# Engaging students: Completing the square

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?

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 ax2+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

1. #### howardat58

/  February 27, 2016

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

2. #### Joseph Nebus

/  February 28, 2016

Oh, I love the pictorial version of this.

3. #### howardat58

/  February 28, 2016

I 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, 2016

While 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?