It will only get worse! or better depending on your point of view
My daughter has a math worksheet of polynomials to do over the Christmas holiday. When doing it she asked why does “10^0 = 1” ?
Why is that? It isn’t obvious looking at exponent values:
10^3 = 10*10*10 = 1000
10^2 = 10*10 = 100
10^1 = 10
10^-1 = 1/10
10^-2 = 1/10 * 1/10 = 1/100
but how do you do something zero times? I couldn’t recall off the top of my head, so later I sat down to doodle on it. And I remembered how to sort it out.
Welcome to the tricks of math, where you use something else you know to figure out something you don’t know (yet).
First the rules of exponents…
10^2 * 10^2 = 10^(2+2) = 10^4 = 10*10*10*10 = = 10,000
So multiplying exponential numbers (of the same base) is the same as adding the exponents.
How can we use this to figure out zero? How about this expression:
10^2 * 10^-2
Expanding it out we see this:
10 * 10 * 1/10 * 1/10 = 1
and since
10^2 * 10^-2 = 10^(2-2) = 10^0
we now have proved that
10^0 = 1
It also sort of makes sense logically, after all, what is between:
10^1 = 10
10^0 = ?
10^-1 = 1/10
So there you have it Jenny… welcome to your first math proof.
December 22nd, 2009 at 8:58 am
Much ado about nothing!
http://en.wikipedia.org/wiki/0_(number)
December 22nd, 2009 at 9:33 am
We are 10^0 on this.
December 22nd, 2009 at 10:55 am
The first question of the day had to do with dividing a fraction by a fraction (1/3 / 1/2 = 2/3). WHY does multiplying the numerator by the reciprocal of the denominator work? We could PROVE it works, but trying to physically show it by folding a piece of paper into thirds and then dividing it by 1/2, we couldn’t get it to show 2/3 (the correct answer). Some things don’t translate easily with Montessori methods.
December 22nd, 2009 at 11:43 am
1/3 / 1/2,
Here is the algebraic solution:
a / b = a * 1 / b = a * (1/b)
so, what you really need to understand is that
1/ (x/y) = y/x
which is very easy in algebra: just multiply by (x/y) and simplify.
I don’t know a tactile lesson for this off the top of my head but this is how I would explain it:
Division is like “breaking it into parts.” When you divide by 1, you are breaking it into one part, so it stays the same. When you divide by 3 you are breaking into 3 parts so you get a third. When you divide by a number smaller than one, you are saying that the parts are bigger than the original. So, division by one third is like saying it needs to be three times as large.
December 22nd, 2009 at 2:59 pm
Only a few math procedures have a physical analog, like dividing some physical object into component parts. For all the others, the student has to learn the procedure and gradually see when and how it applies to a specific problem. For example, if you paid $40 for something that had a 6% tax, how do you find the cost of the item before the tax? You DIVIDE the final cost by (1+ tax rate) 40/1.06
December 22nd, 2009 at 9:23 pm
#5. Your example is flawed. No such thing as a 6% tax. Way too low. Why settle for 6% when you can steal 10%?
December 22nd, 2009 at 9:33 pm
http://www.cut-the-knot.org/proofs/index.shtml
I didn’t incude the above earlier. Look at “Halving a Square” #9 to see how Socrates taught the “proofs” issue. BTW calculus depend on breaking things into vanishingly small component parts.
December 23rd, 2009 at 12:53 am
Most mathematical innovation since Newton is non-intuitive. Calculus makes vast amounts of sense once you learn it, but it isn’t obvious.