Answers
Counting On A Clock
1) 15 (mod 12) = 3
2) 29 (mod 12) = 5
3) 7 (mod 12) = 7
4) 36 (mod 12) = 0
5) -4 (mod 12) = 8
6) 10 (mod 4) = 2
7) 12 (mod 7) = 5
8) 35 (mod 12) = 11
9) 2 (mod 25) = 2
10) 8 (mod 4) = 0
11) -5 (mod 10) = 5
12) -13 (mod 6) = 5
Challenge Problems
a) 5, 8, 11, etc. (There are other correct answers.)
b) The remainder of 4 and 7 when divided by 3 is the same, 1. (Answers may vary on this question.) Positive numbers: 10, 13, 16, etc. Negative numbers: -2, -5, -8, etc. (There are other correct answers for both positive and negative numbers.)
Modular Arithmatic
1) 2 + 7 (mod 4) = 1
2) 3 + 8 (mod 7) = 4
3) 1 + 1 (mod 9) = 2
4) 2 + 3 (mod 5) = 0
5) 4 + 0 (mod 6) = 4
6) 7 - 2 (mod 2) = 1
7) 8 - 4 (mod 3) = 1
8) 4
9) 1
10) 3
Challenge Problems
2 * 3 (mod 4) = 2
4 * 2 (mod 7) = 1
3 * 4 (mod 5) = 2
Modular Multiplication
1) 3 * 2 (mod 4) = 2
2) 7 * 3 (mod 5) = 1
3) 2 * 2 (mod 4) = 0
4) 10 * 3 (mod 7) = 2
5) 0
6) 0
7) 0
Challenge Problems
a) 2
b) 2
c) Multiplicative inverse of 3 in mod 7 is 5. Multiplicative inverse of 3 in mod 5 is 2. Not the same. This is because they are in different mods or they are dealing with different divisors or different cycle lengths (answers may vary).
Multiple Operations
1) 5 * 3 + 2 (mod 6) = 5
2) 2 * 1 + 4 (mod 3) = 0
3) 3 * 4 - 5 (mod 2) = 1
4) 5 * 0 + 2 (mod 8) = 2
5) 5 * 0 - 2 (mod 8) = 6
Linear Equations
1) 2
2) 4
3) 3
4) 8
5) 1
6) 2
7) 2
8) 1
Graphing
Y = 2x + 3 (mod 5)
| X |
Y |
| -5 |
3 |
| -4 |
0 |
| -3 |
2 |
| -2 |
4 |
| -1 |
1 |
| 0 |
3 |
| 1 |
0 |
| 2 |
2 |
| 3 |
4 |
| 4 |
1 |
| 5 |
3 |
Students' plots should match the plot that the model makes.
The graph repeats because it is in mod or can't go above a certain number (answers may vary).
Predictions
| X |
Y |
| -10 |
3 |
| -9 |
0 |
| -8 |
2 |
| -7 |
4 |
| -6 |
1 |
| 6 |
0 |
| 7 |
2 |
| 8 |
4 |
| 9 |
1 |
| 10 |
3 |
Predictions should be repeats of the original graph on each side of the plot.
Y = 3x + 2 (mod 4)
| X |
Y |
| -4 |
2 |
| -3 |
1 |
| -2 |
0 |
| -1 |
3 |
| 0 |
2 |
| 1 |
1 |
| 2 |
0 |
| 3 |
3 |
| 4 |
2 |
Predictions
| X |
Y |
| -8 |
2 |
| -7 |
1 |
| -6 |
0 |
| -5 |
3 |
| -4 |
2 |
| 4 |
2 |
| 5 |
1 |
| 6 |
0 |
| 7 |
3 |
| 8 |
2 |
Solving Systems of Equations
Number 1
| X |
Y |
| -5 |
2 |
| -4 |
3 |
| -3 |
0 |
| -2 |
1 |
| -1 |
2 |
| 0 |
3 |
| 1 |
0 |
| 2 |
1 |
| 3 |
2 |
| 4 |
3 |
| 5 |
0 |
Number 2
| X |
Y |
| -5 |
0 |
| -4 |
1 |
| -3 |
0 |
| -2 |
1 |
| -1 |
0 |
| 0 |
1 |
| 1 |
0 |
| 2 |
1 |
| 3 |
0 |
| 4 |
1 |
| 5 |
0 |
Plots should match the model.
3) Solutions are X = -3, -2, 1, 2, 5
4) Yes there is a pattern. Two solutions next to each other and then a gap of two before the next solution (count three and that one is a solution). X = 6, 9, 10, 13, 14, etc (There are more correct solutions.).
Number 5
| X |
Y |
| -5 |
2 |
| -4 |
0 |
| -3 |
1 |
| -2 |
2 |
| -1 |
0 |
| 0 |
1 |
| 1 |
2 |
| 2 |
0 |
| 3 |
1 |
| 4 |
2 |
| 5 |
0 |
Solutions are X = -5 and X = 5.
There are not enough points to find a pattern and predict other solutions.