3
Exponential Equations
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Exponential Growth—Interest
You will need
• Interest.tns
You have $600 that you want to save, and you know that it can earn money for you.
When you let someone else use your money, they must pay you rent. That rent is
called interest. In this activity, you will explore different types of interest.
INTRODUCTION
A relative agrees to pay you interest at a rate of 10%. This means that 10% of the
$600 will be added to your savings each year. This is called simple interest.
Q1
How much will your savings increase each year?
A bank advertises that it will pay you 6% interest. The bank’s interest is compounded.
In other words, each year the bank adds to your savings 6% of the growing amount
of savings, not just 6% of the original $600.
Q2
How much will your savings increase the first year? The second year?
Your goal in this activity is to decide whether the bank’s offer can ever be better
than your relative’s offer.
INVESTIGATE
1. Open the TI-Nspire document
Interest.tns on your handheld. Go
to page 1.2. You will be making a
spreadsheet showing your earnings
over 50 years.
2. First enter a formula that will generate
the years. The formula for year looks
like this:
⫽ seqn(u(n ⫺ 1) ⫹ 1, {0}, 50)
3. To calculate the simple interest, enter
the formula
⫽ seqn(u(n ⫺ 1) ⫹ 60, {600}, 50)
The u(n ⫺ 1) in the formula tells TINspire to use the previous value of the
variable. The 600 tells TI-Nspire to use the
value 600 for the initial case (here, it is the
0th case).
Q3
What does the 60 in the formula do?
Exploring Algebra 1 with TI-Nspire™
3: Exponential Equations
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Exponential Growth—Interest
continued
4. For compound, set up a formula to
represent the bank’s offer. You can use
the formula
seqn(u(n 1) + .06 u(n 1),
{600}, 50)
Q4
Explain in your own words what this
formula does.
Q5
Will your savings ever earn more
money with the bank’s offer than with
your relative’s? Explain.
You would like formulas that give you the value of your savings for any year,
without having to find the values for all previous years. You will use scatter plots to
find these formulas.
5. At the top of page 1.3, make a scatter plot of the (year, simple) data.
6. The data points appear linear, so you’ll
be looking for an equation of the form
simple a b year. Choose Plot
Function from the Actions menu, and
enter the equation f(x) a[1] b[1] x. Use the sliders at the bottom of the
page to adjust a and b to fit the data
points as well as possible.
Q6
What values of a and b give the best fit?
How do these values relate to $600 of your
initial savings and to the 10% simple interest rate?
7. At the top of page 1.4, make a scatter plot of the (year, compound) data and plot
the line from page 1.3.
Q7
Can you adjust a and b to make the
graph fit the compound interest data?
Why or why not?
Because compound interest cannot be
modeled by a linear equation, you will
look for another type of equation that can
model it.
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3: Exponential Equations
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Exponential Growth—Interest
continued
8. Perhaps this curve is part of a parabola.
Go to page 1.5 and make another scatter
plot of the (year, compound) data. Plot
the function f(x) ⫽ c[1] ⫹ d[1] ⭈ x2.
(To get the exponent, press l or
q.) Adjust the c and d sliders to
match the data as well as possible.
Q8
How well does this graph model the
data? Do you think compound interest
growth can be modeled by a parabolic
graph?
9. Another curve is the graph of an
equation in which year is in the
exponent. Go to page 1.6 to explore an
equation of the form compound ⫽ m ⭈
n year. You will need to plot the function
f(x) ⫽ m[1] ⭈ n[1] x. This is called an
exponential equation. To adjust n, you
may need to zoom in on values near 1.
To zoom, use the
Window/Zoom menu
or drag the ends of
the axes.
Q9
Q10
What values of m and n give the best fit
for the data? Do you think compound
interest can be modeled by exponential equations?
When compound interest in this situation is modeled by an equation of the
form A ⫽ mnx, how do you think m and n relate to the $600 initial investment
and to the rate of 6%?
EXPLORE MORE
Insert a new Lists & Spreadsheets application as page 1.7. What formulas give the
following sequences?
2, 4, 6, 8, . . . , 16
3, 6, 12, 24, . . . , 1536
1, 1, 2, 3, 5, . . . , 89
Exploring Algebra 1 with TI-Nspire™
3: Exponential Equations
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Exponential Growth—Interest
Activity Notes
Adapted from Exploring Algebra 1 with Fathom by Eric Kamischke, Larry Copes, and Ross Isenegger.
Objectives: Students will learn that a linear equation
models simple interest and an exponential equation
models compound interest. They will relate the values of
a and b in the equations A a bx and A ab x to the
principal and interest rate, respectively.
Q5 The savings in the bank passes the savings with the
relative in year 18.
Q6 a 600, the initial amount; b 60, the amount of
interest being added each year. Introduce the terms
principal for the initial amount of $600 and interest
rate for the 10%.
Activity Time: 35–50 minutes
Materials: Interest.tns
Mathematics Prerequisites: Students should be able to
calculate a percentage of an amount, write a percent as a
decimal, evaluate equations by substitution, and substitute
variable expressions into a formula.
TI-Nspire Prerequisites: Students should be able to
open and navigate a document, use formulas in the Lists
& Spreadsheet application, create scatter plots, plot an
equation on a scatter plot, adjust the scale of an axis, and
insert a new page. (See the Tip Sheets.)
TI-Nspire Skill: Students will insert formulas using the
seqn command.
Notes: As you facilitate student work, probe for
understanding of the quantity 1.06, especially in Q9. You
might have pairs who finish the main activity first prepare
to present the Explore More questions and answers.
For a Presentation: As you lead a class discussion using a
presentation computer or projected handheld, emphasize
Q5–Q9.
INTRODUCTION
Q7 No, the compound interest graph is not linear.
Q8 The compound interest graph is not quite parabolic.
It fits well for the first 20 years but does not increase
fast enough after that.
Q9 The best value for m is the vertical intercept, 600. The
best value for n is 1.06. An exponential equation
fits well.
Q10 m is the principal, $600. n is 1 more than the
interest rate.
DISCUSSION QUESTION
• You entered several formulas at the beginning of this
activity. What does each part of the formula do?
Here is a quick look at the formula = seqn(u(n 1) 60,
{600}, 50):
defines a formula
seqn
creates a sequence
u(n 1)
the previous term in the list—
u(n 1) and u(n 2) are the
previous two terms.
60
what to do to previous term
{600}
the initial term of the sequence.
(The first and second terms would
be {a, b}.)
50
the number of terms to list
Q1 10% of $600 is $60.
Q2 Year 1: 6% of $600, or $36; Year 2: 6% of $636,
or $38.16
INVESTIGATE
Q3 The 60 adds 10% of $600, or $60.
Q4 The formula starts with 600. It takes that value,
multiplies it by 0.06, and adds that to the value to get
the next value. It does this 50 times.
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3: Exponential Equations
EXPLORE MORE
seqn(u(n 1) 2, {2}, 8); seqn(u(n 1) 2, {3},
10); seqn(u(n 1) u(n 2), {1, 1}, 11). Students
could also find these formulas using cell operations and
fill down.
Exploring Algebra 1 with TI-Nspire™
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Inverse Variation—Boyle’s Law
You will need
• Boyles Law.tns
The amount of time a scuba diver can safely spend under water depends on the
amount of air in the diving tank. Your friend has asked for your help in determining
how much air is in a particular tank.
You can see that the volume of the tank is 0.340 cubic feet, but you also know
that the tank is pressurized to pack more air into a smaller space. As the air leaves
the tank, it expands. You can see that the tank is full and that the pressure gauge
reads 3500 pounds per square inch (psi). To determine how much air the tank will
supply while your friend is diving, you need to know what the volume of the air will
become as it is released from the tank and as the pressure becomes the normal air
pressure of 14.7 psi. Boyle’s law relates these two qualities.
INVESTIGATE
1. You have found some data on the
Internet and put it into a table to
help you look for patterns. Open the
document Boyles Law.tns and go to
page 1.2 to see the data.
2. Go to page 1.3 and create a scatter
plot of the (volume, pressure) data
at the right of the page. The data
are not linear, but perhaps they are
exponential.
Click on the point
representing
volume 48 to see its
values. You can also
look in the table.
Q1
As the volume is halved from 48 ft3 to
24 ft3, what happens to the pressure?
Q2
Does the pressure approximately
double when the volume goes from
30 ft3 to 15 ft3? Does the pressure
always double when the volume is cut
in half? Show the evidence for your
conjecture. You can use either the
scatter plot or the table to find this
information.
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3: Exponential Equations
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Inverse Variation—Boyle’s Law
continued
Q3
If the equation were exponential, would the pressure double when the volume is
cut in half?
3. Because the data do not appear to
be exponential, you decide to try
modeling with a power equation of
the form y ⫽ k ⭈ x b. Unlike in an
exponential equation, the variable
pressure will be in the base rather than
in the exponent. Add the graph of the
power equation to your scatter plot.
Use the equation f1(x) ⫽ k[1] * x^b[1] .
4. Adjust the k and b sliders to fit the data as well as you can.
Q4
According to your model, what pressure is needed to reduce the volume
below 10 ft3?
You may have found that a good value for b is approximately –1. Using the
definition of a negative exponent, it can be simpler to write the equation as
1
pressure ⫽ k ( _____
. The coefficient k of this equation is called the constant of
volume )
variation. Note that k ⫽ pressure ⭈ volume. Finding the value of this constant in
particular situations is important for answering pressure-volume questions like the
one in this activity.
5. To see this relationship in the data table, add a new variable constant with the
formula pressure ⭈ volume.
6. Add a new Data & Statistics page by pressing c and choosing Data &
Statistics.
To plot a value, choose
Plot Value from the
Actions menu.
62
Q5
Create a new dot plot of constant and
plot the value mean(constant) . How
does this mean relate to the values of
the sliders? Explain.
Q6
What is the constant of variation for
the scuba tank you are examining? Why
might it be different from the constant
of variation in the table?
Q7
What volume will the air have when it
is released from the tank and the pressure becomes 14.7 psi?
3: Exponential Equations
Exploring Algebra 1 with TI-Nspire™
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Inverse Variation—Boyle’s Law
continued
EXPLORE MORE
1. To explore more equations of the form
xy ⫽ k or the equivalent, go to the plot
of y ⫽ k ( _1x ) on page 2.1. Explore the
graph for the values of slider k near 0.
Describe the graph when k ⬍ 0, k ⫽ 0,
and k ⬎ 0.
2. In Explore More 1, you studied a graph
in which the product of x and y was
the constant k. Now explore graphs in
which the sum of x and y is the constant k. You may need to add a new Data &
Statistics page after 2.1 to do this. Be as specific as you can about what is always
true about the graph and what changes as k changes.
3. What if the difference were always k? What if the quotient were k? Try this. Are
you surprised by the results?
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3: Exponential Equations
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Inverse Variation—Boyle’s Law
Activity Notes
Adapted from Exploring Algebra 1 with Fathom by Eric Kamischke, Larry Copes, and Ross Isenegger.
Objectives: Students will model inversely proportional
quantities with equations of the form y ⫽ k ( _1x ), xy ⫽ k, and
y ⫽ kx⫺1. They will investigate the change in one variable
as the other doubles, relate y ⫽ k ( _1x ) to a linear model, and
explore the graph of xy ⫽ k.
Q1 The pressure roughly doubles from 29.125 psi to
58.8125 psi.
Activity Time: 30–40 minutes
Materials: Boyles Law.tns
Mathematics Prerequisites: Students should be able to
solve literal equations and evaluate a formula.
TI-Nspire Prerequisites: Students should be able to create
a scatter plot, plot a function on a scatter plot, use
TI-Nspire to calculate a mean, and change the scale of
sliders and scatter plots.
TI-Nspire Skills: None
Notes: Boyle’s law states that if a gas is kept at constant
temperature, the pressure and volume are inversely
proportional, or have a constant product. Robert Boyle
published his findings that pressure times volume is
constant in his 1662 article “A Defense of the Doctrine
Touching the Spring and Weight of the Air.”
Q2 As the volume goes from 30 ft3 to 15 ft3, the pressure
roughly doubles from 47.0625 psi to 93.0625 psi.
Some students might say pressure is not quite
doubled as volume goes from 15 ft3 to 30 ft3. The table
demonstrates that the ratio of the pressures is very
close to 2 in every case:
Volume
(ft3)
Pressure
(psi)
Volume
(ft3)
Pressure
(psi)
Ratio of
pressures
48
29.1250
24
58.8125
2.019313
46
30.5625
23
61.3125
2.006135
44
31.9375
22
64.0625
2.005871
As you facilitate student work, look for students who
have complete answers to Q2 and Q3. Have them share
their results with the class. A variety of answers to Q6
can also be shared; students taking physics will know that
temperature is an important factor.
42
33.5000
21
67.0625
2.001866
40
35.3125
20
70.6875
2.001770
38
37.0000
19
74.1250
2.003378
36
39.3125
18
77.8750
1.980922
34
41.6250
17
82.7500
1.987988
INVESTIGATE
32
44.1875
16
87.8750
1.988685
30
47.0625
15
93.0625
1.977424
28
50.3125
14
100.4375
1.996273
26
54.3125
13
107.8125
1.985040
24
58.8125
12
117.5625
1.998937
1. These data are Boyle’s original. The Internet source
is given on the last page of the document. Boyle’s
methodology and his published findings are
intriguing reading.
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3: Exponential Equations
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Inverse Variation—Boyle’s Law
Activity Notes
continued
Q3 No. If the equation were exponential, the pressure
would double for constant changes in the volume, not
for proportional changes.
4. Finding the best values is tricky. It requires fine motor
control, the ability to adjust the scale of a slider, and
visual estimation skills. This model, with k ⫽ 1253
and b ⫽ ⫺0.97, matches the graph remarkably well,
which might make you wonder whether Boyle’s
data were fudged. There is some evidence that
other historically important data can be statistically
shown to be too close to the predictions to have been
produced by experimentation.
EXPLORE MORE
1. When k < 0, the branches are in the second and
fourth quadrants. This graph is called a hyperbola. Its
branches approach, but never cross, its asymptotes,
which, in this case, are the x- and y-axes. When k ⫽ 0,
the graph is the x-axis with the point (0, 0) removed
(because _10 is undefined). When k > 0, there will be
two branches of the graph, one in the first quadrant
and one in the third.
2. The equation is y ⫽ k – x, so the graph is a straight
line with y-intercept k and slope –1. It crosses the
y-axis above the origin when k > 0, at the origin when
k = 0, and below the origin when k < 0.
3. If the constant difference equals x – y, then y ⫽ x – k;
if the constant difference equals y – x, then y ⫽ x + k.
The graph of each is a straight line with slope 1; k is
either the y-intercept or its opposite. If the constant
quotient equals _xy , then y ⫽ _1k x ; if the constant
Q4 Using pressure = 1253 · volume⫺0.97, a pressure of
quotient equals _xy , then y ⫽ kx. In both cases, the
graph is a line through the origin with slope either k
or its reciprocal.
160 psi will decrease the volume below 10 ft3.
Q5 If slider k is set to the mean value, 1408, the graph
goes through the data points. Doing a power
1400.9
regression on the data yields the equation y ⫽ ____
x .
(Choose Regression from the Actions menu.)
EXTENSIONS
1. Have students research the pressure experienced by
divers. How deep would a diver need to be in order
to be subject to twice the pressure experienced at
sea level? [A diver at depth 10.3 m under fresh water
experiences a pressure of about 2 atmospheres—
1 atm for the air and 1 atm for the water.]
Q6 pressure ⭈ volume ⫽ constant, so (0.34)(3500) ⫽ 1190.
Possible reasons the constant may differ from that of
the table include differences in temperature and the
nature of the gas being compressed.
___
Q7 About 81 ft3 volume ⫽ _____
pressure ⫽ 14.7
(
constant
Exploring Algebra 1 with TI-Nspire™
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)
3: Exponential Equations
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Inverse Variation—Boyle’s Law
Activity Notes
continued
2. Pose this problem: A data analyst uses the
TI-Nspire to fit the exponential equation volume ⫽
46.41(0.98032)pressure–29 to the data in the table. Make a
convincing argument for why either the exponential or
the power model is better. You may want to discuss the
vertical intercept and its role in the model, as well as
the halving time for volume. [The equation volume ⫽
46.41(0.98032)pressure–29 goes through the point (29,
46.41), which is approximately pressure and volume in
the first case. The volume is decreasing approximately
2% per increase of 1 unit of pressure at the beginning
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3: Exponential Equations
of the experiment, but it decreases more slowly in the
later values. Not only does a graph of the form pv ⫽ k
fit the points better, but it also ensures that the graph
has no vertical intercept (as pressure decreases to 0,
volume expands to infinity). The exponential model
implies that when there is no pressure, the volume
will be about 82.6, which is physically incorrect. In
the activity, volume is halved as pressure is doubled
in Boyle’s model, as opposed to having a fixed halving
period as in the exponential function model.]
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Exponential Models—Radioactive Decay
You will need
• Radioactive
Decay.tns
• paper plate
• protractor
• supply of small
counters
The particles that make up the atoms of some elements, like uranium, are unstable.
Over a period of time specific to the element, the particles change and the atom
eventually becomes a different element. This process is called radioactive decay.
EXPERIMENT
In this experiment your counters represent atoms of a radioactive substance.
1. Take a paper plate and draw an angle from
the center of your plate, as illustrated.
2. Count the number of counters. Open the
TI-Nspire document Radioactive Decay.tns
on your handheld and go to page 1.2. Enter the
number of counters as the number of “atoms”
after 0 years of decay. Pick up all of the counters.
3. Drop the counters randomly on the plate. Be
sure the counters are scattered evenly over all
parts of the plate—do not aim for the center. Counters that fall inside the
angle represent atoms that have decayed. Decide how you are going to handle
counters that fall on the lines of your angle and that miss the plate—they need
to be accounted for also.
4. Count and remove the counters that fall inside the angle—these atoms have
decayed. Subtract the decayed atoms from the previous value and enter the
number of counters remaining after 1 year of decay. Pick up the remaining
counters.
5. Repeat steps 3 and 4 until you have
fewer than ten atoms that have not
decayed. Each drop will represent
another year of decay. Enter the
number of atoms remaining each time
on your handheld.
Exploring Algebra 1 with TI-Nspire™
3: Exponential Equations
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Exponential Models—Radioactive Decay
continued
INVESTIGATE
You’ll use your handheld to make a model for your experiment. Let x represent
elapsed time in years, and let y represent the number of atoms remaining.
6. On your handheld, make a scatter plot
of the data. To do this, go to page 1.3 and
add the appropriate variables to the plot.
Q1
What do you notice about the shape of
the graph?
7. Calculate the ratios of atoms remaining
between successive years. That is,
divide the number of atoms remaining
after 1 year by the number of atoms
remaining after 0 years, and so on. To do this, go to page 1.2. Arrow to the top of
column C and type the variable name, ratio. In cell C1, type the formula = b2/b1
and press ·. To fill in the rest of the values, highlight the first cell and choose
Fill Down from the Data menu. You will see the cell surrounded by dashed lines.
Press ¤ to highlight all but the last row of data you entered, then press ·.
You should have a ratio for each row except the last.
Q2
How do the ratios compare?
8. Find a representative ratio by calculating either the mean or median. To use the
Calculator application, go to page 1.4. Type either mean(ratio) or median(ratio) .
68
Q3
Which representative ratio did you choose and why?
Q4
At what rate did your atoms decay?
Q5
Write an exponential equation that models the relationship between the amount
of time elapsed and the number of atoms.
3: Exponential Equations
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Exponential Models—Radioactive Decay
continued
9. Graph the equation with the scatter plot: choose Plot Function from the
Actions menu and type the equation.
Q6
How well does the equation fit the data? If the equation does not fit well, adjust
the two values until you are satisfied. To do this, double-click the equation on
the screen and edit the values. Record your final equation.
Q7
Measure the angle on your plate. Describe a connection between your angle and
the numbers in your equation.
Q8
Based on what you’ve learned and the procedures outlined in this activity, write
an equation that would model the decay of 400 counters using a central angle
of 60⬚.
Q9
What are some of the factors that might cause differences between actual data
and values predicted by your equation?
EXPLORE MORE
Hint: Look in the
Actions menu
on page 1.3.
Perform an exponential regression with the handheld and compare this equation
with the others that you found.
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3: Exponential Equations
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Exponential Models—Radioactive Decay
Activity Notes
Adapted from Discovering Algebra by Jerald Murdock, Ellen Kamischke, and Eric Kamischke.
Objectives: Students will write exponential equations that
model real-world decay data.
INVESTIGATE
6. Here is a graph of the sample data.
Activity Time: 50–60 minutes
Materials: Radioactive Decay.tns, paper plates,
protractors, counters (Skittles, M&M bits, candy corn,
dried beans, etc.). Optional: Radioactive Decay Sample.tns
Mathematics Prerequisites: Students have had some
experience with exponential equations and function
notation.
TI-Nspire Prerequisites: Students should be able to open
and navigate a document, graph an equation, create a
scatter plot, and use the grab tool. (See the Tip Sheets.)
TI-Nspire Skills: Students will find the mean or median
and type formulas into cells of the Lists & Spreadsheet
application.
Notes: There is a TI-Nspire document with sample data
for your use. All the screen shots below are with the sample
data. You may choose to have students work collaboratively
with a partner or as a whole class.
EXPERIMENT
1. Students should use protractors to make an angle
of less than 90°, but not too small. They can
approximate the center by using a ruler to find the
midpoints of several diameters.
3. The objective is to have the counters spread evenly
and quickly. An acceptable plan would be that
counters on the line and counters that fall outside the
plate but lie within the extended rays of the angle are
accepted as being within the angle.
4. Remind students that they are interested in the
number of atoms that do not decay.
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3: Exponential Equations
Q1 Students should notice a curved, exponential pattern.
7. You may have to assist students with this step. Some
may just type in the formulas for every cell, which is
not an efficient approach.
Q2 The ratios should be approximately the same. For the
sample data, they range between 0.73 and 0.9.
Q3 Students could give reasons for selecting the mean,
the median, or another value. In these sample
data, the mean is about 0.802 and the median is
about 0.804.
Q4 For these sample data, using 0.802, the rate of decay is
19.8% per year. If you are not using the sample data,
you might use this opportunity to discuss various
student answers. You may need to encourage students
to look at their constant multiplier in the form of
(1 ⫺ r). That is, a ratio of _34 would be (1 ⫺ 0.25).
Q5 For this sample, y ⫽ 201(1 ⫺ 0.198)x.
Q6 For the sample data, the equation does not appear
to fit very well. Students may find that the multiplier
A seems to adjust the curve’s position vertically and
r seems to change the steepness of the curve. The
equation y ⫽ 201(1 ⫺ 0.22)x fits the sample data better.
Exploring Algebra 1 with TI-Nspire™
© 2009 Key Curriculum Press
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Exponential Models—Radioactive Decay
Activity Notes
continued
EXPLORE MORE
To find and graph the exponential regression from the
Data & Statistics application, choose Regression from
the Actions menu, then choose Exponential Regression.
For the sample data, the regression equation is y ⫽
177.22(0.80)x. This equation is close to the predicted
equation, with a smaller A-value and a very similar r-value.
Comparisons to students’ adjusted equations will vary.
Q7 To help students with this question, ask “What is
the ratio of your angle measure to the whole plate?”
The ratio of the angle measure to 360⬚ should be
68
approximately the same as r. In the sample data, ___
360 ⬇
0.19, which is close to the r-value used in Q5.
(
60
Q8 y ⫽ 400 1 ⫺ ___
360
x
)
Q9 Factors given might include how evenly the counters
are distributed on the plate, what you do when
a counter is on an angle line, and how you treat
counters that fall outside the plate.
Exploring Algebra 1 with TI-Nspire™
3: Exponential Equations
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Half-Life—Pendulums
You will need
• Pendulums.tns
• soda can or bottle
of water
• string
• meterstick
In this activity you will do an experiment, write an equation that models the
decreasing exponential pattern, and find the half-life—the amount of time needed
for a substance or an activity to decrease to one-half its starting value.
EXPERIMENT
1. Make a pendulum with a soda can half-filled
with water tied to at least 1 m of string—use the
pull tab on the can to connect it to the string.
Place the meterstick underneath or behind the
pendulum so you can take readings.
You may have to
collect data for every
fourth or fifth swing
to get an accurate
reading.
2. Pull the can back about 1 m from its resting
position, then release it. Measure how far the
can swings from the resting position for as
many swings as you can.
Resting
position
3. Open the TI-Nspire document
Pendulums.tns on your handheld and
go to page 1.2. Enter your data in the
Lists & Spreadsheet application.
Q1
Go to page 1.3 and make a scatter plot
of your data. What type of pattern does
the graph seem to show?
INVESTIGATE
4. Go to page 1.4. Here you will see a
scatter plot of data on 19 pendulum
swings collected using a CBR2. These
data show the full swing of the
pendulum.
Q2
Describe this graph as fully as you can.
How does this graph compare with the
graph of your data? Explain.
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3: Exponential Equations
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Half-Life—Pendulums
continued
5. The maximum swings from the sample
data are on page 1.5. Go to page 1.6
and make a scatter plot of these data.
Choose Function from
the Graph Type menu
to graph. Double-click
the equation to edit it.
Q3
Does this graph look similar to the
graph of your data? Explain.
Q4
Find an equation in the form
y ⫽ A(1 – r)x that models the sample
data. Graph this equation with the
scatter plot. Adjust the values until you
have as good a fit as possible. What is your function?
Q5
Find the half-life of your data. To find the half-life, approximate the value of x
that makes y equal to _12 A.
Q6
What does the half-life mean for the situation in your experiment?
Q7
Find the maximum distance after one half-life, two half-life cycles, and three
half-life cycles. How do these values compare?
Q8
Write a summary of your results. Include descriptions of how you found your
exponential model, what the rate r means in your equation, and how you found
the half-life.
EXPLORE MORE
Perform an exponential regression with your handheld: go to page 1.7, which shows
a Data & Statistics scatter plot, and choose Regression from the Actions menu.
Compare this equation with the one you found.
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3: Exponential Equations
Exploring Algebra 1 with TI-Nspire™
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Half-Life—Pendulums
Activity Notes
Adapted from Discovering Algebra by Jerald Murdock, Ellen Kamischke, and Eric Kamischke.
Objectives: Students will write exponential equations that
model real-world decay data and find the half-life for the
equation.
Activity Time: 50–60 minutes
Materials: Pendulums.tns, soda cans or bottles of water,
string, metersticks
Mathematics Prerequisites: Students have had some
experience with exponential equations and function
notation.
notice that the top points of the graph look like
their graph.
5. Discuss with students any patterns they see with the
data table.
Q3 If students collected data accurately, this graph should
resemble their data. They may note that this graph
shows every swing, whereas theirs shows every five
swings or so.
Q4 Sample equation:
TI-Nspire Prerequisites: Students should be able to open
and navigate a document, graph an equation and modify
it, create a scatter plot, and use the grab tool. (See the Tip
Sheets.)
TI-Nspire Skills: None
Notes: You have four options for this experiment,
depending on your time constraints and classroom setup.
Option 1: Have students collect data, enter the data into the
handheld, analyze the data, and compare them with the
sample data. Option 2: Do a demonstration for the whole
class and collect data using the meterstick, enter the data
into the handheld, analyze the data, and compare them
with the sample data. Option 3: Demonstrate the basic
experiment so students understand the situation, then go
right to the sample data. Option 4: Discuss the experiment,
then go right to the sample data. You could, theoretically,
also collect data similar to the sample data using a CBR2
or other data collection device. However, the process of
manually capturing the maximum swing points requires
patience and precision, and may not be worth class time.
You may choose to have students work collaboratively with
a partner or as a whole class.
Q5 Using the sample equation, the half-life is about
82 swings. One way to find this is to graph y2 ⫽ _12 A
and find the intersection with y1 ⫽ A(1 ⫺ r)x. To find
the intersection, choose Intersection Point(s) from
the Points & Lines menu and select the two functions.
Q6 Sample answer: The half-life of the pendulum swing
EXPERIMENT
Q1 Students will probably recognize an exponential
pattern.
Q2 The location of the pendulum bob is harmonic, but
its maximum distance from the resting position is
roughly exponential. Students should notice the
sinusoidal pattern (in their own words) and that the
graph is decaying, or getting smaller. They should
Exploring Algebra 1 with TI-Nspire™
is approximately 82 swings. This means that on the
82nd swing the pendulum’s maximum distance is half
of its original maximum distance.
Q7 Using the sample equation: approximately 0.393 m,
0.197 m, and 0.098 m. With each consecutive half-life,
the value of y will be half the previous value of y.
Q9 Student summaries will vary.
3: Exponential Equations
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Half-Life—Pendulums
Activity Notes
continued
DISCUSSION QUESTIONS
• How does the shape of the graph in Q2 correspond to
the situation with the pendulum?
However, if students saw a greater domain and range for
the graph using the method given above for Q5, they may
think the fit is quite good.
• You might want to introduce the equation
x /t
y ⫽ A _12 , where t is the half-life. Students can see
that the graph of this equation is similar to that of
their equation in the form y ⫽ A(1 ⫺ r)x. Ask them to
think about why the graphs are the same.
( )
EXPLORE MORE
The regression equation y ⫽ 0.78(0.99)x is probably similar
to the equation students found. Neither equation seems
to fit the data remarkably well over this small set of data.
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