The wholesale price of a hat is $9.00. It is marked up by 75%.

What is the final retail price?

Since this is Apprenticeship & Workplace Math, most of my students have struggled with math for years. They likely have failed this very course at least once before. So, I am attempting to simplify things as much as possible. I am teaching them one mathematical tool that they can use for almost every problem they will encounter in the course... ratios and proportions.

It works something like this (for the above problem):

__75__=

__x__

100 $9.00

Then cross-multiply and divide to get the markup amount

Add the markup to the wholesale price and

*voila!*you get the retail price

I know what you're thinking. The quiz question is less than stellar to begin with. And teaching procedure-based mathematics completely misses any real conceptual understanding. But, remember, my main goal here is to give my struggling students one procedure that they can learn well and use for almost every problem they see.

You would think this would be appealing to my students. But the resistance is incredible. They want to use various other techniques they half-learned last year and sort of remember. Most of the time the results are disastrous. But occasionally there is a flash of brilliance.

Hank answered the above question correctly. At least the final answer was correct. But the steps he used to get there were baffling to me. Numbers appeared as if by magic. But somehow the final answer appeared at the end. It took me a while to figure out what he was doing. So, I decided to ask him to explain himself to see if

__he__understood what he was doing. And he did.

His reasoning went like this:

- I know 10% of $9.00 is 0.90. I just moved the decimal point to the left.
- Then I multiplied 0.90 by 7 because 7 x 10% is 70%. I got 6.30.
- Then I divided 0.90 by 2 because 10% divided by 2 is 5%. I got 0.45.
- Then I added 6.30 + 0.45 because 70% + 5% is 75%. I got $6.75.
- Then I added $6.75 + $9.00 to get the retail price. It is $15.75.

Wow! I think this demonstrates a pretty good grasp of numbers, percents and multiplication. Not to mention a fairly unique problem solving strategy. I never would have thought of this. Maybe I shouldn't be surprised when students do stuff like this. But I still am.

I do believe there are some potential problems with this method:

- This approach is much less efficient than my proportion method
- It is much more difficult when the markup is 63.7%
- With so many calculation steps, this method opens the door to simple arithmetic mistakes
- One simple strategy that works for all problems is easier than trying to remember several unique strategies that only work for specific problems

So, how did I respond? I told Hank what great thinking this was. I congratulated him for demonstrating strong problem solving skills. I explained the potential problems with his approach. Then I told him to do it my way.

And I admit... a little piece of me died when I said it.

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