An Introduction to the Symbolic Math Guide
15
Purpose of Symbolic Math Guide (SMG): To help students learn algebra and some aspects of calculus by guiding them as they develop correct textbook-like solutions. SMG can be used when a student first learns a topic or as quick review.
SMG is a tool for learning algebra that provides additional help to support textbooks and teachers.
It is not a powerful computer algebra system that can solve large classes of difficult problems. If
you wish to expand (x + y)
of the TI-89, TI-92 Plus, Voyage™ 200, TI-Interactive!™, or Derive™.
Features:
• Work is organized by problem sets. SMG comes with sample problems. Teachers, publishers,
and students can create their own sets.
• Students work problems by choosing from problem-solving options that SMG provides based
on the problem and the current solution step.
• Students learn strategy in the absence of arithmetic mistakes and lower level algebra.
• Students learn to think about math objects, how to classify them, and what transformations
apply to them at various stages of solving problems.
• The transformations suggested are designed to focus on the new material being learned.
or do other complicated problems, you should use the Home screen
A Brief Guided Tour
You will need:
• TI-89/TI-92 Plus or Voyage™ 200 with the Advanced Mathematics operating system version
2.08 or higher)
• Symbolic Math Guide (ver 1.5 or higher).
• Problem Set Tourps.
Let’s start solving problems with Symbolic Math Guide!
1. Press ´ to start the calculator.
2. Press O and highlight Symbolic Math Guide.
3. Press ¸ ¸.
4. Choose Open and then press ¸.
5. Open Tourps, which contains the problems we will use for this guided tour. Note that the
first problem is ready for you to work.
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An Introduction to Symbolic Math Guide
3⋅x+1=x−2
MAIN PROCESS: For each step
A. Think about what you want to do next.
B. Press †to see a list of possible transformations.
C. If you see what you want to do, press the corresponding number and follow the instructions.
Anticipate the result produced by your choice.
D. If you don’t see what you want to do, choose the best choice offered or choose a different part
of the current math object (See below after Problem 5.).
E. If you need help understanding the goal of solving a problem, press [‰:1].
Problem 1: Solve linear equation
for x.
SMG feature: For maximum learning, students should anticipate the result of each
transformation. Writing each step down with pencil and paper helps some students. When in the
TIME TO THINK mode, SMG gives students extra time to write their next step by waiting until
they press
¸
to show the result. Students will probably not want to use this mode when they
are using SMG to review material previously learned. To turn TIME TO THINK mode ON and
OFF, go to Format screen (ƒ: 9).
1. Press † and choose a transformation that will accomplish this goal. For example, press †
and choose subtract ? from each side and type x when prompted.
2. Press ¸ to clean-up the result.
3. Move the 1 to the right-side. Press † and choose subtract…
4. Press ¸ to clean-up the result.
5. Press † and choose divide…
6. Press ¸ to clean-up the result.
7. Press C repeatedly to scroll through your solution.
SMG Feature: As you have just seen, SMG will show the details of a transformation. To clean-
up the result of a transformation without choosing a transformation from †, press ¸. This
allows students to focus on the current new material without having to go through the details of
previous work. If you want your students to handle the details themselves, have them choose
appropriate transformations from †.
8. To move to the next problem, press ˆ:1.
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An Introduction to Symbolic Math Guide
=+⋅
≠
Problem 2: Simplify monomial
32
yy ⋅ .
1. Press [ˆ:1] if you haven’t already.
2. Press † and choose A^U • A^V -> A^(U+V).
3. Press
4. Press the up cursor to go up to the original math object
¸
to clean up the previous result.
32
yy ⋅ .
5. Choose a different transformation. The old transformation and result are replaced.
SMG Feature: SMG has ‘quick keys’ for certain frequently used transformations. For example
press « to add ? to each side of an equation.
SMG Feature: SMG implements domain constraints. When a transformation changes the domain
of definition, new conditions are displayed as part of the result.
6. To move to the next problem, press ˆ:1.
Problem 3: Solve linear equation bxc
3 for x.
1. If you are not sure of the goal of solving this problem, press ‰:1. We want to move the 3 on
the left side to the right side.
2. Press quick key |. Type 3. Press ¸.
3. Press ¸.
4. Press e. Type c. Press ¸. Dialog box “This action …” shows up.
5. Press ¸ to continue. Please note that the Problems statement has been modified to
include the constraint
0
c .
6. Press ¸ to simplify the result.
7. Press ˆ:1 to move to the next problem in the current problem set.
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An Introduction to Symbolic Math Guide
(
)
dx
dx
Problem 4: Compute the derivative
1. Press † and apply derivative of composition.
2. SMG provides a dialog box which requests definitions for the outside function, f, and the
inside function, g. Enter the information or press ƒ for Help (SMG will provide possible
choice for the two functions.) Make sure that the choice is the one you want. Change it if it is
not. Press
3. Think about the result (or write it down). Then press
4. Compare results and explain any significant differences.
5. Press † and look for a way to transform the remaining derivative. Choose, for example,
d
¸
twice to continue to the display.
1−⋅→rr
xrx
.
d
cos x
4
¸
to see SMG’s result.
6. Press ˆ:1 to move to the next problem in the current problem set.
Next, consider several examples that use subexpression selection.
Experts solve algebra problems by recognizing that they can perform a certain transformation on
a particular part of a math object and that this transformation moves toward the goal of the
problem. For example, one might want to factor the left-hand side of an equation when trying to
solve the equation.
REVISED MAIN PROCESS: For each step
A. Think about what part (or whole) of the object you want to change; select it.
B. Press † (Trans).
C. If you see what you want to do, press the corresponding number.
D. If you don’t see what you want to do, choose a different part of the current math object. [See
SUBEXPRESSION SELECTION PROCESS immediately below.]
E. Press † again to see what you can do with the currently selected math object.
F. If you are not sure what you are supposed to be doing, check ‰:1 for a hint or for the goal of
this particular type of problem.
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An Introduction to Symbolic Math Guide
SUBEXPRESSION SELECTION PROCESS: [This information is available under ‰:3. ]
A. To start press … or †: enter subexpr selection.
B. To move to a smaller part of the object press D.
C. To move to an object at the same level press B or A.
D. To move to a larger object press C.
E. To exit subexpression selection press … or F4: exit subexpr selection.
Problem 5: Compute the derivative )cos()sin(
d
dx
2
xx
⋅
1. Press † and choose derivative of product.
2. SMG provides a dialog box which requests definitions for the functions f and g. Enter the
information or press ƒ for Help (SMG will provide possible choices for the two functions.)
¸
Press
3. Press …. Press
twice to continue to the display.
B D
to select
d
dx
)sin(2x
.
4. Press † and choose derivative of composition.
5. SMG provides a dialog box which requests definitions for the outside function, f, and the
inside function, g. Enter the information or press ƒ for Help (SMG will provide possible
choice for the two functions.) Make sure that the choice is the one you want. Change it if it is
not. Press ¸ twice to continue to the display.
6. Press † and choose basic derivatives. Press ¸ to clean-up.
7. Press ˆ:1 to move to the next problem in the current problem set.
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An Introduction to Symbolic Math Guide
x2−3x=4
Problem 6: Solve quadratic equation
.
• Press … to try subexpression selection. Press B followed by A. Press D. Try your own
2
arrow choices. Press … or N to exit subexpression selection. If
x is selected, how do you
select 4?
• Before trying to solve this problem, under ‰:1, find the goal of solving quadratic equations.
Students should be able to factor the left-hand side of this equation once it is in the proper form.
1. Press | to subtract…, Type 4. Press
¸ ¸
.
2. Press … to select the left-hand side.
3. Press † and choose factor. Note how the left side remains boxed so that you can tell what
part of the equation was factored.
4. Press † and choose A⋅
⋅B = 0 -> A = 0 or B = 0.
⋅⋅
5. Press †and choose solve linear equation(s). Note that students don’t need to figure out
how to use SMG to finish this problem. A teacher could require students to provide the
details, in which case they finish by selecting the individual equations and choosing
appropriate transformations.
Let’s try to solve this in a different way. The quadratic formula is useful for solving quadratic
equations.
SMG Feature: Students can perform legal steps that are not necessarily the best steps and
students can use different methods to solve a given problem. So SMG makes it easy for students
to go back to any previous step and try a different transformation. Press C to get back to the step
you wish to change. Then choose a new transformation.
1. Press C until you get to the original equation. (You could use the 3rd equation.)
2. Check † to see if we can use the quadratic formula now. It’s not there. We need to modify
the equation first.
3. Press | to subtract…, Type 4. Press ¸ ¸.
4. Now, choose quadratic formula from †. Press ¸ to simplify the answers.
If you wish, you can go back to the first step, press 2 C and solve this quadratic equation by
completing the square. (You are on your own.)
5. Press ˆ:1 to move to the next problem in the current problem set.
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Problem 7: Simplify rational expression
3
−
2
+
x−1
16x=
4
2
x
1. Press † and choose factor numerator.
An Introduction to Symbolic Math Guide
2x
x
.
2. Press † and choose divide like factors. Press
¸
.
3. Press ˆ:1 to move to the next problem in the current problem set.
Problem 8: Simplify the polynomial
x2+ x + y +1− (x + y).
1. Press † and choose A-(B+C)->A-B-C.
2. Press † and choose order terms.
3. Press † and choose combine like terms. Press ¸ again to clean-up.
SMG Feature: To give the student flexibility to handle special situations, SMG allows students to
select a subexpression and replace it with an equivalent expression. SMG will test for equivalence
of the two expressions. Select the subexpression to change and press the § key.
4. Press ˆ:1 to move to the next problem in the current problem set.
x
Problem 9: Solve the exponential equation
.
1. Press …. Press D to select 16. Press the § key or choose ‰: Rewrite. Type 4^2. Press
¸.
2. Press … to select the left-hand side; press † and choose (A^U)^V -> A^(U*V).
3. Press † and choose A^U = A^V -> …
4. Press † and subtract 2x from each side. Press ¸. Press ¸ again to clean-up.
5. Press † and select factor right side.
6. Press † and choose A•B=0 -> …
7. Press † and choose solve linear equation(s).
8. Press Š to verify solution.
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An Introduction to Symbolic Math Guide
SMG Feature: To give the student flexibility to handle special situations, SMG allows students to
substitute variables for expressions. Select the subexpression to substitute for and press
‰: Substitute
9. Press ¸ to return to the problem.
10. Press ˆ:1 to move to the next problem in the current problem set.
)()( −+
Problem 10: Simplify difference quotient
xfhxf
h
, where f(x)= x .
1. Press † and choose apply definition of f .
2. Press † and choose rationalize numerator .
3. Press † and choose expand & simplify numerator.
4. Press † and choose divide like factors.
5. Press ˆ:1 to move to the next problem in the current problem set.
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An Introduction to Symbolic Math Guide
Problem 11: Solve the exponential equation 032*2)2(
2
xx
=−+
.
1. Choose ‡:Substitute.
2. SMG provides a dialog box which requests substitution information.definitions. Enter the
information or press ƒ for Help (SMG will provide possible substitution information). Make
sure that the choice is the one you want. Change it if it is not. Press
¸
twice to continue
to the display.
3. Press † and choose factor left side.
4. Press † and choose A⋅⋅⋅⋅B=0 -> ….
5. Press † and select solve linear equation.
6. Press ‡ 4: Back Substitute to back substitute 2^x for u.
7. Press …. Press B to select
x
32 −=
and conclude that there are no solutions for this equation
by pressing † and selecting negative = nonneg ->false.
8. Pressing
¸
will get rid of “false”.
9. Press † and choose A^U = B -> … Press ¸ to clean-up.
10. Press Š to verify solution.
SMG Feature: Symbolic Math Guide lets you simplify expressions and solve equations that
contain functions. When you create a problem that contains a function, a dialog box is displayed
that lets you define the function. Transformation “Apply definition of …” applies the definition
of the defined function into the expression or equation.
11. Press ˆ:1 to move to the next problem in the current problem set.
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An Introduction to Symbolic Math Guide
Problem 12: Compute the indefinite integral x ⋅sin(x)dx
1. Press † and choose integration by parts.
2. SMG provides a dialog box which requests f(x) and g’(x). Enter the information or press
for Help (SMG will provide possible choice for f(x) and g’(x)). Make sure that the choice is
the one you want. Change it if it is not.
3. Press
4. Press
5. Press † and choose
6. Press
7. Press † and choose
¸
until a dialog with f(x), f’(x), g(x) and g’(x) is displayed.
¸
.
⋅→⋅ dxxfcdxxfc ))(())(( .
¸
to simplify.
)sin()cos( xdxx →
.
ƒ
SMG Feature: The key SMG transformations to help students learn to solve indefinite
integration problems are substitution, integration by parts and partial fractions. SMG also
includes sum/difference and scalar product transformations as well as transformations for
indefinite integrals of basic functions.
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