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Thought for the day: One of the most exciting things about "wondering" is that students actively participate in creating the problems that they solve!
Concepts: addition and subtraction properties; number patterns (possibly even / odd numbers; negative numbers)
Beginning
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 wonder how many squares it would take to make bigger “L” shapes.
I notice that I can always make a bigger “L” by joining two blocks (one on each side).
I notice that it always takes an odd number of squares to make this kind of “L” shape. I wonder why.
Exploring
I wonder if I can make the numbers on each side have the same sum.
I notice that when the sums are different, I can sometimes just switch two numbers to make them the same.
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 if there is a pattern to these sums.
I notice that when I make the two sums the same, the corner square is always odd.
I wonder why the corner square is always odd 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.
I notice that when I choose numbers that are not consecutive, I cannot always make the sums the same.
I wonder if I can predict what kinds of numbers to choose so that I can make the sums the same.
Creating
As they notice and wonder, students may create their own versions of the problem by choosing different “L” shapes (or other designs) and numbers. Many of their ideas will flow from things that they have wondered about. For example, they may
Change the number of squares in the "L."
Use shapes other than “L”s.
Change the collection of numbers that they use and figure out the possible sums for each collection.
Search for patterns in the sums and figure out what causes them.
Increase or decrease the numbers by some amount and figure out what happens to the sums.
Figure out why the corner number is always odd in some cases and always even in others.
Try it with negative numbers!
Reflecting and Extending
I notice that math is full of patterns even when I don’t expect it.
I notice that when I don’t know what to do, I can just try something and watch what happens.
I notice that it helps to think of numbers in pairs when I solve this problem.
I wonder what would happen if I made the “L” into a staircase pattern so that there were three or more sums to make the same.
I notice that I can always predict what will happen when I add even or odd numbers.
even + even = even
odd + odd = even
even + odd = odd
odd + even = odd
I wonder what would happen if I used subtraction or multiplication instead of addition.
Notes
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 basic problem suggested by this prompt appears in Linda Jensen Sheffield's book, "Extending the Challenge" published by Corwin Press in 2003. See her book for more ideas.
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 worked on it for three months. We could have continued much longer, but we ran out of time!
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