Lesson 2

Representations of Growth and Decay

2.1: One Fourth at a Time (5 minutes)

Warm-up

This warm-up prompts students to interpret the meaning of each part of an expression that represents a recursively defined geometric sequence. This prepares them for the more in-depth analysis of exponential change in the lesson.

Student Facing

Priya borrowed \$160 from her grandmother. Each month, she pays off one fourth of the remaining balance that she owes.

  1. What amount will Priya pay her grandmother in the third month?
  2. Discuss with a partner why the expression \(160 \boldcdot \left(\frac34\right)^3\) represents the balance Priya owes her grandmother at the end of the third month.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Make sure students can articulate the meaning of each term in the expression \(160 \boldcdot \left(\frac34\right)^3\):

  • 160 represents the initial amount, in dollars, that Priya owed her grandmother.
  • \(\frac{3}{4}\) is the amount of the previous balance that is still owed after Priya makes each monthly payment.
  • The exponent of 3 is the number of months Priya has made payments.

Display a table of how much Priya pays and owes each month for all to see, like the one shown here: 

month amount paid in dollars amount owed in dollars
0 0 160
1 40 (or \(\frac14 \boldcdot 160\)) 120  (or \(\frac34 \boldcdot 160\))
2 30 (or \(\frac14 \boldcdot 120\)) 90  (or \(\frac34 \boldcdot 120\))
3 22.50 (or \(\frac14 \boldcdot 90\)) 67.50  (or \(\frac34 \boldcdot 90\))

Emphasize that we can describe the quantity as changing “exponentially” because it is changing by the same factor each time.

2.2: Climbing Cost (15 minutes)

Optional activity

In this activity, students interpret a given exponential equation in context and use their interpretation to answer questions about the situation and to write a new expression. In doing so, they reason quantitatively and abstractly (MP2). They also recall the meaning of negative exponents, how to use function notation to describe a relationship, and how to evaluate a function at given values.

Launch

If students do not recall or are unclear about what “increasing by the same percentage” means, clarify that the college has applied the same percent increase in tuition every year since the year 2000. If students need help recalling how percent change works, pause the class after 2–3 minutes of work time and select students to share the meaning behind the 1 and 4 in the value 1.04.

Reading: MLR6 Three Reads. Use this routine to support reading comprehension, without solving, for students. Use the first read to orient students to the situation. Ask students to describe what the situation is about without using numbers (The tuition at a college is increasing by the same percentage each year). Use the second read to identify quantities and relationships. Ask students what can be counted or measured without focusing on the values. Listen for, and amplify, the important quantities that vary in relation to each other in this situation: cost of tuition, in dollars, and the number of years since 2012. After the third read, ask students to brainstorm possible strategies to answer the first question. This helps students connect the language of a word problem with the reasoning needed to solve it.
Design Principle: Support sense-making

Student Facing

The tuition at a college was \$30,000 in 2012, \$31,200 in 2013, and \$32,448 in 2014. The tuition has been increasing by the same percentage since the year 2000.

  1. The equation \(c(t)=30,\!000 \boldcdot (1.04)^t\) represents the cost of tuition, in dollars, as a function of \(t\), the number of years since 2012. Explain what the 30,000 and 1.04 tell us about this situation.
  2. What is the percent increase in tuition from year to year?
  3. What does \(c(3)\) mean in this situation? Find its value and show your reasoning.
    1. Write an expression to represent the cost of tuition in 2007.
    2. How much did tuition cost that year?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Jada thinks that the college tuition will increase by 40% each decade. Do you agree with Jada? Explain your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

This is the first example of many in which an expression, \(30,\!000 \boldcdot (1.04)^t\), only approximately models a value— tuition, in this case. In reality, the tuition will probably be rounded, for example, to the nearest 100 dollars and under no circumstances would it have fractions of a cent. Students may be confused at the complicated number when they use a calculator to find the tuition 5 years ago. Remind them that the model does not give an exact value and ask them what they think a good estimate would be. 

Activity Synthesis

Here are some possible questions for discussion:

  • “How can we tell what the percent of increase is?” (It can be interpreted directly from the expression \(30,\!000 \boldcdot (1.04)^t\). The 1.04 is the factor by which the tuition is growing.)
  • “If the equation weren’t given, can we still tell that the tuition is growing by 4% or by a factor of 1.04? How?” (Yes. We can find the quotient of the tuition costs in 2012 to 2013, or from 2013 to 2014. The changes from 30,000 to 31,200 in the first year and from 31,200 to 32,448 are each an increase of 4%.)
  • “Why does -5 make sense as the input needed to calculate the cost of tuition in 2012?” (Since the input of \(c\) is years since 2012, finding the tuition in 2007, which is 5 years before 2012, means using an input value of -5.)
  • “Are the tuition costs changing exponentially? How do we know?” (Yes. Each year it is increasing by the same percentage or by the same factor.)
Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of exponential functions. Display the function and annotate with the meanings and applications of each value and variable. For example, labeling 30,000 with “original” or “starting amount” and labeling the 1 with “the whole,” or 100%, and the 4 with, “plus 4 percent.” The entire value 1.04 can also be labeled, for example, with “4 percent increase” and “growth factor.”
Supports accessibility for: Conceptual processing; Language

2.3: Two Vans and Their Values (15 minutes)

Optional activity

The goal of this task is to review the connections between different ways of expressing a relationship: a description, a graph, and likely an equation. Students begin by analyzing different graphs, finding one that matches a given description, and explaining how they know the graph is a correct representation (MP3). They then use their analyses to evaluate the function at a different input value, compare it to another exponential function, and sketch a graph of the second function.

Monitor for students who:

  • Articulate clearly how the features of the graph enable them to choose the correct graph to represent the depreciation of the first van.
  • Write an expression to answer questions about the value of either van.

Launch

Clarify the meaning of depreciation for students who may not know what it means.

Arrange students in groups of 2. Give them a moment to think quietly about the first question, and then share their response with a partner. They should be ready to explain to each other how they know that certain graphs cannot represent the given function.

Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, invite students to highlight each important element in the problem statement in a different color (starting price, year, and percent depreciation). Invite students to use these same colors to identify corresponding elements on the graphs.
Supports accessibility for: Visual-spatial processing

Student Facing

A small business bought a van for $40,000 in 2008. The van depreciates by 15% every year after its purchase.

  1. Which graph correctly represents the value of the van as a function of years since its purchase? Be prepared to explain why each of the other graphs could not represent the function.

    Graph A

    Coordinate plane, x, years since purchase, y, value in thousands of dollars, curve through points 0 comma 40 thousand and 1 comma 46 thousand.

    Graph B

    Coordinate plane, x, years since purchase, y, value in thousands of dollars, curve through points 0 comma 40 thousand and 1 comma 34 thousand.

    Graph C

    Coordinate plane, x, years since purchase, y, value in thousands of dollars, curve through points 0 comma 40 thousand and 1 comma 20 thousand.

    Graph D

    Coordinate plane, x, years since purchase, y, value in thousands of dollars, curve through points 0 comma 30 thousand and 1 comma 25 thousand 500.
  2. Find the value of the van 8 years after its purchase. Show your reasoning.
  3. In the same year (2008), the business bought a second van that cost $10,000 less than the first van and depreciates at 10% per year. Would the second van be worth more or less than the first van 8 years after the purchase? Explain or show your reasoning.
  4. On the same coordinate plane as the graph you chose in the first question, sketch a graph that shows the value of the second van, in dollars, as a function of years since its purchase.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Students may not be sure how to start calculating the value of the van after 8 years. Ask these students to make a table showing years since 2008 and the value of the van each year.

Activity Synthesis

Invite previously identified students to explain how they chose the graph that represents the value of the first van. Make sure students see that:

  • The $40,000 price of the van appears as the vertical intercept of the graph.
  • The 15% rate of decay tells you that after 1 year, the van loses 15% of its value, retaining 85% of its original value. \((0.85) \boldcdot 40,\!000 = 34,\!000\), so the point \((1, 34,\!000)\) must be on the correct graph.

If students haven’t already shown (in their partner discussions) that they understood why graphs A, C, and D cannot be the right representations, clarify the reasons (as shown in the Student Response).

Conclude the discussion by selecting previously identified students to share how they used expressions to answer questions about the value of the vans over the years. If no students wrote expressions, invite them to do so now. Highlight that:

  • Calculating the decay factor means subtracting the depreciation from 100%. For example, the first van loses 15% of its value each year, which means each year, it is worth only 85% of the amount of the previous year.
  • To find the value after 8 years, we could write an expression (\(40,\!000 \boldcdot (0.85)^t\) for the first van and \(30,\!000 \boldcdot (0.9)^t\) for the second van, where \(t\) represents time in years since purchase) and calculate its value when the input is 8.
Representing, Conversing: MLR7 Compare and Connect. Use this routine to prepare students for the whole class discussion. At the appropriate time, invite students to create a visual display that shows how they determined the values of each van 8 years after purchase. Groups should consider how to display their calculations so that another student can interpret them. Some groups may wish to add notes or details to their displays to help communicate their thinking. Give students 3–4 minutes to quietly circulate and observe at least two other visual displays in the room. Invite groups to discuss what they noticed about what is the same and what is different between their display and the others they observed. Listen for and amplify the language students use to interpret the exponential expressions that represent the situation.
Design Principle(s): Cultivate conversation; Support sense-making

Lesson Synthesis

Lesson Synthesis

The goal of this discussion is to emphasize that exponential functions change by equal factors over equal intervals, and that the equal factor can be observed in all representations of a function. Take this situation, for example: “A car costs $20,000 and loses 20% of its value every year.” Discuss how the equal factors show up in the description, a table, a graph, and an equation representing the situation. For example:

  • In the description, losing 20% of its value every year means that every year the value of the car is multiplied by the same factor, 0.8.
  • In a table (and a graph) showing the car values from year to year, each entry is 0.8 of the previous entry. (Consider drawing arrows from one row to the next, showing the multiplication by 0.8.)

    years since purchase value in dollars
    0 20,000
    1 16,000
    2 12,800
    3 10,340
  • The function \(f(t) = 20,\!000 \left(\frac{4}{5}\right)^t\) represents the value of the car \(f(t)\), in dollars, after \(t\) years. The initial value of the car is \$20,000 and the common factor by which the value changes each year is \(\frac{4}{5}\).

2.4: Cool-down - Flu Outbreak (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

There are lots of ways to represent an exponential function. Suppose the population of a city was 20,000 in 1990 and that it increased by 10% each year.

We can represent this situation with a table of values and show, for instance, that the population increased by a factor of 1.1 each year.

year population
1990 20,000
1991 22,000
1992 24,200
1993 26,620

We can also use a graph to show how the population was changing.​​​​​ While the graph looks almost linear, it has a slight upward curve since the population is increasing by a factor of 1.1 and not a constant value each year.

Coordinate plane, x, time, years since 1990, y, population, thousands. Curve through points 0 comma 20, 1 comma 22, 2 comma 24 point 2.

An equation is another useful representation. In this case, if \(t\) is the number of years since 1990, then the population is a function \(f\) of \(t\) where \(f(t) = 20,\!000 \boldcdot (1.1)^t\). Here we can see the 20,000 in the expression represents the population in 1990, while 1.1 represents the growth factor due to the 10% annual increase each year. We can even use the equation to calculate the population predicted by the model in 1985. Since 1985 is 5 years before 1990, we use an input of -5 to get \(f(\text-5)=20,\!000 \boldcdot (1.1)^{\text-5} \), which is about 12,418 people.

Throughout this unit, we will examine many exponential functions. All four representations—descriptions, tables, graphs, and equations—will be useful for determining different information about the function and the situation the function models.