1) The document explores difference patterns for squares, cubes, and 4th powers in tables ranging from 0 to 102, 0 to 103, and 0 to 104 respectively. 2) It notes that extending the tables to -10, the patterns continue to hold and asks what pattern could be used to predict the differences in a 5th power table without directly calculating it. 3) It has the reader look back at the exploration from a teacher's perspective regarding lower level skills involved, appropriateness for students just beginning exponents, and gains/losses of using calculators versus doing calculations by hand.

1. Claudia Chavez
Math 301A, PDC
Explorations
11 March 2011
Exploration: Difference Patterns (Page 147)
8a.) Squares
02 12 22 32 42 52 62 72 82 92 102
0 1 4 9 16 25 36 49 64 81 100
1 3 5 7 9 11 13 15 17 19
2 2 2 2 2 2 2 2 2
b.) Cubes
03 13 23 33 43 53 63 73 83 93 103
0 1 8 27 64 125 216 343 512 729 1000
1 7 19 37 61 91 127 169 217 271
6 12 18 24 30 36 42 48 54
6 6 6 6 6 6 6 6
c.) 4th Powers.
04 14 24 34 44 54 64 74 84 94 104
0 1 16 81 256 625 1,296 2,401 4,096 6,561 10, 000
1 15 65 175 369 671 1,105 1,695 2,465 3,439
14 50 110 194 302 434 590 770 974
36 60 84 108 132 156 180 204
24 24 24 24 24 24 24

2. -102 -92 -82 -72 -62 -52 -42 -32 -22 -12 02
100 81 64 49 36 25 16 9 4 1 0
-19 -17 -15 -13 -11 -9 -7 -5 -3 -1
2 2 2 2 2 2 2 2 2
d.) extend to -10do the patterns continue to hold in the extended tables?
e.)Find a pattern in the results from pats abc above that would allow you to predict what
you would find if you made a difference table for fifth powers. To veriry the prediction directly
would be a lot of work. Instead, check youor overall patter in a very simple case, first powers.
9. Look back at questions 8 from the point of view of a teacher.
a. what lower level skill are involved
b. would question 8 be appropriate for students who have just begun to work with
exponents or whose subtraction skills are weak? Could question 8 be modified to make it less
demanding, yet still reach the same goals.
c. what would be gained and what would be lost if students did all the calculations in
question 8 on a calculator or computer?