Texas Instruments Voyage 200 Handbuch

8/14/2003 For more info please visit our web site http://education.ti.com Page 1 of 10

An Introduction to the Symbolic Math Guide

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)

or do other complicated problems, you should use the Home screen

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.

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 Oand 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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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 xwhen 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 Crepeatedly 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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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 ¸to clean up the previous result.

4. Press the up cursor to go up to the original math object 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

⋅

3for 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

≠

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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Problem 4: Compute the derivative

(

)

cos x

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 ƒfor 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.

3. Think about the result (or write it down). Then press ¸to see SMG’s result.

4. Compare results and explain any significant differences.

5. Press †and look for a way to transform the remaining derivative. Choose, for example,

1−

⋅→ rr xrx

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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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 Bor 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( 2xx

d⋅

1. Press †and choose derivative of product.

2. SMG provides a dialog box which requests definitions for the functions fand g. Enter the

information or press ƒfor Help (SMG will provide possible choices for the two functions.)

Press ¸twice to continue to the display.

3. Press …. Press BDto select )sin( 2

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 ƒfor 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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Problem 6: Solve quadratic equation

−

•Press …to try subexpression selection. Press Bfollowed by A. Press D. Try your own

arrow choices. Press …or Nto exit subexpression selection. If 2

xis 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 Cto get back to the step

you wish to change. Then choose a new transformation.

1. Press Cuntil 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 2Cand 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 x

−

1. Press †and choose factor numerator.

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 x

+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.

Problem 9: Solve the exponential equation

1. Press …. Press Dto 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 2xfrom 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 Što verify solution.

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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 hxfhxf )()( −+ , 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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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 ƒfor 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 Bto select 32 −=xand 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 Što 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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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 ƒ

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 ¸until a dialog with f(x), f’(x), g(x) and g’(x) is displayed.

4. Press ¸.

5. Press †and choose  ⋅→⋅ dxxfcdxxfc ))(())(( .

6. Press ¸to simplify.

7. Press †and choose )sin()cos( xdxx →

.

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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