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.