Lesson Exponential Decay - Core-Plus Mathematics Project · Exponential Decay This lesson engages ... we have chosen a simple example—the height of a bouncing ball after each

T 4 3 9 U N I T 6 • E X P O N E N T I A L M O D E L S

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Exponential DecayThis lesson engages students in analysis of situations

where some quantity of interest decreases by some constant factor as each unit oftime passes. Since the factor change for a function that decreases is a fractionbetween 0 and 1, exponential decay is likely to be a bit more challenging for stu-dents than the previous lesson on exponential growth.

The experiment at the beginning of the lesson is designed to illustrate visuallyone of the common settings for exponential decay—removal or diffusion of somepollutant. You could derive a mathematical model for the pollution remaining aftern days, or the change from one day to the next, but since there is an element of ran-domness involved in this process, it is probably better to begin looking for an alge-braic rule with the other, more well-behaved, examples that follow.

If you actually do this experiment you will get better results using large numbersof beans of two colors, removing two small cups at a time. Students do not actual-ly have to count the beans which are removed. It is enough for students to under-stand that, if there are two cups of brown beans (the pollution) mixed with eightcups of white, then the two cups removed will subtract some but not all of the orig-inal brown beans. As you add two more small cups of white beans to compensatefor the runoff, students can see that the proportion of brown beans looks less, sofewer will be removed with the next runoff. (If you actually count the brown beansremoved each time, be sure to count the initial number first. In this way, you cantrack the pollution remaining in the container. These data are good to come back toafter students know how to use exponential regression on the calculator.)

To begin developing the arithmetic and algebraic rules that match exponentialdecay, we have chosen a simple example—the height of a bouncing ball after eachrebound. The idea is that if a ball always rebounds to some fraction r (0 � r � 1) ofits drop height, then the height of each succeeding bounce will be related to theprior bounce height by the equation NEXT � r � NOW. Repeated application ofthis relation leads to the exponential pattern that on bounce number n the ball willrebound to rn of its original height. Students might need some help to find this pat-tern, and especially to find a way to express it with algebraic rules.

The example of drug decay illustrates a significant setting for exponential decay.Although we provide students with the rule in this case, it is important for them tosee why the shape of that rule implies decay by repeated multiplication with thefactor 0.95.

By the end of this lesson, students should realize that there are decay analogs forthe repeated multiplication models of Lesson 1 and that the rules take the same gen-eral form (with rate factors between zero and one). They also should have a sense ofthe shape of graphs that arise from rules of the form y � a (bx) when 0 � b � 1.

Lesson Objectives■ To determine and explore the exponential decay model, y � a(bx), where

0 � b � 1, through tables, graphs, and algebraic rules■ To compare exponential decay models with exponential growth models■ To compare exponential models of the form y � a(bx), where 0 � b � 1, to

linear models

Lesson2LESSON OVERVIEW

L E S S O N 2 • E X P O N E N T I A L D E C AY T 4 4 0

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See Teaching Master 166.The straight line shows a constant decrease in the pollutant remaining as the timeincreases. The curve shows large decreases initially, but the rate of decrease slows astime increases.The curve is the pattern that would be expected. Repeat the experiment until the classsees that the curve is the correct graph.The models for these graphs are y � �4x � 20 and y � 20(0.8)x. Students may nothave a good equation to model the cleanup situation, but you should leave the responseopen for now. You may wish to come back to it later in the unit. Students should real-ize that the graph should not be linear.

More Bounce to the OunceMany students are uncomfortable with fractions. As you circulate you may need tohelp them get started. 27 � 2

3� � 18, 18 � �

23

� � . . . . Some will want to use their cal-culators to enter 27 � 0.66. Discourage this by pointing out that the factor 0.66 isrounded off. They could enter on their calculators 27 � 2 � 3 � 18, or 18 � 2 � 3 � . . . , but recording answers in fraction form makes it a lot easier to seethe familiar powers of two in some numerators and powers of three in the denomi-nator. Of course, starting with 15 or some other whole number will result in thesame pattern, but there may not be any powers of 2 readily apparent in the numer-ator. For some students, it may be appropriate to ask if we always will see the pow-ers of 3 in the denominator.

INVESTIGATION

c

b

a

The graphs below show two possible outcomes of thepollution and cleanup simulation.

What pattern of change is shown by each graph?

Which graph shows the pattern of change that youwould expect for this situation? Test your idea byrunning the experiment several times and plotting the (time, pollutant remaining) data.

What sort of equation relating pollution P and time twould you expect to match your plot of data? Test youridea using a graphing calculator or computer.

20

16

12

8

4

00 3 6 9 12 15

Po

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t R

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ain

ing

Days Since Spill

Use with page 440. U N I T 6 • E X P O N E N T I A L M O D E L S

Transparency MasterMASTER

166

Think About This Situation

Master 166LAUNCH full-class discussion

EXPLORE small-group investigation


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1. Bounce Number 0 1 2 3 4 5 6 7 8 9 10

Rebound Height 27 18 12 8 �136� �

392� �

62

47� �

18218

� �22

54

63

� �57

12

29

� �21,,108274

�

a. It decreases by one-third on each bounce. The scatterplot is decreasing, but by asmaller amount each time. Students may not see the �

13

�. Adding a row “Change inRebound Height” to the table may help.

b. NEXT � �23

� (NOW), starting at 27.

c. y � 27(�23

�)x

d. The table values will be smaller; the plot will have the same shape but will not be ashigh, and the coefficient in the equation will be 15 instead of 27.

2. a–c. Responses will vary but should be close to a 0.67 rebound factor.

NOTE: You may wish to use the Texas Instruments CBL or a similar piece of technologyto measure the height and graph the relationship. Give the students plenty of time to prac-tice how to measure the height with the technology.

3. A sample response using a tennis ball follows.

a. Bounce Number 0 1 2 3

Rebound Height 50 30 18 10

b. NEXT � NOW � 0.6, starting at 50.y � 50(0.6)x

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When you are listening to students share their answers for the Checkpoint, youmay want to remind them that the rumor grew much faster when the tree grew bya factor of 3 at each stage than when the factor was 2. Ask which will make thebounce heights decline faster, a bounce factor of �

23

� or a bounce factor of �25

� (as in theupcoming “On Your Own”). Students will have to think about the results of multi-plying by �

23

� or �25

�: Which gives the smaller result? How can this be seen on the graph?

See Teaching Master 167.

■ The rebound heights are always less by some constant factor.■ The change is shown by a gradual decline in the height of the data points.■ The equations relating NOW and NEXT all have the same form: NEXT � b · NOW,

starting at a. They are different because a and b may be different. This differencesare probably due to the different materials from which the balls are made.

■ All of the equations have the form y � a(bx), where a is equal to the original heightof the ball, and the exponent (x) represents the number of bounces the ball has taken.The equations have different coefficients (a) because of the different initial heights.They also have different rates of rebounding (b) due to the type of ball.

The form is the same: NEXT � b · NOW starting at a, or y � a(bx). The patterns in thisinvestigation are different only because b is a rational number less than 1 rather than aninteger. When b � 1, the curve is decreasing. The examples from Lesson 1 were allincreasing curves.

a. Bounce Number 0 1 2 3 4 5

Rebound Height 25 10 4 �85

� �12

65� �

13225

�

b. NEXT � �25

�(NOW ), starting at 25.

y � 25(�25

�) x

c

b

a


167


Different groups might have used different balls and dropped the balls fromdifferent initial heights. However, the patterns of (bounce number, reboundheight) data should have some similar features.

Look back at the data from your two experiments.

■ How do the rebound heights change from one bounce to the next ineach case?

■ How is the pattern of change in rebound height shown by the shape of the data plots in each case?

List the equations relating NOW and NEXT and the rules (y = …) youfound for predicting the rebound heights of each ball on successivebounces.

■ What do the equations relating NOW and NEXT bounce heights havein common in each case? How, if at all, are those equations differentand what might be causing the differences?

■ What do the rules beginning “y = …” have in common in each case?How, if at all, are those equations different and what might be causingthe differences?

What do the tables, graphs, and equations in these examples have incommon with those of the exponential growth examples in the beginningof this unit? How, if at all, are they different?

Be prepared to share and compare your data, models, andideas with the rest of the class.

Checkpoint

Master 167SHARE AND SUMMARIZE full-class discussion

APPLY individual task

n Your Own

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c. ■ The rebound height of the softball is consistently less than what would be expectedfrom a new softball. Furthermore, the ratio of the rebound height from one drop to thenext is not consistent. So the data may indicate that this softball is not top quality.

■ Responses may vary, depending on what height ratio the student chooses. A sampletable of rebound heights for the ball dropped from 20 feet is provided here. For thistable, the ratio from one bounce to the next is the same as the ratio for the corre-sponding bounces in the original table.

Bounce Number 1 2 3 4 5

Rebound Height 7.6 3 1.2 0.4 0.1

d. y � 20(�25

�)x or NEXT � (�25

�)(NOW ), starting at 20.

Sierpinski Carpets

It is apparently not as obvious to students as it is to teachers that if we cut out1 square in 9 there will be �

89

� of the original carpet area left. Students may find thelarge hole in the center distracting. Even those who understand that �

19

� is removedmay have trouble connecting this to �

89

� remaining. Perhaps it is the question “�89

� ofwhat?” that needs to be stated clearly. Expect to see more counting than seemsnecessary, until students convince themselves of the pattern. If students spend toomuch time coloring and counting, you might want to do this as a large group,though this still will leave some students unconvinced.1.

INVESTIGATION

MOREASSIGNMENT pp. 448–454

Students can now beginModeling Task 1 or OrganizingTask 1 from the MORE assign-ment following Investigation 3.



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EXPLORE continued

2. a. �89

�

b. �67

42� � �

89

�

This is �89

� of �89

�, which is �68

41� of the original square meter of carpet.

c. �55

17

26

� � �89

�

This is �89

� of �89

� of �89

�, which is �57

12

29

� of the original.

d. After 4 cutouts: �89

� � �89

� � �89

� � �89

� � �64,,506916

�

After 5 cutouts: (�89

�)5� �

35

29

,,70

64

89

�

3. NEXT � NOW(�89

�), starting at 1

a. After 10: (�89

�)10� 0.31 m2

b. After 20: (�89

�)20� 0.09 m2

After 30: (�89

�)30� 0.03 m2

4. y � (�89

�)x

a.

b. 6 cutouts are needed in order to get more hole than carpet.

(�89

�)6� (�256

321

,,14

44

41

�) ≈ 0.49

NOTE: Some groups may need help understanding that there is more hole than carpetremaining when y is less than 50%.

See Teaching Master 168.The patterns are the same. Slight differences are due to a larger value for the base in theSierpinski problem and a smaller value for the coefficient.The Sierpinski pattern represents an exponential decrease, whereas the problems inLesson 1 represented an exponential increase. In one case, the base is less than one, andin the other it is greater than one. All of the graphs show a changing rate of change. Besure students understand that exponential decrease occurs when the base is less thanone, and exponential growth or increase occurs when the base is greater than one.

b

a

Summarize the ways in which the table, graph, andequations for the Sierpinski carpet pattern are similar to,and different from, those for the following patterns:

the bouncing ball patterns of Investigation 1;

the calling tree, king’s chessboard, and bacteria growthpatterns of Lesson 1.

Be prepared to share your summaries of similarities anddifferences with the entire class.



168

Checkpoint

Master 168

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X�0

Y10123456

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SHARE AND SUMMARIZE full-class discussion

See additional Teaching Notes on page T481F.


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a. The cutout number is in L1 and the area of the remaining carpet is in L2.

b.

c. NEXT � NOW � (�89

�), starting at 9d. y � 9 � (�

89

�) x

e. Six cuts are needed.■ In a table of values, you can look for a y value that is less than 4.5.■ In the plot of the data, you could use the trace function to determine when the value

of y is less than 4.5.f. The data show the same patterns but these answers are all larger because the second car-

pet was larger. In the NOW-NEXT equation, we start at 9 instead of 1; the coefficient inthe other equation is 9 instead of 1. The answer to Part e is still 6 cuts.


n Your Own

L1

L2(1)�9

L2 L30123456

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L1

L2(12)�

L2 L35678910

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MOREASSIGNMENT pp. 448–454

Students can now beginModeling Task 2 or ReflectingTask 4 from the MORE assign-ment following Investigation 3.


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Medicine and Mathematics

Students often mix up how much insulin is lost every minute (5%) with how muchis left (95%). As you circulate, be sure they are writing carefully what multiplying by0.95 does. You may wish to ask questions such as the ones below.

What variables are we tracking in the graph? (Amount of insulin remainingagainst time)What does the equation say about these variables? (If you get vague answers forthis one, such as “It is decreasing by 95%,” you may be able to stir up some reac-tion if you deliberately misinterpret this. “The number of diabetics is decreasingby 95%? The amount of blood is decreasing by 95%? The amount of insulin isdecreasing by 95%?” All of these are, of course, incorrect. The amount of insulinin the blood is decreasing by 5%, and the amount remaining is 95% of what itwas, for every unit of time.)You may see a similar problem occurring in contexts which develop the concept

of a gain of 5%, resulting in a next value of 105% of whatever the current value is.The implied 100% “start” may be the missing idea.

1. About 14 minutes2. Approximate values of the data from the graph are in the table below.

x 0 3 6 9 12 15 18 21 24 27

y 10 8.75 7.5 6.5 5.75 4.9 4.2 3.75 3 etc.

The equation fits the data fairly well but generally gives values a little less than those onthe graph. 10 is the initial amount of insulin taken and 0.95 is the fraction of the originalinsulin left after 1 minute.

3. NEXT � NOW(0.95), starting at 10.

INVESTIGATION



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4. The calculations tell you both the amount left after a particular time, and that the amountleft is continuously decaying.a. The amount of insulin left after a minute and a half is 10(0.95)1.5, or about 9.3 units.b. The amount left after four and a half minutes is 10(0.95)4.5, or about 7.9 units.c. The amount left after eighteen and three-quarters minutes is 10(0.95)18.75, or about

3.8 units.

5. a. Time in Minutes 0 1.5 4.5 7.5 10.5 13.5 16.5 19.5

Insulin in Blood 10 9.3 7.9 6.8 5.8 5.0 4.3 3.7

b. These values are close for the first few minutes but then are slightly lower than thevalues on the graph.

c. After approximately 13.5 minutes, the insulin is half gone.

See Teaching Master 169.NEXT � NOW � b, starting at a.

The value of a tells us the beginning value of y. In the table, it is the value of y when xequals zero, and it is the value of the y-intercept of the graph.The value of b tells us how the values of y are changing. Each successive term can beobtained by multiplying the preceding term by b. Assuming the x values change by one,it is the constant ratio between two consecutive y values in the table. If b is greater thanone, the graph and table values will be increasing; if b is less than one and greater thanzero, the graph and table values will be decreasing.In both linear and exponential equations, a provides the value of y when x is zero. In bothtypes of equations the value of b will tell us if the values are increasing or decreasing.For linear equations, if b is greater than zero the y values will be increasing, but for expo-nential models the value of b must be greater than one for the values to be increasing.Also, linear models change by a constant amount (b), whereas exponential modelschange by a constant factor (b).

JOURNAL ENTRY: Imagine that you are a medical examiner assigned to the following case.A toddler died mysteriously at 12:05 on a Monday afternoon. An autopsy revealed that theyoungster’s system contained 2 teaspoons of ethylene glycol, the primary ingredient inantifreeze.

When questioned by the police, the parents were certain that the child had not been leftalone, except for a short time early the previous morning when the parents were folding thelaundry, around 7 or 8 o’clock. After some mathematical calculations, you conclude that itwas not likely that the child had drunk antifreeze while unattended. (The metabolic half-lifeof ethylene glycol is approximately 4 hours.)

Consequently, you continue the investigation. Searching through a medical database, youfind references to a rare disease that causes the body to produce a chemical similar to ethyl-ene glycol. Using further tests, you conclude that this disease was indeed the cause of death.

Write a report explaining how you came to the conclusion that it was unlikely that thechild had drunk antifreeze. Include in your explanation any calculations you may havemade.

d

c

b

a


169


In this unit, you have seen that patterns of exponentialchange can be modeled by equations of the form y � a(bx).

What equation relates NOW and NEXT y values of thismodel?

What does the value of a tell about the situation beingmodeled? About the tables and graphs of (x, y) values?

What does the value of b tell about the situation beingmodeled? About the tables and graphs of (x, y) values?

How is the information provided by values of a and b inexponential equations like y = a(bx) similar to, anddifferent from, that provided by a and b in linearequations like y = a + bx?

Be prepared to compare your responses with those fromother groups.

Checkpoint

Master 169

SHARE AND SUMMARIZE full-class discussion

NOTE: Responses will vary,but students should be able toprove that it would have beendifficult, if not impossible, forthe toddler to drink enough(256 tsp) antifreeze Sundaymorning for two teaspoons ofethylene glycol to be presentin the youngster on Mondaymorning.

CONSTRUCTING A MATHTOOLKIT: Students should adda summary of the Checkpointresponses for Parts a–c totheir Toolkits (TeachingMaster 200).


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a. Time 12:00 1:00 2:00 3:00 4:00 5:00

Amount left (mg) 300 180 108 64.8 38.88 23.33

b.

The plot shows that penicillin decays at a decreasing rate.c. y � 300(0.6)x

d.

The half-life of penicillin is between 1 hour 15 minutes and 1 and a half hours.e. The amount of penicillin left in the blood will be less than 10 mg after about 6.7 hours.


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Documents

Lesson Exponential Decay - Core-Plus Mathematics Project · Exponential Decay This lesson engages ... we have chosen a simple example—the height of a bouncing ball after each