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 1: Representations of Linear Equations

Increasing my students' representational fluency has been something I've been working on for a while.  Our second semester started last Monday, and to start my Algebra 1 students off easy, I had them do a four-fold poster of a linear relationship to review what we did last semester: situation, equation, table, and graph.  They did the work fine overall, but quite a few students had more troubling questions than I had expected (i.e. "how do you make a table?").   I guess it just shows that we have to keep going through these different representations and their connections again and again.

We have started the new unit - working with linear equations - in which students have to write the equation of a line given its slope and a point, or two points, or a point and a parallel line.  In the past, I have done this only algebraically (except for the initial explanation of concepts); this time around, the students will have to practice the problems both algebraically and graphically.  And, more importantly, the skills tests will require them to show mastery with both methods.  Let's build those connections!

Here are the first few lessons in the unit.

Lesson 1 (Representations of Linear Functions)
Lesson 2 (Graphing Practice)
Lesson 3 (Write the Equation of a Line)
Lesson 3: Keynote / Powerpoint

And some snippets from the worksheets to illustrate what I am talking about:

 

Algebra 1: Solving Equations

I am beginning the planning stages of our unit on solving equations in Algebra 1. In my past experiences, some students pick this up very quickly, no matter how you teach it, while other students struggle mightily. I want to try some alternate approaches this year, to really reach those students who have not been able to learn this skill in the past. I remembered an order of operations approach that I read about in the NCTM magazine a few years back. I can't recall the name of the article, but a little google searching found me this document that is even better than what I remembered.

Our students in Numeracy already work with bar modeling to solve word problems, so this seems like a natural extension to solving equations. I like this approach because it helps focus on the idea that the variable is a given quantity that must be determined, instead of focusing on the steps that isolate the variable. It also might help with those difficult to master "converting verbal sentences to algebraic equations" problems. Here are a few examples of how this might look. I know the diagrams are a bit confusing at first, but I think they would make more sense to students as they watch them get created and do them by themselves.



I also like the other representation discussed in the article. This is the original order of operations process that I had been searching for. I like this because it gives a very clear framework for solving equations - reversing the order of operations.


When you look at each stage, you can draw equal signs between the boxes. These would be equivalent to the intermediate statements in the traditional "do the same thing to both sides" approach.

So for the unit, I am thinking that we would spend two or three lessons on bar models to build the concept of what we are actually trying to do (find the value of the unknown amount). Then, spend a couple lessons on the order of operations representation to build an understanding of the process for isolating the variable. Finally, transition to the traditional approach, which is clearly the fastest and cleanest way to solve an equation of the three. This would take more time, of course, but the hope is that it would build a more enduring understanding.

Has anyone tried these methods with their students?

Edit:
I wonder now if it would make more sense to start in with modeling sentence/word problems with the bar model method, and not start by saying that we are "solving equations". That way, more students would be engaged with the material, and we could eventually use the bar models to develop the equations.

This way, the unit doesn't start with the problem "solve (3/5)x = 45", which will stop most kids dead in their tracks, but maybe with something like "It took Sandra 45 minutes to finish 3/5 of her homework. How long will it take her to finish it all?", which kids might have more of an entry to. After we solve it, we can then discuss how to represent it as an equation.

Edit 2:
I also need to think about how to incorporate the balance idea and preserving equality... Kids don't always know what the equal sign really means. Maybe in the transition time from the box method to the traditional method?

Edit 3:
(Written on 10/27 - at the end of the unit)
On reflection, the problem was not having enough time to really devote to the two alternative methods. Both did show a lot of promise, but we weren't able to really practice either enough for it to really stick with students. The bar model method really worked to help students set up and solve word problems, so I think I will stick with that next year. Give it some more time so that it really sinks in and can be used to get a deeper understanding of fractional coefficients. I will probably save the GERMDAS method for individual tutoring with students who are not having success with the traditional balance method. Less fights... more differentiation.

All of these lessons have been added to the box widget on the left.

Algebra 2: Polynomials and Factoring

We just finished our first week of the second semester. The previous two weeks have been our Intersession period, where all students and teachers do totally different classes. This year, I did an algebra review class, helped organize our junior "boot camp" to help get them more ready for the college application process, and taught an anime class.

But now it's back to normal school, algebra 2, and time to start learning about polynomials. The first lesson was not that exciting, as we spent a lot of time learning all the needed vocabulary. But we also did learn about end behavior of polynomial functions, both graphically and algebraically. The next two lessons were more interesting, as we looked at the zero factor property from a graphical perspective, and then we learned how to sketch a polynomial function when given its linear factors graphically. This is scaffolding for the number line model lesson that will happen on Monday, which will allow students to solve factored form polynomial inequalities like (x - 3)(2x + 5) < 0. This isn't in the algebra 2 standards, but this kind of analysis will push them to understand functions more deeply, so I think it is worth the time.

In lesson 3, students worked as a class to generate sketches of product functions by multiplying the linear factors. They really caught on, and were able to easily get through the first problem pictured in this post. It was great to watch them work together as a class so well. The goal is that, on Monday, they will be able to understand and solve the second problem in this post.

Here are the files:
Lesson 1 (Classifying Polynomials / End Behavior) word / keynote / quicktime
Lesson 2 (Zero Factor Property) word / keynote / quicktime
Lesson 3 (Curve Sketching - Graphically) word / keynote / quicktime
Lesson 4 (Curve Sketching - Analytically) word / keynote / quicktime

Algebra 2: Evaluating Functions

We're beginning our trek into the deep waters of representational fluency - my favorite part of the algebra 2 curriculum. If my students only retain one thing from my course, I'd like it to be the ability to move back and forth between equations, graphs, and tables with (relative) ease. So, today we evaluate functions in all these forms. Tomorrow, we do the same, but toss in composite functions. I think the Keynote animations are really powerful here - especially when working with graphs. Though they know which axis is which, students tend to get all turned around when trying to read graphs this way. I think the vertical lines and moving points help them see what they are looking for.

Lesson 2 (Evaluating Functions)
Lesson 2 Keynote
Keynote Quicktime

Next Lesson: Representational Fluency

The composition lesson went fairly well, though I could have used about 5 more minutes to finish the lecture. Students seemed comfortable with finding things like f(g(2)), but I definitely lost many of them when I tried to finish with finding f(g(x)) as an expression. We'll definitely need to review this a couple of times.

In the next lesson, after reviewing the homework, the Do Now will focus on practicing these concepts (as well as reviewing absolute value inequalities).

Then, I'm going to squeeze in a mini-lesson on using the TI-83+ to graph inequalities and absolute value inequalities. In this case, I mean graphing things like y = (x < 5), where it returns 1 if true and 0 if false. This is a pretty cool way to generate a graph that looks like the number lines we shade by hand. It can solve absolute value problems the same way: y = (abs(2x+1) > 3). After learning this technique, students will check their answers from the Do Now by graphing. I'm hoping that this doesn't take too much time...

Finally, I will give students a handout on Representational Fluency. I've learned from my experiences teaching AP Calculus in previous years how important it is for students to be able to move comfortably between equations, graphs, tables, verbal descriptions, arrow mappings, etc - especially when it comes to the concept of functions. This sheet focuses on graph and table representations of functions - students have to figure out things like f(2) and f(g(-2)) from these representations. I'll post this on ILoveMath .

Update:
I'm trying to decide if I should do some work on function decomposition. This is clearly a skill that students will need for Calculus (i.e. working with the chain rule). I also think that decomposing functions might help them understand better what composing functions really means. But I'm also worried about overloading them, and I wonder if they need more time to digest function notation and composition first. It's not in the standards, as far as I can tell, so I wonder if students are expected to understand this idea before getting to Calculus. Any ideas?