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Solving Word Problems Using Units CancellationOne of my least favorite types of math problems when I was in high school was the type of problem in which two or three people are working on a job together, such as the problem below:
If Larry can mow a lawn in three hours, Curly can mow a lawn in two hours, and Moe can mow a lawn in five hours, how long will it take the three of them to mow 62 lawns working together?
The equation for this problem is simple enough, and is shown below:
x/3 + x/2 + x/5 = 62
Once you have this equation, it's a simple matter of finding the least common denominator (in this case: 30) and adding the fractions:
(10x + 15x + 6x)/30 = 62
31x/30 = 62
x = 62 * 30/31
x = 60
Part of the reason I disliked these problems was because I honestly can't remember anyone explaining to me why I set up the equation at the beginning. Being the kind of student who liked to know the why's and wherefore's of my solutions, this aggravated me. Somehow the rationale for putting their 'speeds' in the denominator eluded me. At the time it simply didn't dawn on me that what I was thinking of as speeds (3, 2, and 5) were not really speeds at all!
Since then, I've discovered that one of the simplest ways to get students to see that this equation makes sense is by comparing the 'mowing' speed to the 'speed' of a car. First, students will easily recognize that a car speed is generally written in miles per hour (or kilometers per hour). This unit of measurement has distance in the numerator, and time in the denominator (miles/hours)
But in the lawn mowing problem, we don't have a typical 'speed' unit because there is no distance measurement. So we propose the following: the 'distance' unit in the numerator of 'miles per hour' is really analogous to the number of lawns mowed by the boys. In other words, instead of speaking in 'miles per hour', we'll speak in 'lawns per hour'. So what are Larry, Curly, and Moe's speeds?
Larry: 1 lawn every 3 hours = 1/3 lph
Curly: 1 lawn every 2 hours = 1/2 lph
Moe: 1 lawn every 5 hours = 1/5 lph
Now the original equation makes much more sense!
In x hours, the number of lawns mowed by Curly can be expressed as:
x hours * 1/3 lph = x/3 lawns.
Similarly, the number of lawns mowed by Curly and Moe in x hours are x/2 and x/5 respectively. Now the reason for the equation becomes very clear, and in the process, we have forced students to look at the cancelation of units (hours canceling against lawns per hour).
There is nothing earthshattering or suprising in this solution; the suprising thing is the number of students who will plug numbers into the equation at the top, never understanding why they are doing it, or even worse, having an incorrect understanding of why they are doing it.
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