Dan Meyer wants to know what to do with rational expressions. Here is an example:

Of course, the students have a ready answer to this.

But … the answer is 48 miles per hour. Experimentation with a meter stick and toy cars can eventually lead to the formula:

Where *a* is the first speed taken, and *b* is the second. This is the harmonic mean of the two numbers.

Simplifying:

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John Armstrong, on June 12, 2009 at 1:17 pm said:Problem: what do you mean by “half”? A perfectly valid interpretation is spending one hour driving at 40 mph and another hour at 60. Then it takes a total of 2 hours to get there, for an average speed of 50 mph.

Hyun-jae, on October 18, 2009 at 10:31 pm said:Yeah thats what Ithought too. I thought I was wrong for a second there…

Jason Dyer, on June 12, 2009 at 2:17 pm said:Good point! I might consider leaving the text as it is and having the students argue that point before laying down the exact meaning.

Samantha, on December 8, 2009 at 10:59 pm said:That means it would have taken the same amount of time to drive the same distance even though he was traveling at a slower speed…

Samantha, on December 10, 2009 at 11:29 pm said:Just kidding

Ali Turk, on December 16, 2011 at 12:03 am said:You actually sound very right Samantha. I agree with you 110%.

Anonymous, on October 22, 2011 at 4:16 am said:what is the solution of 90% and 30%

Outline from my talk on Lessons from Game Design for the Classroom « The Number Warrior, on January 21, 2012 at 6:05 am said:[…] Have them work on average speed problem. […]

Laurie Gael, on May 26, 2012 at 8:07 am said:How did you experiment with toy cars and a meter stick to derive that formula?

Jason Dyer, on June 1, 2012 at 9:53 am said:They can collect data and test out their guesses on a small scale (do the same problem on a 1 meter journey where the speed changes at the 50 cm part). The same thing can be done on paper but with props it is more fun!

As far as deriving the formula goes, at some point they do need to get their hands dirty with the math, but they can do it by looking at a version of the formula with the numbers substituted in and change them to variables instead.

Anonymous, on June 12, 2012 at 4:05 pm said:could you work this out and show me exactly how you got 48 mph? because i do not completely understand how youve gotten this solution

kiki lowery, on June 14, 2012 at 9:41 pm said:Divide 100 by 2 and that’s half of your trip. Use the formula Average Rate=Total distance/Total time, For the first half of the trip which is 50miles you would put, Average Rate= 50miles/40mph=1.25hours

For the second half of the trip you would put Average Rate=50miles/60mph=.83hours. Add the hours together which equal 2.08 hours, Go back to the orignal formula, Average Rate=100miles/2.08=48.07=48mph. So the correct answer is 48mph