A few weeks ago I asked students in my Algebra class a problem akin to the typical

A car leaves Philly for DC at 3 going 60mph and another leaves DC for Philly at 4 going 65mph, if the two cities are 138 miles apart how long until they cross paths and how far from Philly will they be when they do?

Nobody had any idea how to think about motion like this. Despite having done a few problems over the course of the unit, they got lost trying to think this through. It didn’t help that up until I asked them on the test I had been making them translate this stuff into a system of equations.

**What went wrong?** Well, students lacked a clear understanding of the problem in its context. Other problems in which they needed to set up a system of equations were fine.

This stuck; they get the situation and could probably talk like an expert about how they set out to solve the problem. With the motion problems they don’t have this familiarity and easily accessible framework.

To try to put a point on it, the problem was: my students lacked a framework for understanding a problem and consequently the ability to translate the problem from situation to mathematical calculations. To solve this involved some backpedaling on my part. First, I was arbitrarily forcing the tool on my students to solve this problem, and, in this case, I was missing the point. Kids didn’t understand how to think about objects moving together and apart, especially with respect to their velocity.

We needed some good visual fodder and time for students to get their hands dirty. I found an interested student to volunteer to bring a Flip in and shoot a bunch of scenes of kids walking back and forth. We spent a day taking these shots. We came back in and watched them, watched them and drew on them. Kind of like Dan’s “Graphing Stories” except the goal of graphing was to get kids to see that in different situations it is appropriate to add or subtract the speeds of two objects. We also noticed that if two objects cross at a given point, the sum of their distance traveled is each to the total distance between them originally.

Lights started to come on for them, and our class discussions improved dramatically. Kids started to explain why if two objects are approaching each other you can add their velocity, and that if you know the distance between them it becomes easy to think about when they’ll cross and exactly where on the route that will take place.

We worked through this problem set and students had some good discussions. But I dragged it out too long, and didn’t have an answer key ready. That has been fixed, and the final page has answers to the activities.

We spent a few more days talking about different problems. They worked through one which I collected and scanned, through I apparently didn’t get a double-sided scan so quite a few of these are cut off, but I still think it helps to see what they were doing. **Edit:** *I’m going to check and see if it’s cool if I post work before it shows up here.*

**A few take aways.** First, I shouldn’t be forcing a tool or strategy, say systems of equations, for solving problems onto students as a general rule. Instead of viewing problems through the lens of the current unit, it’s important to constantly be assessing whether or not students have a general familiarity, comfort and ability to work within the environment of a given situation. Before you get to any of the heavy lifting of solving problems in a tough context, create a solid understanding. Role play, demonstrate, enable them to perform the “back-of-the-napkin” calculation that demonstrates an ability to translate the context from a situation into calculation. Despite that I was teaching a unit on systems I shouldn’t force students to apply them blindly to every problem they see in the unit, and when past knowledge is sufficient for the task at hand, I should be praising them for choosing a successful strategy and make sure that others notice the connection as well. If I want to teach systems of equations I *need to choose problems and situations that make them necessary and useful*. In many cases, this is a place where our textbook fails us. The problems are often contrived, shallow, and/or easy to solve by much simpler strategies. In a few cases they get it right, this is where you want to spend your class time if you want your kids to appreciate the fancy new mathematical tricks they’re learning and not yearn for the old days.

Finally, this comes back to WCYDWT. We shot those videos, and I’ve been working on editing them. I think there will be a general wow effect when students see the final product, but for this class I don’t think that will translate into more meaningful work, we’ve spent our time in this context, they’re familiar now. As I’ve been editing them I can’t shake the feeling of being frequently rewarded when a calculation adds a meaningful piece of info into the scene or turning pixels roughly into meters and determining a subjects’ speed by calculating the difference in location over time. The math feels really useful. It’s hard to get that by having them watch the video. I had it by *making it*.

The magic of WCYDWT is when students are What-Can-You-Do-With-This-ing. Taking a situation and opening up a bag of math on it. Finding meaningful calculations, bringing ideas out of their repetoire and putting them to work to do something meaningful, interesting, and rewarding. That’s a hard thing to engineer. But it sure seems like a good way to spend your class time.

I make it that the cars get together 642 miles past Philly at 4 AM. (I have not checked this carefully.) But the one hour head start that the first car gets it will be only 78 miles from Philly when the second one starts out. The second car is 60 miles behind to start and only gains at 5 mph, so it will catch up in 12 hours, if they keep going on I-95 they’ll be somewhere near Bangor.

I like walking-around and tactile activities, and I agree that that’s one element that the WCYDWT idea seems to lack. A good extension of your concept can be when you swim in oceans, or anything that has a current. The kids should be able to make the discussion bridge that links what you have done and those types of problems. (And then maybe you can bring in Dan Meyer’s WCYDWT article of some-woman-swimming-across-some-ocean.)

Another non-math connection that I like, that always wows me still, is when you’re moving at/approaching the speed of light. If Person A is moving towards Person B at the speed of light, and Person B is also moving towards Person A at the speed of light, relative to each other they are still only moving at the speed of light!! Completely amazing, from a mathematicaly standpoint.

Nick,

I love your idea of filming the students walking towards and away from each other to let them visually and physically experience the concept of relative movement. My students also struggle with this concept. I usually have them draw pictures or diagrams of the situation but I like your idea and plan to try it this year. Many of the students need more than just a R x T = D chart to set up their equations in these types of situations. Thanks for sharing your idea.

Another struggle my students have is with water current and wind speed problems. Some of them cannot grasp the concept of the constant rate of the vehicle being added or subtracted from the wind/water rate. I’ve tried to connect it to the idea of walking down the aisle inside a moving train. This made it click for some but not all.

Jim