The first, deals with completing the square. The article gave a long list of things the tiles could be useful to help teach. Having been tutoring for awhile, I've realized that many students do not know what it is to complete the square or get confused about how. So I wanted to put it to the test and see how the tiles would be able to show the concept of completing the square.

I did a quick Google search to find some problems that require completing the square:

- x^2 - 4x + 6 = 0
- -x^2 - 2x - 5 = 0
- 4x^2 + 4x - 3 =0

Then I completed the square to find the solution, so I could compare when working with the algebra tiles.

- (x - 2)^2 + 2 = 0
- -(x + 1)^2 - 4 = 0
- (2x + 1)^2 -4 = 0

So I began, and at first, I was confused. I collected all the tiles I would need, but somehow could not form a rectangle/square. It simply was not possible.

After a moment, I realized, to complete the square I will have to create a square with the tiles. So first, after placing the x^2 tile, you have to divide the x tiles evenly on either side.

Then I filled in the units to complete the square, with 2 of the six leftover. So when writing the solution you get (-x + 2)^2 + 2 = 0. Which is equivalent to the solution above. As I solved the others, I realized it got easier as I went along. You just have to be careful with the units (that the total tiles is the total units you had in the original equation). Both ways, visual and symbols seem like fine ways to teach it. It seems like it would be helpful for visual/struggling learners and interesting to others. But potentially a waste of time for students who already grasp the idea symbolically, if they must do a lot of work using the tiles.

Here are the visual representations of the other 2 equations:

My second question, was if algebra tiles could make realizing if a given binomial can not be factored easier? So again, I Google searched and found two equations that could not be factored and got out my tiles.

- x^2 + 2x + 5 = 0
- x^2 + 4x + 1 = 0

After setting up both, the answer is yes! If the equation can not be factored, you will have more or less units than what is necessary to create a rectangle with the tiles. Since students are sometimes asked to factor an equation, or say if it can not be factored, this method seems really helpful. Students can see the solution quickly, whether it can be factored or not. It also allows students to pay attention to the units of the equation when deciding if something can be factored. When they face larger equations, which the tiles would not work well for, students know where to look.

So both questions resolved, I recommended the use of Algebra tiles. The algebra tiles help to introduce ideas of factoring (which can feel and seem abstract) in a more concrete way. Students can see, visually, why something works and how to fix the problem. They become more comfortable with the ideas of factoring and gain insight into recognizing when things can and cannot be factored. So hopefully, as a result, factoring will not feel like a daunting task, but an achievable one.

I like the concrete nature of this post and the great illustrations. Were you using virtual tiles?

ReplyDeleteTo make an exemplar, I'd just like to see you flesh out your conclusion a bit more. You'd recommend them because... (how do they convey the factoring idea, why is it helpful)

Otherwise: clear, coherent, complete, content: +

I really enjoyed this post. I didn't even think of using the algebra tiles for completing the square. I will definitely use this post as a resource to teach my students how to complete the square. It would be a great hands on activity and visual aid if the students are having difficulty grasping the concept algebraically.

ReplyDeleteI dislike algebra tiles, probably because I learned operations without them. Furthermore, my mind wants to understand the various tiles' value in terms of their geometry, not just their labels. This is not insignificant because the whole point is to visualize, right? For example, look at your first example. It's clear that the value of x^2 is a square of side length x. However, if that's the case, how can x be a rectangle? That makes no intuitive sense at all. My mind wants x to be a LINE, not a rectangle. Do you see what I'm saying here? And then those little squares, what are they? Is each little square supposed to represent the number 1? If so, now I'm completely perplexed because you've used rectangles (because a square is a rectangle) to represent UNLIKE terms: x^2, x and 1. That does not help me understand what is happening, it throws an additional, nonsensical translation of the situation into the mix. It's like translating Spanish to English by first translating it into Chinese then to English, and asking the leaner to pick up on the Chinese.

ReplyDeleteI never found the idea of completing the square difficult because all you're doing is using algebra to transform a quadratic equation into a situation where the perfect square of some expression equals a constant. You can then solve it (if there is a solution) by square-rooting both sides.

There are concepts that lend themselves to enhanced understanding by using geometric representation. Algebra tiles are not among them because the geometry of the figures doesn't match the stated value. A line can't be a rectangle and a small square shouldn't be an arbitrary constant, unless that constant is the square of something. I understand the rationale behind algebra tiles, I just think they're an impediment and not an enhancement to mathematical understanding. It is for these reasons that I never use algebra tiles to explain concepts that make perfectly good sense with simple algebra. Cheers.