Geometrical verifications for Algebraic formulae

Question

What is the importance of using approaches related to Geometric Algebra in teaching,is it only useful when introducing Algebra to the students or can it be helpful to improve creative skills in mathematics?
As an example if we consider the completing square method used by Al-Khuwarizmi the father of Algebra, he used it only for special cases such as when coefficients are positive. Is it sensible to use the same approach in the cases of negative coefficients by assigning negative values for the sides of squares or do we have to use this approach only to introduce the concept of completing square?
Is it not much sensible to rearrange the quadratic equation
$x^2 - 6x + 8 = 0 $ as
$(-x)^2 + 6(-x) + 9 = 1 $
and draw squares of sides $-x , 3$ ?

Answer 1

In my experience,

• geometry can lead you into interesting questions that can be answered geometrically in special cases, and

• algebra can help you tighten up the rigor of your answers and generalize your results.

For instance, here's an example of how I would combine geometric and algebraic approaches to teach an algebraic concept. I'll illustrate on a simple concept, expanding binomials.

1. Motivate the end result using geometric intuition.

The quantity $(a+b)^2$ represents the area of a square with side length $a+b.$ Draw a picture of said square...

...and illustrate how it can be partitioned into

• $2$ squares of dimensions $a \times a$ and $b \times b,$ and
• $2$ rectangles of dimensions $a \times b$ and $b \times a.$

This would suggest that

\begin{align*} (a+b)^2 &= a^2 + b^2 + ab + ba \\ &= a^2 + 2ab + b^2. \end{align*}

2. Tighten up the rigor using algebra.

Sure, this picture makes sense when $a$ and $b$ are positive numbers. But what about if $a$ or $b$ is zero or negative?

Granted, if we come up with a situation like that and plug it into our formula, the formula seems to come out to a true statement. But how can we be sure it will always work? And how can we make sense of the fact that it does in fact still work?

To answer these questions, we observe that the same result can be obtained by writing the exponentiation as a multiplication and then applying the distributive property:

\begin{align*} (a+b)^2 &= (a+b)(a+b) \\ &= a(a+b) + b(a+b) \\ &= a^2 + ab + ba + b^2 \\ &= a^2 + 2ab + b^2 \quad \checkmark \end{align*}

3. Demonstrate the utility of algebra.

What if we wanted to compute $(a+b)^3,$ or $(a+b)^4,$ or a higher power? It's tricky to draw a picture of a cube, and impossible to draw a picture of a cube with $4$ or more dimensions. What do we do?

We just use the algebraic approach:

\begin{align*} (a+b)^3 &= (a+b)(a+b)^2 \\ &= (a+b)(a^2+2ab+b^2) \\ &= a(a^2+2ab+b^2) + b(a^2+2ab+b^2) \\ &= a^3 + 2a^2b + b^2 + ba^2 + 2ab^2 + b^3 \\ &= a^3 + 3a^2b + 3ab^2 + b^3 \end{align*}

Answer 2

I agree with the general idea of Justin Skycak, and I think you'd be doing the students a disservice if you don't show them some kind of geometric pictures to go with this stuff.

In your specific example, I strongly disagree with anyone who says you shouldn't do the "completing the square picture" with negative sides. Why? Well, first of all, if you show them how to do it pictorially with positive numbers, the first time they encounter an equation with negative numbers they're going to draw a picture. So the question is almost unavoidable.

At this point you have two options. 1) Tell them negative side lengths make no sense and not to draw the picture, removing a tool they have to solve the problem. And 2) say "well, I've never seen a negative side length and they don't exist in 'the real world', but let's see what happens if we finish the problem." Sure enough, you'll get the right answer. Option 2 should be accompanied by algebra, of course, as well as trula's suggestion of deducting rectangles.

The problem with option 1 is that when they learn complex numbers, the teacher will pretty much go with option 2 for square roots of negatives. If the student prefers option 1, which they've been taught before, well, tough luck kid, that was last year and math is capricious and arbitrary!

As educators, we have to be consistent. If we tell them that the beauty of math is that we can create new, fantastic objects as long as they're consistent and help us solve problems, we have to put our money where our mouth is and entertain the idea of negative side lengths.

Answer 3

I personally had a lot of fun in high school trying to extend the geometric picture for completing the square to include negative numbers. I ended up inventing a system of signed areas which was equivalent to the sign conventions followed with determinants! My teacher was, thankfully, knowledgable enough to be able to make this connection for me. I don't think it would be good required material, but it could be enriching stuff for a math club to play with.

Answer 4

There are always a number of students who understand the geometry and like it and profit from it, others get more confused and like the algebraic approach. so why not use both. But sides should not be negativ, instead you deduct the rectangles. Also visualize earlier the three binomial formulas with squares and rectangles.

Answer 5

I think it will hurt more students than it helps, if you spend much time on this sort of thing. (Unless you have very strong students, who also are not being highly accelerated...thus having time/capacity for enrichment.)

In equations, the key idea (from pre-algebra on) is the concept of the equals sign and how you manipulate the terms on each side, to analyze problems. Do not underestimate how many people are weak at this and suffer from lack of equation manipulation "muscles".

Note: this is NOT to deny that there aren't some cool insights from geometry. But you need to think about the constraints of time and audience. And the need to prepare/move students for/along the whole "algebra to calculus" pathway.