## Monday, February 10, 2014

### Why Precision Matters

The following home workshop was used with preservice teachers in an effort to highlight the importance of attending to precision when doing mathematics investigations.

One of the focuses (foci?) of the workshop was on Attending to Precision -- Standard #6 of the CCSS Mathematical Practice Standards.

One natural way this arose in the workshop and follow-up activity was in considering the tools we had available. The angle ruler we use has a 5-degree increment scale with an overlay of 1-degree increments. These markings are quite small and close together, so it can be challenging to obtain an accurate angle measure that is precise to the nearest whole degree. A protractor, being somewhat larger, can make it easier to get accurate results at this level of precision (nearest whole degree). There was an element of strategic competence / phronesis underlying that conversation.

But a more subtle connection to Attending to Precision arose in the follow-up activity -- it was not one I had anticipated.

Here's how it went:

Each student brought to class a drawing of a hexagon (not necessarily convex) that they had measured the vertex angles.

Their angle sums varied from 690 to 742 degrees. I was surprised -- that's quite a spread, and one I had not anticipated.

 The "dissect-into-triangles-method" (source .pdf)
We recorded the results in a chart on the front board, and then used the dissect-into-triangles-method to prove the angle sum must be 720.

We looked back at our chart. How were our measurements so far off? There were only six angles, but they were off by as much as 30 degrees. We couldn't have measured that inaccurately, right?

The results were the same, more or less. Some students began to question whether our proof was misleading in some way, and that the conjecture was more of a rule of thumb than an invariant fact.

Upon closer inspection, we realized our drawings were flawed -- or more to the point, they were drawn imprecisely. Have a look at the hexagon below.

First let's clarify: The figure is a hexagon, or at least it would be if drawn more precisely. A hexagon is a polygon with six vertices. This is a hurdle that takes some modeling to overcome, because most students imagine a regular hexagon when they are asked to consider a hexagon.

By the way, Christopher Danielson has a nice series of posts on the hexagon hierarchy.Worth checking out.

The fact that this hexagon was not particularly well drawn really makes a difference in our angle-measurement context. The sides are mostly curved inward slightly, and that makes the corresponding vertex angles somewhat smaller -- in some cases, significantly smaller.

Have a closer look at the angle I've measured:

The angle at the intersection of the poorly drawn green lines varies from the correct angle by roughly 21 degrees! That's an error of nearly 25% of the correct measure.