Wednesday, March 6, 2013

Taking Precautions - A Life Application from the Substitution Method

One of my favorite lessons to teach in Algebra 1 is the lesson on the substitution method of solving a system of equations. If your unfamiliar with the method, but interested, you may watch my lesson of it on youtube. As we talked about it this week, the lesson took a more spiritual turn than I had expected.

One of the first steps in the substitution method is to work to isolate one of the variables in one of the equations. Often you have some choice in which variable you'd like to isolate.  For example, in the system below, which variable would you choose to get alone?
2x + 3y = 11 and x + 5y = 16

If you chose the x in the second equation, then I would agree with you, and that's what I suggest to my students. The choice is so obvious to me, that I struggle sometimes to understand why my students don't see that. 

When addressing some students attempts at solving this with the substitution method, I approached a student who had a completely wrong answer (it's x=1 and y=3 by the way) and started looking through their work to find their mistake(s). After seeing tons of fractions and negative signs, I realized that they had made life much more difficult for themselves then they needed to. They had tried to solve the first equation for y, which produces the ugly y= -2/3x + 11/3 which then needs to be substituted into the other equation and multiplied by 5. Somewhere in the work that follows they must have made a mistake or two, but I couldn't make heads or tails of it with all the horrible fractions.

I told them that they're mistake was choosing to solve for y first, instead of finding x. They disagreed and told me "You said you can choose any of them!" which was correct, but not the full truth. I said you COULD choose any of them, but there is usually a wiser choice. 

In sharing this experience with the rest of the class, I told them how I choose the variable that doesn't have a coefficient in front of them, because I won't have to divide. Division brings fractions, and fractions is just one additional difficulty that can trip us up. I told them how in similar situations, I'll choose an equation with all positives instead of negatives to work with because those negative signs are just one more potential pitfall. 

This lead me to recite the verse from Hebrews saying "Let us throw off everything that hinders, and the sin that so easily entangles, and let us run with perseverance the race marked out for us."  I told them of how sometimes the decisions we make that seem relatively arbitrary often have important consequences, and can put us in paths that might trip us up. Over the years, I remember many times making conscious decisions one way or the other to try to avoid temptations or opportunities to fail: 
  • I tie knots on the ends of my shoe and sweatshirt strings to make sure to keep them from sliding through
  • Our family does not have cable television. Part of that is because we can't afford it, but even if we could, we decided long ago we did not want that to be a temptation because we knew that we would easily fall victim to it and watch hours of television that we don't REALLY want to. We do watch tv, but we decide what we're going to watch and either watch online or go somewhere and watch with others.
  • In high school and college, I struggled with looking at inappropriate material online, and so many weekends when my roommates would be going away, I made them take internet cable away with them, so that I couldn't possible give in to the temptation even if I wanted to. Now web services like SafeEyes, CovenantEyes, accountability partners, and having a centralized computer instead of computers in bedrooms are strong suggestions to help provide internet safety.
  • On a lighter note, I put my keys in the fridge on leftovers that need to come home. Or on my guitar case, so that I won't forget to bring my guitar home from school. I've even once put my keys on my daughters car seat so that I wouldn't leave her behind. (Unfortunately, I have read of parents who had left their child behind in the back seats of cars on hot sunny days because they were sleeping.)
  • I always lock my car with my car keys button, and never lock the car with the button on the door. One time locking my keys in the car at Michigan Adventure and my wife travelling out to rescue me is enough.
  • I now have a toothbrush and toothpaste in my lunch box. While I still haven't developed the habit I'd like (and need) I have increased the likelihood of me making that wise choice to brush after school because my stuff is always there
  • I've removed my facebook shortcuts and apps from my phone and computer. Why I haven't just eliminated my account entirely, I don't know -- but now I have to purposefully choose to visit it instead of just going there out of habit and ending up an hour later wondering what happened. At the same time, I've made it easier to find my Bible.
What sort of things do you do to help yourself remember to do good things, or avoid falling into trouble?

Monday, March 4, 2013

Square Roots as Line Segments

Each of the square roots from 1 to 17
 are shown as the colored segments

Image by Rocky Roer
Made in Geogebra
One thing I wish my students and I had a better grasp of was square roots. We work with integer numbers well enough, but often struggle to make any sense at all of other perfectly valid numbers like square roots. It is unnatural to think of them as numbers, or lengths at all, and perhaps additional understanding could be gained from seeing them like that?

One thing I like to do is realize that each square root really is just a number. Many of them get a bad rep simply because they can't be written in fully written in decimal notation, but they can easily be drawn. Each of the square roots can be constructed quite easily, as illustrated in the picture to the right.  The first 17 square roots are drawn there, simply by drawing a unit length, constructing a perpendicular segment, drawing a unit circle, finding an intersection, and repeating. I stopped at 17 only because if I went further they would have started overlapping and I didn't like the way that looked.

Another thing I have found a little enlightening is a variation in simplest radical form. In PreCalc we've been simplifying expressions like tan(30). That process produces:
That last expression of √3/3 seems devoid of meaning, but is the required "simplified radical form" called for. I have found some enlightenment coming from describing that instead as:
This helps me to have an idea of that length -- it's just a fraction of the √3 length. Other trig segments like √3/2 and √2/2 and such could also be expressed as 1/2*√3  or 1/2*√2 and it's a step towards increasing my understanding of what those segments really are.

In our last unit, sometimes my students found values like sin(15) which is an especially ugly exact value:
Written as 1/4(√6) - 1/4(√2) is not significantly better, but helps me to realize how this number could be constructed. I could take a fraction of the yellow green segment and take away a fraction of the red segment above and make it.  Or alternately, 1/4 (√6 - √2) suggests take the yellow-green segment and chop off a red segment. Save that piece cause you'll use it many times (for secant and cosecants of 15 and 75 for instance) but take a fourth of it for now.

Now if only this understanding would help us avoid the temptation to "simplify" √6 - √2 and make it √4.
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