What are my professional goals? Many days, I’ve got precisely three: I want kids to feel curious, then frustrated, then *ohhhhhhhh*.

Math is pretty great for this. It’s full of puzzles and mysteries. Why do the angles in a triangle always sum to the same thing? How many moons would fill the sun? Why is it so hard to roll double sixes? There’s plenty here to excite curiosity, to elicit frustration, and to satisfy the intellectual itch for *ohhhhhhh*.

But there’s a common problem: too often, kids can beat the puzzles without feeling *any* of those things.

I find this, for example, with lines in the coordinate plane.

The suite of coordinate games should fill weeks, or months. But there’s a cheat code: y = mx + b. In my class, lots of students arrive already knowing this formula, and it means they can solve would-be-tricky problems (“See this line? Give equations for five more that never touch it!”) with boredom and ease (“Uh…. y = 2x + 1, y = 2x + 2, y = 2x + 3… do you want me to keep going?”).

Do they understand what the cheat code means, or why it works? Rarely. But that doesn’t bring my vanquished puzzles back to life.

So I need new puzzles. Ones that work for kids who have never heard of “y = mx + b,” but which also challenge students who have.

I’m still growing my portfolio, but here’s some of it:

How much of high school math would be easier if students understood that graphs express relationships between variables?

Answer: Basically all of it.

My hope is that questions like this can build the groundwork early.

It’s natural to see this and think, “Wait, a hyperbola? That’s way too hard. You’ve got to walk before you run. And you’ve got to run a mile before you run a marathon over hot coals.”

But I see it a different way: Before you make an earnest study of lines, you’ve got to study some *non*-lines.

If I were trying to teach you about animals, I might start with cats and dogs. They’re simple, furry, familiar, and lots of people have them lying around the house. But I’d have to show you some other animals first. Otherwise, the first time you meet an alligator, you’re gonna be like, “That wet green dog is so ugly I want to hate it.”

Equations aren’t etched in stone. They’re flexible and mutable as dry-erase marker. All those fun manipulations we use to *solve* equations can be used to reconfigure and redecorate them, too. And this quick task opens up two fun conversations:

Fun conversation #1: “Hey, check out how the same x-y pairs work in all these equations!”

Fun conversation #2: “Which of these equations is best for generating points on the line?”

Time has smiled on this little puzzle. Not only does it build intuitions about slope, but its room for freedom and ambition (“I’m going to do a point where the x-coordinate is a million!”) brings out interesting ideas and puts student conceptions on display.

Sometimes, I want puzzles with simple numbers. They’re more forgiving, inviting varied approaches and intuitions.

But sometimes, I want puzzles with hard numbers. These ones are rigid as brick. They demand greater technical competence, and usher students towards strategies that are more abstract and general – strategies they can carry forward.

What a wonderful philosophy. I hope to achieve something similar. It’s exceedingly difficult, though. We’re constrained by a few factors: The relatively-rigid “standard” math curriculum that does things largely from cultural inertia. (Radian measure before you’ve done calculus? Why?) The scattered pre-existing knowledge and skills. The overreliance of earlier teachers on talismanic reasoning and ritual performance. And looming over all of them, the lack of time to do it right.

But I do find it helps to keep the goal in sight, even knowing I’ll probably never quite get there.

I have found radians to very accessible and intuitive once students firmly “get” that the circumference of a circle is the same length as approx 6.28 (2pi of course) radii. Then you just need lots of examples of circular motion connecting the distance a point travels along the circumference to the rotation from the center. A key idea in computer animation, too.

I am a harpsichordist and I used to use your “ugly wet green dog” method when I taught Baroque continuo. Accompanying from a figured bass (a bass line and some numbers indicating chords) can be a challenge for classically trained keyboardists. (Good pop musicians do that kind of stuff all the time, though.) So I would say, fine, let’s learn using unfigured bass!

Teaches the ears a lot very fast! And they realize that Figures Are Your Friends.

I remember a being slightly blown away when I learned that with the equation of the line in “standard form” the coefficients of x,y were normal to the direction of the line. Once I learned this, slope-intercept seemed really boring.

Point-slope is probably more useful than slope intercept. And it gives an introduction to translations of the graph, with a singular method to do so. Better that than one algorithm for lines (adjust b) a second for parabola y = (x-h)^2 + v, and a 3rd for the rest of the conics (x-h)^2 + (y-v)^2 = r^2

I quite like the intercept-intercept form of a line.

Parametric form of a line… Vector form of a line… simultaneous equations of a line. You could spend a couple of weeks just talking about different equations of lines.

And the all have the same graph.

Regarding the graph of: {x,y|xy = 12}

How about “draw a set of triangles using the coordinate axes for two of the sides, one vertex at the origin and area 6.

Pretty.

I was a fool for the quadratic formula ib my younger years.

Unfortunately I never got past the frustration phase of mathematics. Good philosophy and enthusiasm, perhaps things would’ve been different if I was in high school with this though lol

I’m almost 61 years old. I hated math in highschool. I still hate math. I wish you had been my math teacher.

an approach like this could really excite pupils with a math phobea

this was so cute! i love this approach!

Love it. How do you keep students from falling into formulaic thinking once they are exposed to “standard forms” of lines and and such? My experience is that I can get students to think through these quite well in the beginning of a unit but then after they see things like y = mx + b and the like many of their reasoning brains fall off the cliff!