Des-blog

The best mathematics tasks have layers.

Consider Screen 2 from Adding Integers, a delightful new task—and this week’s Friday Fave—at teacher.desmos.com.

Your challenge is to use some of the numbers 3, 7, 2, 6, 1, and 8 to make two groups with the same sum. That’s the first layer. In most sixth-grade classrooms, this is a fairly routine task that can quickly develop into a challenge to use the most numbers possible. To be fair that’s the original challenge in the text, but if we’re being honest does anybody really read anything anymore? (Also that first layer is really helpful for lots to get started; if we have to meet too many constraints on the first try, some of us freeze up.)

It turns out that the maximum number of cards is five. But how many ways can you do it? Here are two.

And now can you prove that the maximum number of cards is five? Is five the most because we just haven’t yet figured out how to use all six, or is it impossible to use all six?

The best math tasks have layers.

There’s more to the activity, of course. On later screens the cards are negative. You can specify your own sets of cards for your classmates to think about in the Challenge Creator. And these are all worthy of everyone’s time. But the Friday Fave longs to linger in the layers of the second screen.

But if you’re here for the integers, that’s great too! Here are some additional tasks to bring out numbers both greater and less than zero.

Smallest Solution

Inequalities on the Number Line

Point Collector

The Fave is in a playful mood this week. Sure math can be hard work, but it can also be about play and creativity. To that end, here are a few playful activities you’ll find at teacher.desmos.com.

First up is Polygraph, in all its many forms. Many definitions of play involve informal engagement and a development of sophistication over time (which also sounds a lot like learning, and this is not a coincidence!). These characteristics are exactly what we designed Polygraph to support. As Polygraph Continuity (an activity for Calculus) demonstrates, this kind of math play is not limited to the primary grades!

Next is Des-Pet. Use functions to draw your pet’s face. Make it simple or complex; serious or silly. These characteristics are entirely up to you.

All versions of Marbleslides are playful, of course. And the Fave thinks of nearly every screen as an opportunity for playful self-expression. But Marbleslides: Rationals brings creativity, delight, and joy to precalculus. And the final screen where we invite students to Build a Marbleslide that’ll make your classmates chuckle and think, “Oh that’s delightful.”? Pure play.

And that brings us to Tile Pile, which is a fun—if not especially playful—activity until you get to the end, where we challenge you to fill a square with each of four differently-shaped tiles.

This little widget is a ton of fun to play with. You can make patterns and keep track not just of whether each tile works, but the different ways each tile works. Our field research indicates that the Z tile is especially challenging.

Can you fill the square with the Z-tile? Reach out and tell us about your ideas! We’d love to play along.

While the Fave frequently refers to the Desmos Activity Building Code while going about the daily work, the Fave has been thinking a lot recently about offering students interesting ways to be right.

Last week, the Fave featured Challenge Creator, which is chock full of interesting ways to be correct. But there are plenty of other structures in place in a wide range of activities for giving students this opportunity.

For example, we frequently ask students to decide which of two (or more) responses is correct, as in this example from Land the Plane.

A typical class will have students using the y-intercept, students using the slope, and students using both to reason about this task. These students will use formal and informal language, which then provides a launching point for interesting conversation in a classroom even if everyone is right. Some typical responses might include:

• Roman because 8 makes more sense as the starting point
  
• Roman is right, because the equation should have a negative slope and a positive y intercept.
  
• Roman. You’re going down 2 , so the slope is -2
  
• Roman is correct, because his coefficient is negative and so is the slope of the line on the graph.
  

Four correct answers in an interesting mix of ideas and words; all at a teacher’s fingertips in the dashboard.

We frequently present examples and non-examples in order to introduce a new mathematical idea to students. In Marcellus the Giant, we show scale giants and non-scale giants and then ask students to make some conjectures about what makes a giant a scale giant.

The collection of observations in a typical classroom will include angles, side lengths, some generalized ideas about doubling or tripling of lengths. There will be informal language about what the giant “looks like”. None of these will be a full and complete definition of the idea of similarity, but each of them contains a kernel of truth on which the activity and subsequent instruction can build.

A third example of offering interesting ways to be right is a favorite of the Fave: Which One Doesn’t Belong?

Because each of the four options has at least one reason to be the one that doesn’t belong, and also because students are creative people with clever ideas, you’ll get lots of interesting right answers to a well-designed Which One Doesn’t Belong? set such as the one in Inequalities on the Number Line.

Tapping into the minds of students is fascinating work, and it’s what makes offering students interesting ways to be right this week’s Friday Fave.