Des-blog

This week’s Friday Fave is a feature in Desmos Geometry: Transformations.

In order to appreciate the transformations tools, you’ll need to think of a transformation as a function. A function has inputs and outputs, where the set of all possible inputs is the domain, and the set of all possible outputs is the range.

Functions come in families. If you’re thinking about algebra, you can talk about quadratic functions in general, but to find an output for a given input, you’ll need to specify which quadratic function you’re using—perhaps by specifying coefficients for your squared, linear, and constant terms.

That’s how transformations work in Desmos Geometry. Say you want to reflect a triangle. You’ll first need to specify the reflection, you’ll need to specify a line. Once you’ve defined the reflection, geometric objects are your domain. Select anything you’ve constructed, and apply the transformation to see its output.

Rotations work the same way. Specifying a rotation requires two things: a point around which to rotate, and a degree measure for the rotation. If you apply the transformation more than once, each output becomes the next input.

Once you’ve begun thinking about transformations as functions, you may find yourself using these simple objects to make complicated and beautiful things, such as those below (click through on each heading to get to a version you can play with). That’s what makes Desmos Geometry transformations this week’s Friday Fave.

Square tiling

Kaleidoscope

Triangle tessellation

Hexagons

While you’re thinking about transformations, here are some terrific activities using transformations in geometry and algebra.

Transformation Golf: Rigid Motions

Marbleslides: Periodics

Function Transformations: Practice with Symbols

Card Sort: Transformations

The Friday Fave is thinking about sequels this week.

Michael Fenton spoke about sequels a while back, and you should hear what he has to say on the matter.

https://www.youtube.com/watch?v=IreYFX6d3bg

Michael’s sequels are additional questions about a context within a lesson. And it is also worth thinking about when the next lesson ought to be a sequel to the current one.

Consider the case of Polygraph: Lines. Students play an engaging game that requires them to notice and discuss features of graphed lines for which they may not yet have words. As teacher, you introduce some of those words as the game moves along. But even if you don’t play Polygraph again, the game itself has created a rich space for asking additional questions, and for moving students further along in their mathematical journeys.

That’s where Polygraph: Lines, Part 2 comes in. In this sequel to the original Polygraph, students encounter increasing, decreasing, steepness, and intercepts in the context of thinking about situations that could arise in playing Polygraph.

Students sketch, write, and see each others’ responses in the service of better understanding the algebra of lines. All of it made possible by the original context of Polygraph. That’s what makes Polygraph: Lines, Part 2 this week’s Friday Fave.

And here are two more activities with sequels:

Polygraph: Hexagons and Polygraph: Hexagons, Part 2

Function Carnival and Function Carnival, Part 2

We rarely ask students to do impossible things.

Or more to the point, we rarely ask students to design impossible tasks. After all, why would you?

The Friday Fave has an answer to that question actually. When you ask a student to design a task that’s impossible to solve, you engage in the student a particular kind of critical thinking. Success requires analyzing the structure of the task in a different way from solving a possible task.

Success requires considering what the solutions to the possible tasks all have in common, and then doing the opposite of that.

All of which brings us to this week’s Friday Fave: Linear Slalom.

In the standard version of Linear Slalom, we ask you to send a line through sets of slalom poles.

Once you’ve got the hang of it, we ask you to design an impossible slalom.

Think about this for a moment. What makes a linear slalom task impossible? Your mind may go to the vertical line test. But we actually let you write an equation for a vertical line, so you’ll need more insight than that.

Horizontal gates on the same horizontal line will work, as will vertical gates on the same vertical line. Is that all? What’s the general principle here, and why?

There’s a lot of math in just this one screen of this delightful activity, and there’s lots more math to be done as students head to the Challenge Creator to design possible slaloms for each other to solve.

But designing an impossible task is what makes Linear Slalom this week’s Friday Fave.

If parabolas are more your thing, try out Parabola Slalom.

And if you’re looking for more work with lines, here are two more lovely activities.

Card Sort: Linear or Nonlinear

Polygraph: Lines