# Two Trains…

How many of you have seen a problem like this one?

I’m a fan of taking a problem like this, one that you would assign for homework (in the “application” section of the exercises….and one that very few students even attempt….and someone will ask you to take it up next class) and bring it to the start of my lesson. I’ll teach our concept/idea through this problem. But we can’t just throw this problem up on the board and say “Let’s solve it”……because no will want to. There is no drive for any of us. Like Dan mentions here….who cares!

Who cares about the trains travelling…who cares that they are even trains….they could be bicycles, or cars playing chicken….but is changing the context really going to change how engaging the problem is to students? Dan argues no. I agree.  Before you read about this lesson check out this post on Real vs. Fake world….and the Circle Square lesson on 101qs.com which was an inspiration for changing the Two trains problem around.

Here’s my go at this one:

Show them this video:

ask What do you notice? What do you wonder?

Have students guess WHEN the two dots would meet?

Have them guess on WHERE the dots will meet?

Have a discussion on what will be needed to determine the times and distances. Spend some time here on speed. Go over the relationship between distance, time, and speed.

Show them this image and have them makes some guesses on where the dots are now.

then reveal

Calculate the speeds of the dots. Have students go back to their original guess on time and find how far each dot would travel.  Who in the class is closest? Did anyone guess right?

Now help them generalize…

Create the equations

If our lesson is on solving this using an algebraic technique we can teach them that here. Or maybe we want to show them the graphical solution. Either way we have taken the tougher question from homework that no one cares about and used it to set up and teach a skill.

and finally,

I’m sharing this lesson now (before I teach it) with you hoping to get some feedback. Writing these lessons here also help me work out the details. This is week 4 of the #MTBos blogging initiative and its focus is lessons. I won’t get a chance to teach a lesson this week. Our school had final exams and then PD days in preparation for second semester. Good luck to all those starting up again!!

## 5 thoughts on “Two Trains…”

1. Hey Jon,

I absolutely agree with this. It seems counter-intuitive but this is much more real world than the real world examples we so often use.

I made this webapp that you might be interested in as an extension to your lesson. It’s the same question with some tweaks and some variable speeds.

https://tube.geogebra.org/m/2599517

Let me know what you think.

1. Jon Orr

Thanks so much for this! I’ll definitely put it to use. Amazing.

2. I notice in the video and the images, you have grid lines as your background. However, I also notice that nothing you present starts or ends on those grid lines. For example, you indicate that the starting and ending positions of the dots are 0 and 100 cm, then ask the students to guess the location at 3.87 seconds. I think that might be easier to do if the starting and ending marks were on grid lines. Students could then calculate the distance of one “block,” and extrapolate to determine approximate distances for the dots. Sure, being able to adjust the grid in your mind is a great skill to have, but is that the skill you are trying to assess or teach with this problem?
The other question that came to mind related to the ultimate question you asked. “When do they meet?” If they were trains, the point of meeting would be when the front of the engine of the one train met the front of the engine for the other train. In your problem, you seem to really want to ask the question, when do the dots overlap? To me, “meeting” would be when they start to touch each other.
I don’t know if it would be easier to change the notation and the video so that the edge of the dot is at the starting point instead of the middle, or change the question asked. It reminds me of a physics problem where we assume all objects are a single point with no dimensions.
Hope that helps as you flesh out this problem.

1. Jon Orr

Thanks so much Chad for the feedback. I chose that background without really thinking about the grid at all. Thanks for pointing it out….and I’ll definitely line it up. You are also right, we want to know when the dots overlap…not first touch. I’ll make that clear to students after we take their wonderings. Thanks.

1. Glad I could help! I think the students will like this much better than trains.

Too bad you couldn’t do something like Iron Man and Superman coming at each other…