MidSchoolMath National Conference 2016

Squares can fit together in a variety of interesting patterns. Some, like grids and checkerboards, are simple but versatile visual tools for understanding topics like multiplication and area. Others, like the Fibonacci Spiral or the Squared Rectangles described by Martin Gardner, are more intricate, and lead to more advanced concepts.

In this session, we will present a series of interesting diagrams which help lead students to write equations and expressions, first with numbers and then with variables. We begin with diagrams which are puzzles: can you figure out the sizes of the squares without measuring, using only their relationships to each other?

Teachers and students alike have found these puzzles intriguing and motivating. Solution methods range from visual estimation, to guess and check, to numerical reasoning, to algebraic reasoning. All of these solution methods then lead into the second set of diagrams, where our goal is more focused: write expressions to show the relationships between the different square sizes. We consider numeric diagrams first, and then generalize these to algebraic diagrams. We will also have some challenge problems that relate to systems of linear equations, tilings of the plane, and other advanced topics.