## Shock and Disbelief in Math Class Last year I realized that Pre-Calculus is really a class about moving from the particular to the general. We take particular skills and ideas students are comfortable with – like solving a quadratic equation – and generalize them to as many mathematical objects as we can – solving all polynomial equations. As we worked our way through polynomials, we wanted to move from reasoning about particular quadratic equations like y = x2 + 2x + 1 to reasoning about all quadratic equations: y = ax2 + bx + c. For homework, the students had to graph about twenty quadratics with varying a, b, and c.

Then we got together to discuss the results in class. They remembered that a controls the “fatness” or “narrowness” of the parabola and sometimes flips it upside down. They remembered that c moves the parabola up and down. They weren’t totally sure what b did. A few students adamantly maintained that it moved the parabola left and right (with supporting examples). After about fifteen minutes of back and forth, we decided to go to Desmos and just animate b.

Shock and disbelief: the vertex traces out what looks like a parabola as b changes. Furious math and argument ensue. Ten minutes later, a student has what seems to be the parabola the vertex traces graphed in Desmos. Is it the right parabola? Why? Can we prove that? (We could and did!)

Previously: WTF Math Problems. I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.

1. #### education realist

December 13, 2016 - 9:16 pm -

I teach parabolas as three different operations.

1) Take a line, square it, stretch it, and move it up and down. So sketch the line and the x-intercept is the line of symmetry. Very visual. (Vertex)
2) Take two lines and multiply them together. The product will be zero where the x-intercepts of the lines are zero, because zero is the death lord, Shiva the destroyer of multiplication. And because we know about the line of symmetry, we know its right between the zeros. Then plug in ot find y. (looking for a way around that.) Also very visual.

3) This is the new one: a parabola in standard form is the sum of a parabola centered at the origin and a line. Not visual. But the one thing that’s clear is that the origin plus the line’s y-intercept is going to be the y-intercept. And a bit of investigation shows that there’s a clear and predictable slope from the vertex of a parabola to the y-intercept (aka, rate of change). This last has really been useful in unlocking the mystery of graphing in standard form, other than negative b over 2a. But it’s also given me a much clearer sense of how the linear term affects the parabola. The kids, too.

• #### William Carey

December 14, 2016 - 4:51 am -

Katrin,

I suppose it depends on what you want to reveal about the quadratic. For an Algebra I or II class, probably the completed square form reveals more about the parabola. But for a Pre-Calculus class, what I’m trying to do is ruthlessly move from the particular to the general. And the vertex form doesn’t really generalize.

I don’t know that the completed square form generalizes to higher degree polynomials. Is there a completed cube form? A quick google search suggests “no”. That makes the completed square form, in some ways, a dead end.

But if you look at the properties of the coefficients of standard form, you can do similar things for higher degree polynomials. There’s a cubic equation that describes the curve traced out by the extrema of y=ax^3+bx^2+cx+d is surprising and interesting (and requires a metric crap-ton of algebra to work out; if you want to get kids to practice algebraic manipulation, make them yearn to know things that require it in great volume). Why is it cubic? Why isn’t it lower degree? It’s also a nice segue to Calculus, because it makes the location of maxima and minima a headache that kids want aspirin for.

2. #### James Key

December 14, 2016 - 4:13 pm -

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.

3. #### CCSSIMath

December 14, 2016 - 5:13 pm -

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.

4. #### Yongyi

January 6, 2017 - 5:22 pm -

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

5. #### Ethan Weker

January 18, 2017 - 8:27 am -

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.