Solving Algebraically

New hashtag, folks! #physicsteacherprobz. And entry #1 is the student who solves a problem by substituting values first.

What do my kids do when they encounter this?

A simple pendulum has a period of 2 seconds and length of 1 meter. What is the acceleration due to gravity in this environment?

For the non-physics peeps reading, a simple pendulum has a period (T) given by this equation:
bad-but-right-solution

Kids are gonna solve for g like so:

bad-but-right-solution
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To me, that’s so ugly. Substituting numbers in at the beginning and calculating intermediate answers. Kills me to read tests solved this way.

I prefer my physics students solve equations algebraically before substituting numbers for two reasons. One: compounded rounding errors can blow the result. Two: algebra errors are really hard to spot, making the justification of this answer harder to follow. [Of course, the reason "because your teacher wants it that way" isn't sufficient. -MHG]

There is long term value is in describing a relationship in terms of any variable, no matter how the relationship was stated when you were introduced. Right? (Right?)

Solving the pendulum equation for requires a student to apply algebraic rules of inverse functions. Maybe the kids aren’t comfortable with this symbol manipulation. Numbers are concrete, so kids like to plug ‘em in soon as possible.

In all my time in the math classroom, we never had equations of more than one variable, so I think the kids are stunned when they see that simple pendulum equation.

How do I get the kids to see value in solving an equation algebraically before substituting values? #physicsteacherprobz

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5 thoughts on “Solving Algebraically

  1. I think I have an answer here. Rapid fire. Ask students to give you values of T and L and you very quickly David Blaine style can tell then the value of g. I play this rapid fire game and then it justifies solving the literal because it’s so much quicker than solving the equation every time for various values.

    Works for me when I teach solving literals, or often any time I need to justify writing an equation to represent a situation when it’s way easier for one specific set of values.

  2. I always tend to plug in first and then solve myself, but I agree with your two points. I want my physics students to see the value in solving for any variable. So in my algebra class this year, we are going to spend at least two or three days on literal equations. I think it’ll be time well spent later in math and in physics (convenient that I teach both:). Do you get to teach the same group math and physics?

    Marshall – I like that rapid fire idea! I’ll probly use that next week when we do literal equations in algebra.

    • It is convenient when you get to teach the kids both math and physics, unfortunately I don’t. Funny you report substitute then solve as your approach — several colleagues tell me they do the same. I’d be ok with the harder-to-grade papers if the kids made few mistakes in the algebra with numbers part. But I keep seeing stoopid math errors the kids know better than.

      These kids are strong in the algebraic arts. Though a colleague suggested my freshmen (it’s a physics first school) aren’t developmentally ready for symbolic solutions, I think we can do it.

      I’m gonna try the rapid-fire idea of Marshall’s. Also, I’m considering having kids do some problems that just ask for a solved equation — no numbers. Practice is good, right?

  3. I totally get wanting students to be able to solve literal equations (one of my next few topics in fact). However, I would’ve thought everyone else’s students were also more comfortable doing algebra ‘with numbers’ rather than just symbolic manipulations. Perhaps, you should do a couple ‘with numbers’ first then claim to get bored of doing the same things with different numbers (exclaim “This isn’t math class!” in exasperation for dramatic effect even) then claim you’re going to rearrange the formula to highlight the variable of interest so that future problems go a little faster.

  4. I am so glad to see another teacher feel the same way about this! When I taught physics, my students always looked at me as if I had two heads when I told them to solve for the variable first. This was always a particular issue when the variable is in the denominator because the vast majority of kids don’t solve it correctly. It takes some time, but I eventually convinced most of them to solve literally first. Now as an algebra teacher I make special emphasis of solving literal equations.

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