Tag Archives: math

Preparing to Take Physics: 3. What are significant figures and why are they significant?

When I first learned about significant figures, back in high school, they seemed totally arbitrary and annoying to me. Now, though, I know they’re incredibly important. The rules seem silly at first, but actually they make sense. So, if they seem dumb, please, please stay with me.

Ok, here we go. Let’s say I give you a ruler that only has inches marked on it. (So, none of the little tick marks in between, just 1 inch, 2 inches, etc.) Now, I ask you to measure the height of a cup I have. What answer might you give?

4 inches?

4.5 inches?

4.56712 inches?

Really, you can only say for sure that it’s close to 4 than it is to 5 or 3. You might be able to guess that it’s halfway between 4 and 5, so you could say 4.5 pretty reasonably, but you definitely couldn’t say 4.56712. Because how would you know that?

In science, we’re measuring stuff all the time, and not only do we need to know what we’ve measured, but we also need to know how precisely we’ve measured it. Like, is the acceleration due to gravity really 9.8, or were we only able to measure it to one decimal place? (Actually, it’s different in different places, and even points in slightly different directions if you’re near a big mountain or a pile of gold.)

For example, I was out in the woods one day as a search and rescue volunteer, looking for crime scene evidence, and I was pacing out distances. If you’ve never paced out a distance, when most people take two steps they go a pretty constant distance, maybe 4 feet, so you can use pacing to measure how far you’ve gone. Well, I was out in the woods with the other members ESAR, and we were pacing along looking for evidence, and I found a piece of trash. I reported it to my teach lead and he asked me how far I was from the place we started. I have a 4.5 ft pace, and had gone 5 and a half paces, so I did the math in my head and then, without thinking, said something like 24.75 ft. The team lead was laughed and was like “oh yeah? wow.” Because, here I am being incredibly imprecise, wading through bushes, and I’m telling him that the piece of trash I’ve found is exactly 24 ft and 9 inches away. But, that’s what my math told me.

So, the problem is, when we do math with imprecise numbers, we sometimes get things that look more precise than they are.

Like, if I take 1 and divide it by 3. I get 0.333333333333333333333333333333333333333…

The first two numbers (1 and 3) are integers, I have no idea if they’re 1.4, 1.6, 3.1, or what, but when I divide them, I get something that looks like I’ve measured it out to the infinite decimal place. Silly.

So, we have to watch out for situations where math spits out numbers that don’t mean anything. 

After that first 3, the other 3’s don’t mean anything at all. 1 divided by 3 might be 0.32 or 0.34447. Because, I don’t know whether the 1 was really a 1 or whether it was actually 1.1 but I didn’t have good enough tools to measure that carefully.

Does this make sense?  The key here is that 4 is different from 4.000000000 because if I just say 4, then that might actually be 4.1 or 3.9 or 4.12321111.

Because really, in the real world we don’t often have things that are precise, we only have measurements (unless I say something like “I have 2 dogs.” That’s probably exact. But, I’m 5′ 4″ tall, but I have no idea how many nanometers that is.

I hope that wasn’t boring. I realized partway through that maybe no one cares about sig figs, except that you’re made to calculate them. If you just wanted to learn how to calculate them, you can look through this short powerpoint I made that summarizes all the rules and shows some examples. Feel free to steal it and use it however you like (I mean, within reason):

Significant Figures

Summer Math Review Puzzle: Algebra I #1

One of my students asked if I would write her some math puzzles to help her review Algebra I over the summer before she goes into Algebra II. I thought I’d post them here, too! Hope you like it!

SummerAlgebraPuzzle1

Real World Physics Applications – Thermodynamics

There’s something kind of awesome that happens when you realize that things you’ve learned in the classroom might actually have real-world applications.  Like, when you go to a foreign country and can  sort of communicate with the locals because you took Spanish 101.

I was watching Shark Tank recently, and these guys came on with this thing they invented.  I don’t know anything for sure about how it works, but I could make some educated guesses, and I think that looking at their invention is a really interesting way to introduce thermodynamics.

I couldn’t find the clip of them on shark tank, but here’s their kickstarter campaign video, which according to youtube raised $306k in 37 days!

Anyway, it’s a pretty genius invention and I kind of want one. They’re these big metal things shaped like coffee beans and you put them in your coffee.  They immediately cool it to the perfect (still hot) temperature for drinking, and then keep your coffee at that temperature for several hours.

Being a coffee drinker, extreme appreciater, addict, myself, I was like “wow, that’s genius.”

Here’s my guess at how it works: The inside is some metal that has its melting point at a good temperature for coffee.  When you pour coffee over it, the excess heat goes into melting the metal. (This phase change absorbs energy, (called Joules, hence the name Coffee Joulies.)

Then, as the coffee starts to be cooled by the outside air, the metal inside starts to phase change back, solidifying and transferring heat back to your coffee.  (Phase changes can absorb tons of energy, and they will stay at a constant temperature while they do so.)

So simple!

We could even guess at what metal or material might be inside.  It would be something with a melting point around the perfect coffee drinking temperature.

The National Coffee Association says this temperature is 180-185 Fahrenheit, which is 82-85 Celsius. But, drinks.seriouseats.com says 110-120 is ideal, which is 43-49 C.  Coffee Joulies says they keep your coffee at exactly 140, or 60 C.

Then we can check the melting points on the periodic table. Sodium (Na) is 98 C and Potassium (K) is 63 C, Rubidium is 39 C.  Potassium is basically right on, so it could conceivably be made of that.

I just checked their website for more clues. They call it PCM (for potassium? Maybe it’s a mixture of potassium and something else.) Other than that they don’t say what it is, only that it’s natural, edible, and already found in food (potassium again?)

Anyway, fun mystery, and an awesome invention!  And, a really awesome example of how you can use physics to invent useful things.