Building Block Numbers

From my file cabinet, this wordless worksheet.

This is an activity I definitely got from one of Bob Wirtz’s books. It follows the one I shared a few days ago (Sum and Product.) The idea is that you are to cross the above lines with vertical lines to create some target number of intersections. You cannot stop the lines partway. 

Presumably, the target numbers appeared elsewhere. On the board? On another sheet? Or perhaps students chose their own numbers in a warm-up phase of the activity? In any case, here is a solution if the target is 14: 

Can you get 17 intersections? If yes, how? If not, why not?

Once again, this is a way to make arithmetic basics interesting. (2 ⨉ 4) + 6 = 14, but we can’t get 17 with this particular layout. Because 4 and 6 are even, we can only get even numbers when we add them. Elsewhere on the worksheet, we have relatively prime numbers of horizontal rows (e.g. 3 and 5), which allow for many more possible targets.  

You may have recognized here the problem explored at a greater depth in the so-called McNuggets problem, which appears in three locations on my website: Building Block Numbers, In Addition…, and a full lesson plan co-authored with Amanda Cangelosi. A condensed solution to the general problem appears in a letter I wrote to The Mathematics Teacher in 2020.

— Henri