In this project, students attempt to create all of the possible array representations for each of the numbers smaller than a given number(which may depend on the amount of time they have to devote to this project).
For example 6 = 2 x 3 and 6 = 1 x 6, so two possible arrays for 6 are:
The objective after they have these arrays is to classify the numbers into groups based on whatever patterns they see in the structure of the arrays.
What observations do you think students might have when doing this project?
This idea comes from Malke Rosenfeld. There are other ways of arranging the tower, so while the diagram above should give you some ideas as to what a multiplication tower is, you and your students should adapt this project. Additionally, once it is built, you can use the tower to look for patterns in multiplication.
A number of years ago, I had a 9th grade math class where I decided we would investigate when the decimal representation of a fraction repeats, and when it terminates. We also decided to investigate to see if there was a pattern in the number of repeating digits when the decimal representation of a fraction does repeat. It took us about two weeks to conclude that we weren’t going to find a pattern without a lot more work, and so we abandoned the investigation, but I still remember the process quite keenly.
If you decide to try this investigation with your own students, you may find this arbitrary precision division calculator (we quickly ran into the limits of our computer’s calculator when doing this investigation) useful: http://davidwees.com/divider/