# Monthly Archives: June 2017

I’m returning to my ‘blog after a prolonged absence.  My next several posts will explore ideas I shared and learned at the USACAS-10 conference hosted at Hawken School last weekend.

Finding equations for quadratic functions has long been a staple of secondary mathematics.  Historically, students are given information about some points on the graph of the quadratic, and efficient students typically figure out which form of the equation to use.  This post from my Curious Quadratics a la CAS presentation explores a significant mindset change that evolves once computer algebra enters the learning environment.

HISTORIC BACKGROUND:

Students spend lots of time (too much?) learning how to manipulate algebraic expressions between polynomial forms.  Whether distributing, factoring, or completing the square, generations of students have spent weeks changing quadratic expressions between three common algebraic forms

Standard:  $y=a*x^2+b*x+c$

Factored:  $y=a*(x-x_1)(x-x_2)$

Vertex:  $y=a*(x-h)^2+k$

many times without ever really knowing why.  I finally grasped deeply the reason for this about 15 years ago in a presentation by Bernhard Kutzler of Austria.  Poorly paraphrasing Bernhard’s point, he said in more elegant phrasing,

We change algebraic forms of functions because different forms reveal different properties of the function and because no single form reveals everything about a function.

While any of what follows could be eventually derived from any of the three quadratic forms, in general the Standard Form explicitly gives the y-intercept, Factored Form states x-intercepts, and Vertex Form “reveals” the vertex (duh).  When working without electronic technology, students can gain efficiency by choosing to work with a quadratic form that blends well with given information.  To demonstrate this, here’s an example of the differences between non-tech and CAS approaches.

COMPARING APPROACHES:

For an example, determine all intercepts and the vertex of the parabola that passes through $(10, 210)$$(5, 40)$, and $(-2, -30)$.

NON-TECH:  Not knowing anything about the points, use Standard form, plug in all three points, and solve the resulting system.

$y=a*x^2+b*x+c$
$210 = 100a+10b+c$
$40 = 25a+5b+c$
$-30 = 4a-2b+c$

Use any approach you want to solve this 3×3 system to get $a=2$, $b=4$, and $c=-30$.

That immediately gives the y-intercept at -30.  Factoring to $y=2(x+5)(x-3)$ or using the Quadratic Formula reveals the x-intercepts at -5 and 3.  Completing the square or leveraging symmetry between the known x-intercepts gives the vertex at $(-1,-32)$.  Some less-confident students find all of the hinted-at manipulations in this paragraph burdensome or even daunting.

CAS APPROACH:  By declaring the form you want/need, you can directly get any information you require.  In the next three lines on my Nspire CAS, notice that the only difference in my commands is the form of the equation I want in the first part of the command.  Also notice my use of lists to simplify substitution of the given points.

The last line’s output gave two solutions only because I didn’t specify which of x1 and x2 was the larger x-intercept, so my Nspire gave me both.

The -30 y-intercept appears in the first output, the vertex in the second, and the x-intercepts in the third.  Any information is equally simple to obtain.

CONCLUSION:

In the end, it’s all about knowing what you want to find and how to ask questions of the tools you have available.  Understanding the algebra behind the solutions is important, but endless repetition of these tasks is not helpful, even though it may be easy to test.

Instead, focus on using what you know, explore for patterns, and ask good questions.  …And teach with a CAS!