What *are* quadratics?

Earlier today I was in a discussion with @mathhombre on Twitter about what is needed in an College Algebra course (roughly equivalent to Alg2+Precalc for high school).  I came to the (perhaps too radical) idea that lines are simulateously one of the most applicable concepts to a person’s “real life” but also one of the most useful tools for accessing higher math.  There are entire areas of study about linearization as a tool to simplify complicated cases not just in pure math, but economics, physics, engineering, etc.

Any that is just the prelude.  If we are going to talk about lines, what happens to the darling of any high school level algebra course?The quadratic function / The parabola.  parab product thumbConsider f(x) = x^2 – 2x – 15.  Factoring, we obtain f(x) = (x+3)(x-5).  A subtle idea that might be skipped over: f(x) is the product of two lines.  Literally lines y = x + 3 and y = x + 5.  Do our strudents face the idea that a linear factor and a line are the same?  Consider the product of the values of the lines — point by point.  Try it with Geogebra.  The f(3) = (line1 at 3)*(line2 at 3) = (6)(-2) = -12.

We also know the Fundamental Theorem of Algebra will guarantee n roots for a degree n polynomial.  But those n roots may have non-real parts, for example g(x)=x^2 + 4 does not factor over the reals  g(x) = (x+2i)(x-2i)

So can our “line*line” idea survive these new types of models?  And where can we see these complex lines?  The product relationship still holds.  g(3) = (3+2i)(3-2i) because the imaginary terms will cancel since complex roots always come in conjugate pairs.  g(3) = 3*3 + 3*2i + 3*-2i + – 2i*2i = 9 + 4 = 13.  But wait, lets slow down.  3 + 2i is a point.  A point on the line x+2i.  It is a line hidden to our view because we lack the dimensions on our plane to see it.  We have only the 1D real line as our inputs.

So lets consider the imaginary part of our domain.  Click the picture to see a larger view:

angled view 1That’s g(x) = x^2 + 4.  The red axis is the standard x-axis.  The green axis is the standard y-axis.  But the blue axis represents the imaginary part of our domain.  A normal classroom might be used to plotting 3+2i on a plane, but do we often make note that its a nominally different plane than the one we graph functions upon?  3+2i will be on the plane that passes through the red and blue axes.  I chose to view it this way to keep y outputs as close as possible to our usual view.

angle view 2 angle view 3Ok so lets see 3+2i and 3-2i and their output product 13.  I included two views here since it is difficult to get a grasp of the 3D situation.

These lines y = x+ 2i and y = x – 2i are kind of abuses of notation.  They should be specified that they are lines in space, namely y=x limited to the level planes Im = 2 and Im = -2.

There’s certainly more to do with exploring this idea, especially making the presentation more robust.

What really appeals to me is the symmetry of a quadratic’s complex conjugate roots is similar to the symmetry of a separate quadratic’s real factors around the axis of symmetry.  Even more, the idea of the imaginary number i as a 90 degree rotation fits puzzle pieces together.  The complex roots are 90 degree rotations around the axis of symmetry of a reflection of the parabola (reflect across a horizontal line passing through the vertex).  This is the part that holds some promise, but that I haven’t quite explained in my own head yet.

I invite you to play around with the Geogebra applet I used to explore this.  It is set up to start with x^2 – 8x +18.  Find the complex roots algebraically first, then see if it meshes with the visuals in the applet. Thoughts?angle view 4

The Lines are Not Always Parallel! A geogebra approach to Alternate Interior Angles

alt int angle image bigStudents are frequently confused by us harping on the importance of things that seem obvious: because we hardly ever show them the cases where the theorems are false.  One of the areas this comes up is Angles in Parallel Lines.  (The whole course of Geometry may feel like this, actually).  We teachers might feel like, “how do kids get confused here? its so easy and obvious!”  That’s right… so easy and obvious that we teachers have lost sight that we only show them these narrow cases.

We spend time proving that certain pairs of angles on parallel lines are congruent or supplementary.  Over and over and over… but always parallel lines!!  Maybe maybe we make an offhand comment about “if these lines weren’t parallel then the angles wouldn’t be congruent!”.  But those words hardly paint the picture for student.

But think of the infinite number of line that are not parallel!!

GeoGebra-logoSo, let us turn to Geogebra.  My philosophy in making this applet (and most of my applets) is to loosen some restrictions in our normal presentation so that more of the “sample space” can be explored.  Lets consider all lines cut by a transversal, not just parallel lines.  Then how many students will not only (1) appreciate parallel’s special nature and (2) make conjectures about congruent angles all by themselves?   This is much in the same vein as Michael Serra‘s Discovering Geometry book: set up a situation so that the students actions will lead them to make those conjectures.

The applet here lets students drag the line to all positions, and illustrates the red and blue angles, giving no measurements.  We want intuition here, not measurement.  Students might already have some idea “yeah they look close enough”.

The check box “Compare Angles” allows them to update their intuition when faced with a little more precision.  This kid who wasn’t able to articulate anything before might now realize in what realm we are trying to explore.

The check box “Compare Lines” allows them to make a connection to the properties of the pairs of lines.  Again, this is some non-verbal feedback and prodding to a student.  The student must assimilate this and then can provide the verbalization themselves.

Finally, I added in a little game at the end.  It will spit out a score based on how close to congruent the angles are.  (The formula is arbitrary, but spikes up very high when you’re close).  Let me know how this plays out with your students!  Do they get invested in beating each other’s high scores?  If so, notice that in trying to beat one-another, they must know that they want to make the lines “more parallel”.  Here we can embrace the difficulty of being precise when doing this with a trackpad or touchscreen.  Will your students say “hey if I could make them parallel, that should be an unbeatable perfect score!”  (I actually think Geogebra will spit out “infinity” score if it gets close enough– a happy consequence of the data structures they’re using)