First estimate as a sanity check

Is the train going faster than walking speed (3-5 mph)? It appears to be, but it doesn't appear to be going at freeway speed either, so probably in the range of 5 to 60 mph. 

Although not very precise, that range will help determine if we've screwed up the calculation badly later.

Jeff Allen

Miles per hour, that's distance/time

We need to measure how long it takes the train to travel a specific distance. The "rate" of speed is the ratio of distance traveled divided by the time required to cover that distance. This is something almost every student has an intuitive understanding of, but they get bogged down when trying to manipulate that understanding formally.

For example, I start by randomly calling on students and asking them questions like "If you traveled for 2 hours at a speed of 25 miles per hour, how far would you have gone?" or "If you traveled 100 miles and it took you 10 hours to do so, what was your average speed for the journey?" They all "know" the relationship between distance, rate and time, but they're not in the habit of treating their understanding mathematically. 

Jeff Allen

Measurements

So, we measure a clearly marked distance along the track.

I asked students to use the edges of the whiteboard frame, about 51-3/4". It will be easy to see when the nose of the locomotive reaches these points. 

Then we use a stopwatch to time the interval as the train travels this distance:

In this case, it took 16.36 seconds. 

So, the train went 51.75" in 16.36 seconds. Some students were uneasy about that shift from mixed fractions to decimals. We didn't have to think about how we would have handled the time if the interval had been greater than a minute!

Jeff Allen

HO scale

What fraction of "real" size is that little man? Half size? One student guessed one thousandth. The actual value is quite cockamamie, at least from the perspective of the uninitiated: one eighty seventh! 

This just isn't the time to break it to them that the actual value is closer to 1/87.1, especially since I'm trying to convince them that mixing fractions and decimals in the same expression is unconventional. 

So, the scale distance traveled by the train is 87 times greater than the 51.75" we measured. Fingers are now flying over the calculators. 4502.25 inches! How many feet is that? I'm delighted to hear that most students understand that there are 12 inches to the foot, and most of THEM can see that dividing by those 12 inches will convert scale inches to scale feet.

375.1875 scale feet. I ask them to do another sanity check: does that whiteboard look to be a bit bigger than an American football field, roughly, from the perspective of our scale man? It does, roughly, we're ok. 

Do we need to carry that ".1875" through the rest of the calculation? That's a good conversation to have. Let's call it 375 scale feet. 

Jeff Allen

feet per second to miles per hour

So, the train is traveling at a scale 375 feet in 16.36 seconds. Don't we need to multiply the time by 87 too? Another worthy discussion, more subtle, but no we don't. 

Now the question is how many miles is 375 feet, and how many hours is 16.36 seconds? 

Most of the students have heard that there is some large, random number of feet in a mile, and it is a source of irritation to them that they might have to commit that number to memory when all over the rest of the world students are free from that mental burden. Although I sympathize, the fact is that reality is messy like that. 5280 feet per mile. We spend a few minutes trying to reason through the decision of whether we need to multiply our 375 by 5280, or divide it one way or the other. Three possible operations, only one makes sense: we can see that 375 feet is less than a mile, so whichever value is less than one must be it. 

0.071058238636363 miles, according to the student with the largest display value on her calculator. 

How many hours is 16.36 seconds? A similar discussion ensues that enables us to convert first to a small number of minutes (less than one), then to a very small number of hours.

0.004544444444444 hours. 

Every student has their own calculator, each slightly different, so I decide not to discuss the memory capabilities that would save them some keystrokes and preserve accuracy. Instead we talk briefly about how much accuracy is appropriate for what we're doing. In particular, when we're dividing such small values we need to be careful to maintain similar relative precision. 

0.071058238636363 miles per 0.004544444444444 hours works out to about 15.75 miles per hour! That fits our original sanity check! We did it!

Jeff Allen

Pretty smart. I am still

Pretty smart. I am still trying to figure out where you bury the survivors of two planes that crash on the international border of a third country..

Quick estimate method

Jeff --

Here's method of determining scale speed that might be fun for your kids to use without a calculator:

Count the number of 40' cars that pass a given spot in 5 seconds and multiply by 6.

Partial cars can be estimated as 1/4, 1/2, etc.  Longer cars can be counted as 1-1/4, 1-1/2, etc.

Seconds can be counted by saying, "one thousand and one, one thousand and two", etc.

Multiply by 6 and the result should be pretty close to the measured speed.

Then there might be the opportunity to get into averages, means, distributions, etc.

Wish I'd had a middle school teacher who used trains for math.

Draw a picture of what you know

Jeff - this is probably already in your bag of tricks, but here goes.

When I started taking engineering classes, there were several semesters of "applied technical math" required for all of the engineering disciplines.

Day 1, Class 1 - the professor said start by drawing a picture and writing down everything you know before trying to solve the problem.  He went on to pound this into our thick skulls every day for the remainder of the first semester.

Probably the most useful lesson I learned.

gs

Processing feedback in real time...

You guys are awesome. Don, it's midday and I've got one more class coming in this afternoon and several of my ace math students are in it - my challenge to them will be to show why your "rule of thumb" method works! Perfect. 

It's a great example of what we call "low floor, high ceiling" activities. Making an estimate using that method is readily accessible to all, and motivated/curious students can take it further.

I've read that locomotive engineers "back in the day" (before speedometers) had similar rules of thumb for figuring their speed using mileposts or regularly spaced telegraph poles. 

gs, that's some sage advice and I certainly emphasize making diagrams as a tool for understanding a situation. Working with young minds has significantly altered how I make and use sketches for teaching - it has reminded me that some drawing conventions are innately comprehensible but others must be learned. But either way, the process of trying to draw an idea has a tremendous effect on how students conceive of it. 

Jeff Allen

A favorite distance, rate and time math problem

Two trains are headed towards each other on a single track, each traveling 50 mph. They start 100 miles apart.

A fly on the pilot of one of the locomotives darts ahead at 80 mph until it reaches the pilot of the oncoming locomotive. It instantly reverses direction and returns to the first locomotive, continuing to fly at 80 mph, then reverses again and again until it is crushed in the horrendous collision of the two trains. 

What total distance did the fly travel during this episode? 

Jeff Allen

A more interesting question...

Where did the fly get the infinite amount of energy required to instantaneously reverse at full speed?  Or does the fly have no mass?

Restate the problem with the acceleration and deceleration rates of the fly and you'd have a good puzzler for a more advanced class.

Jeff, your students are lucky to have you as a teacher!

80 Miles?

Since the trains would collide in 1 hour and the fly is going 80 miles an hour, he would have flown 80 miles.  Of course, this is simplified for taking avoiding the acceleration/deceleration of the fly, but that is more a physics class assignment, and I don't remember any of my high school physics even getting that complex.  

Pretty smart. I am still

the funny part was, I told this joke once to someone from an Eastern European military and he looked at me for a moment and said, "we bury at sit of crash, then burn remains of plan, then cover with trees so no one knows!"

And he meant it.

Pictures are good

Same experience in e-school. Draw a diagram first!. I do remember freshman physics when student answers would be all over the map for the same problem. People would give obviously wrong answers with confidence, because 'it's what was on the calculator'. The proff's reaction to that would give even the densest student pause. 

Math and Trains: two fun things that go great together.