Visual Mathematical Laws

Inspired by the following blogpost, I decided to make my own visual representation of mathematical laws. Art being my tied for favorite subject with math, the idea of visual representations sounded not only fun to create but potentially really helpful for visual learners. The visual representations take away the daunting idea or confusion variables bring to many students. If students are confused about variables or just have a general dislike, with the visual representation they are still receiving the same information just in a more understandable way. Along with this, students should be brought to discover or shown how the laws have come to be, since understanding the laws is more important and helpful than just remembering them.

In addition, to me, the visual representation is more interesting. Students see numbers and variables all the time in their math classes - of course! Color, on the other hand, not so much. So maybe students will have a tendency to recall the laws better because of the uniqueness with which they were presented. Plus it's always fun to see math and art come together! - even if in such a small way.

Through the process of making the laws, I became more aware of how some of the laws work. Before I just took it for what I was told, memorizing but not really seeing the connections. The colors helped me to make the connections and clearly see where each piece is coming from. It's difficult to say if I would personally ask my students to create their own - it might take some students a lot of time and they may not see the purpose in it. I would however in going over the laws with a class encourage them to use colors in place of the variables and have a poster of the laws in this way displayed in the classroom.

So for mine, I decided to link it to those seen in the blogpost(which focused on laws of exponents) and visually represent the laws of logarithms :)
They are as follows:

• Logarithm to Exponential
• Canceling Exponentials (2nd and 3rd)
• Product
• Quotient
• Power
• Changing Base
• BONUS!

Working with Algebra Tiles

After using them in class and reading an article advocating for the use of algebra tiles in classrooms, I was left with two main questions.

 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:

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

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

1. (x - 2)^2 + 2 = 0
2. -(x + 1)^2 - 4 = 0
3. (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.

1. x^2 + 2x + 5 = 0
2. 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.