Comments on: Shock and Disbelief in Math Class

By: Ethan Weker

Ethan Weker — Wed, 18 Jan 2017 16:27:46 +0000

I’m late to this discussion clearly, but can’t wait to start playing with all these ideas! I’ve taught parabolas as products of lines in the past (including with this Desmos activity: https://teacher.desmos.com/activitybuilder/custom/56c39561a72db32f0a6f0e28), but there’s clearly so much more rich content that I can and should get into here. I especially love the description (from EducationRealist – I feel like I should know who you are) of a quadratic in standard form as the sum of a quadratic with a vertex at the origin and a line – that should be a nice way to add to/expand my previous activity.

By: Yongyi

Yongyi — Sat, 07 Jan 2017 01:38:35 +0000

In reply to Yongyi.

And here’s a Desmos graph showing that my formula in the cubic case works! In fact it goes through both extrema of all the graphs!

https://www.desmos.com/calculator/3s3obopabq

By: Yongyi

Yongyi — Sat, 07 Jan 2017 01:22:48 +0000

I’ve proved that the graph of the minima of a general polynomial of any degree traces out a polynomial of the same degree! There’s a lot of algebra and mathematical notation in the proof, so I’ve written it up in LaTeX and you can see the proof here: http://latex.artofproblemsolving.com/miscpdf/huzvyugx.pdf?t=1483752108866

By: William Carey

William Carey — Thu, 15 Dec 2016 01:16:15 +0000

In reply to CCSSIMath. Sweet. Now talk through how to get your students to that for the cubic over their lunch break. Go.

By: CCSSIMath

CCSSIMath — Thu, 15 Dec 2016 01:13:04 +0000

If a quadratic function is given by y=axÂ²+bx+c, then the x-coordinate of the vertex is âˆ’b/2a. That’s basic.

If you substitute âˆ’b/2a for x and solve for y, the y-coordinate of the vertex becomes âˆ’bÂ²/4a+c.

Since the coordinates of the vertex are (âˆ’b/2a, âˆ’bÂ²/4a+c), if you leave a and c constant and vary b, then the y-coordinate changes as the square of the x-coordinate, in other words, a parabola.

The result that seems to be impressing people is not so remarkable if you make an effort to do simple math.

Yes, desmos can be used to stimulate an effort to prove this result, but it might instead be used to confirm the result, which is more in line with the scientific process.

Incidentally, not that many seem to give too much credence to international assessments, but in nations such as Japan, analyzing quadratic functions is in the standard curriculum for 9th grade.

By: James Key

James Key — Thu, 15 Dec 2016 00:13:24 +0000

Since we are on cool facts about quadratic functions, here’s an insight I learned from James Tanton: all the textbooks teach that the axis of symmetry has equation x = -b/(2a), but the reason is seldom explored in a satisfactory way. Check this out:
y = ax^2 + bx + c = x(ax+b) + c

From this we can read off the solutions (0,c) and (-b/a, c), and the symmetry line lies halfway between any two points with the same y-coordinate, so its equation must be x = -b/(2a). Neat stuff! This is a great way to graph parabolas that are presented in standard form. But the cool thing for students to learn is the *technique,* not the formula.

By: Randy Revels

Randy Revels — Wed, 14 Dec 2016 21:00:08 +0000

This same thing happened in my Algebra 1 class the last time we studied transformations of parabolas. So cool. :-)

By: Laurie Hailer

Laurie Hailer — Wed, 14 Dec 2016 16:33:50 +0000

Yes, I love that Desmos function. I was amazed when I first saw that and show as many students and colleagues as I can. I think it’s really cool that you are proving that in Precalc. Go Math!

By: William Carey

William Carey — Wed, 14 Dec 2016 16:11:30 +0000

In reply to Anna Scholl. A good question. The other question I'm wrestling with is this: there are cubic and quartic formulas, but per Abel-Ruffini, we know there's no quintic formula. Is the result that the extrema of a curve are functions (of the same degree!) of the coefficients of the curve a sufficiently weaker result such that you could express the coordinates of the extrema of a sixth degree polynomial as another sixth degree polynomial?

By: Dan Anderson

Dan Anderson — Wed, 14 Dec 2016 15:50:45 +0000

In reply to Anna Scholl.

Cool question Anna. It led me to look for an answer (to a different question). I was wondering if when you change the second coefficient in a cubic, if the “vertex” (really the inflection point) of the cubic also traced out a cubic graph. like how the second coefficient in a quadratic traces out another quadratic. Here’s what I came up with: https://www.desmos.com/calculator/imsllrnfvn
I think I’ve made a deeper connection between the neat fact that William and Dan shared, and calculus. Thanks for the thought Anna, even if I mostly ignored it!