x^0 and x^1 Power

The explanation involves consistency of the mapping rule that multiplication of numbers can be mapped to addition of their logarithms .
(Logarithm being the power with which to raise a base to produce a particular number)

Thus 4 x 1/4 = (2^2) x (2 ^ -2)
........> USING log( base 2).....>
2 + -2 =0

So it follows that 1 (4 x1/4) has a log equal to zero and this is true for any base of logarithms b

Thus b^0 =1

I would use powers of ten.... a concept we are given at an early age. The pattern is pretty easy to see.

10^-2 = .01
10^-1 = .1
10^0 = 1
10^1 = 10
10^2 = 100

You could then use number places 214 = 2 * 10^2 + 1 *10 ^ 1 + 4 * 10 ^0.

And after that, powers of two would be an interesting topic.

A practical application of powers of ten is the timescales over which evolution takes place. You could include a discussion of how carbon dating works.

ebrown,
The 10's pattern is the traditional way of learning it but it does not explain
that it works for any base.

Exactly, which is I would then go to powers of 2.

That's how humans learn. You start with the familiar, then you build on that finding patterns in different examples.

I tried googling this and the slightly easier explanation given refers to the law of exponents.
n^a x n^b = n^(a+b)
e.g
n^2 x n ^3 = n ^ 5 which is easily demonstrated.
we then ask what is
n ^ 3 x n ^ 0 ? which by the rule must give n ^3.
thus n ^0 must equal 1.

Ok fine.

But now find a connection between that example and a student's real life.

The concept of consistency is indeed an abstract concept but a very important one. The "explanation" of the meaning of powers 0 and 1 of all bases (not just 10) can only demonstrated by consitency of rules of combination. Application is not "explanation".
Alternatively it is by showing inconsistencies like violation of the angle sum for triangles that new mathematical models (non-planar surfaces) can be introduced. The interest of students in new models can then be promoted by illustrating their utility. (e.g.Einstein's "space is curved")

At what age do you propose explaining the curvature of space? Getting any real understanding of this requires Differential Calculus.

Having a wrong understanding of a scientific concept is worse than having no understanding at all. What you end up with is the silly pop-culture understanding of Relativity that is pure fiction masquerading as "science" with a few science terms thrown in to make it deceptive.

Children of 10 upwards easily absorb the fact that "there are no straight lines in nature". Natural examples of circles and parabolas are easily demonstrated ( I used to do it) and practical games trying to flatten orange peel are useful for explaining why shortest routes are not straight lines on rectangular maps.

The idea is to prepare the ground and plant a few seeds.

We may never agree on this Fresco. I think we have a fundamental disagreement.

"There are no straight lines in nature" is exactly the kind of pop-science that drives me crazy. That is certainly not a line you will hear in any serious conversation between real physicists.

I have studied General Relativity... and I don't even know what "no straight lines in nature means". It certain has nothing to do with the differential equations that are behind the revolutionary ideas of Einstein and Heisenberg et al.

The reason this is educationally counterproductive is that it puts the teacher in the position of mystic... telling students strange facts that are so beyond their level of comprehension, that it turns into something almost spiritual.

Of course this is not how real physicists and mathematicians grasp the subject. In graduate school students are ready to tackle the ideas, not as mysticism, but as powerful mathematical arguments expressed as differential equations. Sure there is a sense of fascination, but it isn't magic and it at this point isn't beyond a graduate students ability to comprehend.

We were also supposed to teach about the solutions to differential equations for electron orbitals (a QM phenomenon). It was ridiculous, the kids had the picture of electrons as flowers with absolutely no idea of what it meant. When I talked to them... they understood it was ridiculous.

Education is to teach kids how to think and understand. They should be given things they can grasp, master and then express.

A great topic that kids get so much out of is projectile motion. When they have algebra, can understand that horizontal motion is independent of vertical motion (an introduction to the concept of being "orthagonal") and then can make calculations that predict the distance traveled by real projectiles.

Kids should be taught things that they can master-- so then they are not being taught to accept when they are told no matter how incomprehensible. Instead they are being taught to think critically, reason and to apply past knowledge to develop new ideas.

Teaching over the heads of your students is worse then useless.

I was satisifed when I got it working backwards, thusly, from a college prof, who worked in terms of fractions.

It's easily enough to show that x^z/z^y = x^(z-y). Fer instance, 2^3/2^2 = 8/4 = 2 = 2^(3-2) = 2^1, and 2^5/2^3 = 32/8 = 4 = 2^2 = 2^(5-3). Works for all numbers. Now work out 2^2/2^2. Clearly the answer is 1. Per earlier examples, demonstrated ad infinitum, that x^z/z^y = x^(z-y) is true, then if z=y, then x^(z-y) = x^0 = 1.

Worked for me, anyway.

Maybe kids don't like the height of the pedestal that "real physicists" place themselves on !

This is a textbook explanation.

This is a clearer reply.

You have a point.

Another good reply.

I also searched online and no easy explanation for this concept was found. This is why math teachers never bother to explain how these ideas came to be. They simply tell the class to memorize the rules.

I'm curious. Exactly who, what and where do you teach ?

The goal of a good physics teacher is to raise the kids up to the pedestal of real science.

You don't get there with misleading or untrue pop science.

Besides, kids get enough of that dumbed down, bad science from TV.