The goal of this activity is for students to follow directions to build several right triangles in order to become familiar with the process and form initial opinions (MP1), such as whether or not the instructions will always work. The following activity builds on the work done here, asking students to write and then reason about expressions representing the different sides built from the two chosen integers.

Monitor for students making labeled sketches of their triangles and using the converse of the Pythagorean Theorem to check that the triangle is an actual right triangle.

The purpose of this discussion is to record a variety of triangles that work using the directions and at least one that does not work. Additionally, make sure the converse of the Pythagorean Theorem (if for a triangle with sides , , and , then the triangle must be a right triangle) is stated during the discussion since students will need to use it in the following activity.

Select 2–3 previously identified students to share their reasoning about whether or not the instructions will always work. Record for all to see the integers students pick and the triangles they do, or do not, create.

If not brought up, ask, “Are there any restrictions on the two integers chosen at the start of the instructions?” (They cannot be the same value since that makes one of the sides of the triangle equal to 0. They also must be positive since side length is always positive.)

Building on the work done in the Warm-up, in this activity students write expressions for the different parts of the instructions and then use the Pythagorean Theorem to show why the instructions work for making right triangles. The goal of this activity is for students to recognize that the equation they create is an identity and that, for certain restrictions on the two chosen integers, the identity can be used to generate Pythagorean Triples.

Monitor for students who answer the last question by proving that is equivalent to in different ways, such as:

• Use the distributive property on each expression to rewrite them without parentheses.
• Rewrite one of the expressions in order to prove its equivalence to the other.

Use this activity if students need more practice rewriting expressions.

The purpose of this activity is to bring together identities and the work students did earlier with rational equations. For each question, students first rewrite the fraction using the given formula and then prove that the formula is an identity.

The whole-class discussion at the end of the activity should focus on how students showed that the formula is an identity. Monitor for groups who use different methods to share during the discussion. For example, they may identify different common denominators when adding together the fractions in the expression on the right side of the formula.