Monday, 7 July 2014

Cross About Multiplying

You know what I hate? When numbers get needlessly rounded off.

But you know what else I hate? Cross multiplying.




CONTEXT


This is actually a post I wrote in late October 2013. I didn't publish it then because I wanted to add a couple images, to better explain what I was talking about. Then I saw a much better post about Reasoning Proportionally (by @NicoraPlaca), so my urgency for this post dropped way off. Then I got busy. But I (almost) always publish what I start! Because you should never allow someone else posting something to keep you from putting your two cents in! (No, seriously, you shouldn't. Other people should definitely report on conferences like CMEF and OAME. It's not just me, right?)

Anyway, in October, I was teaching 4C math. Specifically the strand "Geometry and Trigonometry". This means I have to at least address "cross multiplying" - by the time students hit Grade 12, they've seen it before somewhere. Thus I do my best to emphasize how "it only works when you cross through an equals sign!" (Don't you DARE try to do this with division, I will cry!!!) And I admit -- the alternative way of solving, namely multiplying each side of the equation to create a common denominator (or simply multiplying by the denominators), tends to confuse more than anything. Especially when students have this other "perfectly good method".

But "cross multiplying" is SO easy to misuse, and basically fails to illuminate what's really going on! Which you may recognize as a complaint against most of the shortcuts in Tina's "Nix the Tricks" book. But here, I wasn't sure (at first) how to make the alternative a more appealing option.


FIXATE ON THE UNKNOWN


In retrospect, my solution seems surprisingly simple. Always put the unknown in the upper left corner. After all, what we're dealing with here is proportions, effectively a rectangle in four parts. Allow me to illustrate the procedure:


You need to convert imperial to metric? Unknown "x" (for metric) in the upper left. This defines the top as metric. Now put the conversion rate on the equation's right side, keeping metric on top. Your KNOWN value should now align with imperial on the bottom. Proportion is set.

Multiply out that known value on both sides.

You need to convert for a scale diagram? Unknown "x" (for your measure) in the upper left. This defines the top as the diagram. Now put the scale factor on the equation's right side, keeping the top as the diagram. Your KNOWN value should now align with real world measure on the bottom. Proportion is set.

Multiply out that known value on both sides.

You need to use the Sine Law, given angles? Unknown "x" (for your side) in the upper left. This defines the top as sides. Now put in the corresponding side-angle on the equation's right (don't forget a "sine"). Your KNOWN value should now align with "sine" of the angles in the bottom. Proportion is set.

Multiply out that known value on both sides.


This is now faster than cross multiplying (or at worst equivalent), since with the latter you often have to re-divide to get "x" alone. Granted, for something like Sine Law you may still need to inverse sine in another step (when angles are the top), but in my mind "multiplying out the denominator" is a lesser evil as compared to a "cross multiply".

This method also forces you to think about how you're setting up the problem, which ideally students should be doing anyway. As it said in Nicora's post, proportions don't apply to every situation. It can also help with estimating, towards asking if the student wants an answer that's larger or smaller.

Now, has doing proportions this way been a SUCCESS? Honestly, I'm not sure. But if even one less person is "cross multiplying" in the end, I'll be happy.


POST MORTEM


I altered that post (which is now #150!) very slightly since the initial writing. Notably I used to say "upper right", but "upper left" makes more sense. It now being 8 months later, I'd also like to say that I have more conclusive proof after the exam... but I don't. For one thing, conversions weren't a big part of the final evaluation. For another, I let them use a help sheet, so maybe they were copying off that. Still, I feel good about this idea, enough to (finally) throw it out there. Feel free to disagree.

Tomorrow: A bit more depth on use of Sine Law.

2 comments:

  1. I love it! Cross-multiplication is a morass ...

    ReplyDelete
    Replies
    1. Thanks! Yeah, it is... I have a personal dislike, mostly because use of cross multiplying leads to MISUSE. (I do allow it if students seem to know what they're doing.) My latest avoidance technique is that, if you already have a proportional relationship set up, but the unknown is on the bottom, to flip everything (take the reciprocal of both sides). Tends to come up with sine law.

      Delete