Students will learn what Pythagorean triples are, and solve some problems involving them. Students should also be shown that all Pythagorean triples can be generated from the primitive Pythagorean triples. After that, show students the formula for generating Pythagorean triples, as seen here. Here's an interesting problem: Have students mentally find the perimeter of this trapezoid:

Next, give your students this challenge: In the figure below, \(\overline{AB} = 9.\) Mentally determine the area of the purple figure.

Here's the solution: First notice that \(\triangle DFC\) is a 3-4-5 triangle. We know \(AECD\) forms a rectangle, so \(\overline{DC} = \overline{AE} = 5.\) From this, and the fact that \(\overline{AB} = 9,\) we know \(\overline{EB} = 4,\) thus \(\triangle CEB\) is also a 3-4-5 triangle. Relocating \(\triangle CEB\) to \(\triangle DFC,\), the purple figure becomes a rectangle, the area of which, is \(\overline{AE} \cdot \overline{AD} = 15.\)

Next, give your students this challenge, then this one, then this one, and finally, this one.