Lesson | Loading...

Unit 4, Lesson 3

Understanding Rational Inputs

$Course Icon$

Integrated Math 3

3.1

Warm-up

Standards Alignment

Building On

Addressing

Instructional Routines

None

Materials

None

Activity Narrative

This Warm-up is a review of rational exponents, preparing students to explore exponential functions with rational input values, by focusing on two basic fractional exponents, and , and some of their properties.

Launch

None

Activity

None

Select all solutions to . Be prepared to explain your reasoning.
Select all solutions to . Be prepared to explain your reasoning.

Activity Synthesis

Briefly survey the class on whether each expression is a solution to the given equation. If there is a disagreement about whether an expression is a solution, ask students with contrasting views to share their reasoning so agreement can be reached. For an expression that everyone agrees is a solution, invite 2–3 students to share their reasoning.

Make sure students recall that a number raised to an exponent of , as in , is . Highlight that squaring each number, and , results in :

Likewise, a number raised to an exponent of , as in , is defined to be . Cubing each number, and , results in :

3.2

Activity

Instructional Routines

Graph It MLR5: Co-Craft Questions

Materials

To Gather

Graphing technology

Activity Narrative

In this activity, students construct an exponential function to model population growth. Then they interpret the function when evaluated at certain rational numbers. They do so quantitatively, in terms of the population, and abstractly, in terms of an expression with a fractional exponent (MP2).

As students work, identify those who express the population of a given year as an exponential expression and those who use a calculator to estimate a numerical value. Both approaches are important: the first, because it gives an opportunity to interpret a fractional power, and the second, because it gives a concrete sense of the population growth.

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.

3.3

Activity

Instructional Routines

None

Materials

None

Activity Narrative

In this activity, students are given a graph of an exponential relationship to produce an equation and to reason about the context. Students then express the change in the function in terms of its decay rate, recalling the connection between decay factor and decay rate.

Launch

Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Check in with students after 2–3 minutes of work time to provide feedback and encouragement. Encourage students who are stuck to look for ways in which the two points can provide information about the initial value and growth rate.
Supports accessibility for: Attention, Social-Emotional Functioning

Activity

None

Lesson Synthesis

The goal of this discussion is to summarize how fractional exponents and exponential functions come together in this lesson. Display the graph, and tell students it is the function , which models the population of a certain bacteria , in thousands, hours after being counted.

<p>Points P and Q and line on grid.<br>
</p>

Here are some questions for discussion:

“What are the coordinates of and what do they mean in this context?” ( or about . 15 minutes after the hour, the population is about 1,189.)
“What are the coordinates of and what do they mean in this context?” ( or about . 1 hour and 15 minutes after being counted, the population is about 2,378.)

If students notice that the value of the function at is twice the value at , invite students to reason about why that is true. (The interval between and is 1 hour, which means the outputs are related by a factor of 2.) Reasoning more deeply about why this is true is a focus of a future lesson, so it is okay to skip discussing this feature of exponential functions now.

Explain to students that it is useful to move between exact and approximate values for different purposes. Exact values such as are useful for revealing mathematical structure, but can be difficult to interpret in real situations. Approximate values such as 1.26 are useful for understanding and measuring in real situations, but can hide the mathematical structure and are limited to the precision chosen for rounding. When a value is approximate, it can be helpful to use notation such as to note that the given value is not exact.

to access Cool-down.

Student Lesson Summary

Some exponential functions can have inputs that are any numbers on the number line, not just integers.

Suppose the area of a pond covered by algae , in square meters, is modeled by , where is the number of weeks since a treatment was applied to the pond. How could we use this equation to determine the area covered after 1 day?

Well, since is one week and each week has 7 days, is 1 day. So after 1 day, the algae covers square meters, or about 181 square meters. Using a calculator, we know that the expression , which is equivalent to , is about 0.906. This means that after 1 day, only 91% of the algae from the previous day remains.

This information can also be seen on a graph representing the area. The point at marks the area covered by the algae after 1 week. Point marks the covered area after of a week, or one day.

<p>Coordinate plane, x, time, weeks since treatment, y, area, square meters. Curve through 0 comma 200, a point P at fraction 1 over 7 comma unknown y value and a point at 1 comma 100.</p>

The graph can be used to estimate the vertical coordinate of and shows that it is close to 180.

Launch

Arrange students in groups of 2. Introduce the context of population growth in a video game. Use Co-Craft Questions to orient students to the context and elicit possible mathematical questions.

Display only the problem stem, without revealing the instructions. 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 “predict,” “each year,” and “growth factor.”
Reveal the instructions, and give students 1–2 minutes to compare the instructions to their own questions and those of their classmates. Invite students to identify similarities and differences by asking:
- “Is there a main mathematical concept that is present in both your questions and the instructions provided? If so, describe it.”
- “How do your questions relate to finding the population in years that are not 10-year increments?”

Graphing technology is needed for every student.

Activity

None

In a video game, Jada is building a moon base to support a growing population and to deal with challenges. Jada’s base has a population of 54,500 in the year 2240, and between 2240 and 2270 the population of the base grows exponentially by about 60% per decade.

Find the population of Jada’s moon base in 2250 and in 2260 according to this model.
The population is a function of the number of decades after 2240. Write an equation for .
1. Explain what means in this situation.
2. Graph your function using graphing technology, and estimate the value of .
3. Explain why we can find the value of by multiplying 54,500 by . Find that value.
Based on the model, what was the population of Jada’s base in 2258? Show your reasoning.

Student Response

to access Student Response.

Building on Student Thinking

If students do not yet correctly graph the function, consider asking:

“How did you choose the horizontal and vertical dimensions for your graph?”
“How could adjusting the horizontal and vertical dimensions help you see how the population grows between 2240 and 2270?”

Activity Synthesis

Focus the discussion on the meaning of and in this context and on how to reason about these values. Display a table for all to see, like the one shown here, to fill in during the discussion:

time (decades since 2240)	population	approximate population
0	54,500	54,500
0.5
1		87,200
1.8
2		139,520
3		223,232

Select previously identified students to share their responses for the population of Jada's base with the inputs 0.5 and 1.8, written as expressions ( and ) and as numerical values (about 68,938 and 127,003). Help students see that:

We can think of the population in 2260, 2 decades after 2240, as because the 60% increase is applied twice. Although we can’t write the same kind of expanded expression for the population 0.5 decade after 2240, we can use properties of exponents to reason about its value.
Here are three different ways to write the growth factor for the population 5 years after 2240: , , or .

Conclude by asking students to write three different ways to express the growth factor for the population 18 years after 2240 (, , or ).

Lesson | Loading...

Settings

Audience

Assessment Access

Lesson 3

Understanding Rational Inputs

Warm-up

Standards Alignment

Building On

Addressing

HSN-RN.A.1

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Activity Synthesis

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Lesson Synthesis

Student Lesson Summary

Student Response

Building on Student Thinking

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Student Response

Building on Student Thinking

Activity Synthesis

Building Toward

Standards Alignment

Building On

HSF-LE.A.1.a

Addressing

HSF-LE.A.2

Building Toward

Standards Alignment

Building On

Addressing

HSF-LE.A.2

Building Toward