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Unit 6, Lesson 7

Using Negative Exponents

Algebra 1

7.1

Warm-up

5 mins

Exponent Rules

Standards Alignment

Building On

Instructional Routines

None

Materials

None

Activity Narrative

This Warm-up prepares students to work with expressions involving negative exponents. Students have studied exponents and their properties in grade 8 and encountered negative exponents at that time. The goal here is to review the fact that for integer exponents and and a non-zero base , this property holds: .

Launch

None

Activity

None

Student Task Statement

How would you rewrite each of the following as an equivalent expression with a single exponent?

Activity Synthesis

Review, if needed, the conventions that and , emphasizing that these conventions guarantee that for any two integers and .

7.2

Activity

15 mins

Coral in the Sea

Instructional Routines

5 Practices Extend It

Materials

None

Activity Narrative

Up to this point in the unit, students have written and interpreted equations representing exponential change that were meaningful for non-negative input values. In this lesson, they interpret the meaning of a negative value in a context. The independent variable is time, , so negative values of refer to times before is 0.

For the last question, to find the time when the coral had the given volume, students may:

Calculate the values of when is -3, -4, -5, and so on or extend the table of values, if they previously created one, until they reach 37.5 or a value close to it.
Repeatedly divide by 2 (or multiply by ) the value of when is -2 until it approaches or hits 37.5.
Use the equation to find the value of and then reason about from there.

Have students present in this order to support moving their understanding from concrete to abstract.

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

7.3

Activity

Optional

15 mins

Windows of Graphs

Instructional Routines

Graph It

Materials

To Gather

Graphing technology

Activity Narrative

The goal of this activity is to address a skill important to successfully using graphing technology: choosing an appropriate graphing window. Students are given three graphs representing a relationship from an earlier task. They comment on their effectiveness in representing the relationship and consider ways to adjust the graphing window to improve the information that the graph shows.

Choosing an appropriate window for a graph that represents a situation characterized by exponential change can be challenging. Because exponential functions eventually grow very quickly, if the window is too small or too large, then the function may not be visible or may not show features that are interesting. Students could use an interactive graphing tool to experiment and to decide on an appropriate graphing window. It is also helpful, however, to consider the context, the initial amount, and the growth factor.

7.4

Activity

15 mins

Ghost Town Population

Instructional Routines

Graph It

Materials

None

Activity Narrative

In this activity, students use a description and table of values to write an equation that represents the situation. Then, they interpret the equation for negative values of the exponent and produce a graph. They also interpret the numbers in the equation in terms of the context.

For the final question, only an estimate is possible. Expect different answers that range between 1897 and 1898, as a more precise answer cannot be given right now. Look for students who use the graph and try to extend it to values in between integer years. Invite them to share during the discussion.

Throughout this activity, students are building skills that will help them in mathematical modeling (MP4). They don't decide which model to use, but they make connections between representations In the Activity Synthesis, they consider what input values for the model do and do not make sense for their equation.

Launch

Launch this activity by asking students what they know about ghost towns or if they have ever visited one. Invite 3–5 students to share what they know. Make sure that students understand that ghost towns were once flourishing towns that are now abandoned. Ghost towns can happen due to natural or human-caused disasters. For example, one might happen if a necessary resource, like water, is exhausted, or for economic reasons, like factories shutting down. As a connection to local history, if there are any ghost towns nearby, consider sharing the reasons that the town was abandoned.

If students continue to use graphing technology while working on this task, they are likely to enter their equation and see a smooth curve. It is not the intention that they try to sketch a curve at this time. Draw their attention to the instructions that say to plot the points.

Action and Expression: Provide Access for Physical Action. Provide access to tools and assistive technologies such as a graphing calculator or graphing software.
Supports accessibility for: Visual-Spatial Processing, Conceptual Processing, Organization

Lesson Synthesis

We looked at some situations where negative values representing times are meaningful. Review how we can use equations to understand a quantity before and after a certain time.

Suppose represents the population, , of a colony of bacteria, days since Monday. Ask students to respond to these prompts quietly.

Think of a possible value for . What does it represent in this situation?
For your value of , determine the corresponding value of . What does it represent in this situation?
Describe in words what the model says about this bacteria colony.

Invite students who chose different values to share their responses. Be sure to select at least one student who chose a negative value for . For the last prompt, highlight relevant vocabulary like "initial amount" and "growth factor." If time permits, draw a set of axes for all to see, and plot points corresponding to the values shared by students.

to access Cool-down.

Student Lesson Summary

Equations are useful not only for representing relationships that change exponentially, but also for answering questions about these situations.

Suppose a bacteria population of 1,000,000 has been increasing by a factor of 2 every hour. What was the size of the population 5 hours ago? How many hours ago was the population less than 1,000?

We could go backward and calculate the population of bacteria 1 hour ago, 2 hours ago, and so on. For example, if the population doubled each hour and was 1,000,000 when first observed, an hour before then it must have been 500,000, and two hours before then it must have been 250,000, and so on.

Another way to reason through these questions is by representing the situation with an equation. If measures time in hours since the population was 1,000,000, then the bacteria population can be described by the equation:

The population is 1,000,000 when is 0, so 5 hours earlier, would be -5 and here is a way to calculate the population:

Likewise, substituting -10 for gives us (or ), which is a little less than 1,000. This means that 10 hours before the initial measurement the bacteria population was less than 1,000.

Launch

After analyzing three attempts at graphing, students are instructed to use graphing technology to create a version that is better. Provide access to graphing technology (Desmos is available under Math Tools). Students may need instructions or a refresher on how to change the graphing window in the technology they are using before they can be successful with this task.

Engagement: Internalize Self Regulation. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. Invite students to choose and respond to one of the graphs and then to report back to the whole class.
Supports accessibility for: Organization, Attention

Activity

None

Student Task Statement

The volume, , of coral in cubic centimeters is modeled by the equation where is the number of years since the coral was measured. Three students used graphing technology to graph the equation that represents the volume of coral as a function of time.

<p>Two points on coordinate plane, origin O.</p> — A

For each graph:

Describe how well each graphing window does, or does not, show the behavior of the function.
For each graphing window that you think does not show the behavior of the function well, describe how you would change it.
Make the change(s) you suggested, and sketch the revised graph using graphing technology.

Student Response

to access Student Response.

Building on Student Thinking

Some students might not know how to begin to gauge the fitness of a graphing window. Encourage them to consider whether the quantity of interest is increasing over time or decreasing over time, and how this information might be conveyed by the graph. For example, this can help students decide whether the vertical intercept should be near the top or bottom of the graph.

Students might also make a table of values to find the -values when is 1, 2, and 3. Prompt them to discuss whether those points show up on each of the three graphs and whether those points are needed to show how the quantities are changing—and, if so, how to adjust the graphing window to show the points.

Are You Ready for More?

Student Task Statement

A town’s population decreased exponentially from the late 1800’s until the mid 1900’s, when the last residents left the town, leaving it a ghost town.

1. The population of the town, , for a number of years since 1900, , is shown in the table. What is the growth factor? That is, what is in an equation of the form ? What is ?
2. Find the population of the town when is -1 and -3. Record them in the table.
, years since 1900 , population

0 1,500

1 1,350

2 1,215
What do , , and mean in this situation?
The town’s population was at its greatest in 1895. What was that population? Explain your reasoning.
Plot the points whose coordinates are shown in the table.
Based on your graph:
1. When did the town have about 2,000 people?
2. What was the population in 1905?

, years since 1900	, population


0	1,500
1	1,350
2	1,215

Activity Synthesis

The goal of this discussion is to highlight some aspects of mathematical modeling. In particular, considering the reasonableness of a model and choices about how to visualize a model by graphing points or a line.

To demonstrate how the model is less reliable when using inputs that are far from the data, have half the class calculate and interpret and the other half calculate and interpret . They will find that in 1970 the population is 0.94 (or 1 if they round to the nearest person) and in 1830 the population is 2,393,953. Neither of these are reasonable.

Here are some questions for discussion:

“Explain if it would be reasonable to use this model to:
- “Estimate the town’s population in 1910.” (Yes, that year is close to the data given in the table and the Task Statement said that the pattern continues until the mid-1950’s, so it would be reasonable to use this model to estimate the population in 1910.)
- “Estimate the town’s population in 1890.” (No, the town’s population was at its greatest in 1895, but using an exponential model would put the population in 1890 as greater than the population in 1895.)
- “Estimate when the town’s population will be 0.” (No, that year will be far from the data given in the table, and models are less reliable farther from the known data.)
“When calculating the town’s population in 1895, your answer was a decimal. Why was it reasonable to round the answer?” (Population is a measure of people, so it can only be a whole number.)

Conclude with a discussion about input values between the integers, which later lessons will expand on. Ask students to recall their answer to the value of when the town had 2000 people and if they think non-integer inputs make sense in this context. (Yes, because numbers between the integers are different times in the year.).

If students still have access to graphing technology for this activity, tell them to enter their equation and see the curve for themselves or display these graphs for all to see.

A graph showing $p$ over $t$ with the graph of points from $p=1500\boldcdot(\frac{9}{10})^t$ every $\frac{1}{2}$ year.

A graph showing $p$ over $t$ with the graph of $p=1500\boldcdot(\frac{9}{10})^t$.

Explain that if the population is plotted at time intervals smaller than 1 year, the graph may look like the first graph. Another option is to graph a relationship where a quantity changes continuously using a curve. Students will focus on non-integer domain values in upcoming lessons, so there is no need to discuss the meaning of non-integer inputs in depth at this time.

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Lesson 7

Using Negative Exponents

Warm-up

Exponent Rules

Standards Alignment

Building On

8.EE.A.1

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Task Statement

Activity Synthesis

Activity

Coral in the Sea

Instructional Routines

Materials

Activity Narrative

Activity

Windows of Graphs

Instructional Routines

Materials

To Gather

Activity Narrative

Activity

Ghost Town Population

Instructional Routines

Materials

Activity Narrative

Launch

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

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Addressing

Building Toward

Standards Alignment

Building On

Addressing

HSA-CED.A.2

Building Toward

HSF-IF.B.4

Standards Alignment

Building On

Addressing

HSN-Q.A.1

Building Toward

Standards Alignment

Building On

HSA-SSE.A.1

HSF-IF.B.4

Addressing

Building Toward

HSF-IF.C.7.e