Here is a run down of my experiences (the pros and cons) of the different strategies I've used and my experiences.
Algebra Tiles: Algebra tiles incorporated with the "balance scale" method has always been my go-to for solving equations. It's tried and true and I'm comfortable with it. Before introducing equations, algebra tiles are great for modeling simplifying expressions. Students can visualize that 2(x+4) is actually 2 groups of x+4 which is 2 x's and 8. It starts getting a little iffy when tiles are used for problems such as -2(2x-4) and (x+5)-(-2x-3), but for the most part students are successful. For me, algebra tiles start to fall apart when introducing fractions and decimals into equations.
- Great for transitioning into a more algebraic approach. Students see the method of what you do to both sides to isolate the variable. (Adding 2 red tiles to zero out 2 yellow tiles is the same as adding negative 2.)
- Great for visualizing positive and negatives values in equations.
- Great for seeing different paths to solve. For example, in the problem 3(x+4)=12, you can either combine like terms to get 3x +12 in which the first step is to distribute or you can divide the algebra tiles into groups of 3 and notice that x+4 is the same as 4 in which the first step is to divide both sides of the equation by 3.
- Not overly useful for equations with decimals and fractions, although it is possible with a dry erase marker and as long as the fraction partitions aren't too large.
- Students have a hard time modeling subtraction of expressions and often model addition instead.
For the first time last year I used the "working backwards" strategy to solve equations. This stemmed from the amazing MARS task Building and Solving Linear Equations. This task tells the story of x and in doing so, students create some pretty complex equations. It then shows how we can walk the story backwards to determine the value of x. The students were extremely successful with solving equations this way and even my students that typically struggle were solving complex equations with ease. I loved it! However, I realized this strategy falls apart when you have variables on both sides of the equation or 2 variables on the same side. Because of this major limitation, I chose not to use this method this year.
- Solidifies order of operations in equations. Students really see what comes first, second, etc...
- Allowed for students (even my lowest) to be successful in solving pretty complex equations with 1 variable.
- This strategy is limited to equations with 1 variable. I don't see a way for it to work with a variable on both sides or even more than one variable on the same side.
Double Number Line (Clothesline)
I'll admit, I'm a little obsessed with number lines this year. It hasn't been without it's bumps and I've learned a lot about how to implement them next year, but I've used number lines with just about everything: Operations with rational numbers, proportions, and then I tried double number lines to solve equations. It didn't go great, but I think that was due to my implementation rather than the method itself.
This year, I used Algebra tiles to start talking about expressions and equations. When it was time to incorporate fractions and decimals, I decided to try the double clothes line I first saw from Andrew Stadel, Jon Orr and Matt Vaudrey. The problem was that the students didn't have enough time to get comfortable with expressions on the clothes line before I started throwing in negative variables and non-integer numbers. In an hour long class period, they got 3 problems completed. It was actually a great problem solving opportunity, but didn't lend itself well to the actual method for solving. The students quickly went back to the standard method of isolating the variable in which they felt more comfortable.
- Can be used for pretty much any type of equation.
- Helps solidify number sense while solving equations. Students are actually thinking about the numbers and their relationship to the other numbers while placing them on the clothes line and not just going through a routine.
- Gets the kids up and is kinda fun!
- Much more time consuming than other methods, in my experience. This could change for classes that spent more time solving equations on a double number line before incorporating fractions. However, just the process filling out the number tents is time consuming.
- Students didn't naturally gravitate to wanting to solve the problems this way. When given a choice, they chose algebra tiles. Again, this could be an implementation issue.
I have learned so much on a personal level about solving equations since joining Mtbos. It's been fun for me to play around with different methods and see how they connect. I'm not sure, however, this has been beneficial to my students. I definitely need to rethink things for next year. Is it best to stick to one method? Show all of the methods and let students choose? How much time do I spend on the concrete methods before moving into more abstract problems in which the models start to fall apart? I have vague answers to each of those questions, but I wonder what others do that is successful. Feel free to comment or hit me up on twitter if you have the answers. :)