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Unit 3: Reasoning and Explaining (MP2 and MP3)

# 3.4.5 Student-Generated Math Challenge Led by a student with special needs, the class developed their own challenge: What happens when adding odd and even numbers?

Max, Spencer, and Haley worked on this challenge during their own time.

Max:

Along the way we discovered that theories have been tested; some were successful and some were not. A successful one was even + even = even and odd + odd = even, but even + odd = odd.

even plus even:  4 + 4 = 8 (2, 4, 6, 8) even
odd plus odd:      3 + 9 = 12 (2, 4, 6, 8, 10, 12) even
even plus odd:    4 + 9 = 13 (2, 4, 6, 8, 10, 12, 14) odd

Spencer:

The theory we had was that if you plus a number plus itself, no matter what, you will get an even number.

1785 + 1785 = 3570 … even!
1853 + 1853 = 2706 … even!

The other theory we had was that if you add an even plus an odd number, you will get an odd number no matter what.

19 + 24 = 43
18 + 17 = 35

Haley:

odd + odd = even

1 + 1 = 2         3 + 3 = 6         5 + 5 = 10       7 + 7 = 14       9 + 9 = 18
1 + 3 = 4         3 + 5 = 8         5 + 7 = 12       7 + 9 = 16
1 + 5 = 6         3 + 7 = 10       5 + 9 = 14
1 + 7 = 8         3 + 9 = 12
1 + 9 = 10

Conjecture: For odd and odd will always equal to even.

even + even = even

0 + 0 = 0         2 + 2 = 4         4 + 4 = 8         6 + 6 = 12       8 + 8 = 16
0 + 2 = 2         2 + 4 = 6         4 + 6 = 10       6 + 8 = 14
0 + 4 = 4         2 + 6 = 8         4 + 8= 12
0 + 6 = 6         2 + 8 = 10
0 + 8 = 8

Conjecture: For even and even will always equal to even.

odd + even = odd

1 + 0 = 1         3 + 0 = 3         5 + 0 = 5         7 + 0 = 7         9 + 0 = 9
1 + 2 = 3         3 + 2 = 5         5 + 2 = 7         7 + 2 = 9         9 + 2 = 11
1 + 4 = 5         3 + 4 = 7         5 + 4 = 9         7 + 4 = 11       9 + 4 = 13
1 + 6 = 7         3 + 6 = 9         5 + 6 = 11       7 + 6 = 13       9 + 6 = 15
1 + 8 = 9         3 + 8 = 11       5 + 8 = 13       7 + 8 = 15       9 + 8 = 17

Conjecture: For odd and even, it will always be odd.

This student-generated challenge fits the descriptors of Ball’s reasoning community, stating how public knowledge (e.g., the definition of odd and even) becomes the foundation for new knowledge under construction (e.g., developing rules for adding odd and even numbers). View this final odd/even video of three English learner students describing viable arguments.