Sorry for the silly title but when the temperature approaches 100, I start becoming delusional!
Algebra teachers know that the equations of horizontal and vertical lines (the coordinate axes in particular) are stumbling blocks for students and creative educators and desperate students often resort to clever mnemonics and other memory aids to recall these. I invite readers to share their favorite. The teachers in my department (former department that is -- be kind, I'm adjusting to retirement) became enamored of HOY-VUX. I'm not sure who originated it but this person deserves credit! The name is silly (reminiscent of horcruxes from Harry Potter), the students laugh at it, but when the test comes around, they write it at the top of their paper. Here's how it works:
HOY: Horizontal, slope 0, Y=...)
VUX: Vertical, Undefined slope, X=...)
Now I know that others out there have their favorite ways of teaching this so PLEASE SHARE!
Believe it or not, the above was not the intent of this post but it's probably more interesting than the technical stuff to follow! This discussion is intended for Algebra 2 students and beyond. A full treatment requires some vector analysis but I will avoid that for now. Unlike most of my offerings, I did not set this up as a worksheet but you'll get the idea. You may want to bookmark this and save it for when 3-dimensional graphing comes up in the curriculum.
Start with a horizontal number line: <------------------|---------------->x
Ask students to plot x = 3 on the line. No ambiguity here, right!?!
Thus, the 1-dimensional graph of x = 3 is a POINT! Easy, so far.
Now draw both coordinate axes. Plot the point at 3 on the x-axis, ask a student for both of its coordinates and ask if it satisfies the equation 1x + 0y = 3.
Confirm this: 1(3) + 0(0) = 3.
Students generally treat x = 3 as an exceptional case of the equation of a line, but having both variables may help them see it isn't that special (other than its slope of course!).
Ask students to verify that (3,1), (3,2), (3,-1), (3,-2) all satisfy the equation 1x + 0y = 3.
Have them plot these points.
Ask the class (I didn't feel like writing this in the form of a worksheet today) to verify that (3,k) satisfies this equation for any real value of k. Students need to understand this significance of the zero coefficient of y.
This should help them to recognize that the graph of x = 3 is a vertical line. Don't get me wrong. Understanding this does not necessarily lead to getting it right on a test! They still need survival gear (mnemonics) for that! HOY-VUX to the rescue!
BUT THERE'S MUCH MORE TO THIS!
Point out that in the equation 1x + 0y = 3, we see that the resulting line is PERPENDICULAR to the axis with the non-zero coefficient (x-axis here) and PARALLEL to the axis whose coefficient is zero (the y-axis in this case). We're not proving anything here or explaining why this is true, just making an observation that we will generalize later.
Ok, so where's the plane in all of this?
In 3-dimensional space, we examine the equation 1x + 0y + 0z = 3. We can still graph the point corresponding to 3 on the x-axis, the vertical line graphed above (y can be chosen arbitrarily) and now z can be any real number. Corresponding to each variable whose coefficient is zero, the graph will now be a plane PARALLEL to that axis and PERPENDICULAR to the axis whose coefficient is not zero. Thus, our graph is now a plane parallel to the y- and z-axes (therefore parallel to the yz-plane determined by these axes) and perpendicular to the x-axis. Of course, software like Mathematica, Derive, or even freeware available on the web will help students visualize this better. Cardboard or Styrofoam models are also highly effective here. The more the students construct these models and label the axes and planes, the better they will be able to make sense of all this.
So what is the graph of x = 3? All of the above!
Now have your students analyze the equation y = 3 following this model!
Monday, July 9, 2007
x = 3: It's a Point! No, It's a Line! No, It's a Plane! It's... It's SuperPlot!
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4 comments:
I use position vs time graphs, where slope is speed. Convinces me (every year) that the slope of a vertical line should be undefined -- not sure it really does that well with the kiddies.
I like your x=3 business. I haven't done that. However, all of my students have been exposed to one-dimensional circles, and a few of them can even explain their relation to the old-fashioned, two dimensional kind, maybe even in more than one way.
thanks jonathan--
I like your 's-t' graph model! Not only would the slope (speed) be undefined (or perhaps infinite speed is required to rise vertically), but consider that the position graph in this case is not a function. In fact,for some particular time, like t = 3, the particle would have infinitely many positions. And we complain about being in TWO places at the same time!
The word you should introduce your students here is hyperplane. As you know, a hyperplane is a linear subspace of 1-less dimension than the ambient space. A hyperplane for the line is a single point, a hyperplane for the plane is a line, a hyperplane for 3-space is a plane, and a hyperplane for 4-space is a 3-plane. Any linear equation (except for impossible ones: 0x + 0y + 0z = 1, for example) on n-space gives a hyperplane.
In fact, generally any single equation on R^n gives an n-1 dimensional locus. Any 2 equations give an n-2 dimensional locus, and so on. This can fail for basically three reasons:
1. The equations are dependent. x + y = 3 and 2x + 2y = 6 are equivalent equations; they correspond to the same line in R^2, and so on.
2. You are working with a field that isn't algebraically closed. x^2 + y^2 = -1 has no solutions in R^2, and x^2 + y^2 = 0 is just the origin, but z^2 + w^2 = 0 becomes two complex lines in C^2. Notice the variable change here. x and y have overloaded meanings.
3. Most interesting and accessible for your students is the case where the equations are incompatible in R^2 or R^3, but if one goes to projective space they become compatible again. Consider the two lines in R^2, x + y = 1 and x + y = 2. Obviously, they never meet; these lines are parallel. Now, define RP^2 as the set (x,y,z) ∈ R^3/~ where
(x,y,z) ~ (ax,ay,az) for any a ≠ 0. There is a 1-1 mapping of R^2 into RP^2 by (x,y) → (x,y,1). But, how do we ever get a zero 3rd coordinate?
Take the equation x + y = 1, and replace x by X/Z and y by Y/Z. You see, the partial inverse of that mapping is (X,Y,Z) → (X/Z, Y/Z, 1) (if Z isn't 0).
So, x+y = 1 becomes X/Z + Y/Z = 1, or X+Y=Z.
Similarly, x+y = 2 becomes X+Y = 2Z. How can these both be true? Only if Z = 0, and then we have X+Y = 0, and so (X,Y,Z) = (X, -X, 0). X can't be zero here, and we end up with (1, -1, 0) by the equivalence relation. These two parallel lines in R^2 meet at a point on the line at infinity in RP^2.
Where would your students see this? At the art museum! During the Renaissance, when artists discovered the principle of perspective, they discovered that when one draws parallel lines on a canvas, they all head toward a particular point on the canvas, the point at infinity for that family of parallel lines. Some artists, starting with Filippo Brunelleschi, discovered a painstaking way of drawing the proper perspective. You can probably make up a slide show for your students. You might attract some students who cared for art more than mathematics.
I think I know the name of the student who made up HOY VUX. HE was a student at Ramapo and is now ready to graduate college. He was a student of Mrs. Pappas. Is he truly the creator? Maybe he could put it on his resume! He is my son! Thanks.
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