Thank you for saying this, and for the reference to

*Nix The Tricks*. I've placed an order for 2 on Amazon and downloaded a copy to get started. I never did teach tricks, and have tried to win students over from the use of them; I hope that this gives me some ways to be more effective in doing so. ...and maybe I will discover my own foible(s) somewhere.

**Long digression to set up my desired comment below:** I have been involved in the development and teaching of a new course for those students who are well below the prerequisite point for College Algebra but must have College Algebra (and perhaps more) for their degree programs. I have found that the vast majority of them are very quick to pick up on Algebra but are not good at reliably arriving at a solution or answer. Part of it is a reliance upon tricks; besides all the other problems with tricks, tricks trap their solution-seeking within their quick-recall memory (very prone to error and to missing subtlety) rather than employing their thinking ability. I show the students this video

Science of Thinking and compare it to the method of 'Gun' versus the method of the harpist. I try to get them to approach our class like the harpist. Your goal is important, but *how* you approach your goal is much more important. [Not mathematics, right? but these students need to be convinced that this time with Algebra will be different, and that they *can* do it.]

**The actual comment that I wanted to make:** By the end of this Fall semester's class it occurred to me that my view of our relationship had become that they were my apprentices. I was letting them in on the secrets of a trade and developing in them a craftsmanship. The secrets of the trade are not a basket of tricks, but of course the actual properties *and* how to strive to be faithful to those properties and use those properties to provide the seeds for a needed idea.

Next time, after they get to know me and what I'm after, I plan to let my students in on the apprentice analogy (I think).

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Edward Thome

Chair and Associate Professor of Mathematics

Murray KY

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Original Message:

Sent: 12-12-2022 14:31

From: Lane Walker

Subject: Visualizing the Quadratic Formula

I absolutely agree with not having students memorize the quadratic formula without seeing how it is derived. Since so many math teachers struggle to get through the proof perfectly, I have not yet had a class that I expected to re-generate the proof; but we work through it in a whole-class discussion.

As a contributor to Nix The Tricks and a huge fan of "13 Rules that Expire," I don't recommend even teaching things like "cross products," so thank you for bringing that up!

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Lane Walker, Ed.D

High School Math Teacher

Fuquay-Varina, NC

Original Message:

Sent: 12-12-2022 13:28

From: Tayah Rendina

Subject: Visualizing the Quadratic Formula

Hi Lane and Edric,

I think this visual is fun for students to see and could help with memorization. I am a first-year algebra 1 teacher getting ready to teach quadratic formula.

While this is helpful in making it stick, I do like what Edric discusses in deriving the formula from the standard form of a quadratic equation. I think I will definitely have student discover these connections before showing them this visual. For our state testing, students will be given the formula on their reference sheet. I want them to understand WHY it works and how to reason through finding solutions.

Thank you for sharing!

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Tayah Rendina

Salisbury MD

Original Message:

Sent: 11-28-2022 22:26

From: Edric Cane

Subject: Visualizing the Quadratic Formula

However students memorize the formula, I like them to also think of the version where dividing by '2a' appears twice: -b/2a + or - (Sq rt of (b^{2} – 4ac))/2a. This gives meaning to the formula: the first term (-b/2a) is the x-value of the axis of the parabola; + or – a positive second term is the distance on either side of that axis to the parabola itself.

Given an example of a quadratic equation, y = ax^{2} + bx + c, I want students to use it to visualize the parabola:

With x = 0, y = c: 'c' is the parabola's y-intercept.

Y = bx + c is a line through c that is easy to visualize ('c' is y-intercept, slope is 'b'). It is tangent to the parabola at c.

'ax^{2}' is the curving factor that applies to the y = bx + c line: it makes the line curb back on itself changing it into the parabola.

If we cut out a paper parabola for a = 1, for instance, it is easy to position it on a graph tangent to line y = bx + c. This immediately tells us where the roots would be (if there are real roots). It immediately shows that, if 'a' and 'c' are positive, any real root must both be negative. If 'a' is positive and 'b' negative, we immediately know that both roots, if real, must be positive. Just by looking at the equation, students can learn to visualize the graph of the parabola and immediately see many of its essential characteristics.

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Edric Cane

Math teacher, retired

Author,

Teaching to Intuition: Mathematics.

Workbook: Making Friends with Numbers: Let learning Multiplication Facts teach you Math.

edriccane@aol.com

(916) 973-8569.

Carmichael, CA 95608

Original Message:

Sent: 11-25-2022 22:02

From: Ancy Mariam Joseph

Subject: Visualizing the Quadratic Formula

Thank you Lane, for sharing this.

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Ancy Mariam Joseph

Engineer | Educator for 8 years | Math Department Lead

Single Subject Math Credential (CA)

Graduate Student @Notre Dame School of Education

San Francisco Bay Area

CA

Original Message:

Sent: 11-16-2022 22:06

From: Lane Walker

Subject: Visualizing the Quadratic Formula

Great teachers don't miss opportunities to bring in all the senses to make something "stick." The proliferation of Quadratic Formula songs on Youtube bears witness of that. But when students confuse b and c because they rhyme, a song doesn't necessarily cover all the bases for memory. Here's how I cover the visual component:

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Lane Walker, Ed.D

High School Math Teacher

Fuquay-Varina, NC

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