Constructing and Deconstructing Equations

I am ready to move my algebra 1 students into solving equations, now that they have done such a good job with patterns.  I looked through the different FALs available on the MAP site and Building and Solving Equations 1 caught my eye, as I experimented with using this method to teach equations in the past.  I don't think I did a great job of it last time, and I think I can definitely improve and hopefully make it work for the students.

The most common problem I see with students who struggle with solving equations is the order to do the steps in.  Usually, they can figure out that, in a 2-step equation, you always get rid of the subtraction/addition part first, but multi-step equations are a real issue.  And, when they get the dreaded "reverse style" equation, like 8 - 2x = 2, then everything can fall apart.

The idea of the deconstruction method is to really have students focus on the skill they do know - order of operations - to figure out what steps to take to solve.  And, it goes beyond just saying "reverse PEMDAS", to really concretely having students walk forward and backwards through all the steps.

Yesterday, I had students practice simply constructing equations.  I gave them x = 6, as suggested in the FAL, and we built one-step equations off of that using different operations.  Then, we took those new equations (like x + 5 = 11) and added a second step with a different operation.  Eventually, we built through 4-step equations.  And, we practiced checking by substitution.  The FAL includes sample student work and we analyzed that.  One student's work is correct, but can be improved by clearly showing the steps.  The other student's work is incorrect, and checking by substitution at each step reveals where the error is.  (The error has to do with fraction addition, so I didn't really discuss it with most students, since that would have been a bit too overwhelming in this lesson.)

At the end of the lesson, I could see that students were starting to get it, but still not totally clear.  So today, as a class, we built two more equations.  Then, I began the process of deconstructing them.  Here is how the board looked after those two problems:

After this, I continued with the activity in the FAL.  Each student picks two solutions, and constructs a four-step equation around them.  Then, they exchange problems, and have to deconstruct each others' equations to see if they could find the original numbers.

Students were working really hard on it, but were definitely struggling with a variety of issues - order of operations, basic calculations, how to organize their work, and so forth.  But, on the positive side, most students knew which operations they should be undoing at each step.  So, progress!  I'm going to keep working with them on this for a few more lessons.  Then, I'm going to give them a set of two-step equations, and my hypothesis is that they will be able to blaze through them, like a batter taking the donuts off the bat.  Hopefully, pushing them really hard at the outset will pay off later.  I'd appreciate hearing other teachers' thoughts on this method for teaching equations.

Here are the handouts we're going to practice with tomorrow.   (doc)   (pdf)

Linear Patterns in Algebra 1

I've been working with my Freshmen on linear patterns for the last couple of weeks.  I've been really amazed by how well they've taken to them, and how quickly they were able to start figuring out the function rules.  I've been using lesson ideas from The Pattern and Function Connection by Fulton and Lombard.

First, I gave students matchstick patterns and boxes of toothpicks, and had them build the patterns.  This was important, because often students would draw the picture incorrectly (not seeing where toothpicks were repeated, for example) and would not quite see how the pattern was growing.

Then, I gave them other types of patterns like this one, and asked them to describe how they were growing, and to determine the number of items in the 30th step and in the n-th step.  I think that building a concrete connection between the variable "n" and the "step number" will be invaluable as we move into solving decontextualized equations.  I was really excited when the first student figured out that this pattern could be thought of as a center square surrounded by "groups of n squares" and explained it to the rest of the class.

After this, we moved into organizing the steps of the pattern into a t-table, and then plotting them on a set of axes.  We defined "n" as the step number, and "f(n)" as the function value, meaning the "number of items at step n".  I worked through one example with them, but then I found that most students took off on their own and blazed through the whole set of patterns I gave them.  During the following lesson, we repeated this work, but I introduced the concept of "rate of change" and "starting point" and we talked about how these values are seen in the different representations of the pattern.

I really liked the following diagrams from the book:

I decided to build a poster project around it.  So in the following lesson, student pairs picked a new pattern to work on, figured out all of the representations, and then made a poster to illustrate what they learned.  I asked them to indicate in one color how the rate of change was portrayed in each representation, and in another color the starting point.  Students worked for a lesson and a half, and did a great job.  Here are a few examples of their work:

Today, I gave them some function rules, and they had to then work backwards to generate a pattern that could fit that rule.  They struggled with this quite a bit at first, and it showed me that, although they are doing really well, they are far from having fully internalized the connections between the representations.

On Monday, I will give students a pattern, and tell them how many pieces are in the last step.  From there, they will have to figure out what the step number is.  I think this will be a good scaffold for understanding how to solve equations.  For now, I am going to treat it like a problem solving activity and am interested to see what they will come up with.

Below are some of the handouts I made to supplement the excellent materials in the Fulton book.
Pattern Project handouts (doc) / (pdf)
Working Backwards handouts (doc) / (pdf)

IB Math SL - Functions

I'm teaching IB Math SL for the first time this year.  For those unfamiliar with the IB program, it is a two-year program that covers pre-calculus and calculus topics.  I'm doing year one right now, which means my juniors will be with me again next year.  I feel bad for the students I had last year in Algebra 2 - they'll have me for 3 out of 4 years in high school!  I only teach one section of the class, but so far I am really enjoying it.  The students are very motivated and positive, and the textbook for the class is actually pretty decent, so it is a good resource.  I do still supplement it with other resources, however.  This past week we've been reviewing functions (operations, inverses, etc.) and I modified some of my old work to use with them.

One activity that I use every year is the Functions Relay Race.  It really pushes students to use all of the different representations of functions at the same time, and it helps them see "f(x)" as a single value that can be operated on.  I also target typical problem areas - absolute values, negative exponents, etc.  I give each team the main handout that has all of the functions on it, and then they get the problems one round at a time.  I usually give a small prize when they finish a round.

• Functions Relay Race (doc) / (pdf)
• More Functions Practice (doc) / (pdf)

And then here is a handout that helps students review/explore rational functions.  I especially like the point-by-point division that shows how a hyperbola is generated when you divide two lines.  Although, students are definitely confused by the directions and I always have to show them what I mean.  I've rewritten the directions several times, and I can't figure out a way to phrase them that works.  I think the problem is probably not in the directions themselves, but that students don't really have a solid understanding of the connection between the graphical representation and the symbols used.  In any case, feel free to check it out and let me know what you think.

• Rational Functions (doc) / (pdf)

The Exponential Curve, Phase II

I haven't posted anything for a couple of years now, and I think I'm finally able to start up again.  I'm at a very different place in my life now, compared to when I first started this blog, and, thankfully, compared to when I stopped writing.

I am really excited by the move toward Common Core, and the abandonment of the CST tests.  I've felt my spirit and teaching practice withering and dying with each new modification I made to try and cram more standards in faster and more efficiently.  When students would ask why they need to learn math, I would give a standard reply about problem solving abilities and critical thinking in all areas of life, and this felt more and more cynical as I compared what I was saying to the actual content I was delivering.

This year, in my intensive algebra 1 class (for students who are well below grade level), I've quit worrying about skills lists.  I'm focusing on using high-quality problems and resources.  I'm trying to actually do what I say, and engage them with activities that require thinking, explaining, justification, problem solving, and persistence.  And since the new Smarter Balanced tests, according to David Foster, will be only 31% material at levels 1 and 2 of Webb's DOK, and 69% at levels 3 and 4, I feel like I can justify my new approach to any skeptic.

Just yesterday, I spent nearly 30 minutes playing the Game of 21 with them, a quick and easy misere game.  (The first person says 1, 2, or 3.  The next person increases by 1, 2, or 3.  Alternate turns.  You can't go past 21, and the first person to say 21 loses).  It was great to watch them struggle with it.  Of course, I made them start, and so I won game after game.  But then some students started realizing that I was going to win as soon as I said 16, and after a while longer, they realized I would win when I said 12.  Some students were frustrated, some continued to challenge me blindly, and some were clearly paying close attention and trying to develop a strategy.  I love the moment when the first kid tells me that, no, *I* need to go first.  And then when they finally beat me, it's a great moment!  At this point, I stopped the game without discussing the strategy, and told them to play it against a friend or family member for homework.  We'll pick it up again on Monday and see if more can beat me.

We've been working on finding out the rule for patterns, using only visual examples, building them with toothpicks, etc.  So I decided to start them on the non-linear Growing Staircases POM from SVMI for the second half of the class (A pattern where you start with one square, then add two, then three, etc).  First, I spoke with them about what perseverance means, and why it's so important.  I told them that we were going to all get through level C of the POM, but we were not going to finish today, or even by the next class.  That real math problems take longer than an hour to solve (or the 15 seconds that they are accustomed to).  They mostly completed level B (figuring out how many squares it would take to build a 10-step staircase), and they used counting and other patterns to do so.  Level C (find a rule for the number of squares in an n-step staircase) is going to be a lot harder, and I'm looking forward to seeing what they come up with.

I decided to do a lot of patterning work with them before we even talk about solving equations, using their visual experience with developing a rule as leverage for understanding.  When they see something like 4n + 3 = 51, I want them to think something like "That's 4 groups of n-blocks, and then 3 blocks more, giving a total of 51 blocks.  So the 4 n's would have to make 48 blocks.  So each one would be 12!".   Right now, their pattern work ends with coming up with a rule.  Soon, I'll start asking them to figure out what step number it would have to be, given a specific total.  I'll treat this as a problem-solving exercise now, and eventually formalize the steps into the standard solving equations algorithm.

I've also been drawing heavily upon the excellent work found in the Formative Assessment Lessons at the Mathematics Assessment Project .  These are really meaty lessons that incorporate excellent group tasks, and really help push students to use the CCSS Math Practice standards.  So far, in this algebra class, I have used Positive and Negative Numbers in Context and Interpreting Algebraic Expressions.  The one caveat I have is that the amount of time it takes to run one of these lessons is well underestimated in their notes.  The expressions FAL took my intensive class four lessons to complete, not the optimistic 1 lesson with a 20 minute follow-up!  But of course this will vary from class to class.

Aside from this, I am also teaching Algebra 2, and I am using CPM instead of whatever NCLB book we have sitting in our library.  This will be the subject of other posts, as my students and I wrestle with a completely different way of teaching and learning math.

And, as if this weren't enough, I am teaching the first year of IB Math SL (which is roughly equivalent to the first year in a pre-calculus/calculus sequence).  This class is filled with really positive, engaged, motivated students, and it's a joy to teach, so I shouldn't complain too much about the extra prep!

So hopefully this post will be the first of many more to come.   Being part of the mathtwitterblogosphere is invaluable, and I want to start adding my voice to the mix again.