I posed the question with a rubric.
Can a unit fraction always be written as the sum of two unique unit fractions?
Rubric
5: Precise proof that demonstrates all cases (abstract, general rule)
4: Reasonable argument that demonstrates some cases (numeric, gives examples)
3: Gut level or weak argument
2: Does not present an argument
1: No evidence of understanding
Students played around with a few unit fractions and after a few minutes we had a couple of them.
Shortly, we had a student come up with an hypothesis:
which was soon followed by another student example:
Uh-oh, that doesn't fit the pattern.
"Does this example wreck our hypothesis?"
This led to a nice conversation on whether this new example and our hypothesis can coexist. It was interesting to see how many students initially thought the hypothesis was wrecked.
We tested a few more examples and shared results--all confirming our hypothesis.
Then I asked, "So where does this put us on the rubric?"
And a student asks, "What has to happen for a 4 to become a 5?"
In other words, when does a numeric (quantitative) argument become abstract [1]?
Had to pause. This one is worth it. So we discussed simple example:
I quickly came up with the question and answers 1, 3, and 4. At lunch I added 2, which really added to the conversation for 6th period.
Which answer provides the stronger argument? Most saw 4 as the strongest and agreed 1 was the weakest. But very few saw 2 on the same level as 4. Then one student says, "I see that 2 and 4 are similar but 4 is just kinda strung out."
Yep, the kid has a feel for brute force vs. elegance. Love it.
By the end, we agreed that 2 and 4 were more abstract and 3 was more quantitative. What about 1?
Well, 1 was what they would've considered a great answer a few months ago.
[1] This is what prompted my question about SMP 2 on Twitter.
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