Two missiles are approaching one another moving at 9,000 mph and 21,000 mph, respectively. How far apart are they 1 minute before they collide? 9,000/60 gives 150 miles per minute. 21000/60 gives 350 miles per minute. Put those together and you get 500 miles in two minutes. Because we want to get one minute, divide 500 by 2. The missiles are 250 miles apart before they collide. -A presentation by two students.

The class buys it. I express confusion: Isn’t there just one minute we’re talking about? Exactly! The whole class responds—they’re so pleased I finally get it—That’s why we have to divide by 2!

By the norms of the class, if the community buys the argument, the presenters can submit the paper. It still has to pass review, however, and I’m the reviewer.

The goal, of course, is to avoid the completely useless path of simply telling them the answer is 500 and they’re just wrong. As far as I can tell, it is precisely as obvious to them that the answer is 250 miles as it is to me that the answer is 500 miles. Their answer *makes sense* to them, and my answer doesn’t. That’s what I need to reverse.

So I offer my burden up to the blogosphere: What question can I ask my two junior colleagues that will help them see what I’m talking about? Or, as a distant second choice: what’s a clear way I can explain to them why it’s 500 miles and not 250?

Try:

From the viewpoint of a radar range finder on missile A, how fast is missile B approaching?

There is a conceptual error here, but in this case there is a more glaring numerical issue in saying that they were 250 miles apart when one of them travels further than that to the collision point. Did they draw a diagram?

Ask them look at a simpler similar question. Suppose one missile is stationary, traveling at a rate of 0mph or 0 miles/minute and the other is traveling at 9000mph or 150 miles/minute. On minute before the collision, are they 75 miles apart?

I’d approach it by asking how far away they are when they collide–totally obvious, 0–then backtrack. How far did the first missile go in the previous minute? How far did the second one go in that same minute? So, how far apart did they have to be a minute ago?

I’m having trouble following the words. I’d ask for a diagram. Using multiple representations usually clears up confusion in my classes.

I’d say, “So in that last minute missile I traveled how far? (150miles.) So then missile II traveled 100 miles, right?” Also, as Nate said, a picture would be instructive here.

What if you asked, “So the first plane travels 350 miles per minute. How far does it travel in two minutes? 10 minutes? Can you draw a diagram of where both planes will be after 1 minute? After two minutes?”

The most general question: Did you check your work? How?

I think most of these responses are missing the junior-colleague tone you’re going for, but that Nate’s contains the idea that’s gonna work.

It’s enough to get them to see that there’s a problem with their idea; you don’t need to get them to the right idea. They will do that themselves on the strength of the cognitive dissonance once they see that something making perfect sense to them can’t be true.

“Wait. I’m lost. If the missiles are 250 miles apart,

and one of them is going 350 miles a minute…?”“

… won’t they hit in less than a minute?”Bingo.

Hm. Appealing to the internal contradiction is probably the best tack. I’d also be curious about their addition of the denominators of the fractions… X miles/ A minutes + Y miles/ B minute = X+Ymiles / A+B minutes. I’m wondering how they imagine the meaning of this… “So when you say that the two missiles travel 500 miles in two minutes, does that mean that the first missile travels 150 miles in the first minute, and the second missile travels 350 miles in the second minute?”

Thanks, everyone – these have been really helpful. So many ways to create that perplexing cognitive dissonance … and all for such a simple problem.