What secrets are hidden in squares? While the Pythagoreans are best known for their triangle theorem, they were in fact religious cult who believed in the god of mathematics. Pythagoras and his followers believed that everything in the universe could be described as a ratio of two numbers...and tenet that was quickly challenged.

In this lesson, students use concrete models to explore square numbers and square roots and confront the philosophical and moral questions posed by the existence of irrational numbers.

### Students will

• Develop a geometric understanding of square root; reason about and find square roots of rational numbers
• Identify the "square root of 2" as irrational
• Use rational approximations of irrational numbers and locate them on a number line

### Before you begin

Students should know that a square’s length is the same as its width, and they should understand how to find the area of a square by counting the unit squares that comprise it. Students will need to order and compare decimals. It’s helpful, though not essential, for students to know how to calculate a square’s area by multiplying its length and width; this relationship between area and width is explored and reinforced throughout the lesson. Students need not have any experience with square roots or the notation for squares or roots before doing this lesson: in fact, square roots and their notation are formally defined only at the end of the lesson, after students have plenty of experience with them in a geometric context.