Teaching Mathematics with Technology: Exploring PatternsAuthor(s): George W. BrightSource: The Arithmetic Teacher, Vol. 36, No. 3 (November 1988), pp. 56-57Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41193505 .

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Teaching mathematica coith Technology

rz ; ; _ . . i Outputs I

[ExptormgPattern^^^^b

rz ; ; _ . . i boxi Outputs

box* I One of the more exciting outcomes for a student to CZI C3 ГП I experience, or for a teacher to observe, is seeing pat- I terns common to a collection of objects or data. Com- i 1 | | i 1 I puter output and calculator displays can be used to I 1 I 1 I lend opportunities to look for patterns. Some students I are more successful with the graphics output, where-

[ j 1

1 I as other students see patterns in the numbers more I 1 I readily. In either situation, encourage students to pre- I I » П I I I diet what the output will be and then to test their pre- ' ' I dictions. This activity will engage them more in the ■ ' I mathematics of the activity than simply using a trial- I and-error approach. I

Ä . Questions to ask I Logo

Ä Programming

. I 1 . What patterns do you see in the drawings? For I

Here are two Logo procedures that draw boxes: BOX2, what happens when you reverse the order I TO BOX1 :AMOUNT of the inPuts? I

REPEAT 2 [FD :AMOUNT RT 90 FD 2 * 2. Can you make a square with BOX1 ? Can you I •AMOUNT RT 90] make a square with BOX2? I

тгк davo CK|ATU IA -г 3- ̂ hat Pattem of inputs would be needed for BOX2 I тгк T°REPEAf

davo 2E[FD CK|ATU

1ENGTH IA -г

RT 90 FD :WIDTH RT 90] fOr ¡t t0 draW the Same kind °f *"» aS B0X1 ? I

END I _ Ä From Procedures to Pictures I _ From Pictures to Ä Procedures I

Have the students type in the procedures them- I Type these procedures into Logo before class. Give selves. Then ask them to predict what the drawings I the students only the names of the procedures and will look like for the inputs given previously. Ask stu- I the number of inputs each requires. Challenge them dents to act out the turtle's movements as part of this I to find out what kinds of pictures each procedure prediction process. Be alert for misconceptions in ori- I draws. Use input like the following: entation, direction, or distances; for example, confu- I . . sion of right for left or confusion about angle size. I

BOXI §0X2 Variations I 5 5 10 10 10 1. Change the multiplier in the second FD command I

10 5 20 10 20 in the definition of BOX1. For example, use FD 3 I 15 5 30 10 30 * :AMOUNT or FD 4 * :AMOUNT, and so on. I 20 5 40 10 40 2. Use addition instead of multiplication in the sec- I

Prepared by George W. Bright, University of Houston, Houston, TX 77004 I

56 Arithmetic Teacher

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ond FD command of the definition of BOX1 . For I example, use FD :AMOUNT + :AMOUNT. Start witn ° Start with ° start with ° ■

3. Use multipliers in both FD commands in the defi- Press + Press + Press + I nitionof BOX1. For example, use FD 3 * presso press 2 press 3 ■ :AMOUNT and FD 7 * :AMOUNT as the two FD press = press = press = I commands. I

4. Interchange the two FD commands in the defini-

Ones Digits of the Numbers I bone of the two boxes. Ones Displayed

Digits bv of the the

CalcTüalcT Numbers

I 5. Use a multiplier in one or both of the FD com- ■

mands in the definition of BOX2. For example, 5 2 3 ■ use FD 3 * WIDTH as the second FD command. ° 4 6 ■ This change will require inputs like 15 5 to make 5 6 9 ■ BOX2 draw a square. ° 8 2 ■

0 5 ■ BASIC I * I Here are three programs that will generate number ■ 4 ■ patterns: ■ 7 ■ 10 FOR N = 1 TO 5 S ó I 20 PRINT N, N + 1, N + 2 ó ■ 30 NEXT N ■ ■ 10FORN = 1TO5 • I 20 PRINT N, N + 5, N + 10 ■ ■ 30 NEXT N Questions to ask I 10FORN = 1TO5 ■ 20 PRINT N, 2 * N, 3 * N How many different digits does each keystroke se- ■ 30 NEXT N quence produce before it begins to repeat? Щ

Can you write down other keystroke sequences that Щ Sample runs repeat only two digits? Four digits? All ten digits? ■

234 2712 246 Start with 1 Start with 1 Start with 1 I 345 3813 369 press x press x press x I

5 6 7 5 fu Î5 5 fu 15 Press5 Press2 Press3 I press = press = press = В

Questions to think about ■ Ones Digits of the Numbers ■

What patterns do you see? Displayed bv the Calculator ■ Where is the smallest number in the output? The ■

largest number? à q I What is the difference between the largest and small- 5 8 7 ■

est numbers in each row? g -j Щ What is the difference between the largest and small- 2 3 I

est numbers in each sample run? 4 I How could you predict these differences just by look- 8 . I

ing at the programs? . . I

Calculator Questions to ask: I Here are some keystroke sequences to repeat. For How many different digits does each keystroke se- ■ each sequence, write down the ones digit of the dis- quence produce before it begins to repeat? ■ play after each press of the equals sign. Start with 0 Can you write down other keystroke sequences I when the "+" key is used, and start with 1 when the which repeat only one digit? Two digits? Four dig- I "x" key is used. its? All ten digits? m Щ

November 1988 57

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