Khwarizimi's Visual Algebra and its Influence Today

I have always struggled with any mathematics that is inherently visual, and so when I missed a class about Khwarizimi's visual representation of completing the square and had to look online to teach myself about it, I knew I was in for a challenge. However, as an education major interested in new ways to present difficult mathematical concepts, I was interested in this visual algebra and how such a tool could be utilized to teach completing the square to students. I was also interested in examining its similarities to the "box method" way of teaching factoring. So, the questions I wish to explore today are: How beneficial are these methods of factoring? How can we utilize visual tools such as these in teaching? What can mathematicians such as Khwarizimi teach future teachers about new ways of presenting concepts?

Khwarizimi's Visual Representation of Completing the Square 

I'm not sure about my other fellow educators, but, to me, both in the classroom and where I work as a tutor, explaining the completing the square method to students has always been very difficult. While it is obvious to me that this method directly relates to getting an equation into vertex form, this isn't very obvious to most students, and the approach they are taught is more often linear and taught in a set of steps rather than in a way that makes sense to students. 

So, could a technique such as Khwarizimi's be a good way to teach such a concept? 

For those of you unfamiliar with Khwarizimi's method, I have it outlined below: 

Consider the function x^2+10x=39. We start with a square with an area of x^2, such that each side is length x. Then, we add to this 10x, by adding four rectangles with a width of 10/4 and a length of x along the side of the square we just created, for an area of 5/2x for each rectangle. These rectangles are in purple above. Finally, we complete the square by adding four small squares of length and width 5/2 in the empty spaces left by the purple rectangles. These small squares each have an area of 25/4 and are drawn in red above. The outside square, then, has an area of 39+ (25/4*4), which is equivalent to 39+25=64. Then, the sides of the outside square are 8. But each side is x+5/2+5/2=x+10/2=x+5. So, x+5=8, and x=3. 

I guess a good question would be: What do I think of this? 

I think Khwarizimi's method is pleasing to the eye and, visually, makes a lot of sense. It even makes sense during the construction, and I can see how I can relate the areas of the squares to 39 due to my work in Euclidean geometry. However, while I think the model is student-friendly, I think the explanation of this visual algebra still leaves a lot to be desired. While this may be an error of my own since I was unable to attend class for this lesson, I wonder: How could we explain such a model to students? What kind of model would help students see why we complete the square? This also doesn't relate to the vertex equation at all, so is it possible we could create a different sort of visual model illustrating this relationship? The possibilities are endless! 

If anyone has any ideas, I would love to read them in the comments. I think completing the square is one of the most difficult quadratic concepts to teach, and I would love to find a wonderful model I could use to build student understanding in a way that isn't just teaching the steps necessary to perform the operations. 

A Related Approach: The Box Method 

One of my favorite ways of teaching students how to factor quadratic equations is by utilizing the box method. I help teach an entry-level mathematics course twice a week, and during one of my sessions this week, I taught one girl how to use this method. Her eyes lit up after I showed her and I could tell she understood. This understanding associated with a visual representation is very powerful for students, so I love showing students how to use this method (along with the AC-method) so that they can see that there are several ways of approaching the same problem. 

For anyone unfamiliar with the box method, it is illustrated and explained below: 

Consider the function 6x^2+11x-10. As per usual with factoring, we want to multiply the first coefficient and the last coefficient together, find factors of it, and see which factors add up to our middle term. So, we multiply 6 and -10 to obtain -60. We want to find factors of -60 that add up to give us 11. These terms happen to be -4 and 15. 

Now, we utilize our box. In the first box, on the top left, we put our first term, which is 6x^2, and in the last box, on the bottom right, we put our constant, which is -10. Since we already found the two terms that add up to be 11x, we put them in the remaining two boxes, giving us -4x and 15x, respectively. Now, I like to think of this box as a Battleship grid. So, for example, the number on the left side of 6x^2 multiplied by the number on top of 6x^2 must multiply to obtain 6x^2. So, I would ask my students "What do 15x and 6x^2 have in common?" They would answer 3x, and I would put 3x on top of the grid so that, if you move along the grid, you see that 3x is a common factor of 15x and 6x^2. Then, I would ask my students "What do 6x^2 and -4x have in common?" They would answer 2x, and I would put this on the other side 6x^2 to show this common factor. We can see that 3x*2x is 6x^2, so those are definitely our factors for the first box. 

Continuing on in a similar manner, we obtain the outside numbers of 2x+5 and 3x-2. These are our factors, such that 6x^2+11x-10=(2x+5)(3x-2). (If this explanation didn't make any sense to you, there's a pretty good video explaining it here.)

So, how does this relate to Khwarizimi's visual algebra? 

I think this kind of approach is very similar to what Khwarizimi wanted to do in his explanation for completing the square. It represents fairly complicated mathematics in a way that is approachable and accessible for students and allows them to visually see where the factors come from. While I think Khwarizimi's example still has a little ways to go before it can be accessible for students who are learning this material for this first time, it is still a step in the right direction to grasping such difficult material. 

Mathematicians as the Teacher's Helper 

I think it is all too easy for education students to dismiss mathematicians as only relevant to those who are majoring in pure mathematics. To be honest, I was fairly skeptical myself when I entered this course, and I feared that a lot of the material we learned would not be relevant to me, since I will be teaching mathematics rather than utilizing it in a way many of my classmates will be, such as in data analysis or actuarial sciences. However, I think that pure mathematics and teachable mathematics go hand-in-hand. We, as educators, truly need to have a firm understanding of mathematics in order to teach it, and I think it is important that we are able to examine the work of famed mathematicians to see how their work affects how we teach. 

The visual algebra of Khwarizimi is a great example of this. While I'm not sure if Khwarizimi thought about the positive impact his visual representation could have on classrooms when he developed it, understanding such a visual representation makes it easier for teachers to present material in more than one way so that we can continue to eliminate the gap in understanding between students. We must learn to teach mathematics in more than one way, even if this way isn't our instinctual approach. In order to do this, we turn to mathematicians as our helpers, and we look at their work. Mathematicians can help us to see things in a new way, much like students, and we must pay attention to those new ideas. 

I have learned so much from the mathematicians I have studied in this course, and I hope to continue to learn more and to implement all I have learned in my own classroom. As I end this journey and I enter teacher assisting in the next few weeks, I hope I can remember what I learned here, and use it to become an educator any mathematician would be proud of.