Des-blog

As soon as we released Polygraph, teachers asked for two things:

(1) Quadrilaterals, and

(2) The ability to make their own.

Quadrilaterals

This was the easy part. Today we give you two (TWO!) versions of Polygraph: Quadrilaterals. There is the basic version and the advanced one. You and your students can play either one, or both. If your goal is to lay the groundwork for the standard hierarchy of quadrilaterals, Polygraph: Basic Quadrilaterals is a great place to start. If your goal is to challenge your students to notice and describe subtle but important differences among geometric figures, Polygraph: Advanced Quadrilaterals is right in your wheelhouse.

Enjoy.

Building your own Polygraph

Given that one of the Desmos guiding principles is Trust Teachers, we knew what was the right thing to do when teachers asked to build their own Polygraph versions.

And now we have done it, so you can have at it. Welcome to Custom Polygraph.

You pick a name for your version of Polygraph. You make 16 graphs using our familiar graphing calculator interface. You save it. You use it in class. Easy peasy.

Anything you can put in the usual Desmos graphing calculator interface you can put in a Polygraph example: graphs, images, points, etc. Want to make your own polygons? Graph line segments and turn off the axes (or leave them on: coordinate geometry, anyone?)

You can tweak your custom Polygraph before or after you use it in class—it’s totally editable.

You can send your friends the link and they’ll be able to use it, too. (Don’t worry—they can’t edit yours; they’ll have to make their own if they want to build on your ideas.)

Click here to get started.

Words are fun.

Words are powerful.

Vocabulary lists—for most of us—are not so much fun.

Whether it is French class (-ir verbs) or Geometry (types of triangles), most of us have memories of long lists of words out of context and feeling as though we have no hope of remembering them all, nor any purpose for doing so.

And we also know well the pleasure of having just the right word handy at just the right time—what the French call le mot juste.

We have designed Polygraph to foster the pleasure and the power of words without the drudgery of the lists.

Here is how it works.

Each student has a partner. One student is the picker, the other is the guesser.

The picker picks one thing from a set of things. Depending on the version the teacher chooses, these things might be graphs of lines, or of parabolas, or of rational functions. They might even be hexagons!

The guesser sees the same things on her screen that the picker sees on his, but scrambled to prevent location being a clue.

Here the picker has selected a graph and will soon see the guesser’s questions on the screen.

The guesser types a question. The picker can answer yes, no or I don’t know. A well written question helps the guesser to eliminate one or more options. Desmos provides the tools to keep track of questions and eliminated options.

In the next task, we ask students to analyze questions like those the class asked. We ask students, Given these two hexagons (or lines or parabolas or…), which of these questions would be the best one to ask? and The answer to this question wasn’t helpful—how can you improve the question?

Picking and guessing create the need for words—students want to describe the subtle differences they see in the mathematical objects in front of them. Sharing questions provides opportunities to spread words among students—students read how their classmates have described these differences. Whole class conversation led by the teacher helps to formalize and cement the vocabulary—the teacher can introduce the standard words that describe these same differences.

Beth Herbel-Eisenmann has studied the development of informal vocabulary in classrooms. She has described how formal mathematics vocabulary can meaningfully arise from less formal vocabulary that students use to describe mathematical objects (e.g. dented hexagons become concave; slanty-up lines have positive slope, etc.)

With Polygraph, Desmos provides tools for doing this. Words should result from a need to describe our world—this is where they gain their power. There is no better time than now to develop your students’ power of expression, so click through and set up a game of Polygraph tout de suite!

(Shout out to Brenna Magnuson, Jennifer Carlson, Mona Yusuf, Ruth Pieper and Brandon Schwab—all current or former elementary education students at Normandale Community College who helped to design the hexagons in this lesson.)

From arithmetic to algebra—students need to make this transition.

Too often, students may view these two worlds as discrete and disconnected. Arithmetic is about numbers and answers. Algebra is about letters. An important challenge for teaching is to connect these two—to help students understand that algebra helps us to express the ideas of arithmetic. Algebra makes the structure of our computations clear.

This is a two-way street.

We want students to use their knowledge of computation to inform their algebra understanding, and we want them to see that representing their ideas with algebra can save a lot of computation time.

We don’t want to just tell students that algebra is useful. We want students to experience the power of algebra.

Our latest Desmos classroom activity will put the power of algebra in the hands of students by asking them to design parking lots.

The activity begins simply enough: Space three lines to make a four-space parking lot. Be sure to space those lines equally! They see the cars park in these spaces as soon as they get it right.

Pretty soon, the size of the lot is changing, and so is the width of the lines. Eventually the number of spaces changes too. Students notice these changes. Importantly, they notice what doesn’t change.

We provide the tools for students to notice: They drag the lines to the right places to make equal-sized parking spaces. At this point, they are working with their intuition.

Then we provide students the tools to describe what they notice—first with numbers, and then with algebraic symbols. Continuing a theme we developed in Function Carnival, we also provide the tools for students to see the consequences of their ideas. Allotting too little space between lines will result in scraped fenders, broken side mirrors, and angry drivers. Allotting too much space means there won’t be enough parking spaces.

For students to experience the power of algebra, they need to see their equations in action. Students need to see that their equations work under certain conditions, and that they fail under others. And when the equations fail, students need to be allowed to try again.

As their algebraic power grows, students will become masters of Central Park.

Click through and try Central Park yourself