A Collection of Ideas and Activities

David A. Young

These activities are a collection of ideas designed to either exploit the capabilities of the TI-82 Graphics Calculator, explore "change" in a mathematics class, do stuff, and/or all of the above. All these activities would be useful with students as they prepare for calculus and/or life, and, in some cases, useful once the student has made it into calculus. Some of the activities may be done on a TI-81 Graphics Calculator, or a spreadsheet. The instructions, when specific, will be for the TI-82.

1. Mapping The Doubling Function

In an effort to visualize the effect of iterating the function f(x) = 2x and its orbits [ A First Course In Chaotic Dynamical Systems Theory and Experiment , by Robert L. Devaney, pp. 24-25, 1992, Addison-Wesley Publishing Company ] , you can use the sequence mode or, from the home screen, use the "ANS" key. If we are interested in the unit circle, centered on the origin, we can start with a "seed" of some angle (x), representing a point on the circle. If you choose 72 as the starting angle (seed) then we have f(x) = 2x which becomes f(72) = 2*72 = 144. This then becomes f(144) = 288, and so on as we iterate. When the value of the function exceeds 360, we will want to use modular division on the value { f(x) mod 360 } to get the adjusted angle. This will give you the whole number remainder from normal division. If we continue the iterations with the next value, 288, we get 576 which should be 216 since 576 mod 360 is 216, or 576-360 = 216, which is actually the method we will use on the TI-82. If you could use the mod function on the TI-82, you would continue with larger values of x, but only report values less than or equal to 360. This could lead to a rounding error, which we will also experience in a variation of this method. To do this from the sequence mode on the TI-82, place the calculator in the appropriate mode by pressing , and highlighting the "seq" option on the function line, then pressing also select the "Dot" mode. [The up/down/left/right keys are used to move the cursor around the screen.] Next you will need to place the appropriate function in the "Y=" menu and select an appropriate set of values in the "WINDOW" menu before you graph. Press and key in the function U_n = 2U_n-1 - (360*(2(U_n-1 )>360) ) and turn the V_n line off. To get the U_n-1 symbol press . Now select a window to view the graph in. Use these values: U_n Start = 72; _n Start = 0; _n Min = 0; _n Max = 25; Xmin = 0; Xmax = 25; Xscl = 0; Ymin = 0; Ymax = 375; Yscl = 0. Make sure you're in the "Time" graph mode under "FORMAT" in the "WINDOW" menu. Now press and you will see the orbits. As you trace, you will see a pattern. This may also be seen in the "TABLE", and you may extend the number of iterations by increasing _n Max and Xmax equally. You would now experiment with different seed values, which would be placed in U_n Start.

This same pattern can be seen from the home screen when using the "ANS" key. Clear the screen and key in 72 as your seed. Now create the following code: (2Ans)-(360(2Ans > 360)) and press enter. The value of 144 appears. If you repeat the iteration by continuing to press the enter key you will see the same pattern: 72, 144, 288, 216, 72, .... If you wish to start with another seed, simply key it in and press , then recall the code you used by pressing until it appears on the screen, then press and find the orbits of that value. You might want to try the following values as a seed: 8, 23, 30, 15, and 57.

If you wish to look at the same problem when one rotation about the circle, 360°, is referred to as 1, then we would have 72° become 72/360 which is 1/5 or 0.2. To look at this problem we could enter our seed, now in terms of a fraction of a rotation, from the home screen and use the following code: fPart(2Ans)Frac. This will generate a series of fractions that will show a cycle of 4. Experimenting with other fractional rotations may show a cycle, with some seeds resulting in the aforementioned rounding error. Some values to experiment with: 1/4, 1/45, 1/7, 1/11.

This idea can also be investigated in the sequence mode of the TI-82 by returning to "Y=" and changing the U_n line to equal fPart (2U_n-1 ). You will need to reset your window as follows: U_n Start = 0.2; _n Start = 0; _n Min = 0; _n Max = 25; Xmin = 0; Xmax = 25; Xscl = 0; Ymin = -0.5; Ymax = 1.5; Yscl = 0. When you graph and trace you will see the pattern. If you experiment with different seeds in U_n Start you will get different orbits. The patterns, in some cases, will degenerate due to computer error; to see this you will need to extend _n Max and Xmax equally.

You may also participate in the TI-82's ability to perform more than one operation during one execution, using the ": " option. You can use the "Pt-On(" operation with "PRx(", and "PRy( " and the fact that R = 1, and = (2Ans)-(360(2Ans > 360)), if you are looking at angles in degrees. This will plot the points (x, y) as they would appear on the unit circle, assuming you have the appropriate range and have already entered the seed angle. You will need to press and to get each iteration and use "ClrDraw" when you change seed angles. Make sure you have all the "STAT PLOTs" off as well as all the functions in "Y=" from the function mode. Since we are using angles, the mode should be set to degrees, "Degree".

2. Regressive Gender Mathematics

An article by Nicholas D. Kristof titled "Chinese Turn to Ultrasound, Scorning Baby Girls for Boys" from the New York Times, dated July 21, 1993 reports that pregnant women in China are bribing their doctors to find out, from an ultrasonic scan, if they are going to have a boy or a girl. The bribes range from $35.00 to $50.00, with an abortion thrown in for free if the baby will be a girl. One village reports that they have only had one girl born there in the last year. This trend is reflected in data from the Chinese census. Listed below are the data for the number of males born for each 100 females each year, for several years.

Males per

Year 100 Females

If we look at these data, it seems natural to ask if there is a pattern so that we may predict the number of males to females in the future. When you plot these data on the TI-82, you will see with the scatter plot what looks like a quadratic, or an exponential. By using various regression techniques and looking at residuals, you will not see a good fit. This may be related to a change in the dynamic since 1953 and/or 1964. If you only look at the data after 1964, there really isn't enough data to examine. You could find additional, more current, data and continue the analysis.

Questions to investigate:

• What if the trend continues? What are the implications to society, lifestyle, and work?

• What are the data for your city (state, country, other countries) of the number of males and females born each year?

• Would a decreasing, or increasing, birth rate have any effect on the data, or your predictions?

• How many doctors in your city use the ultrasonic scan and how do the future parents react? Any difference in reaction by the mother or father?

• Why do people have the ultrasonic scan done? Would you have one for your pre-kid? Are you male or female?

• Who gives more of the ultrasonic scans, male or female doctors?

3. Algebra On The TI-82

Just as the TI-81 opened up the minds of both students and teachers in Algebra I, as well as changed the curriculum, so will the TI-82. The following are some ideas to help start, and stoke, the fire.

Order Of Operations

A major problem with students' understanding of algebra is their lack of a feel for "Order of Operations". The TI-82 has a logic menu that contains, among other things, the operator "and". If you use this, from the home screen, you can create a series of statements that represent the steps of simplification of an expression. If one strings these statements together with the "and" statement, the calculator will report that the simplification was correct, or not. The "and" will report a value of "1" if the statements on either side of it are both true, "1". If either side is false, the "and" reports "0", which is false.

Suppose that you have the expression 33 - 3(5+4) to simplify. The following code, on the TI-82, would be a representation of the steps in simplification.

(33-3(5+4))=(33-3(9)) and (33-3(9))=(33-27) and (33-27)=(6)

Once these symbols are keyed in and the user presses , the TI-82 will report the value of "1" for true. That is, the simplification was done correctly. This exercise can be expanded to more sophisticated expressions. Some additional benefits of this activity include: giving the student experience with working with parentheses, and having the student become familiar with the TI-82 keyboard.

Another way to investigate this concept, as it occurs in simplifying expressions with variables, is to place, in a list, the values of two or more equivalent expressions (the original and the simplified forms). For a simple example, let us look at 4(X + 2). We would expect the student to write this as 4X + 8, probably because we want her to undo this when we teach factoring. To do this on the TI-82, use the "seq(" option under the "LIST" menu. This is obtained by pressing the keys, and selecting option 5. The code we will use to place some values of X in list L₁ is: seq(X,X,1,25,1) -> L₁. We then need to place a list of values into L₂ and L₃ for each of the expressions, 4(X+2) and 4X + 8. We could do this using the "seq(" command in the home screen, as before, but let us try a different method that is quicker. Go to the list menu and select "Edit". Move the cursor to the L₂ column and up into the heading of the list. Now key in 4(L₁+2) and press enter. This will fill the L₂ column with values of the expression, using the listed values of L₁ for X. Repeat this technique for the L₃ list using 4(L₁)+8. Now, if the student moves up and down the list, she will see the values in L₂ and L₃ are the same. This is only a sample of values, but it shows, for that domain, that the expressions are equivalent. This technique can be used for additional forms of an expression. There are 6 lists, 5 that the student could use. You could also create different X values, in L₁.

Value of a Function

The TI-82 can be used in several ways to show the value of a function. This concept seems to elude many algebra students, resulting in problems in later courses. If we have the expression "3X² - 5X" and we are interested in its value when X=4, such as in the situation where X is the score on the AP Calculus Examination when there are "3X² - 5X" hours of study per week. On the TI-82 you can place the expression in the "Y=" menu and with a good window: Xmin = -8.8; Xmax = 10; Xscl = 1; Ymin = -10; Ymax = 25; Yscl = 1, you can trace to the point that X=4 and see that the value of the function is 28.

Another method would be to get the table of values by pressing . Moving up or down the X column, you will see different values for X and the value of the function, assuming that you have placed the expression in the "Y=" menu, as before, and depending on your table set-up, "TblSet". Again the student will see a value of 28 when X is 4.

The next method is performed in the home screen using the TI-82's function notation. One may create the expression f(x) on the calculator and collect values such as f(4). To do this, we will look at Y₁(4), and press enter. Again we assume that the expression is in the "Y=" menu under Y_1. The key-strokes, from the home screen, are as follows: . You will see the Y₁ on the screen. Now key in the value desired, 4, in parenthesis so that you have Y₁(4). If you then press you will get the value of the function, 28. This is different from the TI-81, which would, given the same key-strokes, multiply the value of the function, Y₁ , for whatever value was stored in X times 4. That result would have been f(x)*4.

The final method suggested is also created from the home screen. The student will key in the expression 3X² - 5X, replacing each X with the number of interest, in this case 4. The screen should look like this, before enter is pressed: 3(4)² - 5(4). This results in the same value, 28, as before.

If a student is exposed to these four methods, and is allowed to explore, with calculator in hand, the idea of the "value of a function" will have been presented in several different styles to her. This might result in a greater, and longer, understanding of the concept.

Undefined Values and Domain Restrictions

The TI-82's "TABLE" can be used to help show a student the undefined values of a function. If the function is placed in "Y=" and the "TblSet" is on the correct step, "Tbl", the table will show "ERROR" at those difficult points. The "nice" range on the TI-82 graph is a multiple of 94 for X and a multiple 62 for Y. But this is not needed when using the table. In the case of

f(x) =

we will find that when x is 5, the table reports the error.

The "TABLE" can also be used to search for needed values, as is a common need in the Computer Intensive Algebra program from Penn State. The student would place a function in the "Y=" list, preferably a function that has some meaning, and then use the table to search for a particular value, either in the X or Y column. If f(x) = 5x² + 4, where x is the number of puppies sold at a market in a day, and f(x) is the length, in centimeters, of their left ears, when placed end-to-end, the question might be "How many puppies do you need to sell to get enough ears to reach the nearest dogwood tree, which is 251 centimeters away?". If you set your "TblMin", under the "TblSet" menu, to 0, and set the step, "Tbl" to 1, the table will reveal that the value is between 7 and 8 puppies. If you return to the "TblSet" menu and change "TblMin" to 7 and "Tbl" to 0.2, your table will zoom in on the value needed. Repeating this you should find a x value near 251 cm using 0.01 for "Tbl". The discussion might lead to the prudence of reporting your answer in parts of puppies, and if this is OK, were the parts sold with the ears or not!

Solving Equations with Variables on Both Sides

When students are learning to solve equations that have variables on both sides of the equal sign, the TI-82 lists can be of use. If your problem is to solve 2X - 10 = 11 - X, you can place these in the list and have them evaluated for a series of X values. From the home screen on the TI-82, use the following code: seq(X,X,1,25,1) -> L₁: 2L₁ - 10 -> L₂ : 11 - L₁ -> L₃ followed by enter. This will place a set of X values in list 1, and appropriate values for each of the expressions in list 2 and 3. By looking at the lists the student will see that the two expressions have the same value, 4, when X is 7, so the solution, and the value the student would get when she checks it, appear in one fell swoop. If you select a set of expressions in the original equation that don't have integer solutions between 0 and 26, you will need to repeat the above code, changing only the L₁ list. You might want to look at the two expressions between 7 and 8. You might do this after seeing the list for values of X from 1 to 25, that is, seeing that the two list did not have an equal value, but noted a "change" between values of X at 7 and 8. From the home screen press , and this will bring back the code listed above. If you have done some other calculations on the home screen, you may need to call back "ENTRY" several times to get the right code. Once you have the code retrieved, edit it by moving the cursor to the segment "seq(X,X,1,25,1) -> L₁" and strike over it, leaving that segment looking like: seq(X,X,7,8,0.1) -> L₁ . The rest of the code should remain unmolested, and when you are ready, press . This will recalculate, as before, with a new domain

Again, this visualization might open up the students to some insight to the world of algebra.

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