by Michelle Youngs
The importance of "piddling around" has been explored in this magazine before (see Piddling Around. May/June 1996), and this particular activity is a practical example of the benefit of such piddling. The Puzzle Corner this month is a new puzzle which I developed while "piddling around" with an activity that I happened to be looking at.
The problem I was working on is called Arranging Rectangles and can be found in the May/June 1993 issue of the AIMS magazine, or online (www.aimsedu.org) in the Puzzle Corner section, April 1998. The puzzle involves cutting out seven shapes and arranging them to form a rectangle. The pieces in Arranging Rectangles have a total area of 18 square units, therefore it is possible to create two different rectangles --a 2x9 rectangle and a 3x6 rectangle. As I considered this puzzle, the wheels began to turn. "How boring that there are only two possible solutions," I thought, "there should be at least four!" With this idea in mind I set about to create my own puzzle which would have at least four possible rectangles that could be made from one set of shapes. After piddling around with a sheet of graph paper and a pair of scissors, I arrived at this puzzle which has eight pieces that can be arranged to form a 2x18 rectangle, a 3x12 rectangle, a 4x9 rectangle, and a 6x6 rectangle (square).
In Recreating Rectangles, students are challenged to cut out the eight pieces on the page and put them together to create a rectangle. If they are careful in their search for solutions, they will discover that there are four different sized rectangles which can be made using the eight pieces. As the facilitator of the activity, you should not provide students with this information, but allow them to discover it on their own. The extra challenge that is presented simply makes the problem a little more difficult by taking away the freedom to flip the puzzle pieces. All four rectangles are still possible with this restriction.
Once students discover a solution, they should record it on grid paper. Depending on how many solutions an individual student discovers, he or she may need more than one sheet of grid paper, so have some extra copies of this page on hand. You may also want to make colored pencils or markers available for students to use when they record their answers so that they can easily distinguish between the different pieces in each solution. Students should work alone to discover and record as many different solutions as they can. If desired, this problem can be spread over several days so that they can work on it in sections rather than spending a lot of time on it all at once.
I hope you and your students enjoy this Puzzle Corner.