Lesson 14

This warm-up refreshes students' memory of strategies for working with percent change situations, preparing them to apply this understanding to situations of exponential change.

As students work, look for the different strategies they use to answer the questions. For example, students might think of the first question as \((0.2) \boldcdot 160\), as \(\frac{1}{5} \boldcdot 160\), or as \(160 \div 5\), all of which can be calculated mentally. Others might draw a tape diagram to represent the quantities and their relation to each other. Invite students with contrasting strategies to share later.

Also notice the different ways students approach and express the percent decrease situations to help inform the activities ahead.

Select students to share how they approached the questions. Record their responses for all to see. Focus the discussion on the last two questions. Note the different ways students expressed 20% more than $160 and 25% less than $192 and why the expressions are equivalent. For example:

• \(160 + (0.2)\boldcdot 160\)
• \(160 \boldcdot (1.2)\)
• \(160 + 160 \div 5\)
• \(160 \boldcdot\frac 65\)

• \(192 - (0.25)\boldcdot 92\)
• \(192 \boldcdot (0.75)\)
• \(192 - 192 \div 4\)
• \(192 \boldcdot\frac 34\)

If any students drew representations like tape diagrams and used them to reason, it is worth taking the time to make connections between the diagrams and the numerical expressions.

In this activity students solve problems about percent increase and decrease. Students may use a variety of calculations. Monitor for students who write any of the following expressions for the first problem:

• \(12,\!000 +0.08 \boldcdot(12,\!000)\)
• \(12,\!000 \boldcdot (1 + 0.08)\)
• \( 12,\!000 \boldcdot (1.08)\)

While all of these are accurate, the third is the most helpful for the work students will do in the next several lessons.

Monitor for students who write any of these expressions for the second problem:

• \(8.50-0.3\boldcdot(8.50)\)
• \(8.50\boldcdot (1 - 0.3)\)
• \(8.50\boldcdot (0.7)\)

The third of these expressions will be the most useful in future lessons as it only involves multiplication.

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, using addition or subtraction, to make the calculations. But the expressions using multiplication will be very important moving forward.

Select previously identified students to share their expressions for each problem, in the same sequence as shown in the Activity Narrative. Help students make the connections between the different forms, clarifying them in terms of properties of operations.

For the third problem, highlight these three expressions, all of which represent the sale price of the shoes in dollars. Ensure that students understand why they are equivalent and how to write the percent increase and decrease in one step, using multiplication, as shown in the last expression.

• \(79 -0.35 \boldcdot(79)\)
• \(79 \boldcdot (1 - 0.35)\)
• \(79\boldcdot (0.65)\)

In this activity, students continue to represent percent increase and decrease, but are specifically prompted to write expressions that only use multiplication. For example, to express the price (in dollars) of a $500 scooter after a 35% discount, students may write \(500 - (0.35) \boldcdot 500\), which involves subtraction. They can express the same quantity using only multiplication, by writing \(500 \boldcdot (0.65)\).

When a quantity changes, it is intuitive to look at the difference of successive values. Students are accustomed to this from their prior work with linear functions. For exponential functions, however, it is the quotient of successive values that is of interest: by what number do we multiply one value to get the next value (what is the common factor)? Describing change using only multiplication is useful in this sense: an expression that uses only multiplication can help us represent exponential situations concisely.

For the scooter, if the discounted price were again discounted by 35% this could be expressed as \(500 \boldcdot (0.65)^2\). While the scooter price is unlikely to be discounted by 35% indefinitely, the technique is important for representing exponential growth or decay that results from repeated percent increase or decrease.

Present the scooter sale example and ask students to write different expressions for the same scenario. Record their expressions for all to see. Highlight the expression that uses a single multiplication.

Arrange students in groups of 2. Ask them to think quietly about the questions before discussing their expressions with their partner.

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 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 \(150 \boldcdot (1.08)^2\) 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?” (\(150 \boldcdot (1.08)\boldcdot (1.08)\boldcdot (1.08)\) or \(150\boldcdot (1.08)^3\))
• “How can we express the population size if the population continues to grow by 8% for \(n\) years?” (\(150 \boldcdot (1.08)^n\))
• “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.)