# Is this a good way of teaching this lesson?

I do some online tutoring. I was given a problem by a student and I did not do a very good job of explaining it. The problem is not difficult if seen in the right way. The difficulty is in seeing it the right way. Afterward I thought of an approach that I would like to test out here. The basic strategy is this. I am going to present a problem like the original one. Then I am going to present a similar problem that I think is intuitively easier. I will use Socratic method on this second problem and then return to the original.

Original type of problem:

If Joe can paint a room in 6 hours and Sally can paint it in 3 hours, how long does it take for them to paint the room if they do it together?

Now consider this modified version of the problem.

Joe can paint 3 rooms an hour and Sally can paint 2 rooms an hour.

1. When they work together, at what rate can they paint rooms?

2. How long will it take them working together to paint 15 rooms?

3. How long will it take for them working together to paint 1 room?

Back to the original problem.

1. What are the individual rates at which Joe and Sally paint rooms expressed as rooms per hour?

2. What is their combined rate?

3. How long will it take for them to complete one room?

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## 6 Answers

If Joe can paint a room in 6 hours and Sally can paint it in 3 hours, how long does it take for them to paint the room if they do it together?

Three hours, if Joe stops bugging Sally while she paints.

Actually, Joe’s meager contribution makes a noticeable difference.

I think for the socratic method wouldn’t work if the student doesn’t understand the underlying basic concepts in the first place. You have to step the questions back even further to to lead the students to think about how work effort relates to the task, and think about the ratio.

1. “Even though Sally works faster than Joe, Joe makes a contribution to the job and as a result the room gets painted faster than if Sally works alone.” That would let the student know that the answer is less than 3.

2. “The rate at which Joe and Sally paint can be compared to each other as a ratio. If you compare Joe to Sally, Joe does in 6 hours what it takes Sally 3 hours. 6/3 = 2 Joe takes two times longer than Sally. Likewise if you compare Sally to Joe, Sally takes 3 hours to do what it takes Joe 6 hours. 3/6 = ½ It takes Sally half as long as Joe to do the same work.”

From that point, you can then direct the student to the correct answer by asking questions. But you have to levelset the playing field for the student first so they have a chance of being led to the right answer. There’s nothing more frustrating than someone asking you questions for which you don’t have a base understanding.

I like the wording of the modified set. It give the hint that they must calculate rates rooms per hour. That helps the students along.

Joe is 1/6 r/hr + Sally ⅓ r/hr = 3/6 r/hr = ½r/hr 2hrs

They should get the hint.

@worriedguy brings up an interesting idea. Perhaps the first question to ask yourself is, what is missing that is stopping the student from understanding the problem, and the answer to that is how to break down a word problem. If you can lead them to understand that when they se the word “per” in a word problem, it means that they are going to see a fraction.

Switching your mental model from *rooms per day* to *days per room* is big conceptual leap for the student who hasn’t encountered it before. It seems like a simple inversion or reversal, lulling you into a sense of symmetry. Yet they behave completely differently when combined. Rooms per day add linearly, while days per room combine in a way that requires algebra. (Reminiscent of electronic resistors combining differently depending on whether in series or parallel.)

I’d stress the *difference* between using direct versus inverse quantities, and that this mathematical problem is an example of how changing the way you think about something (in this case, the concept of a person’s painting speed) can simplify the solution.

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