What kinds of problems allow elementary-age students an opportunity to experience, dare I say it, the exhilaration of proof? Certainly, there are several classics, which can give young students a sense of the beauty, elegance, and thrill of discovering a proof. The Seven Bridges of Konigsberg is one such classic problem many elementary school students can figure out.
The ideal context is developmentally appropriate, or can be adapted as to present the portion of the context that is manageable for the group, and most importantly provides students with a true taste of what mathematician experience when they are faced with a structure so interesting and surprising they can’t resist trying to explain the how and why!
A few years ago, while working with a group of 2nd graders, I presented an interesting problem that appeared in a New York Times NumberPlay article.
The puzzle asks which numbers can you draw a square made up of that many squares. The squares need not be the same size. See the article for a few examples, or read on to see the results from my students.
This group met each Friday for 30 minutes. At the end of the first session, most agreed that 2, 3 and maybe 5 were not possible. So I asked them to try to find other "squareable" numbers at home, and bring the results in for the next week. Students spent the following two Fridays investigating which numbers were squareable.
Finally by the fourth Friday we had a strong feeling that only 2, 3, and 5 were impossible. I asked them to work together to demonstrate which of the first 25 numbers were squareable. By the end of that session we had the following chart, each student participated in the creation of one or more "squareable" numbers.
In the final session, I showed them the chart and asked them "so what about the next five numbers, are they 'squareable'?"
Students quickly started raising their hands to explain how you could make 26, 27, 28, 29 and 30. The methods they used were so similar that we decided to name the methods as a kind of "operation".
Students proceeded to explain two ways of adding three squares, they dubbed these ways the "banana split" operation and the "gigantor" operation.
One student offered her "banana split" operation which cuts an existing square into four quarters (each a square).
“Banana Split” Operation on 4 to make 7
One student directed our attention to 23, which is basically a 20 with three squares added to make a square four times the area. We called this the "gigantor" operation to formalize it, and decided we could do this with any square we've made to show the "squareability" of that number plus three.
"Gigantor” operation on 20, to make 23
As students wrapped their heads around these two operations, we proceeded with our original question for the day, what about the next 5 numbers are they squareable? Here’s the roughly the discussion that ensued:
Whole-Class Discussion:
me: "Great, now what about 30 - 35, and the next 5, etc?"
students: Yes, they are all possible!
me: "At what point can we tell that the rest of the numbers will be squareable?"
students: "What do you mean?"
me: "When is the earliest number you could stop, and build the rest using your 'banana split' or 'gigantor' operations?"
students: "Eight!"
me: "why eight?"
students: "because using the banana split operation you could build 9 from 6, 10 from 7, and 11 from 8, and so on"
me: "Awesome! let’s record your proofs on paper!"
students: "huh?"
So once I felt confident that they all understood this reasoning, I asked them to write their explanations down on paper. Most struggled to put it into words, many just made diagrams with numbers. Perhaps it was the limited amount of time, or that they were just not ready or practiced in this kind of written task. Unfortunately, our group didn't meet for a few weeks after that, and the kids were ready to move on to another problem.
Nevertheless, I think it’s really important to give time for students to express their mathematical thinking in words and pictures, and honor wherever students are in their journey of mathematical writing, celebrating even half formed “proofs” as a step in that journey. Just as we don’t expect young artists to have perfect renditions, (don’t we hang up our kids’ finger paintings all the same) so should we showcase and celebrate all steps our students take towards developing their mathematical reasoning.
Here are a few examples of what kids made to try to express their proof for why every number greater than 6 is squareable…
While the proof that 2, 3, and 5 are impossible requires quite a bit more experience. These young students were able to collectively come up with an explanation for why there is an infinite set of "squareable" numbers, starting at most 6. That was thrilling for me to watch, I could see the excitement in their eyes as they had the "AHA!" moment. What's more thrilling (in mathematics) than coming up with a simple and elegant argument that applies to an infinite set of numbers?!
Connected Follow Up Problem
We followed it up with a "Coin Problem," which involves similar reasoning to solve. This provides a natural opportunity for students to utilize Mathematical Practice 8 "Look for and Express Regularity in Repeated Reasoning." This generalized problem can get quite complicated, but the basic example is accessible to early elementary. Here is what we started with.
"If you had an unlimited number of 3 cent and 5 cent coins, which values could you make? Which values would be impossible?"
Here is the graphic organizer I gave them: Coin Problem
Try this one out for yourself, and see if you can figure out where students could employ a similar kind of reasoning as in the squareable number puzzle.