Show a few passes of this car crash simulation, but stop before there is a crash:

http://www.geogebratube.org/student/m46112?mobile=true

Time intervals of 8 & 11 make for some close calls. Hopefully some disagreement about if or when there will be a crash. Repeat with time intervals that have a common denominator (like 10 & 25). Hopefully some think the first crash will be at 250 (10 x 25).

You could ask the same question again. But you could also ask things like:

1) Choose some time intervals (less than 20 sec) that create frequent crashes. Choose some time intervals (less than 20 sec) that create few crashes.

2) Set the time intervals so that the third crash happens after 24 seconds.

3) Can you set the time intervals so that the time between crashes isn’t constant? (no you can’t)

4) Set the time intervals so that every 3rd car from the right crashes and every 5th car from the top crashes.

5) Can you set the time intervals so that every 4th car from the right crashes and every 8th car from the top crashes (no you can’t).

I live on the north side of Chicago near the Howard Street station on the CTA, which serves as a hub for different lines connecting nearby Evanston with downtown Chicago. The Purple Line runs north into Evanston, the Red Line runs south into downtown.

Also, there’s a plaza just outside the station with a Dunkin’ Donuts.

So, suppose that northbound Purple Line trains depart the station every 12 minutes, and southbound Red Line trains depart the station every 15 minutes. You see a Purple Line train leaving the station just as you arrive. You MUST catch the next Red Line train in order to get to work on time. You want a coffee from Dunkin’ Donuts, and it will take 5 minutes to get a coffee and get onto the station platform.

Do you have time to get a coffee?

(The time at which trains start leaving at these intervals is purposefully omitted. Also, I just kinda picked the numbers out of my head because I’m in a bit of a hurry at the moment.)

Then show first 4 min if this vi hart video

http://m.youtube.com/watch?v=i_0DXxNeaQ0&desktop_uri=%2Fwatch%3Fv%3Di_0DXxNeaQ0

And get out 2 tuning forks. Tell them the frequency of each fork and ask if they will sound good or bad together. Repeat for more pairs of forks

Perhaps an intersection. Every 30 seconds a car passes through going N. Every 25 seconds a car/dog/ball crosses going E. Pretty easy to animate on geogebra.

Two BART trains leave 16th street station at 6:00 am, one going to Richmond and returning every 25 min, one going to Concord and returning every 30 min. If they both return at the same time they will crash. Will they crash? If so at what time?

I guess settling a debate or averting disasters might provide more motivation and engagement, but that’s just my hunch. If the math is settling a debate that students actually have with each other or others then I’m certain it will increases engagement.

@Burton It reminds me of sitting in the left hand turn lane trying to figure out how often the turn signals will match up on the cars.

Right there with you. I gave that a shot awhile ago. I don’t like how it turned out, I think, because the answer comes along too quickly. I like that the boats are on 25 and 30 minute intervals, which pushes the answer out to 150 minutes.