Applications for Living — Geometric Reasoning

We are taking a test in our Applications for Living class, and I am struck by two things.  First, students have made major improvements in how they deal with converting rates (like pounds per second into grams per hour). Second, how bad geometric reasoning is, before and after our work on it.

Just about the simplest idea in all of geometry is ‘perimeter’.  Students have very little trouble with a rectangle as a stand alone object.  This problem created a speed-bump:

As a class, we ‘passed’ on that item (in terms of proportion with correct work).

However, we struggled with this problem:

We did not pass on this item, as a class.  The most common error, of course, was counting the ’12 inches’ (which is completely internal to the figure).  Not as many included the ‘8 inches’, which is also internal.  We always say that perimeter is the distance around a figure, but that is not internalized as strongly as the “2L + 2W” rule.

A bonus question on the test looks like this:

This problem combines the reasoning about perimeter with some understanding of right triangles as components of shapes.  A few students got this one right.

We spent parts of 3 classes working on our reasoning and problem solving.  These compound geometric shapes are common objects in our environment (at least in the USA).  I’d like to think that our students would be able to find the amount of trim or edging to install.

We are a bit too eager to pull out a formula for perimeter (where it is never required for sided-figures); when we talk about circles, it’s not connected well enough to other ideas like perimeter.  One of the problems we did in class caused a lot of struggle:

We used this problem as a tool to work on reasoning about perimeter (and area).  Much scaffolding was needed; since we only spent 3 classes on geometry, we did not overcome prior mis-conceptions in most cases.  Our better results with dimensional analysis (rate conversions) is due mostly to the fact that students had few things to unlearn.

Let’s do a little less variety in geometry, with more focus on reasoning.  Formulas are fine for area and all-things-circular, but have no business in the perimeter of sided figures.

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