## Properties of Kites

20 Nov

So kites aren’t specifically listed in CCSS-M.

Congruence G-CO 11

Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

But look for and make use of structure is one of the Standards for Mathematical Practice. And the Mathematics Assessment Project task Floor Plan has kites in it. And my students haven’t thought about properties of kites before. So they enjoyed thinking about a figure that was mostly new to them.

We used the Math Nspired activity Rhombi_Kites_and_Trapezoids as a guide for our exploration. I gave students about three minutes to play with with this page.

Then I sent a Quick Poll to assess what they had observed.

What else do you notice about the kite?

Some students noticed that diagonal KN is a line of reflection for the two triangles that it creates.

Some students noticed that diagonal IG decomposes the kite into two isosceles triangles. Since the kite is two isosceles triangles, we can deduce even more properties.

What if we construct both diagonals?

We get two pairs of congruent triangles.

The diagonals are perpendicular.

One of the diagonals bisects the other.

I get why kites aren’t explicitly listed in the standards but might still show up in the tasks. But I still want to provide my students the opportunity to think about the structure of a kite while they’re not alone on an assessment.

I’ve had conversations with teachers about some of the Geometry Nspired documents giving away too much of the math. One teacher thought that students should construct the kite instead of it already being made. I think it would be great to have time for my students to construct every one of the quadrilaterals according to their definition and then observe the resulting properties. But I’ve decided I’m not going to spend the time that it takes to do that. Instead, my students constructed the parallelogram from the definition. We proved those properties – and then we observed properties of other quadrilaterals that were already constructed. I’m not convinced that full-time modeling is the way to go in my classes yet, but I want to get to modeling. And if I’m going to get there, I can’t do everything else.

A week after this lesson, students successfully worked through the Floor Plan.

And so, whether or not we have the best plan, the journey continues….