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My son's in 4th grade, so I've been seeing a lot of Common Core stuff over the last few years.

I used to really hate Common Core.

Then things started to click. Turns out the problem isn't with Common Core in and of itself. The problem is that no one knows how to teach it correctly.

This video (https://www.youtube.com/watch?v=aLCO6pG6JDY) is a fairly accurate description of the counting-up method of subtraction that's taught as part of Common Core. It seems ridiculous, doesn't it?

Well, it IS ridiculous, when you do it like THAT.

The problem isn't with the technique, though. Believe it or not, the technique is actually used MORE OFTEN than traditional subtraction with borrowing in the real world.

"I don't believe you, Coda!"

No? Well, check this out:

"39, 40, 50, 60, 70, 80, 90, a buck, two bucks, three bucks, three twenty-five. That's two pennies, six dimes, two dollars, and a quarter; your change is $2.87, have a nice day!"

Why don't they teach it that way in schools? >_< Parents could actually help their kids with their homework!

Every step of that counting-up process is a trivial thing to do in your head. It's all small, round numbers until the very last step -- 2 + 60 + 200 is trivial!

In my opinion, ideally, they teach basic subtraction and that common core sort of way for mental math. Like "here's how to do it fast on paper" then, "here's a way to think about it in life but not on paper." I was taught normal subtraction and..that common core technique reminds me of learning mental math (but I wasn't good at that and we didn't focus on it too much so I don't really remember too well).
I don't really know what common core is though. I think I was in 5th or 6th grade when it was first implemented (that's 2010 I think). And then in 7th grade I went to a school that could care less about state/national standards.



Momentum is proportional to energy..? Is it related to change in momentum over time equals change in KE over distance (I remembered this as a part of some problem on my test..is it related?)? Oh, is that say, we don't know if an object of a higher temperature influences momentum? Err, I hear about dark matter on occasion but what is that?

Something about Newton's 3rd law was shown on the test--but I didn't remember what law no. 3 was. I can understand that explanation but, using Newton's first law, I'm not sure how it's different from the 3rd law..? Internal forces don't count as an outside force and the internal stuff technically moves but overall the object doesn't move anyway and so it doesn't influence the momentum?

I definitely agree that they need to teach multiple techniques. The INTENT of the curriculum is to equip kids with tools to be able to learn in an efficient way, and that's not well-served by trading out a technique that doesn't work for some kids for a technique that doesn't work for other kids.

Really, it's not so much that they're different ways of answering the question. Rather, the third law is WHY the first law holds.

EDIT: Screw it. Hold on for the other question. I'm going to go change stuff in the admin CP.

If common core is replacing "old-school" subtraction with that though, uh, future math classes are going to be tough.

Ohh, thanks that makes sense now.

Admin control panel..?

If my son's classes are any indication, they're not completely replacing it. He still knows how to do subtraction with borrowing.

Yes, Trisphee's admin control panel. You'll see.

As for momentum and energy... that goes back to E² = p²c² + (mc²)². This is saying that momentum and mass are related in an interchangeable way.

But what if there WERE something else that contributed to that total energy figure? Some internal force inside the object? Then you'd have to add another term to the equation.

You suggested temperature. Well, that one we actually DO know the answer to -- it doesn't, because temperature measures the average kinetic energy of the particles inside the object relative to each other. The same argument concerning Newton's first and third laws applies here -- any motion inside the system would have to be countered by an opposite motion. If it WASN'T countered by an opposite motion, then that would be contributing to the momentum of the system, which is contrary to the definition of temperature measuring motion internal to the system.

BUT!

Given that the object isn't exploding, there must be some force holding it together. It could be atomic bonds, it could be gravity, it could be magnetism, whatever. So does this force contribute energy that's equivalent to momentum and mass?

That's what we don't know. We can't prove it doesn't, but if it does then it's such a small contribution that we haven't detected it yet.

But there could be some other force that we don't know about with a very weak effect that operates over very large scales, so an Earth-based experiment would have a very hard time doing anything with it. If there is, then this might be an explanation for why the universe isn't expanding faster than it is, and finding a way to measure how much of it there is could give us a clue as to the far-future fate of the universe.

Now that I think about it, I didn't really learn math basics from school--school gave me problems and my parents taught me how subtraction and fractions and things worked. I still remember asking teachers what in the world I was supposed to be doing, then giving them blank stares after they responded.

Oh woops, temperature would just make things go faster but they still cancel each other out.
That's really interesting! And kind of mind boggling. If an object were disassembled into bits of protons, neutrons, and electrons, and whatever parts there are, I wonder if they'd have the same mass XD.

Nope! I mentioned this before: The energy absorbed or released by chemical and nuclear reactions does in fact influence the system's rest mass! And we can use E=mc² to describe just how much.

One example: A hydrogen nucleus (1 proton, 1 neutron) has a mass of 2.01410178 amu. A helium nucleus (2 protons, 2 neutrons) has a mass of 4.002602 amu. But that means two hydrogen nuclei (2 protons, 2 neutrons) have a combined mass of 4.02820356 amu. What's the ~0.0256 amu difference? It's the binding energy in the nucleus holding the protons together!

0.0256 amu is roughly 4.25x10^-29 kg.

E = mc[sup]2[/i]
E = (4.25x10^-29 kg)(3.00x10⁸ m/s)²
E = 3.82x10^-12 J

And so we predict that this would be the energy released by two hydrogen nuclei becoming a helium nucleus in a fusion reaction. It seems like a pretty small number, but then you can go calculate that red light (remember, the LEAST energetic visible light) has a kinetic energy of only 2.84x10[sup]-19[sup] J and you see just how powerful that actually is.

It should be noted that sometimes bond energy can have a negative contribution to mass -- this is usually unstable. Further discussion of the concept starts getting into some chemistry stuff, so I'll defer that.

"But Coda, didn't you just say a couple posts ago that the forces binding objects together DOESN'T have an effect?"

I did indeed. I'll confess that I wasn't being wholly precise before, since this isn't a masters-level physics course here.

The difference is that atomic/molecular binding energy is part of the rest mass. It acts just like any other mass: it has inertia and it participates in gravity. You can measure it just fine using either of those principles, and in fact the only way to measure that as distinct from the "base" mass (if such a thing existed) would be to break all of the bonds and measure the difference.

This is so fundamental that it turns out that around 99% of the mass of a proton is the binding energy holding its constituent quarks together! Only around 1% is the rest mass of the three quarks inside (inasmuch as you can say that quarks have rest mass, because they're unstable in isolation).

So really, what I was saying that we don't know is if there's a contribution to the total E of a system that you can't measure by observing its rest mass or its momentum.

Woah! So energy is mass..? Or most of mass is energy? Doesn't that mean mass is literally energy bunched up together O_o? And that what's happening at nuclear power plants is more or less spreading that locked and bunched together energy out?

Ohh woops. When energy is released mass is lost, and taking the stuff apart means releasing all the energy. But then again protons and electrons aren't even masses either (I forgot quarks existed, well I never knew what they were until now ^^;;)...What even is mass, it seems that literally everything might as well be internal forces o_o.

Oh! But why do quarks count as rest mass? Aren't they energy as well x'D.

The internal forces aren't accounted for is because the internal forces are part of the rest mass..? And they, because of Newton's third law, aren't going to suddenly change the object's rest mass?

That's exactly what E = mc² means!

You could look at it that way!

I think the bulb may have just lit up. :D

I'll point out that particle physicists measure mass in electron-volts (1 eV = 1.6x10^-19 J) instead of some unit derived from the gram.

They count as rest mass from the perspective of separating binding energy from everything else, but you're right: that rest mass itself is made of energy! That energy has a charge and a spin and other properties, and it's got the same wave/particle duality that photons have, but it's turtles all the way down.

Strictly speaking, it's not the forces themselves, it's the energy resulting from the forces interacting. But basically, yes.

I say basically because radioactivity violates the simplistic reading. The mass has changed, but no outside force has acted upon an atom to make it give off alpha or beta particles; it's one of the things that currently appear to be truly spontaneous in physics. But the system taken as a whole, if you look at everything as energy equivalence instead of distinguishing between mass and energy, obeys it -- adding up the nucleus, the photon of energy, and the emitted particle means the final system has the same mass-energy as the initial one.

I forgot that radioactivity existed :o. Granted, all I know about it is that carbon dating is used and that's related. And carbon dating has to do with emitting something. And I guess emitting something means losing mass or energy.

Err, so anything internal is not accounted for because it's all part of the rest mass. And for things like atoms bouncing around, Newton's third law says that within the object, that sort of bouncing will be balanced out by another atom because within the object itself there is an equal and opposite reaction..?


Electron volts! I don't know what a volt is but it's interesting they use energy, and that it's electrons. Why not quarks xD?

Random tidbit! When dealing with petty cash, a quarter, a dime, and a nickel equal 40 cents! Seems obvious, but it's amazing how useful this little bit of addition is, and how it's not always obvious when you first start out as a cashier.

Electron-volts are like light-years. You're not measuring electrons or volts any more than you're measuring light or time; it's a measure of energy based on an electron moving across one volt, analogous to deriving a unit of distance from how far light travels in a year.

"Why not quarks" is because you can't measure quarks directly, because they don't occur in isolation; they ALWAYS occur in bound groups and will instantly transform into something else if you manage to break them apart. Electrons are, as far as we can tell, actually fundamental, instead of being made up of other things.

And yes, your conclusion about rest mass is accurate as far as modern science can tell.

Ohh that's interesting. I've been (and still am) confused by the idea of stuff multiplying together to make something. The light example makes it clearer.

Oh woops, electrons are very different from protons and neutrons. Err, but what about protons and neutrons..? Do they matter? Are they part of the "volt"? Not that I know what's going on with the electron volt though :x



Wow, that really is useful when I think about it. I'm not a cashier but the first thing I'd think of for 40 cents would be 4 dimes, which I suppose is a waste of dimes XD.

The only relevant part of the electron here is that it's a single unit of negative electrical charge. Everything else about it doesn't matter for this definition.

A volt... hmm. I guess you could describe a volt as the electrical equivalent of the potential energy stored in an object suspended above the ground. It takes work to pick the object up against gravity; it takes work to pull a negative charge away from a positive one (and the electron-volt measures this amount of work). In either case, you can turn that potential back into kinetic energy by allowing the particle to move freely.

Err why does it matter that it's a single unit of negative electrical charge? Because that way it can be measured easily..? One electron is one unit?

So..a volt is how much stored up mechanical energy there is in an object (I looked at the units just now as ‎kg·m²·s⁻³·A⁻¹)?
Why potential? Why not kinetic? Is it because the particle is not moving freely within the object, but when it gets measured it is moving? (All I know is batteries and potatoes can have "voltage" measured with wires and multimeters, and the kg·m² makes me feel like it has to do with KE, since KE is kg. m2/s2).

One electron is one unit is exactly it -- it's as fundamental of a measurement as you can get, nothing arbitrary about it.

Stored-up electrical energy, not mechanical energy. It's possible to CONVERT between the two; that phenomenon is called "magnetism."

It's potential because it's not moving -- it CAN move, if it's allowed to. Just like a ball sitting on a table isn't moving, but it has potential energy: if you pull the table out from under it, that energy turns into kinetic energy.

A potato battery has measurable voltage because the metals you put in it have different electric potentials -- that is, given an opportunity, the electrons will move from one piece of metal to the other. And connecting a wire (or a multimeter!) between those pieces of metal provides that opportunity. It's like falling off a table; the electrons would much rather be on the lower-potential piece of metal than the higher-potential one.

Without going into too much detail right now because I'm tired... Charging a rechargeable battery is like rolling a boulder up a hill -- you put energy in, and later on, that energy can come back out.