Love tangent first. That’s so obvious now that you’ve proposed it.

Favorite framing of trig is what you end with; one way to think of math is as the study of ‘what else do we know?’

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]]>2). I also love that you’re generalizing to cubic and higher-power equations instead of putting quadratics blinders on kids. Math is more fun when you punch through a wall and see a wide-open space, not a narrow tunnel, on the other side.

3). I love the Step 3 activities and will probably steal them this year. Unrelatedly, I love input-output tables laid out horizontally, so the values of the tables match up with the rough locations of the points on the graph (negative x-values on the left, and positive x-values on the right).

As part of Step 3, do you take some time and try to show kids what quadratic functions can actually model in real life? I use an adapted version of Will it Hit the Hoop, but even just a discussion might do.

As part of the discussion on what quadratics model, would you include the idea that they model situations in which the rate of change is changing at a constant rate? To me, that’s a big idea of quadratics that has to come out in the unit. Though I’m in Algebra 1. Not sure if that’s out of place in 8th grade. If you want resources on that, here are some:

— A quick Desmos activity I use: https://teacher.desmos.com/activitybuilder/custom/57693b4f0a66fa8a6a6e2284

— The larger lesson context of that Desmos activity: (though I’m not sure I still agree with everything in this blog post, it still describes the lesson fairly well)

https://ijkijkevin.wordpress.com/2016/03/24/creating-intellectual-need-for-multiplying-binomials/

–Here is the quadratics unit from the curriculum David Wees works on. You can see the big idea of changing-rate-of-change in the first big idea: https://curriculum.newvisions.org/math/course/algebra-i/quadratic-functions/

4). Step 4, love it.

5). In future years, I think it’s worth teaching this type of equation, and equations like 2(x+3)^2, earlier in the year, when you teach the “unwinding” strategy of equation-solving. The work kids have to do is really more connected to the order of operations than it is to quadratics. When I do this early in the year, we find the positive (or easiest) root only. If you did that, too, then your Step 5 would be simply teaching using the +- sign to get both roots, and also teaching about imaginary numbers/no solution. By the way, my decision to move the topic of unwinding quadratic equations to earlier in the year is 100% the result of my buying into your philosophy that students should get ample time to practice skills independently before having to make a connection between them.

6-9). Love them.

]]>it uses a visual approach to develop quadratic expansion and factorization.

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]]>I really enjoy this approach because, as you said, kids get really familiar with each form before they do a lot of manipulating. In particular, they see how the a coefficient has the same effect in all forms, and they see how each form provides an easy way to find some of the important points of a parabola.

I believe I got this idea from the Carnegie Algebra 1 textbook that my district’s high school began using a couple of years ago. I didn’t use their lessons, but I believe I mirrored their overall sequence of ideas.

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