A Nice Game That Illustrates Vector Addition

NCTM (National Council of Teachers of Mathematics) has a very clever online game that helps students understand vectors. I introduced it to my precalculus students today, and they spent more than 30 minutes completely absorbed in it.

The game involves determining the correct velocity vector for a boat to accomplish several tasks. There is a vector for the water’s current, and you can also have the resultant vector displayed. Here’s a screenshot:


Players can adjust the boat’s velocity vector, as well as the water current’s vector. After some trial and error, my students eventually realized that simply pointing the boat’s vector towards the island won’t work, and they began to add the two vectors at the boat’s bow to get a better idea of where it would end up.

There are several levels of the game, and after playing with it, my students all agreed that they had better understanding of what is involved in vector addition.

Here’s a short video of what it looks like to play:



Cats and Calculus

I’ve done this activity once before, and I wrote about it here, but I’m continuing to tweak it. In the late 1800s, Eadward Muybridge published several time-lapse photo essays of animals and people in motion. Because he used a background with a grid on it, it is easy to see how much distance the subject covers in between each frame (which are snapped at 0.031 sec intervals). This sets up a great lesson on calculating average velocity, and approximating instantaneous velocity!

Muybridge Cat

While I still asked my students to plot the cat’s position vs. time by hand, we also used the desmos online grapher to plot the data as verification of their work. It was great to see the light bulbs go off in my students’ heads as they worked out difference quotients for smaller and smaller time intervals.

Here’s the plot generated using desmos:

Muybridge Data


We had an excellent discussion of how the cat’s motion breaks down into two distinct parts, and how the slope of each corresponds to the velocities of the cat walking and running.

Once again, I must thank  Dr. Nell Rayburn, Professor and Chair of Mathematics at Austin Peay State University for sharing this activity with other calculus teachers in Tennessee. Here are her original documents: Cat PhotosMuybridge Cat WorksheetMuybridge Cat Key.


Volumes, Calculus, and Cucumbers

My calculus students have always had a hard time visualizing three-dimensional objects. So, a couple of years ago, I started using cucumbers to help them understand the theory behind using integration to calculate the volume of rotated solids. Here’s a more detailed explanation of the activity.

Actually cutting the cucumbers into disks, measuring each disk’s volume, and adding them up to approximate the total volume is a great hands-on exercise that reinforces the concept of using an integral to add up lots and lots (an infinite number!) of disks to calculate the volume of an irregular solid. Plus, they always enjoy having a snack after all that brain work!

NAIS Annual Conference 2013: Youth Culture and Social Media

My final post from this year’s NAIS Annual Conference is about three researchers who spoke on the topic of social media.

Soumitra Dutta

We are entering a period when the dominant design of many things is changing radically. For example, for centuries the dominant design of personal timepieces has been something small, usually circular, worn on the wrist. That is no longer the case, as more and more people use their phones for checking the time.

When new technologies become available, they usher in a period of rapid innovation until things settle down into a new dominant design. Before Dodge introduced the enclosed-metal-body automobile, in 1926, there were 75 car manufacturers in America, all innovating. Once the Dodge model became the dominant design, those organizations that could maximize production and efficiency survived. The others could not.

Organizations are designed for production and efficiency, not innovation. Innovation can come from any individual – as teachers our challenge is to encourage creativity. We must be inspired, so that we succeed at raising the aspiration levels of our students.

We are at a tipping point in education where the dominant design is changing. It’s time for innovation!

Alexis Madrigal

Technology has an aura, and we are drawn to use it.

We give way too much credit to technology, and not enough to the social forces surrounding it.

“Are you into tech in education?” is a meaningless question. The two cannot be separated.

Remember – “experts” have a terrible track record of predicting the future! (Why isn’t everybody riding a Segway these days?)

If you are feeling overwhelmed by all the change that is happening, consider the generation that lived through 1890 – 1910. Rapid advances in agriculture, medicine, transportation, communication that dwarf our changes.

We need to promote what Ivan Illich called Tools For ConvivialityA perfect example of this is the game Minecraft. Connected Learning is the future (see Mimi Ito’s work.)

Danah Boyd

There is a new kind of public: networked publics. These are public spaces that are created through technology.

Young people want to be in a public world without being public.

Consider a conversation in a hallway between two people. The default is that the details of that conversation will probably not be shared publicly; it will stay private. Compare that to a conversation online (Facebook, Twitter, etc.). Online communication is public by default; you must choose to make it private.

Adolescents get frustrated, because adults don’t understand when Facebook posts and comments are “private” and when they’re not. Adults don’t get who they are “talking to” with online posts. So they have developed strategies for dealing with the 24/7 lack of privacy.

One teenage girl practices whitewalling: deleting at the end of every day all comments from her Facebook wall, and comments she makes on others’ walls.

Since youth cannot control access to the content, they control access to the meaning, in effect hiding in plain sight.

Young people assume they are under surveillance, so they are figuring out ways to circumvent it. For example, posting lyrics to a song that appear to be innocuous, but have meaning to their friends.

Our task as teachers is to help students choose the “right” networked public spaces to be in, and model good citizenship in them.