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Today I taught solving systems of 3 equations with 3 unknowns. I’ve taught this lesson numerous times over the past ten years teaching Advanced Algebra with many years like last year with 3 periods of it. Here’s the breakdown and how it went this year. Both classes of it went very well.
1st: I talked/asked questions about solving systems of two equations by graphing and why we graph to find a solution. I think that it is to get a visual representation of what’s going on because it’s rarely the best option by hand. We talked about how each line represents a bunch of points that make each equation true, and how the intersection of the lines is the point that makes both equations true.
2nd: I drew some pictures of planes interesting and told them that they were not going to draw the planes, but the same thing was happening in these problems. A plane is the “graph” of an equation in three variables and the place where the three planes cross is the solution to the system.
3rd: I put up a problem that had 3y or -3y in each of the equations and gave them 30 sec. to make a decision about how to solve the problem. (Their only previous experience was solving with two variables by substitution and elimination.) A few we’re thinking about adding all three equations and one student found a number(yes that quickly) to multiply the middle equation so that when they added all three equations two of the variables dropped out.
I’ve never tried solving it that way although I admit I was thinking about liner algebra when she said it. I said that I was fairly certain that it would work out, but let’s go with another option because it may be too difficult for most people to figure out what to multiply the equations by to make that happen each time. I’m second guessing myself if we should have just went for it, but I think I did the right thing.
I showed them how to add or subtract the top pair and bottom pair to get two equations that have the same two variables. Then I let them use their previous skills to solve the system of two variables that resulted on their own and they mostly all figured to plug the two variables they found back into the originals to get the third number.
4: I put up a problem up without matching terms and asked them to try and make this equation nice like the first one. I gave them 30 sec. and then had them share their ideas with a partner for 1 min. We talked about how you needed to choose a variable to focus on and while it was personal preference, there is some skill in choosing one that may be easier. Students were able to solve this with no problem after this initial discussion.
5: I put up a problem where one of the equations was missing an X term. I made all the y terms have 1 or -1 as the coefficient. I gave students 1 min. to come up with an idea on their own to attack this problem. Then they shared their ideas with a neighbor. They came up with two different strategies throughout the class. 1. Eliminate the y’s 2. Use the two equations with 3 variables and eliminate the same variable that’s missing from the other one. I was honestly a little bit surprised by their great thoughts.
A lot of the time when I try to set students up to discover something I end up having to just show them anyway. I was thrilled that they were just getting it. — do these students know something that others didn’t, or did I do something different that set them up? Not sure I know. It just felt like a great day though… Wanted to record this.
6: I put up a problem with z=1 and the other two equations had two and three variables respectively. I told them to use their creativity and that I believed they could solve it without help it they really went for it.
I’ve been doing a lot of work in groups, but for this lesson they did most work on their own.