I made a review lesson for my Algebra 2 students on these topics, to make sure they are really ready before we start performing operations on complex numbers.
Some instruction, some board races, and there you go. Hope you like it.
I also updated my Algebra 1 box with unit 2 files and the first four lessons of unit 3.
Monday, September 21, 2009
Saturday, September 12, 2009
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?
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.
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?
(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.
Monday, September 07, 2009
My goal for this weekend was to complete a rough draft of all the skill items that will be assessed on the first semester final exam. These items are assessed in chunks on the weekly skills tests, and in larger chunks on the 6-week benchmark exams. After each benchmark exam, the plan is to spend a lesson or two on targeted reteaching - any ideas that people have on how to make this effective would be very much appreciated.
I've finished the list, and am interested to hear what other Algebra 1 teachers think about the scope and detail of the items. What would you add? Take away?
Sunday, September 06, 2009
This student finished all of the simplifying expression problems on paper (correctly), and then used the algeblocks for the following:
But can I blame him? Why would a student who can already do a procedure well bother trying to figure out a slower, less portable method?
But then, the students around him who really do need to use manipulatives to help them understand the difference between 2x and x^2 don't want to use the blocks either. They feel stupid and they want to do it the faster way too.
This seems impossible without fully differentiating instruction in the classroom. Which also seems, you know, impossible.
We have our Numeracy class, in which students spend time working on ALEKS. The Numeracy teacher this year is doing small group pull-out during that time to work on specific skill building. I am starting to think we should use the manipulatives in that setting only.
Saturday, September 05, 2009
My algebra 2 students needed more practice solving equations with rational exponents. I was trying to think of an interesting and yet still effective way for them to keep practicing, and then I thought about how the errors they make tend to be the same again and again. That reminded me that I hadn't done an error analysis activity in a long time - and just like that, the lesson was created. I assigned each table pair one problem, where they had to find the mistake, explain it, and do the work correctly. Then each pair was called to present their findings to the class; the class then worked out the problem and if they all agreed that they had found the correct solution, I allowed them to move on.
Students were generally good at finding the mistakes I had made. Will this activity help keep them from making the same mistakes in the future? We'll find out next week.
Here are the twelve problems plus homework.