Des-blog

This past week we asked the Desmos fellows to share a favorite Desmos activity and to tell why it worked for their class and how it helped students learn mathematics. The group generated a great list of activities and ideas, three of which we’ll highlight below.

Exploration

Desmos can help students explore concepts visually in a low risk environment. They can test out theories and adjust their current model of thinking accordingly. In Loco for Loci! we start by asking students to drag the green point in a graph so that it is four units from the blue point. Then we ask student to predict what it would look like if all of their classmates dragged the green point four units from the blue point. The teacher reveals the relationship by sharing the class overlay of all points.

Dylan Kane shares that “Students got to play with some points and make predictions, and I don’t know that a ton of learning happened in that first stage, but they had the chance to get some practice thinking about what a locus is (even if they didn’t have that language) as well as make predictions (most of which lacked precision). Then, at the end, we looked back at students’ work as a class and had a chance to formalize a definition of locus, as well as understand why different prompts resulted in different mathematical objects.”

Class Discussion

Kathy Henderson shares that “One of the most valuable outcomes of a good Desmos activity is when the activity allows students to recognize misconceptions of a topic and then allows class discussion of those misunderstandings.” Match My Parabola supports discussion by allowing students to explore the various forms of a parabola followed by an opportunity to reflect on what they’ve learned using various strategies to justify their thinking.

Connecting Representations and Visual Feedback

Neel Chugh shared that Game, Set, Flat “allows students to see the connections between geometric sequences and exponential functions really well. It also gives the the opportunity to conjecture and test out different ideas supported by great animation.”

How has Desmos helped your students learn mathematics? Let us know on Twitter @desmos.

If you’ve ever found yourself using your fingertips to trace the path of a point along a curve to help yourself or someone else think about the meaning of right and left-handed limits, then this week’s Friday Fave—Limits and Continuity—may just be your ticket.

Take all those continuity diagrams from your Calculus textbook and make them interactive. This was Bryn Humberstone’s mission in developing the activity originally.

All we did was apply a tiny bit of Desmos Activity Polish ™ and make it available to you.

NB: While students can interact with these diagrams equally well on laptops and on touchscreens, we cannot be held accountable for fingerprints they may leave on your devices.

While you’re in a Calculus sort of a way, here are three more faves:

The Intermediate Value Theorem

Average Value of a Function

Sketchy Derivatives

[cross-posted from Dan Meyer’s blog]

Eight years ago, this XKCD comic crossed my desk, then into my classes, onto my blog, and through my professional development workshops.

That single comic has put thousands of students in a position to encounter the power and delight of the coordinate plane. But I’ve never been happier with those experiences than I was when my team at Desmos partnered with the team at CPM to create a lesson we call Pomegraphit.

It’s yours to use.

Here is how Pomegraphit reflects some of the core design principles of the teaching team at Desmos.

Ask for informal analysis before formal analysis.

Flip open your textbook to the chapter that introduces the coordinate plane. I’ll wager $5 that the first coordinate plane students see includes a grid. Here’s the top Google result for “coordinate plane explanation” for example.

A gridded plane is the formal sibling of the gridless plane. The gridded plane allows for more power and precision, but a student’s earliest experience plotting two dimensions simultaneously shouldn’t involve precision or even numerical measurement. That can come later. Students should first ask themselves what it means when a point moves up, down, left, right, and, especially, diagonally.

So there isn’t a single numerical coordinate or gridline in Pomegraphit.

Delay feedback for reflection, especially during concept development activities.

It seemed impossible for us to offer students any automatic feedback here. After a student graphs her fruit, we have no way of telling her, “Your understanding of the coordinate plane is incomplete.” This is because there is no right way to place a fruit. Every answer could be correct. Maybe this student really thinks grapes are gross and difficult to eat. We can’t assume here.

So watch this! We first asked students to signal tastiness and difficulty using checkboxes, a more familiar representation.

Now we know the quadrants where we should find each student’s fruit. So when the student then graphs her fruit, on the next screen we don’t say, “Your opinions are wrong.” We say, “Your graph and your checkboxes disagree.”

Then it’s up to students to bring those two representations into alignment, bringing their understanding of both representations up to the same level.

Create objects that promote mathematical conversations between teachers and students.

Until now, it’s been impossible for me to have one particular conversation about the tasty-easy graph. It’s been impossible for me to ask one particular question about everyone’s graphs, because the answer has been scattered in pieces across everyone’s papers. But when all of your students are using networked devices using some of the best math edtech available, we can collect all of those answers and ask the question I’ve wanted to ask for years:

“What’s the most controversial fruit in the room? How can we find out?”

Is it the lemon?

Or is it the strawberry?

What will it be in your classes? Find out and let us know.