Lesson 2

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.

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: 

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

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.

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.

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. 

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.)

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.

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.

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.

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.