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Unit 6, Lesson 18

Expressed in Different Ways

Algebra 1

18.1

Warm-up

5 mins

Math Talk: Equal Expressions

Standards Alignment

Building On

Instructional Routines

Math Talk MLR8: Discussion Supports

Materials

None

Activity Narrative

This Math Talk focuses on using exponent rules. It encourages students to think about the equivalence of values with exponents and to rely on rules of exponents to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students adjust growth rates to different units of time.

To compare the values with exponents, students need to look for and make use of structure (MP7).

In describing their strategies, students need to be precise in their word choice and use of language (MP6).

Launch

Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:

Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
Invite students to share their strategies, and record and display their responses for all to see.
Use the questions in the activity synthesis to involve more students in the conversation before moving to the next problem.

Keep all previous problems and work displayed throughout the talk.

Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization

Activity

None

Student Task Statement

Decide if each expression is equal to .

Activity Synthesis

To involve more students in the conversation, consider asking:

“Who can restate ’s reasoning in a different way?”
“Did anyone use the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to ’s strategy?”
“Do you agree or disagree? Why?”
“What connections to previous problems do you see?”

Pay particular attention to the last expression, . In order to identify this as equal to students need to work backward and write this as . Another approach would be to work forward and rewrite as The third expression is intended to facilitate this thinking. All of these problems rely on an important property of exponents, .

MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because . . . .” or “I noticed _____, so I . . . .” Some students may benefit from the opportunity to rehearse, with a partner what they will say before they share with the whole class.
Advances: Speaking, Representing

18.2

Activity

15 mins

Population Projections

Instructional Routines

MLR2: Collect and Display

Materials

None

Activity Narrative

This activity allows students to practice interpreting and using exponential expressions to solve problems, as well as writing equivalent expressions that allow new insights into the situation. It examines the percent change of the United States population from its beginning through the Civil War. During this time frame, the population happened to be modeled remarkably well by an exponential function.

Students are given a population model in which time is measured in years. They then write different models in which time is measured in decades or centuries. While the century model is primarily of theoretical interest, the decade model is very appropriate for showing the overall growth rate of the United States during this period, reinforcing the compounding effects of exponential growth.

In the Activity Synthesis, the term growth rate is defined.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

18.3

Activity

15 mins

Interest Calculations

Instructional Routines

MLR8: Discussion Supports Take Turns

Materials

None

Activity Narrative

In this partner activity, students take turns identifying and interpreting expressions that represent different compound-interest situations. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and to critique the reasoning of others (MP3).

The expressions are difficult to calculate, so students will need to rely on structure (MP7). From this point of view, the key part of the expression is what is on the inside of the innermost parentheses. The decimal part of this (0.07) represents the annual interest rate while the denominator (2 and 12) represents how frequently the interest is compounded.

Look for students who focus on the structure of the expressions and what each part of the expression means in context. Invite these students to share during the discussion.

Lesson Synthesis

Today we looked at how exponential functions can be written in different ways in order to be more meaningful. Invite students to articulate how this is so, using an example such as the following.

After taking a dose of 0.2 mg, the amount of medicine, , in a person's body, in mg, can be modeled by three different equations based on the units of time used to measure the change. Let , , and represent the number of hours, days, and weeks, respectively, since taking the medicine. , ,

"What does each equation tell us? What does the 0.996 represent? What do the 24 and 168 represent?" (The equations tell us how much medicine is still in the person’s body at different units of time. The 0.996 represents the growth rate of the function and is related to the percentage that is still left in the body after each hour. The 24 represents that there are 24 hours in a day and the 168 represents that there are 168 hours in a week.)
"Which of the equations, if any, gives us a clear picture of how the body breaks down the medicine? Why is that?" (The equation using weeks as the unit of time may make the most sense because it takes about a week to get to half of the original amount, so this time scale is the easiest to think about.)
" and. What do 0.91 and 0.51 tell us? Do they give us a clearer idea of how the medicine breaks down? How?" (0.91 is the growth factor for the amount of medicine in the body each day. 0.51 is the growth factor for the amount of medicine in the body after a week. It means that each day about 91% of the medicine from the previous day is still in the body and that each week about 51% of it is still in the body from the previous week. It is easier for me to understand 51% as about half each week than it is for me to think about 91% each day or 99.6% each hour.)

to access Cool-down.

Student Lesson Summary

Expressions can be written in different ways to highlight different aspects of a situation or to help us better understand what is happening. A growth rate tells us the percent change. As always, in percent change situations, it is important to know if the change is an increase or decrease. For example:

A population is increasing by 20% each year. The growth rate is 20%, so after one year, 0.2 times the population at the beginning of that year is being added. If the initial population is , the new population is , which equals , or .
A population is decreasing by 20% each year. The growth rate is -20%, so after one year, 0.2 times the population at the beginning of that year is being lost. If the initial population is , the new population is , which equals , or .

Suppose the area, , covered by a forest is currently 50 square miles, and it is growing by 0.2% each year. If represents time, from now, in years, we can express the area of the forest as:

In this situation, the growth rate is 0.002, and the growth factor is 1.002. Because 0.002 is such a small number, however, it may be difficult to tell from this function how quickly the forest is growing. We may find it more meaningful to measure the growth every decade or every century. There are 10 years in a decade, so to find the growth rate in decades, we can use the expression , which is approximately 1.02. This means a growth rate of about 2% per decade. Using for time, in decades, the area of the forest can be expressed as:

If we measure time in centuries, the growth rate is about 22% per century because . Using to measure time, in centuries, our equation for area becomes:

Activity Synthesis

Focus the discussion on how students found the percent increase per decade and per century. Ask students:

"By what percentage did the U.S. population increase per decade?" (, so the population grew by around 36% each decade.)
"By what percentage did it increase per century?" (The growth factor, , is about 21.18. Subtracting 1 for the original 100% of the population gives 20.18. This means it grew by 2,018%.)

Explain that in situations characterized by exponential change by a percentage, people sometimes refer to the percent change as a growth rate. (In previous lessons, this idea was called interest rate in the specific context of interest on savings or debt, but growth rate is a more general term.)

Make sure that students understand how growth rate is related to growth factor. For example, in the equation , the growth factor is 1.031, and the growth rate is 0.031, which can also be expressed as 3.1%.

To connect the key ideas in this lesson and the past few lessons, discuss questions such as:

"Is the growth rate per decade ten times the growth rate per year? Why or why not?" (No. The growth is compounded over time. Between 1790 and 1791, the population grows by 3.1% of 4,000 so, in thousands, that's 12.4, giving 4,012.4. Between 1791 and 1792, the population in thousands will grow by 3.1% of 4,012.4. So for the second year, the growth is 3.1% of 4,000 plus 3.1% of 12.4 from the first year. And so on. So the growth rate per decade is more than .)
"Which growth rate is easiest or most helpful for understanding the population change in that period? Why might that be?" (Perhaps the easiest to grasp is the growth rate per decade, about a third per decade. The yearly growth is relatively small, so it is hard to appreciate. The growth per century is very large.)
"If we use the model by year to find the population in 2017, what would it be?" (About 4 billion people. In thousands, it is . It is different from the actual population in 2017 by more than a factor of 10)
"What might be a reason that the population calculated using the model is so far off from the actual number?" (A variety of reasons, but one factor is that limits in land and resources made it unlikely for the population to continue to grow at the same rate over time.)

MLR2 Collect and Display. Collect the language that students use to describe percent increase. Display words and phrases such as “growth rate,” “interest rate,” and “growth factor.” During the Activity Synthesis, invite students to suggest ways to update the display: “What are some other words or phrases we should include?” Invite students to borrow language from the display as needed.
Advances: Conversing, Reading

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 growth rate. Terms may include "growth factor," "percentage," "increase," and "decrease."
Supports accessibility for: Conceptual Processing, Language

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Lesson 18

Expressed in Different Ways

Warm-up

Math Talk: Equal Expressions

Standards Alignment

Building On

8.EE.A.1

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Task Statement

Activity Synthesis

Activity

Population Projections

Instructional Routines

Materials

Activity Narrative

Activity

Interest Calculations

Instructional Routines

Materials

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Student Response

Building on Student Thinking

Are You Ready for More?

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Addressing

Building Toward

HSF-IF.C.8

Standards Alignment

Building On

Addressing

HSA-SSE.B.3.c

HSF-IF.A.2

HSF-IF.C.8

HSF-IF.C.8.b

Building Toward

Standards Alignment

Building On

Addressing

HSA-SSE.A.1.b

HSA-SSE.B.3.c

HSF-IF.C.8.b

Building Toward