Des-blog

The fave is fond of triangles, and after playing with this week’s activity your students probably will be too.

This week’s fave—Exploring Triangle Area with Geoboards—is newly released and consists of a few warm up screens and a Challenge Creator. This Challenge Creator is a doozy.

How many triangles are possible on a 5-peg by 5-peg geoboard? And how many different values for the area are possible? Who knows? But whatever the number, it’s great enough to allow for a lot of creative triangle building.

Maybe you build a tricky triangle and think to yourself, “No one will be able to reproduce this one!”

But it turns reproducing your triangle isn’t the challenge; building a triangle with the same area as yours is. Indeed, odds are that your classmates will solve that challenge with a variety of triangles.

Along the way, you’ll wonder about the greatest possible area (are you sure it’s 8 square units? How do you know?), the smallest possible area (Are you sure it’s half of a square unit?), and what areas are possible in between?

The focus on student-created challenges with plenty of opportunity for creativity—that makes Exploring Triangle Area with Geoboards this week’s Friday Fave.

This week’s fave is a new feature and a salute to things that work the way you sort of expect and hope that they would.

Let’s say you want a movable point that stays within the bounds of a rectangle. That’s no problem. Use slider limits that match the minimum and maximum values with the rectangle.

But let’s say you want that point to stay within some non-rectangular region. Until quite recently, that was a problem because the limits on your slider had to be constants. Staying within limits that change was not possible.

If you’ve ever tried to solve this problem, you’ve probably typed something like this into your slider limits.

Until recently, we threw an error and told you that you couldn’t use variable slider limits. But now you can, and here’s what it looks like.

Variable slider limits, and syntax that feels natural—together those are this week’s Friday Fave.

And here are a few more graphs that use variable slider limits. Maybe they’ll spark some new ideas!

Fraction Bars

Fraction Shading

Strange Rectangle Tool

One of the most important understandings in calculus is that functions have values which can be positive and negative but that those values are also changing, and that change can be in a positive or negative direction. Slope isn’t just for straight lines!

For example, when you’re getting out of student loan debt, the total value in your bank accounts might be negative, but the rate of change of your money is positive.

Or for another example, the value of the gross domestic product of the United States is always positive and the rate of change of the GDP is almost always positive so it makes more sense here to look at the rate of change of the rate of change. What is the rate of change of the increase? How does it compare to the increase of previous decades or other countries?

Because of the importance of these questions, calculus teachers frequently ask students questions about rate of change. Given a function, what is its derivative? Give a second derivative, what might the first derivative look like?

We were extremely impressed with a functions and derivatives activity developed by Sandi Yoder, especially the conversation it generated in her classroom. (Filmed here!) Inspired by Sandi’s work, we created Functions and Their Derivatives.

We give students a function and its first and second derivative, without revealing which is which. We ask them to label the derivatives accurately and then we give them feedback on their thinking.

But then we bring in a Challenge Creator and invite students to create their own function and label its derivatives. If they do that successfully, they can enter it into the gallery to challenge their classmates.

You get one function from us and then dozens more from your classmates. A calculus class that is social and creative! That’s why we’re here.