use?” (The expression using only multiplication would probably be the easiest to repeat.)

Students may have trouble recognizing when to add or subtract pieces to the original value. Ask them to think about the context and whether each quantity will go up or down.

Select students or groups to share their responses. Ask them to explain how they know that the two expressions they wrote for each situation are equal.

Focus the discussion on the repeated percent increase in the last problem. Highlight the fact that the expression captures the two successive increases of 8%. Discuss questions such as:

• “Suppose the population increases 8% again the third year. How can we express the number of people then?” (, or )
• “How can we express the population size if the population continues to grow by 8% for years?” ()
• “Why might it be helpful to use only multiplication to describe the repeated 8% increase in population?” (It is more efficient and less prone to error. Expressing repeated percent change with addition and multiplication would produce longer and more complicated expressions, unless we calculate the result of the operations. When only multiplication is involved, we could use exponents to express the repeated change.)
• “Is the population in this school growing linearly or exponentially? How do you know?” (Exponentially. It is growing by the same factor each year, rather than by the same number of people.)