Unit 6, Lesson 18
Expressed in Different Ways
Algebra 1
18.1
Warm-up
Standards Alignment
Building On
8.EE.A.1
Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, .
Materials
NoneActivity 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
Supports accessibility for: Memory, Organization
Activity
NoneDecide 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
Advances: Speaking, Representing
18.2
Activity
Instructional Routines
MLR2: Collect and DisplayMaterials
NoneActivity 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).
Launch
Arrange students in groups of 2, and provide access to scientific calculators.
If time is limited, consider asking students to complete the first question together with their partner, divide up the work on the last two questions, and then discuss their responses.
18.3
Activity
Materials
NoneActivity 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.
Launch
Arrange students in groups of 2. Ask students to take turns: The first partner identifies a match and explains why they think it is a match, while the other listens and works to understand. Then they switch roles.
MLR8 Discussion Supports. Students should take turns finding a match and explaining their reasoning to their partner. Display the following sentence frame for all to see: “I noticed _____ , so I matched . . . .” Encourage students to respectfully challenge each other when they disagree.
Advances: Conversing
Advances: Conversing
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.)
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:
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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 .
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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: