In my capstone class for future secondary math teachers, I ask my students to come up with ideas for *engaging* their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Diana A’Lyssa Rodriguez. Her topic, from Algebra: completing the square.

*A2. How could you as a teacher create an activity or project that involves your topic?*

Using Algebra tiles is a great visual way for students to understand completing the square. The students start with the tiles that correspond to the given problem. The unit tiles are then flipped and moved to the other side of the equal sign. The remaining tiles are positioned into a square shape. The corner piece that appears to be missing will be filled unit tiles. What you do to one side, must be done to the other. Therefore the amount of unit tiles added to the square will also be added to the other side of the equation. Find the zero pairs and take them away. Then, find the corresponding tiles that will outline the square, so when multiplied together equals the equation.

Step 1:

Step 2:

Step 3:

Step 4:

Step 5:

Step 6:

Step 7:

Step 8:

*D1. **What interesting things can you say about the people who contributed to the discovery and/or the development of this topic? *

Around 815 – 850 AD, a mathematician Muhammad ibn Musa al-Khwarizmi was hard at work discovering algebra. He was actually the first person to write a text about algebra. His focus for a lot of the text was the dimensions of a square. At the time it was not called completing the square, but Muhammad was the one who came up with it because it is exactly what he did in order to solve the different equations he had at the time. Very similar to the process described using algebra tiles, Muhammad also saw the equations in terms of actual shapes. One of the original problems he tried to solve was x^{2}+10x=39. He looked at x^{2} as a square with length x and width x. He then created a rectangle with length 10 and width x. The area would equal 10x. To make his theory work he broke up the 10x rectangle into two squares with length x and width 5. Muhammad combined the x^{2} square and the two 5x pieces into an L shape. This partial square must equal some square with the value of 39. So he came to the conclusion that he had to fill in what was left of the L shape to make it a square. But in order to do that he had to add that same value to the other side. In this case he added 25 (which is 5×5). Muhammad’s final answer was (x+5)^{2}=64, x=3. This was his method but at the time he couldn’t prove that it always worked. So if and when students participate in the algebra tiles activity, they are partaking in a small piece of history.

*E1. How can technology be used to effectively engage students with this topic?*

This video from Khan Academy is a great tool for completing the square. This video explains why we have to take half of the b value and square it (when looking at ax^{2}+bx+c) to obtain the c value. When the students understand why we do something in math, they are more likely to be interested in the topic. The different colors that are used to write out the process allows the students to organize and understand completing the squares better. This particular video is also just long enough to capture the attention of the students but not so long as to lose it. Also, after hearing the same person explain math all the time, students may not understand it as well as they possibly could. So what is said in this video can easily be explained by the teacher but students sometimes need to hear a different voice explain a concept so they can gain a new perspective on the topic.

*Resources*

- http://www.maa.org/press/periodicals/convergence/completing-the-square-quadratics-using-addition
- https://www.youtube.com/watch?v=jxBkRZZFmBM
- https://www.khanacademy.org/math/algebra/quadratics/completing_the_square/v/ex1-completing-the-square

## howardat58

/ February 27, 2016And the wise guy asks “So what do we do with x^2 + 7x + 2 = 0 ?”. Of course, he knows already !

## Joseph Nebus

/ February 28, 2016Oh, I love the pictorial version of this.

## howardat58

/ February 28, 2016I had another look at the arrangements of the tiles and saw that the green tiles were six unit tiles long, so in fact your student has shown that (6 + 4)^2 = 14.

This is a confusion of equations and identities.

What the student should be doing is to say “Can we rearrange the left hand side of the equation into a sum or difference of two squares, or a squared term and a number, and then see that it gets us a step or two closer to solving the equation”.

## John Quintanilla

/ February 29, 2016While I agree that students should eventually be able to rearrange equations without a visual cue, I disagree emphatically that the only way to teach this concept is by symbolic manipulations. Some algebra students have the requisite abstract skills to manipulate equations without much prompting. That’s the technique that worked for me, but I was a future mathematician. Many algebra students (not all, but many) do not enter their class with this level of comfort with symbolic manipulation. The question then is, What can a teacher do to get his/her students to this point when Plan A doesn’t work?

## howardat58

/ February 29, 2016You are quite right ! My real point, which got overwhelmed by my implication that this needed algebraic manipulation, was about the rearrangement. This of course can be done with tiles, or pictures, but should be done several times, with different lengths of green tiles. It doesn’t need symbols.