Re: [Phys-L] widget rate puzzle ... reasoning, scaling, et cetera

[Philip Keller <pkeller@holmdelschools.org>]

Two approaches came to mind.  The first one is related to a method I 
once found myself using to teach nursing students how to calculate 
various dosages...start with what you know and make incremental changes 
to one thing at a time as you work toward what you want.

Starting from:  5 minutes for 5 machines to make 5 widgets.

We want 100 widgets.  That's 20 times as many widgets.  So if we still 
use 5 machines, it will take 20 times longer or 100 minutes.

Now we are at:  100 minutes for 5 machines to make 100 widgets.

We want to use 100 machines.  That's 20 times as many, which will reduce 
the time to 100/20 = 5 minutes which brings us to:

5 minutes for 100 machines to make 100 widgets.

Should I be embarrassed to admit that this was the first method I used 
to get the answer?

But a second approach:  Time required varies directly with job size and 
inversely with job rate.  So a 20-fold increase in both of those things 
will leave the time unchanged.  This is the kind of thinking I steadily 
encourage my students to use.  But not many of them embrace this.  I 
think one of the barriers is actually number-sense fluency or whatever 
you want to call it.  You have to see 100 and 5 and immediately be 
thinking about the ratio.  That is not a universal among my students.



So for 5 machines
On 12/31/2014 2:15 PM, Carl Mungan wrote:
> >     It takes 5 minutes for 5 machines to make 5 widgets.
> >     So, how many minutes does it take for 100 machines
> >     to make 100 widgets?
> How I would explain it.
>
> Start with given rate data:
>
> Divide output (widgets) by input (machines) => each machine makes 1 widget.
>
> Divide production of each machine by total time => each widget takes 5
> minutes to make.
>
> Repeat for unknown rate data:
>
> Divide output (widgets) by input (machines) => each machine makes 1 widget.
>
> Multiply production of each machine by time to make each output => it take
> 5 minutes.
>
> In each case, the second step is a formula of the form: total time/time per
> item = number of items produced per machine. The first step is a formula of
> the form: total number of items produced/total number of machines producing
> items = number of items produced per machine. So we could equate these two
> ratios.
>
> Somewhat similar concepts arise in converting number of molecules, number
> of moles, number of kilograms, molar mass, number and mass density, etc. In
> both cases, the trick seems to be come up with a formula in words and check
> it by carrying units and by putting in some sample numbers to see if it
> makes sense.