Des-blog

Next week brings a holiday to the US—the home of the Fave—so the Friday Fave is taking next week off.

In the meantime, amuse yourself with a little gem that aims to bridge the gap between informal language and formal mathematical representations. At least, at most and the like are notoriously challenging for algebra students to interpret in symbols, even when they can act on these ideas as they appear in everyday life.

(Thanks to Matt Salomone for the fabulous tweet and image.)

The activity the Fave is speaking of is Absolute Value Inequalities on the Number Line. One of a suite of similarly structured activities, this one has students place points on the number line according to given constraints.

We ask students to predict what everyone’s points will look like in the aggregate. We show them what it really looks like, and ask them to verbalize a comparison between the two. We support students in translating these ideas into formal notation. Distance is the absolute value of the difference of the numbers makes more sense when you’ve already been thinking about the distance between numbers on a number line.

So go ahead and play around. The Fave will be back in two weeks’ time.

And while you’re thinking about lines, here are a few more activities on the theme.

Point Collector

Put the Point on the Line

Graphing Stories

The Friday Fave enjoyed a splendid Zero Derivative of Day Length Day this week, and hopes that you did too. In the Fave’s part of the world, the days are now getting shorter. Maybe in yours they’re getting longer. Either way, we all have Zero Derivative Day in common.

That got the Friday Fave thinking about daylight relationships more broadly, which leads the Fave to remind you of the activity Burning Daylight—this week’s Friday Fave.

If you are a math teacher with any affinity for applications or modeling at all (and that should be pretty close to everybody reading this right now), you probably see many beautiful things to notice in the following image, which compares Miami, Florida and Fairbanks, Alaska. Which is which?

In Burning Daylight, this task launches students into the real question, which is “Which city gets more total daylight?” Calculus is a useful tool for answering that question, but not an essential one.

Students model with trig functions; they develop their own ways of answering that big question, and then we ask a terrific question to get them thinking more deeply about the connections between the math and the reality.

Barrow, Alaska is the northernmost city in the United States. What does the graph of daylight hours look like for Barrow?

While you’re thinking about trig functions, here are a few other opportunities to play with sines, cosines, and all the rest:

Marbleslides: Periodics

Polygraph: Sinusoids

Trigonometric Graphing

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.