Lesson 4

So far, students have seen patterns and relationships involving mostly whole numbers. This warm-up encourages students to observe linear and exponential patterns that involve fractions, preparing them for upcoming work.

Invite students to share their observations and questions. To encourage more students to participate, after each person shares, ask if others noticed or wondered about the same thing.

If not already mentioned by students, highlight how the values in the two tables change from row to row (Table A by equal differences and Table B by equal factors). If no students wonder about whether the values in the two tables will ever agree again, ask, “Will a linear growth and an exponential growth that start with the same value have the same value again at some later point?” Consider posting the question for students to think about throughout the unit.

Students have explored quantities that increase by the same factor (e.g., doubling or tripling). Up to this point, that factor has always been greater than 1. In this lesson, they will look at situations where quantities change exponentially, but by a positive factor that is less than 1. The factor is still called the growth factor (the growth factor can be less than 1) though students should learn that sometimes people refer to it as a decay factor.

Multiplying a quantity by a factor less than 1 means that the quantity is decreasing. Students may naturally think of a decrease as subtraction. For example, a quantity \(x\) that loses \(\frac13\) of its value may be represented as \(x - \frac{1}{3}x\). Because exponential relationships are characterized by multiplication by a factor, however, it will be helpful to view this instead as \(x \boldcdot \left(1- \frac{1}{3}\right)\), or as multiplying \(x\) by \(\frac{2}{3}\).

Ask students to close their books or devices. Pose this question:

Diego has $100 and spends \(\frac14\) of it. How much does he have left?

Give students a minute to think about it, and then ask a few students to share their answer and their reasoning.

Students may have trouble making the connection between the fraction that is being subtracted and the fraction that remains when the subtraction is rewritten as multiplication. Invite these students to think through the steps in Diego's calculation. What does the number 100 represent? What about the \(\frac{1}{3}\)? Once students identify the analogous quantities in the second and third questions, encourage them to revisit the subsequent steps of Diego's calculation.

Ask previously selected students to share their explanations of how a subtraction expression can be written as multiplication of two numbers. Make sure that students understand that a decreasing quantity can be expressed both ways, using subtraction or multiplication.

If time permits, consider asking “There were \(x\) dogs at the park and \(\frac{1}{5}\) of the dogs were on a leash. How many dogs were not on a leash?” The number of dogs not on a leash could be expressed using subtraction as \(x - \frac{1}{5}x\) or using multiplication as \(\frac{4}{5}x\).

Emphasize that it is important to be able to express this situation using multiplication because exponential situations involve quantities changing by a common factor (or being multiplied by a number).

In this activity, students examine a situation where a quantity decreases by repeated multiplication by the same factor. They represent the pattern of decrease with an equation and interpret the different parts of the equation in terms of the context (the depreciation of a car). The definition of growth factor is expanded to accommodate situations involving decrease.

Explain the scenario to the class, or just display the first sentence of the task: Every year after a new car is purchased, it loses \(\frac13\) of its value. Let’s say that a new car costs $18,000.

Ask the class to estimate: Without calculating, what do you think the value of the car will be 4 years after it was purchased as a new car? Poll the class for their estimates.

Clarify to students that they may complete the table with numerical expressions that show how to find the values rather than with the results of evaluating the expressions.

Make sure that students understand the expression for the value of cars after \(t\) years came from multiplying by \(\frac23\) repeatedly (\(t\) times). Tell students that we still say this quantity changes exponentially, though sometimes people use the more specific exponential growth and exponential decay to indicate whether the amount is increasing or decreasing. The multiplier is still called the growth factor, but when it is a positive number less than 1, the result decreases with every iteration. Sometimes people use decay factor to indicate that a quantity that decreases exponentially involves a positive factor less than 1.

Questions for discussion:

• “Why does it make sense to multiply an entry in one row by \(\frac{2}{3}\) to get the entry for the next row?” (Each year, only \(\frac23\) of the car value remains.)
• “Why not write the expression for each year as subtraction, for instance, \(18,\!000 - \frac13 \boldcdot 18,\!000\)?” (We could, but writing subtraction can get unwieldy quickly. For year 2, the expression would be \((18,\!000 - \frac13 \boldcdot 18,\!000) - \frac13 (18,\!000 - \frac13 \boldcdot 18,\!000)\). If we calculate the value of each term it would be simpler (\(12,\!000 - \frac13 \boldcdot 12,\!000\)), but it would still not allow us to see a pattern and write a general expression as easily.)
• “Does the \(\frac45\) in the last equation mean losing \(\frac45\) of the value each year? Why or why not?” (No. Multiplying by \(\frac 45\) repeatedly means each time only \(\frac45\) of the value remains, so the car is losing \(\frac15\) of its value each year.)
• “Will the value of the second car (starting at $10,000 and losing \(\frac{1}{5}\) of it each year) ever be worth more than the first car (starting at $18,000 and losing\(\frac{1}{3}\) of it each year)? Explain.” (Yes. After 4 years, the value of the second car will be $4,096 while the value of the first car will be $3,555.56)

Make sure that students understand the exponential expression for the value of cars after \(t\) years came from multiplying by \(\frac23\) repeatedly (\(t\) times).