Habits of Mind: Looking for Patterns

October, 2010

The CME Project Algebra 1 course opens by encouraging students to look for patterns in addition and multiplication tables. The tables are based on the Cartesian coordinate system, so that at each point (x,y) in the multiplication table shown below there is a circle in which students fill in the product of the coordinates, xy.

After filling in this table, students look for patterns.

Many of the patterns that students find in the multiplication table exemplify basic properties of multiplication. For example, multiplication is commutative; you can multiply in any order you want. (3 x 4 is the same as 4 x 3.) Because of this property, the table is symmetric along its main diagonal. (You can fold it in half and all the entries match up.) Other patterns are less obvious, and students are challenged to explain patterns clearly and to prove that their patterns hold.

An example of a less-obvious pattern is that for the four entries in the table at the corners of any rectangle, the products of the diagonals will be equal. Here are some examples:

To explain this pattern, students use algebraic notation and the way the table is constructed to represent the products at the corners of any rectangle.

Finally, students use the any-order, any-grouping properties of multiplication to show that the two diagonal products are, in fact, equal.