One of the skills that’s key to number sense, and is even more evident in many number talks, is flexibly breaking numbers apart and putting them back together. For instance, when asked to multiply 21 by 15 mentally, students who are most successful think about it as 21×10 + 21×5, or 15x7x3, or 21x5x3, or 15x10x2 + 15.

These skills — formally, factoring numbers and using the commutative, associative, and distributive properties — are often glossed over in impenetrable language, or memorized for a test as 7×5 = 5×7, and then forgotten. And the skills are subtle ones that are hard to teach in a single lesson — they are skills that make math easier, but are rarely absolutely necessary to solve a problem — only to solve it well, or solve it in a new way.

I recently came across two resources that I really like to address these skills. From Don Steward, these puzzles:

and from Visualizing Math (although this was all over the internet when I searched for it, I just saw it first there), the chicken nugget problem:

These both struck me as questions that a) don’t fit neatly into any middle school math objective, b) have embedded in them incredibly rich practice breaking numbers apart and putting them together, and c) are *puzzling*.

This math gets at the hazy, nebulous idea of *concept development *that is so hard to facilitate and plan for. In particular, finding a place for these problems so that students can access them, but still find that sweet spot to develop the concepts that students need to be thinking about as they dive deeper into mathematics.

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howardat58I wouldn’t be overly concerned about objectives as the first problem gives many opportunities to develop “Standards in mathematical practice”. Numbers 1,2,3,4,6 and 7 to be exact. (I re-read your earlier post). Anyway I had some fun doing them myself. Explaining the reasoning could be a nice challenge (if guesswork or sledgehammer methods were not used).

dkane47Post authorDefinitely agree that these should exist outside of objectives — I’m contrasting these activities with the more “objective-driven” approach of many schools’ math departments, including mine. I think these types of activities are often overlooked because they don’t “fit”.

To push a little further, I think these provide an excellent opportunity for MP.8 as well — Look for and express regularity in repeated reasoning. For instance, after solving a number of the multiplication squares, students may infer that they can identify where the odd numbers are by patterns in odd products. Extrapolating beyond that, they have the opportunity to realize that because 5 and 7 are unique in the prime factorizations of numbers 1-9, those positions can be determine uniquely as well. There are many more examples that students can grapple with — and I think they are most likely to struggle with those ideas in contexts that lend themselves to student-motivated problem solving.