Des-blog

The Friday Fave is thinking about the power of mathematical generality this week.

Mathematics is full of universal quantifiers: All, none, and every are common words in theorems and proofs. Every multiple of 12 is abundant (an abundant number is one whose factors sum to a value greater than the number itself). How many integer solutions for a, b, and c are there to a^n+b^n=c^n, for whole numbers n>2? None.

All, none, and every are important ending points of mathematical inquiry, but they are lousy invitations.

When the Fave is told to find ALL solutions, or to prove that there are NONE, it can feel like too much pressure. What if the Friday Fave inadvertently leaves one out at first? Much better to begin by considering a small number of solutions, or wondering whether any exist at all. Save generality for later.

And that, friends, is what Compound Inequalities on the Number Line is all about. Indicate a point on the number line that is greater than -3. That’s your first task.

Then do another, and another. And now what would it look like if you could see all of your classmates’ points? It might look something like this.

But that’s still not ALL points. Don’t worry. Students get there.

It’s just that generality is the stuff theorems are made, but examples are the stuff generality is made of. So we start there. Slowly formalizing students’ informal ideas—that’s the stuff the Friday Fave is made of.

And while you’re thinking about generality, maybe take a peek at these other activities that build the general from the specific.

What Comes Next?

Game, Set, Flat

Sector Area

A long time ago, the Friday Fave taught from a very bad algebra textbook that told students to use a fourth-degree polynomial to model a data set that consisted of calendar years on the x-axis and number of miles of railroad in the United States on the y-axis.

Why a quartic? No explanation was offered.

But the text went on to use this as a cautionary tale about over-reliance on mathematical models.

A student who actually knows something about railroads in the United States is not offered any opportunity to apply that knowledge in this task. Indeed the idea that there should be a single function governing the eras of westward expansion, the rise of the automobile, and the subsequent rise of international shipping via standardized shipping containers is absurd.

But enough about that.

The Fave wishes to point your attention to two modeling activities at teacher.desmos.com that take a different stance on the relationship between the world and mathematical models.

First up is Charge!

How long will it take to charge this phone?

The early data suggest that a linear model is the right one, but pretty soon something strange happens, and it turns out to take longer than a linear model would predict.

If you know something about charging batteries, this activity invites you to apply that knowledge. Maybe you’ll even factor that knowledge in at the beginning at the beginning of the activity in order to make a better prediction.

Next is 400 Meter Modeling. We ask you to build a model representing the relationship between the year and the world record for the women’s 400-meter dash.

A key question when you build your model is this: Do you think a linear model is an appropriate choice here? If you know something about the sport of track, or about the development of human performance, then you are well positioned to argue that a line may not be a very good model for this relationship.

In the cases of both Charge! and 400-Meter Modeling, data that look linear become nonlinear when something about the situation changes. This is one reason that mathematicians use piecewise definitions for functions. A linear model works well to describe one phase of the charging process, but we need a different model to describe the end of the process. Similarly, as people approach some practical limit for how fast they can run 400 meters, we have to expect that improvement in winning times will take a new shape.

Mathematics can inform our understanding of the world, and the world can force us to do better mathematics. The opportunity to explore this relationship is why these two modeling activities are this week’s Friday Fave.

While you’re thinking about modeling activities, here are two more favorites in the genre.

Predicting Movie Ticket Prices

Alligator Investigation

The Friday Fave has been playing with the Desmos geometry tool this week. Tessellations, kaleidoscopes, and many other varieties of fun are to be had over there. Our new transformation tools make such constructions straightforward—you can translate, rotate, dilate, and reflect.

The rotation tool requires a center and a degree measure. The dilation tool needs a center and a scale factor. In both cases, it’s really handy to be able to specify the relevant number directly in the tool.

But what if you want to have the rotation be driven by an angle that arises in your construction? Or have the scale factor be the ratio of two lengths in your sketch? Well, that’s when the rotation/dilation tool comes to the rescue.

A rotation/dilation is defined by three points:

1. A center
2. A preimage point
3. An image point
   

Make those three points collinear, and you’ve got a straight-up dilation.

Put the last two points on the circle centered at the first point, and you’ve got a rotation.

Do anything else, and you’ll end up with something that dilates according to the ratio of the two distances from the center, and also rotated according to the difference between the two points’ directions from the center.

This construction by Sean Sweeney—which is this week’s Friday Fave—is a great way to explore how the dilate/rotate tool works. Go play with it yourself, and be sure to share your creations with us!

While you’re thinking about rotation, here are some great activities for helping your students explore the mathematics of rotations.

Transformation Golf: Rigid Motion

Des-Patterns

Laser Challenge

Polygraph: Transformations