Average cost

[Joshua Zucker]

I was working with a group of students recently and what I thought was
going to be the introduction to a problem-solving exploration of
Simpson's paradox turned out to be an uncovering of some very
interesting conceptions of the meaning and use of "average".

The question I asked was written like this:
Lisa’s lemonade stand sells 20 cups of lemonade for 30 cents each
and 80 cookies for 50 cents each. What is the average price per item?

What answers do you think I would have gotten?

I wasn't very surprised to hear 40 cents. I was rather disappointed
to hear kids coming up with 45 -- can you see where they got that?
Some kids came up with 50 cents, perhaps because they knew about
median or mode. What I was most surprised about was how hard it was
to get anyone to come up with 46 cents. How would you have led them
to it?

I think 40, 50, and 46 cents are all correct answers. On the other
hand, I was definitely interested in 46 cents for the purposes I was
aiming towards. So I started to think about how to ask the question
differently, so that instead of "average" I asked something of more
immediate practical use.

Does anyone have a good suggestion about what question to ask (instead
of using just the word "average") to get kids to come up with 46 cents
as the answer? How about 40 cents? 50 cents?

Thanks,
--Joshua Zucker

[Jennifer Newell]

To get at the 46 cent solution I'm also having trouble finding any single word to replace "average". 

At the risk of making the question too wordy, I might add a scenario such as "For each item that Lisa sells, her buddy Sam sells one ice cream cone. How much should Sam charge for each cone in order to make the same amount of money as Lisa?"

For comparison to the 40 cent solution you could kind of go backwards and say something like ''The next day Lisa decided to charge 40 cents for both lemonade and for cookies. At the end of the day she counted her money and found that she had made the exact same amount as the previous day. Did she sell more items on day one or on day two?". 

This kind of deviates from the initial question but you could then compare the total things sold on both days (100 compared to 115) as a way to emphasize the different situations. 

Not sure if that's really what you were hoping for, but it was a nice little thought exercise for myself!

Thanks,

Jennifer

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[Adriana Sofer]

How about:

"The next day, she decides to charge the same amount per item (cookie or
cup of lemonade). She expects that she will again sell all her daily
production of 20 cups and 80 cookies. How much should she charge per
item if she wants to make exactly the same amount of money as the
previous day?"

Adriana

PS: I haven't introduced myself: Adriana Sofer, Math Teachers' Circle of
Austin; Lecturer at the University of Texas at Austin.

On 6/2/11 5:10 PM, Joshua Zucker wrote:
> I was working with a group of students recently and what I thought was
> going to be the introduction to a problem-solving exploration of
> Simpson's paradox turned out to be an uncovering of some very
> interesting conceptions of the meaning and use of "average".
>
> The question I asked was written like this:

> Lisa�s lemonade stand sells 20 cups of lemonade for 30 cents each

[Richard Spindler, Ph.D.]

Joshua:

I don't have a good replacement for "average", but here is how I look
at this. I don't ask the average question, but because then you are
asking them about something you haven't defined yet. I ask how much
money did Lisa make total? It is 0.3 + 0.3 + ... +0.3 + .0.5 + ... +
0.5 = 20 *.3 + 80*0.5. . Now I might ask, suppose I sold 100 items for
this total amount but didn't know how many of each kind, what might I
consider the price per item? After they do it, then you tell them we
DEFINE this calculation to be the average price.

Rich

[Japheth Wood]

I'm thinking of something along the lines of Jennifer's idea of finding the results of charging the same amount per item but noting that more or less money has been taken in. This provides some motivation for Adriana's question of what (same) amount charged for each item would yield exactly the same amount as Josh's original problem. This really gets at the meaning of average (weighted arithmetic mean, in this case).

I would sequence it like this:

A. How much money is made in the original scenario that Josh mentioned?

B. How much money is made by charging 30 cents per item (the cost of lemonade) and selling the same amounts of each?
C. How much money is made by charging 50 cents per item (the cost of a cookie) and selling the same amounts of each?

D. Note that B < A < C. What would be the effect on  of gradually increasing the cost from 30 cents to 50 cents?

E. What same price should be charged for each item to make the same amount as A?

What do you think of this trajectory of problems?

Japheth

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