In a blog post last year, I introduced my Arranging Addends puzzle. Take a look at the websketch below, built with Web Sketchpad. The goal of the puzzle is to arrange the circles and the six numbers (1, 2, 4, 8, 16, and 32) so that three conditions are met simultaneously: The sum of the numbers in the green circle is 21, the sum of the numbers in the blue circle is 26, and the sum of the numbers in the red circle is 14. The numbers can be dragged into the circles, and the circles can be moved as well. Dragging the point that sits on the circumference of each circle changes the circle’s size.
Students quickly discover that the puzzle isn’t quite as straightforward as it looks. The numbers 1, 4, and 16 are needed to make a sum of 21, but the 16 is also needed to make a sum of 26 (2 + 8 + 16). How can the 16 be in both circles at once? The key insight is to realize that circles can overlap each other so that a number can reside in more than one circle.
Below is a twist on the Arranging Addends puzzles that I’m presenting here for the first time. Now, rather than using addends that are powers of 2, the addends are powers of 3, and there are two of each addend. Try solving some puzzles using this new set of addends. Unlike the original collection of addends where there was just one solution to each puzzle, the powers of 3 often yield multiple solutions.
And below, to continue the theme, is yet one more version of the puzzle, this time with powers of 5.
After you’ve solved some puzzles using the powers of 5, press the arrow in the lower-right corner of the sketch to move to a second model. Notice that in addition to the circles and the addends, this model contains boxes for you to input your answers as a code. How does the code work? Well, as a hint, notice that the 25’s sit above one column of boxes, the 5’s sits above another column, and the 1’s above the remaining column of boxes.
Even if your students have never been exposed to working in different bases, they can still understand how these boxes can be used for record keeping. For example, if the green sum is 60, we can spare ourselves the trouble of saying that we need two 25’s, two 5’s, and zero 1’s and simply write “220” (which happens to be 60 in base 5).
My colleague Scott notes that these codes can help students devise a strategy for putting the addends into the circles. For example, suppose we’re given a puzzle where the green sum is 80, the blue sum is 26, and the red sum is 96. First, we write these numbers in code:
A student might then reason as follows to determine how to arrange the circles and addends:
“Let me do the 25’s first. The circles all need at least 1, so I put 1 in the overlap of all three, and then I need 2 more 25’s for the green/red overlap.”
“Now the 5’s. I don’t put any in the overlap of all three circles, but I put 1 in the overlap of the green/red. Now I just need three more 5’s in the red circle.”
“Finally the 1’s. I only need one of those, in the blue/red overlap. All done.”
Younger students might simply solve the puzzles without the code. Older students can be challenged to use the codes to record their solutions, and can also be challenged both to explain their solution strategies and to find ways to make their strategies more efficient. Some students are likely to recognize the correspondence between the codes and the solutions and invent the strategy described above: reading the solution directly from the code.
An annotated list of all our elementary-themed blog posts is here.