Just For Fun …

We have a traditional intermediate algebra course, and my classes are currently working on factoring.  Of course, these topics are only appropriate if a student is headed towards a STEM-type field; most of my students are done with this class, so there is a basic mismatch.  [That problem relates to the current work on the Michigan Transfer Agreement, which may take intermediate algebra out of the general education mix.]

However, we try to always have fun in class, and my students know that I don’t mind looking at other ideas.  One of those ‘ideas’ happened today; this is not radical, nor important in our class — but it was just plain fun.

We were working on factoring by use of formulas.  This particular problem dealt with a perfect square trinomial, with fractional coefficients.  Like this:

¼(a²) – (2/3)a + (4/9)

I’ve already told students that we are doing this much factoring just because it is on our departmental final; we are looking at them as puzzles.  This problem got us into looking for squares of fractional terms.  We got through it, and showed the factored form.

So, one of the students says:

Can we clear fractions?

Of course, I said.  “What would you do?” The student replied “Multiply by 36”.  Now, we have been focusing on what I call the 3 big rules of factoring — write as an equivalent product, use integers unless the problem had fractions, and each factor must be prime.  Since multiplying by 36 clearly changes the value, we need to do something to ‘keep it balanced’.  The solution is to show a division by 36:

(1/36) * 36[¼(a²) – (2/3)a + (4/9)]

So, we distributed the 36 and factored the resulting non-fractional trinomial … and kept the (1/36) factor in front.    To me, this was just plain fun; I know most students don’t agree — but at least they got to see somebody have fun with algebra.

This particular issue has been a problem; it seems like a few students would ‘clear fractions’ but without keeping the balance on the assessments for this material.  These students tended to be those I expect to do better — willing to think and reason, trying to connect information, etc.  I’ve not felt okay about just bringing up the clearing fractions method, because most students do not think of it in this context.

I just hope that I have more students like this one, who will be willing to ask a good question … and we can have some fun!

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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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