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Unit 4, Lesson 4

Representing Functions at Rational Inputs

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Integrated Math 3

4.1

Warm-up

Standards Alignment

Building On

Instructional Routines

None

Materials

None

Activity Narrative

In this activity, students are introduced to the concept of half-life as the amount of time it takes for a material to decompose into half of its current amount. The context arises often when discussing exponential decay, so a good understanding of the topic is helpful to get a sense of the problems and solutions.

Launch

Display one problem at a time. Give students quiet think time for each problem, and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Activity

None

Many materials such as radioactive elements or large molecules naturally break down over time at a rate that is proportional to how much of the material there is. The amount of material left over time is described mathematically using exponential decay.

To get a sense of how stable the materials are, the length of time it takes for half of the material to remain, or its half-life, is given. Here are some half-lives of a few radioactive materials and graphs of how much of the materials remain after there is 1 gram of the material.

Uranium-235

Half-life: 704 million years

Cobalt-60

Half-life: 5.27 years

Bismuth-212

Half-life: 60.6 minutes

exponential decay graph containing the points 0 comma 1, 60 point 6 comma one half, and 121 point 2 comma one fourth

Which of these materials takes the longest to break down?
If you had a sample of 8 grams of cobalt-60, how long would it take for it to break down into only 1 gram left? Explain your reasoning.

Student Response

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Building on Student Thinking

Activity Synthesis

The purpose of the discussion is for students to get a basic understanding of half-life. Invite students to share their solutions for the questions, then ask:

“Looking at the half-life, what could be a problem with radioactive waste containing uranium-235?” (It will take more than 700 million years before it is reduced by half, so it will be around and radioactive for a long time.)
“Which is the more stable material, cobalt-60 or bismuth-212?” (Cobalt-60, because it takes longer to break down into half of the starting amount.)

4.2

Activity

Instructional Routines

None

Materials

None

Activity Narrative

In this activity, students practice constructing an exponential function and evaluating it at a fractional input value as they translate between decades and years. They also think about how to express a percent increase over a fractional interval of input—specifically, how to express the annual growth rate of a population given the growth rate for one decade. They analyze and articulate why the growth rate for one year is not one tenth of that of one decade (MP3).

Launch

Ask students what they know about the country of Nigeria. Tell students that Nigeria is one of the most populous countries in the world. It has grown at a fairly steady (and relatively high) rate since 1980.

Activity

None

4.3

Activity

Instructional Routines

MLR6: Three Reads Poll the Class

Materials

None

Activity Narrative

This task prompts students to carefully interpret a description and some expressions representing exponential decay. It further reinforces the distinction between how exponential and linear functions change over intervals of time, which was explored in the previous activity.

The decay factor is not given for a familiar unit of input (that is, the waste decays by half every 30 years instead of every year). To correctly represent the decay with functions, students need to notice that the does not correspond to the change in 1 year, and that the decay factor in 1 year is not of .

This activity uses the Three Reads math language routine to advance reading and representing as students make sense of what is happening in the text.

Lesson Synthesis

Tell students that sometimes we want to know how a quantity is changing for different intervals of input (or time, in this case), and those intervals might not be whole numbers. For example, a population of a town , in thousands, is modeled by where is the number of centuries since 1800.

Here are some questions for discussion:

“What is the growth rate or percent increase each century for this population?” (50%)
“There are 10 decades in a century and , so is 5% the growth rate each decade?” (No. The function is exponential, so the percent increase for one decade is not one-tenth of the increase for a century. With a 5% growth rate each decade, the growth factor would be 1.05 each decade, and that would make the growth factor each century about 1.63 (), not 1.5.)
“How do we find the growth factor for each decade? How do you know?” (When the growth factor per decade is raised to an exponent of 10, it must equal 1.5. This means the growth factor each decade is because .)
“What is an approximation of the growth rate or percent increase each decade?” (, so the growth rate is about 4% per decade.)

to access Cool-down.

Student Lesson Summary

Imagine a material has a half-life of 3 hours. This means that the amount of material left after 3 hours is half of what there was when it started. For example, if there are 200 milligrams of the material at noon, then at 3 o’clock there will be 100 milligrams left, and at 6 o’clock there will be 50 milligrams left.

If a scientist has 200 mg of the material, then the amount of material, in mg, can be modeled by the function . In this model, represents a unit of time. Notice that the 200 represents the initial amount of material the scientist has. The number indicates that for every 1 unit of time, the amount of material is cut in half. Because the half-life is 3 hours, this means that must measure time in groups of 3 hours.

But what if we wanted to find the amount of material the scientist has each hour after taking it? We know there are 3 equal groups of 1 hour in a 3-hour period. We also know that because the material decays exponentially, it decays by the same factor in each of those intervals. In other words, if is the decay factor for each hour, then , or . This means that over each hour, the medicine must decay by a factor of , which can also be written as . So if is time in hours since the scientist had 200 mg of the material, we can express the amount of material in mg, , the scientist has as , or .

Launch

Ask students to close their books or devices. Tell the class that they are going to do some calculations regarding how the amount of radioactive waste decays over time after it is produced. Before students read the task statement or do any calculations, ask, “If we start with 100 grams of radioactive waste, how much do you think is left after 90 years (if no additional waste is added)?” Poll the class for their estimates, and display the result for all to see. Students will consider this question again during the whole-class discussion.

Use Three Reads to support reading comprehension and sense-making about this problem. Display only the problem stem, without revealing the question.

For the first read, read the problem aloud, and then ask, “What is this situation about?” (nuclear waste) Listen for and clarify any questions about the context.
After the second read, ask students to list any quantities that can be counted or measured. (the half-life of the material and the amount of the material)
After the third read, reveal the first question and ask, “What are some ways we might get started on this?” Invite students to name some possible starting points, referring to quantities from the second read. (Determine the value of the first expression and compare it to the original amount, figure out how much time it would take to have that amount based on the half-life, then look for ways to figure this out without finding the value of the expression.)

Representation: Internalize Comprehension. Provide students with a graphic organizer, such as a three-column table to record relationships between half-life, years, and the expression connecting them.
Supports accessibility for: Visual-Spatial Processing, Organization

Activity

None

Cesium-137 is a radioactive material found in the waste of nuclear reactors. It has a half-life of about 30 years. Let’s suppose that there are 100 grams of cesium-137 as part of some nuclear waste.

Each of these expressions describes the amount of cesium-137 in the nuclear waste some number of years after it is produced. For each expression, find the number of years it would take to have that much cesium-137 left for this waste.
1. Write a function to represent the amount of cesium-137 left in the waste, years after it is produced.
2. The function represents the amount of cesium-137 left in the waste after 30-year periods. Write an equation to represent the function .
3. Explain why .

Student Response

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Building on Student Thinking

A key point in this activity is that groups of 30 years are a unit for the half-life of cesium-137. Identifying that is the number of grams of cesium-137 left after 1 year is the critical step.

If students do not yet view groups of 30 years as a unit for the half-life of cesium-137, consider saying:

“Tell me how you are reasoning about the expression .”
“How does the scale factor relate to how many years it takes for the amount of cesium-137 to be cut in half?”

Activity Synthesis

Begin the discussion by repeating the earlier question, “How much cesium-137 do you think is left in the waste 90 years after 100 grams is produced?” ” Invite students to share their reasoning, making sure they see that “90 years later” is 3 periods of 30 years, so the amount of cesium-137 will have been halved 3 times to leave 12.5 grams: .
Next, focus the discussion on the distinction between expressing the amount of cesium-137 as a function of time in groups of 30 years (function ) and time in groups of 1 year (function ). Discuss with students:

“What growth factor did you use to write the expression for function ? What is the approximate decay rate each year?” (Raise to an exponent of , which is approximately 0.977, so the growth factor is about 0.977 each year.)
“Why not divide by 30 (or multiply it by ) to get the decay factor for a year?” (Because when that factor is raised to an exponent of 30, it needs to equal , is not .)
“What is an equation that could represent the relationship between and as they are defined for functions and ?” ( because represents 30-year periods and represents years, so when .)
“How can we show that the expression for , which is is equivalent to the expression for , which is ?” (We know that . If we substitute in the expression for , we have , which equals .)
“What would change for the two functions if the waste contained a different material—one with a half-life of 5 years?” (Function could stay as the same equation, but our interpretation would be that a unit represents 5 years instead of 30. Function would change so that the denominator of the fraction in the exponent is 5 instead of 30.)

Time permitting, consider showing the two graphs representing the two functions.

<p>Line and two points on grid. Cesium-137, grams. Time, years.<br>
</p> — Function

<p>Line and two points on grid. Cesium-137, grams. Time, periods of 30 years.<br>
</p> — Function

Here are some possible questions for discussion about the graphs:

“How are these two graphs alike and different?” (They seem to be the same curve. They have the same vertical intercept. The scales for the horizontal axis are different. One curve goes through and the other goes through .)
“On the graph representing , what does the point at mean in the context? What is the corresponding point on the graph of ?” (The amount of cesium-137 in the waste after 1 period of 30 years is 50 grams. This is the same as the point on the graph of .)
“How could these two graphs represent the same decay, if the coordinates they go through are not the same?” (They use different units for the input. 1 unit of the input in is 30 units of the input in .)
“To find the amount of cesium-137 15 years after consumption, what input value should we use for each function?” ( for and 15 for ) “What about for 10 years after consumption?” ( for and 10 for )

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Lesson 4

Representing Functions at Rational Inputs

Warm-up

Standards Alignment

Building On

8.EE.A.1

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

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Instructional Routines

Materials

Activity Narrative

Launch

Activity

Activity

Instructional Routines

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Lesson Synthesis

Student Lesson Summary

Student Response

Building on Student Thinking

Activity Synthesis

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Student Response

Building on Student Thinking

Activity Synthesis

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