A tough unit every year in Algebra II is Systems of Equations. They are extremely hard to make relevant even though the idea behind them is very sound and they have a lot of implications later on down the road. But the problem is how FAR down the road they really come in handy. Now, I will argue that the skill of substitution is a valuable asset. If getting a student to understand that math problems can have an element that's completely unknown called "x" is the fundamental principle of Algebra I, then realizing that equations can have unknown elements "x" that themselves represent something like "5y - 10" is the fundamental principle of Algebra II. You know, aside from a 30 second glance at every parent function under the sun. The elmination method reeks of math voodoo though. Every time I'm like "these two system components can be COMBINED and look, the x terms have canceled!" feels like I just made the whole thing up. Reactions range from "ok, whatever you say" to "hold on, what now?" If you sit down and think about it, where else in mathematics is the strategy "let's smash these two things together and hope for the best?" I know it's valid, you know it's valid. But it feels like such an outlier.
This year there were some successes. Previously I've taught subsitution and elimination as two separate concepts and required one or the other for a particular subset of test problems. This year I realized that this is rather pointless, as it's more important to show students that either method is valid and leave the decision making up to them. If the goal of solving two equations for two unknowns, does it REALLY matter how you get there? And that's where I start to get angry at systems.
Are system problems the worst or what? I don't think there's any genre that has to spell out more things than systems of equations word problems. And the scenarios are so brain dead:
On Monday Byron spend $8.00 on lunch to purchase two tacos and one burrito. On Wednesday he purchased one enchilada, one taco, and one burrito and spent $8.50. On Friday it cost Byron $10.00 for two enchiladas and two tacos. If t represents the cost of a taco, b represents the cost of a burrito, and e represents the cost of an enchilada, which system of equations used to find the cost of one taco, t?
I don't know about you, but the system that helps me find the cost of 1 taco is called the MENU. I'm not sure what's worse, the implausibility of the scenario, or spoon feeding the variable names.
I believe in the skills systems are trying to teach. Here are two or more equations, they involve variables that relate to one another in multiple ways, exploit this fact to determine the value of all the variables. Substitution is a noble thing, reinforcing the concept of f(g(x)). I almost want to teach f(g(x)) after substitution because of the pain f(g(x)) inflicted on my students. Pre-Cal kids get the idea no problem, but asking my Alg II's to do the same was like saying "hey, I'm going to kick your puppy now." Due to time constraints, we only ever run substitution in 2x2 systems which is a rather lame implementation. It's much more powerful in something larger.
And it's these larger systems that are really what we should attack. Because, I hate to tell you, unless your friend Byron despearately needs to itemize his taco expenses, the kind of systems you can handle reasonably with subsitution are rare. If the goal of a standardized Algebra II test is to assess a studnet's mastery of Algebra II, don't you want to give them access to all Algebra II has to offer? I know no one administering that test will be looking over their shoulder making sure they solve something via substitution.
Everyone who's anyone that has to solve a system of equations does not do so by hand. This is a fact I really wish someone had told me when I was a sophomore in college, because someone finally showed me a more realistic context for systems of equations:
This is still a contrived problem, but WAY more interesting than our little taco dilemma from earlier. How do you solve something like this? Well, it's a statics problem. Everything in the x, y, and z directions has to balance. You take the known force, break it into 3D components and set up a system that comprises the cable forces and the weight of the tower (oh hey, it only exists in the z direction, that's handy). When I actually worked this problem 9 years ago, there was a ton of algebra involved to run substitution a couple times and arrive at an answer. A semester later someone finally told me what a matrix was and I felt like a sucker for ever doing these problems the hard way. Then like a semester after that you learn engineers take the answers and multiply them all by 5 to keep from getting sued and you get upset that your TA deducted points for not carrying 5 decimal places. My students should not be shielded from the power of a calculator to run a row reduction operation and solve a 3, 4, or 5 variable system in a split second. Now, I can hear the "back in my day"s and "it's good for them"s cranking up already.
I think we spend far too much time romanticizing these dated solution methods in the name of building a better student. If this were true, you'd still be approximating log, trig, and square roots using tables in the back of your textbook. If our job is to prepare a student for the real world, why oh why should we waste a lot of time on methods they will NEVER see? Look deep in your heart, you know the dark secret. You never used a system, and you know most of them never will either.
Keep subsitution as a concept. Keep systems as a concept. Invest the time for them to understand the algebraic relationships that exist in a system. But scrap all the time spent on old world methods. In the age of Google, knowing where to find an answer is becoming more important than the answer itself. If you are teaching systems, invest the time in running through matrix operations. Even a cruddy TI-84 has intuitive ways for entering them. And what's most important, is it's a universal method for solving a system of any size, even the 2x2 ones presented on a state test. I can tell you that this year I dedicated more of my time to the algebraic methods of systems and far less on the calculator portions. I was very disappointed with the results of both. My less able section of Algebra II in particular had such a low rate of retention when it came to translating a system of equations into something computable by calculator. Don't make them wait until they're 20 to find out there was a better way.
What inspired this? Well for one, my growing realization that one of the reasons kids hate math is that we force them to do it the hard way for unknown reasons. Second, when reviewing material for the final exam, a student asked in regards matrix multiplication "are we going to have to show the process?" to which I realized, if the goal of a final is to assess the end result of a semester's worth of work, and part of that work was understanding that matrix multiplication is more efficient on the calculator, what do I learn about the FINAL state of their knowledge by making them pretend this other method doesn't exist?