Algebra 1: Systems of Equations

We are finally getting to move beyond basic graphing and finding equations of lines.  It was a long slog, but the skills tests show that the majority of my students are starting to get the hang of it.  I always look forward to the systems of equations unit, because it is a chance for students to synthesize what they have been learning all year - and, in a situated context, no less.  My plan this year is to deepen the emphasis on representational fluency and summarizing, to help build all of those neural bridges we want the students to have.  We started the unit Monday, and I was really blown away by my classes today - all of a sudden, I have students doing algebra!  I had them solving systems in pairs, using mini-whiteboards, where one does the graphical solution and the other does the algebraic solution, and then they compare their answers.  They did a great job, and it wasn't until this activity that many students realized the answers should be the same.  I got a couple of those hilarious, indignant "you should have told us!" comments.  Next week is winter break, which doesn't come a moment too soon; however, I'm worried about how much will be lost over the seven days that nobody is asking them about starting points or rates of change.  No matter, it's worth it to have a rest.  Here are a couple of  examples of what we're doing, and the links to the lesson materials thus far.

Lesson 1 (Intro to Systems of Equations)  doc / GeoGebra files / Keynote / Powerpoint
Lesson 2 (Solving y = mx + b Systems)  doc / Keynote / Powerpoint
Lesson 3 (Practice Solving Systems)  doc

Algebra 2: Lines and Systems

I'm trying to cram in a unit on systems of equations and inequalities before break. It's hard, since so many students are still not totally comfortable with graphing linear functions. But we're making progress. We're up to solving 3x3 systems with linear combination, and most of them have got the idea. These problems are huge, and are probably the longest routine problems my students have ever done. This is cool, because it makes them feel smart and accomplished when they get one right. Unfortunately, a single arithmetic or copying error (which happen all the time) can crumble the whole thing, and then the frustration is back again, eating away at their self-confidence. I'd like to get to graphing systems of linear inequalities before break. When we get back, we need to review for the final, but I'd really like to do some work with linear programming problems first. Here are the files from the last few lessons.

Lesson 1 (Linear Functions) / Keynote / Quicktime
Lesson 2 (2x2 Linear Combination) / Keynote / Quicktime
Lesson 3 (2x2 Word Problems) / Keynote / Quicktime
Lesson 4 (3x3 Systems)
Lesson 5 (Systems Practice)

Next Lesson: 3x3 Systems Word Problems (and Thanksgiving!)

I just finished the class - last period of the last day before Thanksgiving break, right after a special double-length lunch with music, dancing, and staff v. student volleyball and soccer. Given all that, the class went amazingly well. They were wound up at first and were having "focus problems" (my favorite euphamism), but when we got to the competition, they really got down to work. So the lesson..

After reviewing the homework, students did a warm-up of solving word problems with 2x2 systems. They still have trouble with converting from verbal statements to equations for things like "my second number is 3 more than 6 times my first number". To help them through those, I ask them to try a specific numeric example (reducing the level of abstraction by that step seems to help them see the relationship). I say "imagine my number is 10 - what is 3 more than 6 times 10?" They say 63, and then I ask them how they figured that out. That is usually enough to get them to write out the equation.

After this, we worked together on a word problem that uses a 3x3 system as its model. In this example, the last equation was in a different form (where one variable was already isolated), and they saw how substitution would let them create a 2x2 system, instead of just using linear combination as we had been doing. I'm trying to get them to be as flexible as possible in their problem solving; there are different methods, and you should pick the method that applies best to the given problem.

This left about 35 minutes for students to get into groups of 3 and do a word problem relay. Each group got problem 1 to start with; when they solved it, they brought it to the front for me to check. If they were correct, they got problem 2, and so on. There were 4 problems - I told them they'd get 5 bonus homework points if they solved all 4. At this point, most of the students worked really well (except for one chatty group) and most groups got to the third or forth problem, but none solved the forth in time.

For homework, I gave them a review packet for the Midterm, which will be next Tuesday, when they return. You can imagine how excited they were about that! It's five pages long, and I suggested that they do one per day to keep their math brains running.

It was a nice day today overall, and this little break always comes just in time to keep us all (teachers and students) sane and friendly.

Next Lesson: 3D Graphing (better than the PS3!)

In this lesson, I will take the class to our computer lab so we can work with the OS X Grapher application. This is the program I've been using to generate all the 2D graphs I give them on worksheets. It can do 3D graphs as well, with nice lighting and rotation. You can even graph implicit functions and relations - I like plugging in random equations to see what kinds of shapes it can make.

We'll start by looking at the 3 axes and see what the planes x, y, and z = 0 look like, as well plotting an ordered triple.


Then, I'll show them what the graph of ax + by + cz = d looks like, and how three planes can intersect at a single point (this being the solution to a 3x3 system of equations).


We'll also look at systems with no solution (i.e. at least two of the planes are parallel) or an infinite number of solutions (all three intersect at the same line).


I think that seeing this will help them visualize 3D graphing much better than me trying to draw it on the board.

During the demonstration, I will be teaching them how to enter functions in to Grapher. After the demonstration, I will put two 3x3 systems on the board. One has a solution and the other doesn't. They must first decide which does by graphing both and determining which has a point of intersection. Once they decide, they must solve the system algebraically, and then plot the solution to see if it falls on the point of intersection of the planes.

After this, we'll go medium-tech (i.e. just using their TIs). I will show them how to use the "stat plot" feature, and we will plot the points (-1, -10), (1, -18), and (5, 14). I will ask them if they can figure out the function of the parabola that passes through those three points. I don't expect them to be able to figure this out on their own, so I will lead them through plugging in the points into the standard form of a quadratic function, which will generate a 3x3 system of equations. We'll solve this together to find that a = 2, b = -4, and c = -16. Graphing the parabola y = 2x^2 - 4x - 16 will confirm the work.

Hopefully there will be time for them to try this on their own as practice. I'll ask them to find the function of the parabola that passes through (-2, 0), (-1, -6), and (3, -10). This is definitely a lot for one lesson, and I've never done this before (our lab is new!), so I'll have to see how it goes.

Next Lesson: 3x3 Systems of Equations

Up next are those pesky systems of three equations with three variables. We are going to work on solving them by linear combination. I've decided not to cover matrices in this class because of the time it would take to teach it to a level that had any meaning to the students. I also don't think that they have much application up through Calculus (at least, not enough to justify taking time away from some of the other critical topics).

The class will start with a review of 2x2 systems, including systems with no solution or an infinite number of solutions. They will solve these algebraically and graphically; I want to see if they can draw their own conclusions first about what algebraic results like "0 = 3" or "0 = 0" imply graphically.

After this, I'll do direct instruction on these ideas, followed by a demonstration of how to solve a 3x3 system by linear combination. The steps I'll give them:

1. Decide which variable is easiest to eliminate.


2. Eliminate that variable from one pair of equations.


3. Eliminate the same variable from a different pair of equations.


4. Write the resulting equations as a 2x2 system and solve it.
   
   
   
   1. If the 2x2 has 0 or infinitely many solutions, the same is true about the 3x3 and you can stop there.
   
   
   2. See if you can use division to simplify the new equations.


5. Plug in the solutions to one of the original equations and solve for the third variable.


6. Check by plugging in the solution to the other two original equations.

After that, it's just a matter of practice grinding through a few systems. Homework will be more of the same, with a +10 point mega bonus extra credit problem (solving a 4x4 system).

In the following class, we will use the Grapher application in our Mac lab to see the planes that are produced by an equation with x, y, and z, and to understand what an ordered triple means. We will also be able to see how planes can intersect and so forth, and maybe play around with some more 3-D graphing. Grapher is a great application - check it out if you can.