Fighting for Understanding versus Doing

A pretty interesting conversation unfolded in Geometry this morning. We are getting ready to explore similarity, so I gave the kiddos a quick assignment on solving proportional equations with one variable. This was meant to be pure review. When we started talking about these problems I, of course, heard talk about cross-multiplying, cross products, and even heard one student exclaim something about the old keep-change-flip idea. I decided to stand firm and talk about why we were able to do what we do with these proportional equations. We started simply with the equation like  $latex frac{x}{5}=frac{3}{7}$ One of my students was taking a vocal lead in discussing cross products and I asked her what equation to write next. She told me to write $latex 7x=15$. I agreed that this was correct and most of my students recognized what she was doing. I then asked them to pause and wrote the following equation $latex frac{x}{35}=frac{6}{35}$ I asked everyone what they thought the value of x had to be in this situation. They all seemed to agree pretty quickly that x must be 6. So I got them to agree that an equation with one fraction on each side AND the same denominator demanded that the numerators would be equal. They all seemed to think I was making too big a deal out of this. I then asked them if I could do the following to $latex frac{x}{5}=frac{3}{7}$. I asked if I could multiply the right hand side by$latex frac{5}{5}$ while multiplying the left hand side by$latex frac{7}{7}$. One student protested that I need to do the same thing to each side of the equation. I, of course, agreed with her but I asked her to look more closely at what I was doing. She agreed that I was doing the same thing even though it looked different. Most of my students still seemed to think that I was making too big a deal out of this.  Next came the payoff. I picked the following problem from the homework: $latex frac{x}{4}+5=frac{x}{5}+4$ I pointed out that our cross product idea was not really a comfortable fit here. My KCF student quickly suggested that we clear the fractions out of the problem by multiplying by 20. I agreed that this would certainly work but asked if I could try something different. So I wrote the following equation:$latex left ( frac{5}{5}left ( frac{x}{4}+5 right ) right )=left ( frac{4}{4}left ( frac{x}{5}+4 right ) right )$ I was immediately met with resistance. I begged for patience and made them a promise. I told them that I would carefully explain why I was doing what I was doing and that if they unanimously decided that they did not like this approach, then I would cease and desist. I pointed out that we were, again, doing the same thing to each side even though it looked different. I made an argument that multiplying by smaller numbers decreased my chances of arithmetic mistakes and I pointed out that this technique made the common denominator for the problem obvious. The equation became $latex frac{5x}{20}+5=frac{4x}{20}+4$. I saw some signs of visible relief as they saw that this was now a pretty easy equation to process. Combining like terms gave us $latex frac{x}{20}=-1$ and a conclusion that $latex x=-20$. I then solved the problem the more standard way by multiplying everything by 20 to begin with. I felt like it was a bit of a triumph when they voted that this new technique did not need to be banned from our vocabulary. I know that this is not revolutionary, but I certainly think that I made some strides here. My students are well-trained in mechanics and they know what works. I want to have serious conversations about the ideas behind why these techniques work.

Back at it again tomorrow!

PS – Thanks to David Wees and Zach Coverstone for valuable assistance in learning some LaTex for this post. I hope it looks right when I hit publish