C A S i n th e C l a s s room :
W h a t a re S o m e G ood W a y s to Ma ke U s e of
C om p u te r A l g eb ra S y s tem s i n Y ou r
M a th em a ti c s C l a s s ?
T I-8 9 In s tr uc t i o n a n d A s s e s s m e n t I d e a s
59th Annual Meeting of the ICTM
Peoria, IL
Session 131
October 18, 2008
Dr. Donald Porzio
Mathematics Faculty
Illinois Mathematics and Science Academy
1500 W. Sullivan Rd., Aurora, IL 60506
630-907-5966
dporzio@imsa.edu
Basic TI-89 CAS Capabilities
Explore how the TI-89 handles the following calculations.
1.
100!
5.
6
9.

2.
1
3
3.
4  3 2  3 54
4.

6.
i 67
7.
 6  i 3  2i 
8.
 3
10.
e-
11.
e i
12.
14.
 
cos  
 12 
18.
sin(cos-1 x)
3 1  2 

15. cos


4


19. sin(sec-1 x)
18  2 3
log 4 8  log3
1 
3


 2 
13.
sin
17.
sin-1(sin x)

1 


5 2
2 2 5
2i
4  3i
 5 
sin 

 6 
16.
sin(sin-1 x)
20.
tan(csc-1 x)
Comments?
TI-89 Menus
Let’s take a walk through some of the major menus on the HOME screen.
Commands in the F2 Menu:
A few items to try:
1.
3.
Solve x 2  1  0
2.
Solve 4sin 2  3   1 with 0    2
Factor the polynomial x 4  x 2  12 in four (perhaps more?) different ways.
Dr. Don Porzio, IMSA
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ICTM 2008, Session 131
Dr. Don Porzio, IMSA
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ICTM 2008, Session 131
Commands in the F4 Menu:
Equation Solving and TI-89 CAS Capabilities
When you use the solve( command on the TI-89 to solve an equation, it jumps straight to the correct
answer without showing any work!! That’s great if all we want is the answer, but what if we also want
to work on developing or assessing a student’s ability to solve equations. For example, suppose you
want to see if students not only can obtain a solution to an equation like 3x – 1 = 4 – 7x, but also
understand the process of solving such an equation. This equation can be entered on the TI-89 and then
solved in the “standard” manner of “isolating” the variable and then simplifying. Once a solution is
obtained, it can be checked by substituting back into the equation using the | key.
In doing this process, previous results were “pulled down” from the TI-89 HOME screen to the “edit”
bar at the bottom by scrolling up until a desired result was highlighted and then pressing ENTER .
Dr. Don Porzio, IMSA
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ICTM 2008, Session 131
One strength of this method is that, when
students do an incorrect step, such as
subtracting 10 from each side of the equation
10x = 5 rather than dividing each side by 10,
the TI-89 makes the error readily apparent.
This is in stark contrast what happens in real
life when a student will swear that subtracting
10 from each side of 10x = 5 produces the
equation x = –5!!
Try these three for yourself right now. (Note: Along with the factor command, other commands
under the F2 (Algebra) menu that might prove useful when solving equations in this manner are the
expand command, 3:expand(, the common denominator command, 6:conDenom(,
and the proper fraction command, 7:propFrac(.
1. 6  2  x   4  11x  5
2.
2 y  3  2 y  3  1
3.
ax 2  bx  c  0
6.
ln x 2  3  7
Try these four later to see how they work.
z2  5 z2  z  2

4. z  2
z 1
z 1
7.
5.
2x  5
Solve the following equation using the solve( command:
bx


ax


b

3x  a

b  2a
.
a
b
a
b
a 2b 2
What values of a and b are not allowed? What is the strange second part of the TI-89’s solution
and where did it come from?
2
2
2
2
Pattern Recognition and TI-89 CAS Capabilities
Have the TI-89 calculate each of the following:
ln(2) + ln(3)
ln(6) + ln(y)
ln(x) + ln(10)
Based on these results, what conjectures might be made about the value of ln(x) + ln(y)? Have the TI89 test your conjecture. Did it work? Why or why not? What modifications might need to be made so
that you get the expected result? How might our conjecture be proven?
The CAS capabilities of the TI-89 provide wonderful opportunities for dialog like this during the
exploration of various standard algebraic rules. Play around with the calculator to see what you might
do to explore patterns for the following algebraic rules.
 
ln x y
Dr. Don Porzio, IMSA
ax  a y
 a  bx
ax 
Page 4
y
a x
x y
ICTM 2008, Session 131
TI-89 CAS Capabilities and the Understanding of
Mathematics as a Language
Consider the following problem situation:
An airline transports approximately 1700 people per week between Chicago and London. A
round-trip ticket for this flight costs $362. The company is doing market research to determine
whether it should increase the fare on this flight. Research indicates that for each $5 increase in
price, 19 fewer passengers will take the flight. What ticket price should the airline charge?
In solving this problem, the biggest hurdle for most students is coming up with an equation to model the
problem situation. The statistical and graphical capabilities of the TI-89 can be used to solve the
problem and to derive an equation that models the problem situation. The symbolic algebra capabilities
of the TI-89 then provides students with a means for discovering how to derive an equation that
represents this and similar problem situations without using data or graph.
To begin, we need to create a TI-89 Data variable. To do this, press APPS and select
6:Data/Matrix Editor. A submenu will appear. Choose 3:New.
A dialog box will appear on the screen. At this point, Type should already be set to Data. Scroll
down to the box labeled Variable and give your Data variable a name (like chicago), then press
ENTER twice to save your selections. A Data variable consists of columns headed c1, c2, c3,
and so on. You can adjust the size of the columns through the Format Menu, which can be accessed
by pressing F1 Tools and then selecting 9:Format or by pressing
| . Here, a cell width of
5 was selected so that four columns of data would appear on the screen.
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ICTM 2008, Session 131
Now you are ready to type in the data. In this example, column c1 (Incrs) contains values for the
number of $5 price increases the airline could impose of the ticket price. Columns c2 (Peopl) and
c3 (Tpric) are the number of people who will still take this flight and the ticket price corresponding
to the number of $5 price increases in c1. Finally, column c4 (Trev) is the revenue generated by
ticket sales corresponding to the number of $5 price increases in c1. Headings were added to the
columns by scrolling to the appropriate cell and typing a title. The data in the figure above at the right
was generated by having students calculate the number of people who would fly, along with the
corresponding ticket price and revenue, for specific $5 price increases.
During this process, students may recognize the
patterns in the data being input and suggest that entries
in each column could be “globally” defined (like a
spreadsheet). For example, column c4 can be defined
as the product of columns c2 and c3, as shown in the
figure at the right. What students do not usually
recognize, at least initially, is that these “global
definitions” form the basis for an equation that
represents the problem situation.
Once the data has been entered, a scatter plot of columns c1 versus c4 can be generated. To access
the statistical plot menu, press F2 (Plot Setup) followed by F1 (Define). If you wish to
change the Mark setting, press  and then
to access the Mark submenu, and then select a
different type of mark. Finally, designate column c1 for the x-values and column c4 for the y-values.
Press ENTER twice to save your selections. The calculator will return to the Plot Setup screen
and show your newly define Scatter Plot.
To draw the scatter plot, first press
F1 (Y=) to access the Y= Editor and turn off any extra
functions or plots that are on. Then press F2 (ZOOM) and select 9:ZoomData. If you wish to
F2 (WINDOW), change the window
adjust window used to view the scatter plot, press
settings to your liking, then press
Dr. Don Porzio, IMSA
F3 (GRAPH).
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ICTM 2008, Session 131
The graph above at the right suggests that the relationship between the number of $5 price increases and
the ticket sales revenue might be quadratic. Performing a quadratic regression on the data can test this
conjecture. To do this, again press the APPS key and select Data/Matrix Editor from the
Apps Desktop, only this time, when the submenu appears, choose 1:Current.
Now press F5 (Calc) and select 1:Calculation Type. This submenu contains the commands
for the various regression models available on the TI-89. Choose 9:QuadReg. Next, designate
column c1 for the x-values and column c4 for the y-values just as you did with the scatter plot.
Finally, scroll down to Store RegEq to, press
, select y1(x), and press ENTER .
Now press ENTER to save your selections. The
STAT VARS dialog box will appear, showing the
results of your regression, which would be the equation
y = –95x2 + 1622x + 615400. At the same time, this
regression equation has been stored in equation in
y1(x) in the Y= Editor. Press
F3 (GRAPH)
to view the scatter plot of the data and the regression
equation graph simultaneously.
Dr. Don Porzio, IMSA
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ICTM 2008, Session 131
We can now use the regression equation graph to
estimate the number of $5 prices increases required
to produce the greatest ticket revenue. Press F5
(Math) and select 4:Maximum. The TI-89 will
ask for a lower bound. Type in an x-value that is less
than the x-value of the maximal point on the graph
(like x = 1) and press ENTER .
The TI-89 will then ask for an upper bound. Type in an x-value that is greater than the x-value of the
maximal point on the graph (like x = 20) and press ENTER . The TI-89 will then indicate the
maximal point on the graph of the regression equation between the input lower and upper bounds for x.
Based on these results it appears that the airline could generate maximal ticket sales by increasing the
ticket price to:
(a) about $404.68 ($362 + $5  8.53684 allowing for any price change based on information
provided) or
(b) $407 ($362 + $5  9 allowing for only an integral number of $5 price increases).
Dr. Don Porzio, IMSA
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ICTM 2008, Session 131
Factoring –95x2 + 1622x + 615400, we see the equation for this problem situation can be written as:
y = –(5x + 362)(19x 1700) or
y = (5x + 362)(1700 – 19x),
which is exactly the equation we would have expected
them to derive (or have derived for students). The
difference here is that the equation should be more
meaningful to them since it was derived from actual data
and in the context of solving the problem.
Here are some other problems that combine the CAS and graphing capabilities of the TI-89.
1.
Graph the function (x) = x2 + bx + 2 for at least five values of b. For each, estimate the
coordinates of the minima of the graph. Use these points determined (and any other minima you
may have estimated) to find a function *(x) such that if (x, y) is a minimum point of the graph of
(x) for some value of b, then (x, y) is also a point on the graph of *(x). Use analytically methods
to justify your choice of *(x). Extensions: Try the same type of activity for the local extreme
values for functions like (x) = ax2 + x + 2 or (x) = x3 + bx2 – x + 1.
2.
Graph each of the following rational functions. Then, use the propFrac( command on the TI89 to write the partial fraction decomposition of each rational function. Describe what, if any,
information the partial fraction decomposition gives you about the graph.
a.
x2
x2  x  2
b.
x2
2 x2  3x  2
c.
3x 2  x  2
x2  x  2
d.
3x2  x  2
2 x2  3x  2
Based on these examples, and others you might wish to try, what conjectures might be made about
the graph of a rational function when the degree of the numerator and the degree of the
denominator of the rational function are equal?
Systems of Equations and TI-89 CAS Capabilities
The TI-89 can solve systems of equations directly
using the solve( command, as the figure at the
right illustrates. However, we can use the symbolic
manipulation capabilities of the TI-89, much like we
did previously, to solve systems of equations in a
manner that can develop, or assess, students’
understanding of the processes involved. Let’s look
at some ways the TI-89 can be used to solve the
system of equations 3x + 2y = 11 and 2x – 5y = 13.
Dr. Don Porzio, IMSA
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ICTM 2008, Session 131
Elimination Method
Begin by multiplying each equation by appropriately chosen values and then adding the equations to
“eliminate” x. Next, solve the resulting equation for y. (I believe at this point, it is acceptable for
students to use the solve( command to determine the solution to a linear equation.)
Once the value for y is obtained, it can be substituted into one of the original equations using the |
key. Finally, the resulting equation can be solved for x. The values obtained for x and y can then be
substituted into the other equation to verify that they are a solution to the system.
Substitution Method
Begin by solving one of the equations for y in terms of x. Then substitute this expression for y into the
other equation, using the | key, obtaining a equation in x. Solve this equation for x. Once the value
for x is obtained, substitute it into the equation for y found previously. These values for x and y can then
be substituted into one of the original equation to verify that they represent a solution to the system.
Dr. Don Porzio, IMSA
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ICTM 2008, Session 131
Solving systems of equations in this manner places the focus on the equation-solving process itself and
not on the algebraic manipulations necessary to complete the process. This makes it possible to assess,
or develop, students’ understanding of the process without the algebra “getting in the way” of students’
success or failure at solving systems of equations.
1.
Solve the following systems of equations in two (or more) different ways:
a.
b.
2.
x + y + z = –1
1 3
 4
y z
2x + 4y + z = 1 3
3 1
 5
x z
Solve the following system of equations:
x – y – z = -15
1 3
 9
x y
a11x  a12 y  b1
a21x  a22 y  b2
(Does this look familiar? Can it be done with a general system for 3 equations and 3 unknowns?)
Series and TI-89 CAS Capabilities
On calculators like the TI-84, you could do summations on a finite number of terms using the sum and
seq( commands (both of which can be found quickly on the TI-89 under CATALOG). On the TI-89,
there is a special ( sum command, which can be found under the F3 (Calculus) menu or under
CATALOG, that also handles such calculations.
What distinguishes the ( sum command from the sum and seq( commands is that it can handle
infinite sums and finite sums with a variable number of terms.
Dr. Don Porzio, IMSA
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ICTM 2008, Session 131
Try a few of the summations shown below and then discuss the implications of hand-held technology
that can perform these types of calculations (such as how such capabilities might be used to help
students better understand certain types of mathematical induction proofs!).
n
n
k
k

ar
 k  k  1
k 1
k 1
n
n
3
k 1
k 1

k
k 0
n
k 
 
2
k 2  

k 1
1
2


k 0
n
 a   k 1d
k 1


1
k!
k 1
n
1
k  k  1
n
k 
 
3
k 3  
k 
 
4
k 4  



 ar
k 1
k 1


k 1
n
1
k  k  2
k 
 5 
k 5
With the last four summations, the results have something to do with other binomial coefficients. The
pattern here can be extended to form one of the neater relationships in Pascal’s Triangle.
Activity: Buying Your First Car
Suppose you are purchasing your first new car. With all the accessories you want, the car will cost
$18,000. The dealership offers you a choice of two deals on a 60-month loan to purchase this car.
Deal A:
Deal B:
$2000 cash back with an annual interest rate of 8.4%
No cash back but an annual interest rate of 3.9%
Which one of these deals will give you the smaller monthly payment?
To solve this problem, we will develop a recursive model for determining the balance due on the loan at
the end of each month. Then we can use this model to determine what our monthly payment will be for
each deal.
Consider the situation for Deal A. Initially, we owe $16000 (after the $2000 cash back). What if we
“guess” that the monthly payment for this deal will be $300. Complete the table below to get the
balance due at the end of the first six months. Note that the annual interest rate of 8.4% must be
converted to a monthly interest rate of 0.084/12 = 0.007.
Dr. Don Porzio, IMSA
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ICTM 2008, Session 131
Month
Payment
1
2
3
4
5
300
300
300
300
300
Balance Due
before Payment
16000
15812
Interest on Balance
Due
16000*0.007 = 112
Based on the information in the table above, write a
recursive model for the balance due, An, on this loan at
the end of the nth month.
Balance Due after Payment
16000+112 – 300 = 15812
1600
An  

if n  0
if n  1
Change the calculator to SEQ mode and enter this equation in the Y= Editor. To view a table of
F4 (TBLSET) and set Independent to Auto, TblStart
values for this equation, press
to 0, and Tbl to 1. Then press
F5 (TABLE). Compare the values for the first six months
from the calculator’s table to the ones in the table above and adjust your model until they match.
Of course, the problem with the recursive model we developed is that we do not know if the $300
monthly payment will pay off the entire loan in 60 months. To determine this, first press
F4
(TBLSET) and change Independent from Auto to Ask. Press
F5 (TABLE) followed
by F1 (Tools), and then select 8:Clear Table and press ENTER to clear the entries from
the table. Now enter 60 for n. The amount you still owe or overpaid will appear in the u1 column.
(How can you tell from the number displayed if you still owe money or overpaid?)
Please record the amount you over or under paid on the loan with a $300 monthly payment in the table
below. Then adjust your estimate for the monthly payment based on this result and change your
recursive model accordingly. Continue this process until you have the monthly payment correct to the
nearest dollar. Record your monthly payment estimates and the amount by which you were off after 60
months in the table below (we will use this later.).
Deal A
Monthly Payment
$300
Amount Over/Under Paid
Repeat this process for Deal B. Recursive Model:
1800
An  

if n  0
if n  1
Deal B
Monthly Payment
$300
Amount Over/Under Paid
Dr. Don Porzio, IMSA
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ICTM 2008, Session 131
Based on the above results, which deal was better and by how much money was it better? If Deal A
was better, then determine the annual interest rate (to the nearest tenth) that would make the monthly
payment for Deal A (nearly) the same as the one for Deal B. If Deal B was better, then determine the
cash back amount (to the nearest $100) that would make the monthly payment for Deal B (nearly) the
same as the one for Deal A.
Extension:
Create a scatter plot of monthly payment versus Amount Over or Under Paid from
your data in your table for Plan A. What do you notice about this scatter plot? Why do you think this
might be true? Now perform a linear regression on this data and determine where this line crosses the
x-axis. How does this value relate to the monthly payment amount for Plan A you determined. If you
aren’t convinced, see if this also works for your data for Plan B. Can this result be proven?
Using AND Not Using the TI-89 During Assessments
No, I did not just contradict myself. Student should and should not be allowed to use the TI-89 during
an assessment. It all depends on what it is you are trying to assess. That is the key issue. If you trying
to assess your students’ understanding of a specific symbolic manipulation technique, then you darn
well better take the TI-89 out of their hands. If you are assessing your students’ ability to turn a
problem situation into mathematical symbols and then correctly solve this problem, then perhaps you
should let them use the TI-89 so they do not make arithmetic or algebraic errors in the problem solving
process that will hamper your ability to assess what you wish to assess.
Using a tool as powerful as the TI-89 as a regular part of assessment will require some rethinking about
what you assess. Clearly, you would not want your students to use a TI-89 to solve the problem
x 2  x  2 x  4 , but you would if you asked them to complete the following:

Describe three (or more) different ways you could solve the equation x 2  x  2 x  4 . Explain
how your solution methods are related to each other. Then choose one of your methods and use it
to solve the equation, being sure to explain the steps you are taking to solve the equation.
Such an assessment gets at more than solving the equation. It gets at understanding how different
representations of the equation are related and how they might be used to solve the equation.
Here is another example of a “different” type of assessment item that I have used in my calculus class.

Suppose (–1) = 0 and that  is the function
graphed at the right.



If h(x) = x4(x), determine if h is
increasing or decreasing at x = –2.
If j(x) = [(x)]4, determine if j is
increasing or decreasing at x = 0.
If k(x) = e(x), determine if k is concave
up or concave down at x = 1.5.
Dr. Don Porzio, IMSA
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ICTM 2008, Session 131
To solve these problems, students can use the TI-89 to determine appropriate derivatives of (x), but
they will still need to understand how to use the information provided in the graph, their knowledge of
functions, and their knowledge of what a derivative really is in order to solve this problem. In other
words, for these problems, the TI-89 is simply a tool that students can use to help them arrive at the
solutions, but it is not going to give them the answers.
Giving students opportunities like these to demonstrate their understanding of the connections between
multiple representations of mathematical ideas and concepts will help in their development into be
flexible thinkers and problem solvers. This does not mean that it is no longer important to assess
understanding of symbolic manipulation skills. These skills are still needed. What it does mean is that
we need to look for new and different ways to assess other aspects of their understandings of
mathematics as well. What follows are my views on the use of multiple representations in the teaching
and learning of mathematics along with a rubric I have been working on (for many years) for assessing
work on problems involving use of technology and different representations. You may find this
information and rubric useful in your own teaching (and I would appreciate any comments and
suggestions you might have concerning the rubric)
WHY USE GRAPHICAL, NUMERICAL, AND SYMBOLIC REPRESENTATIONS
IN THE TEACHING AND LEARNING OF MATHEMATICS? BECAUSE THEY:






are inherent in mathematics;
provide multiple concretizations of a concept;
may be used to mitigate difficulties students have in accomplishing certain tasks;
provide students with more mathematical tools for solving problems (assuming they view the
different representations as tools);
may make mathematics more interesting to students;
can help students develop deeper, more flexible understandings of a concept by having them:
 generate the different representations of the concept and then view and reflect upon common
properties and differences between the different representations,
 “see” the concept or idea from several points of view (i.e., different representations) and then
build a web of connections between the different viewpoints (representations), and
 utilize the different representations to model and solve a problem.
ASSESSING WORK THAT INVOLVES USE OF TECHNOLOGY AND
MULTIPLE REPRESENTATIONS; FOCUS ON STUDENT’S ABILITY:






To use appropriate solution methods (not just their ability to get the correct answer);
To recognize and use different representations (symbolic, numerical, graphical) of a given problem
situations and of concepts related to that problem situation;
To “construct” different representations (symbolic, numerical, graphical) using appropriate
technology (such as the TI-89);
To recognize how the same procedure or concept can be applied to different representations of the
problem situation;
To recognize connections between solution methods that use different types of representations;
To justify their use of various procedures, concepts, processes, representations, and connections, to
solve the problem.
Dr. Don Porzio, IMSA
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ICTM 2008, Session 131
NOVICE LEVEL





Use (at least attempted) of only one type of representation
Use of inappropriate procedures, concepts, or methods
Inaccurate, incomplete, or missing descriptions of solutions
No justification of processes or representations used in problem solution
No evidence of recognition of connections between solutions involving different representations
 Serious mathematical errors present
APPRENTICE LEVEL





Use (at least attempted) of two different types of representations
Use of some appropriate procedures, concepts, and methods
Poorly-developed descriptions of solutions
Vague or unclear justifications of processes and representations used in problem solution
Evidence of vague, unclear recognition of connections between solutions involving different
representations
 Minor mathematical errors present
COMPETENT LEVEL





Use of two (possible more) different types of representations
Use of mostly appropriate procedures, concepts, and methods
Reasonably well-developed descriptions of solutions
Fairly clear justifications of processes and representations used in problem solution
Evidence of at least partial recognition of connections between solutions involving different
representations
 Few, if any, minor mathematical errors present
PROFICIENT LEVEL





Use of at least two different types of representations
Use of appropriate procedures, concepts, and methods
Well-developed, accurate descriptions of solutions
Justifications/explanations of processes and representations used in problem solution
Evidence of recognition of connections between solutions involving different representations
 No mathematical errors present
DISTINGUISHED LEVEL
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




Use of three or more different types of representations
Use of appropriate, possibly unique, procedures, concepts, and methods
Very well-developed, accurate, concise descriptions of solutions
Precise, detailed justifications of processes and representations used in problem solution
Discussion of the non-use of certain methods or representations
Clear evidence of understanding of connections between solutions involving different
representations
Dr. Don Porzio, IMSA
Page 16
ICTM 2008, Session 131