# CAS and Dynamic Geometry Activities that Integrate Algebra & Geometry: Investigate, Discover, Prove

• Tom Reardon

## Abstract

This is a cool way to learn … and teach: have your students investigate a geometry activity – hands-on, interactively, self-paced; then have them test and generate their hypotheses; and then have them prove their results using CAS. They are still doing the thinking, the logic, the investigating, the problem solving, but let CAS assist them to do the difficult mathematics. Lots of great mathematics in these activities. And some surprising, but fascinating results.

We will do as many investigations as time permits using the TI-Nspire CX CAS handheld. Here are the activities that I will choose from:

1.    The Square Problem. Given a square with both diagonals drawn.
Find the midpoints of the 4 "half-diagonals".
Draw the 4 segments connecting each vertex with a different midpoint.
a) Of what type is the resulting figure?
b) What is the relationship between the area of the original square and the resulting
figure?

2.    Midpoint Polygons an Investigation. Given a regular polygon, calculate its area and perimeter. Then construct the polygon by connecting consecutive midpoints of the sides of the original polygon. Is there a relationship between the: a. ratios of the perimeters of the two polygons?  b. ratios of the areas of the two polygons? Is so, state and prove those relationships.

3.    Other Square Problem. Given a square. Construct the midpoints of each side. Draw segments connecting each vertex to the 2 midpoints not on the sides that contain the vertex.
a) What is the best descriptor of the type of polygon that is formed.
b) Is there a relationship between the area of the square and the area of the polygon?
If so, what is it?