Determinants are calling me...

Good Story about a class, and my continuing fascination with determinants.

I go to class today, and it is Algebra 2, which I am never really sure how to approach. We are doing a chapter on lines. Now I had all this students last year for Algebra I and it is an honors course, so they are all pretty strong. I come in to class and put the following question on the board:

Determine if the points (-2, 1) (0, 3) and (7, 12) are collinear

And do you know what all the students asked (Go ahead test yourself, see if you can put yourselves in the mind of an 8th grader)

I hope you guessed "What is collinear?" Cause that would be right.

Now we are at the tony private school, so the smartboard is still on from the last class. I login in and go to wolfram.mathworld.com and look up collinear. I let the students read the first couple lines, and now they are off and running. Get lots of good answers/approaches. Some students find the equation of line using two points and check the third point in the equation to see if it works. Some calculate the slope two different ways and see if they are equal. Some calculate the equation two different ways and see if they are equal. It is all very good.

While the students are working I can help but see the following information just a little bit lower down on the mathworld entry for collinear (and this is not a direct quote)

You can check if three points are collinear if with determinants. For points (x_1, y_1), (x_2, y_2), and (x_3, y_3) create the determinant.

|  x_1  y_1  1 |
|  x_2  y_2  1 |
|  x_3  y_3  1 |

if the determinant equals zero then the three points are collinear.

I had totally forgotten about this, even though I had written a post about determinants two months ago.

Now I am standing in front of my class while they work the collinearity problem wondering, should I go for it. Should I show them determinants right now? Should I use this moment to introduce this strange monster of mathematics? It then occurs to me that we can use this to find equations of lines as well, and that in the next chapter I am going to introduce Cramer's Rule. Cramer's Rule is nothing but determinants, and so I go for it. The kids really liked it, and it is a most amazing thing that the series of calculations does this.

The way to find the equation of the line is to take two points and create the determinant:

|  x    y    1 |
|  x_1  y_1  1 |=0
|  x_2  y_2  1 |

this works because you are looking for all the points that are collinear with the other two.

Another way to look at it is the fact that the determinant

|  x_1  y_1  1 |
|  x_2  y_2  1 |*.5
|  x_3  y_3  1 |

Gives the area of a triangle with those coordinates therefore the three points are collinear if and only if the three points define a triangle with zero area.

So the class went well, and they really had an appreciation for determinants. They liked the way this got them pretty close to standard form. They may even remember it. I showed them two by two determinants as well, we will see if they recognize it when it comes up in about two weeks.

So, after another foray with determinants, I left pondering. I wonder "What is a determinant?" What is it's core? It does so many things: linearity, area, invertibility, etc. I just don't know.