Tag Archives: measurement

The mean ? Which mean ? With interesting ratios.

Playing around with the Harmonic Mean of two numbers I stumbled on an interesting ratio, and looked at the others as well.

Here are the definitions, for numbers a and b

means

If we use m for the mean, then

for the arithmetic mean we have the ratio (b-m)/(m-a) = 1

for the geometric mean we have b/m = m/a

for the harmonic mean we have (b-m)/(m-a) = b/a

and for the RMS mean we have (b2 – m2)/( m2 – a2) = 1

I am quite sure that there is a way of seeing these which ties them all together, perhaps Mr. Joseph Nebus can find it !

The harmonic mean can be used to explain the harmonic tuning of a keyboard instrument (as opposed to equal temper tuning). I am planning a post on this for later.

The formula I gave for the harmonic mean is not the usual one – use a bit of algebra ! – but it is easier to calculate with.

The RMS mean is used extensively in Statistics, Rigid Body Dynamics and Electrical Engineering. The well known 110 volts in your house electric system is the RMS mean of the alternating voltage actually supplied. The Standard Deviation is the RMS average of the distances of the data values from the arithmetic mean value.

A non formal view of these means (the first three) is that the arithmetic mean is about the positions of the two numbers, the geometric mean is about the absolute sizes of the numbers and the harmonic mean is about the relative sizes of the numbers.

if we take the zero, the two numbers, and the harmonic mean the four values have a cross ratio of -1 (see part 3 of the Christmas Tale)

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Filed under abstract, engineering, statistics, teaching

In the future (tomorrow?)

“Mommy, teacher says we all have to get the new eyepad”.

“Why? What’s wrong with the one you’ve got?”.

“She says it’s much better, the screen goes right to the edge, and we can use the Ruler App to measure things”.

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Filed under geometry, humor

Fractions are parts of the same whole, part 3

When is a whole not a whole? (again)

When it’s two wholes (or more) :-

John eats 1/2 of his pizza, Mary eats 3/4 of her pizza. So between them they ate 1/2 + 3/4 of a pizza, or 5/4 of a pizza.

So which whole are we referring to ? John’s pizza ……. No.   Mary’s pizza ……. No.    Both pizzas …….. No.    John’s pizza and  Mary’s pizza and  both pizzas …….. No.

Conclusion: What we are referring to as “the same whole” is an abstract unit of one pizza, and the fractions are measurements using this unit. Wouldn’t it be a good idea to start off like this, with fractions as measurements, and avoid years of misunderstanding, stress and confusion.

Is this so different from adding whole(adjective!) numbers , as when  adding two numbers they have to be counts of the same thing (or whole(!) before it is chopped up).?

Fun arithmetic:        3 apples + 4 bananas = 7 applanas

Desperately fun arithmetic :  1/2 of my money + 1/2 of your money = 1/2 of our money

 

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Filed under arithmetic, fractions

The Classroom of the Future (next year?)

The vision below came to me the other day, and after reading Anthony Cody’s piece I had to put it up

You can find it at  http://blogs.edweek.org/teachers/living-in-dialogue/2014/04/the_classroom_of_the_future_st.html

and it contains what before 1984 we would have called a 1984 scenario:

“In this mode of instruction, these devices become the mediator of almost every academic interaction between students and their teacher, and even one another. Students are assigned work on the device, they perform their work on the device, they share work through the device, and they receive feedback via the device. What is more, the means by which learning is measured—the standardized test—will also be via this device.”

So, it is a classroom in the future. 

“Now, children, we are going to measure the classroom.  How many Ipads  long is the room? And how many wide?”

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