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Unit 5, Lesson 16

Using Graphs and Logarithms to Solve Problems (Part 2)

Algebra 2

16.1

Warm-up

Standards Alignment

Building On

Instructional Routines

None

Materials

None

Activity Narrative

In this activity, students interpret function notation in a financial context. Students are given a brief description of the situation, then asked to describe the meaning of inequalities and an equation.

Launch

None

Activity

None

A business owner opened two different types of investment accounts at the start of the year. The functions and represent the values of the two accounts as a function of the number of months after the accounts were opened.

Here are some true statements about the investment accounts. What does each statement mean?
If the two functions are graphed on the same coordinate plane, what might it look like? Sketch the two functions.

Student Response

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Building on Student Thinking

Activity Synthesis

Invite students to share their interpretations of the given statements. Make sure students understand that the 3, 6, and represent the number of months, and that at some number of months , the balances in the accounts are equal. Discuss questions such as:

“Which account do you think started out with more money?”
“Which account do you think has more money after 6 months?”
“What might be reasonable values for ? How do you know?” (Between 3 and 6. At 3 months, investment account had more money, but after 6 months it has less, so at some point between 3 and 6 months the balance of investment account exceeds that of account .)

Then, select some students to share their graphs and check that the graphs meet the criteria of the three given statements. Highlight that the graphs of the functions could be very different, depending on the assumptions we make about how the accounts are changing.

16.2

Activity

Instructional Routines

MLR5: Co-Craft Questions

Materials

To Gather

Scientific calculators

Activity Narrative

In this activity, students interpret the intersection of two graphs representing exponential equations in context. Then, given the output coordinate of the intersection, they explain why the input coordinate could be found by solving either equation separately, or by setting the equations equal to each other.

This activity uses the Co-Craft Questions math language routine to advance reading and writing as students make sense of a context and practice generating mathematical questions.

Launch

Provide access to scientific calculators. Arrange students in groups of 2. Introduce the context of population growth. Use Co-Craft Questions to orient students to the context and to elicit possible mathematical questions.

Display only the problem stem and related image, without revealing the question. Give students 1–2 minutes to write a list of mathematical questions that could be asked about the situation, before comparing questions with a partner.
Invite several partners to share one question with the class. Record the questions for all to see. Ask the class to make comparisons among the shared questions and their own. Ask, “What do these questions have in common? How are they different?” Listen for and amplify language related to the learning goal, such as “intersection,” “coordinates,” and “solution to the system of equations.”
Reveal the question “In this situation, what does the point of intersection of the two graphs tell us?” and give students 1–2 minutes to compare it to their own question and those of their classmates. Invite students to identify similarities and differences by asking:
- “Which of your questions is most similar to or different from the one provided? Why?”
- “How do your questions relate to finding the intersection of the graphs?”

Engagement: Develop Effort and Persistence. Connect a new concept to one with which students have experienced success. For example, remind students that this task contains skills they have already seen, but perhaps in a combination that students may not recognize right away.
Supports accessibility for: Social-Emotional Functioning, Conceptual Processing

16.3

Activity

Instructional Routines

MLR7: Compare and Connect

Materials

None

Activity Narrative

In this activity, students use what they have learned about exponential functions to solve problems in a real-world situation. They make a prediction about whether two populations that grow exponentially will be equal at some point. They also verify a claim that one country will reach a certain population by a certain time.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5). To use an equation to verify a certain claim about a situation, students need to reason both abstractly and concretely (MP2).

Monitor for students who use these strategies to find when the functions are equal and when one of the functions reaches the target output value.

Input different values for to find when the outputs are close.
Solve an equation, possibly by rewriting into a logarithmic form.
Graph the functions to find the intersection.

This activity uses the Compare and Connect math language routine to advance representing and conversing as students use mathematically precise language in discussion.

Lesson Synthesis

The goal of this discussion is to emphasize that graphs and equations can both be used to compare functions and find when they reach certain values or when they both have the same value.

Display this example: Suppose we have two functions modeling the growth of two types of cells as a function of time in days. The function modeling the number of Type A cells is defined by , and the function modeling the number of Type B cells is defined by .

Then discuss questions such as these:

“How can we tell which cell type is growing faster?” (From the equations, we can see that Type B has a greater growth factor. If we graph the functions, we can see that the graph of B shows a quicker increase per day.)
“If we graph the two functions, they intersect. What do the coordinates of the intersection point tell us?” (It tells us the time and number of cells when there are the same number of Type A and Type B cells.)
“How might we find out on which day there will be the same number of cells of each type?” (We can estimate from the graph, or we can solve the equation .)

If time allows, ask students to solve the equation to determine what day there will be the same number of cells. (about 19.2 days)

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Student Lesson Summary

Graphs representing functions can help us visualize how two or more quantities are changing in a situation. Let’s consider the populations of two colonies of ants.

The population, in thousands, of a colony of carpenter ants and a colony of red wood ants can be modeled with functions and , respectively. Here, is the time in months after the colonies were first studied.

From the equations, we can tell which colony had a greater initial population (carpenter ants, 8.1 thousand) and which had a greater growth factor (red wood ants, ). Will the colony of red wood ants eventually exceed that of the carpenter ants? If so, when might it happen? Graphs representing and can help us answer these questions.

<p>Coordinate plane, 2 exponential functions.</p>

The intersection of the graphs tells us that about 20 months after the study began, the two colonies have the same population, about 15 thousand. After that point, the population of red wood ants is greater than that of carpenter ants. To find out more exactly when the two colonies have the same population, we can use graphing technology to find better approximations for the coordinates of the intersection.

Another way to find the point of intersection is using the equations for the functions. At the point of intersection of the graphs, the two functions have the same -value, so we can write the equation . Then we can solve this equation:

This solution means that about 20.3 months after the study began, the two colonies have the same population. To find the population of the colonies at that time, we can use the original functions to find or .

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Lesson 16

Using Graphs and Logarithms to Solve Problems (Part 2)

Warm-up

Standards Alignment

Building On

HSA-REI.D.11

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Launch

Activity

Instructional Routines

Materials

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

HSF-IF.A.2

Addressing

Building Toward

Standards Alignment

Building On

Addressing

HSA-REI.D.11

HSF-LE.A.4

Building Toward

Standards Alignment

Building On

Addressing

HSF-LE.A.4

Building Toward