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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