Thought for the day: One of the most exciting aspects of "wondering" is that students actively participate in creating the problems that they solve!

Examples of noticing and wondering

I notice that the five numbers are "in a row" (consecutive).
I notice that the squares make an upside-down "L" shape.
I notice that there are five numbers and five squares. (One number could go into each square.)
I notice that you can make the numbers on each side have the same sum.
I notice that when I make the two sums the same, the corner square is always odd. 

I wonder how many ways there are to make the numbers on each side have the same sum.
I wonder how many different equal sums I can make.
I wonder why the corner square is always add when I make the sums the same.
I wonder what would happen if I made the "L" shape have four squares on each side.
I wonder if I could still solve the problem if I added (or subtracted) 1 (or some other number) to each of the numbers.


The idea of filling numbers in a diagram to make sums (or products) come out the same is a common type of math puzzle. The problem suggested by this prompt appears in Linda Jensen Sheffield's book, "Extending the Challenge" published by Corwin Press in 2003. Her book has many wonderful ideas for extending the problem.

Problems like this can go on and on! I once worked with a group of second graders who kept thinking of so many new questions to ask that we kept working on it for three months.

  • We changed the number of squares in the "L."
  • We changed the collection of numbers that we used.
  • We figured out the possible sums for each collection.
  • We looked for patterns in the sums and tried to figured out what caused them.
  • We asked what happened to sums when we increased or decreased numbers by some amount.
  • We tried to figure out why the corner number was always odd in some cases and always even in others.
  • We even tried it with negative numbers!