I’m a huge fan of Dan Meyer’s Will It Hit The Hoop lesson. Looks something like this:

Show them this video:

There’s an obvious question here, and all answers are valid. Every kid has an opinion on whether the shot will go in.

Motivating quadratics is a bit more tricky. I’m not sure what questions are most useful for getting a student to say, “hey that looks like a parabola”, but I’m not sure that’s the best use of time — we’re going to give them quadratics anyway. I think we can jump pretty quickly into the modeling.

Dan’s page has a ton of great resources — he has seven different takes, shots from different distances, some that go in, some that don’t. His takes come with a half-video, like above, a full video showing whether the shot goes in, a .png file for the image, and a Geogebra file to model. Geogebra looks like this:

After students model by dragging the sliders around, they get a chance to see the resolution with the full video:

Great stuff.

**What I did:** I put together a Desmos graph for each take. Graphs are here:

Take 1

Take 2

Take 3

Take 4

Take 5

Take 6

Take 7

And look like this:

Graph is interactive; students play with it to predict whether the shot will go in.

**My thoughts:** These were fun to make. I’m not sure how they fit into the lesson — the Geogebra files were pretty great to begin with.

**Advantages with Desmos:**

It’s easier to drag around two points on a quadratic — low floor to introduce students to the topic

Our Chromebooks don’t have the Geogebra app on them yet, and I’m not sure if it will ever happen

Web pages are easier to distribute to students than .ggb files

**Advantages with Geogebra:**

The picture is clearer — I couldn’t get the high-quality image to load in Desmos

Using sliders to adjust the vertex and a-value reinforce deeper quadratic concepts

Students can see the equation

**I’m torn: **I really want to use this lesson early on in quadratics to give them some meaning in context, and help students gain an intuition for what quadratics look like. To me, that suggests Desmos — vertex form is tough, and would require a lot more knowledge for students to be successful. But I can’t help thinking that the Geogebra version is hugely valuable for the parts of a quadratic, and how they relate to each other. That seems like a concrete goal for the lesson — where the Desmos version is more like “go play with quadratics and understand them better”.

I’ve got a few months to decide. Excited for it either way.

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Michael PershanGreat questions in this post. What I’m thinking about is the word “intuition.” I use this word all the time. But what do I mean by it? What do you mean by it? If you have an intuition for quadratics, what does that mean you can do that you otherwise couldn’t?

dkane47Post authorGreat question Michael — I have similar thoughts on number sense, a big set of ideas I have a vague grasp of and reference all the time.

I think that this lesson, and lessons like it, give students another mental representation for what a quadratic is. If their mental representation of a quadratic is that it’s a function/equation that sometimes you have to factor, sometimes you have to complete the square, and sometimes you have to graph, it’s easy to get lost in the sauce of formulas and calculations, and non-routine problems that are simple if you can visualize the curve become complicated because they aren’t “factoring” or “find the vertex”.

That still sounds vague. I want to think more about this.

jnewman85Thanks for putting together those Desmos graphs–they’re pretty great. It’s been nearly a year since I’ve tried, but I think Dan’s .gbb files didn’t work well on the web-version of Geogebra that was installed on our Chromebooks. (Actually I felt that Geogebra on Chromebooks in general was rather sluggish).

You could have the best of both worlds (well, except for the high resolution images) by just using good-old-fashioned sliders in Geogebra to mimic Dan’s Geogebra file. Then they can also see the equation. Something like this:

https://www.desmos.com/calculator/q1bxryhivv

dkane47Post authorI like it! With seven takes to choose from, maybe a better lesson would be to drag the quadratic for the first 3-4 takes to gain some confidence, and then use sliders like your graph for the last few takes to reinforce those concepts. I think that would be a great lesson, but it’s tricky to place it in a unit and get students to analyze what they’re doing and not just play around with buttons.

jnewman85Yeah, a well-written worksheet with some “What do you notice?” and “What do you wonder?” discussion questions would help the students to place the activity in the context of a lesson. If I find some time (ha!) I might do this since it would work well for where my students are, too.

howardat58Initially the students don’t need to know anything formal about quadratics. the curve is just a curve, but a reasonable one. This is what is good about the desmos approach. I would also try a real real-world test. Throw something across the room, at different angles.

The trajectory will come alive, and matching the stills will seem quite sensible.

Beware: One student is bound to ask eventually “Why is it always a quadratic?”.

dkane47Post authorThat’s an interesting approach — the Desmos lesson definitely lowers the floor so that this could be a lesson to introduce quadratics.

James ClevelandI was able to get the high quality photos in: https://www.desmos.com/calculator/o9fzivrlsz

Has Desmos improved itself since September, or was there some other reason?

dkane47Post authorAwesome!

I think it’s an improvement — I asked Eli about it when I first put these together, and he said it was on the Desmos “wish list” and they were hoping to work on it.

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