Trying to find math inside everything else

Potluck Math

I was talking to one of my co-workers about a “Friendsgiving” she is holding, and how the food bill is getting up there as more people are invited. But some of those people are also bringing food – and everyone is worried about having enough.

I realized this is a very common problem with potluck meals. Everyone wants to make sure they have enough food, so the more guests, the more they make. But think about this –

At a 4-person meal, each person makes a dish that feeds 4. (4 servings). So each person then eats 4 servings of food. (Which seems like a normal amount.)

Now it’s a 20 person meal, and each person makes a dish that feeds 20. So now each person eats 20 servings? That seems unlikely – it’s much more likely that people eat 3-6 servings, for 60-120 servings eaten, leaving 80 servings of food left over.

The problem here is that each attendee is treating the problem linearly, when it would better be modeled quadratically. Of course, this is complicated by the one hit dish that every eats a full serving of, and that other dish that no one eats, and everyone wanting to try a little of everything, so figuring out how much to cook can get complicated pretty quickly.

Hiking and Slope

Last summer, I went on vacation out west to see some National Parks (Yellowstone, Glacier, Craters of the Moon). At Craters of the Moon, all the trails had these lovely signs talking about how steep they were – since one of us hikers had a bad knee, we needed to make sure the trails weren’t too tough. What’s interesting is that they didn’t just talk about the average grade, which many hiking books do (as we learned to our chagrin in Glacier), but also the maximum and minimum. I feel like this is a good opportunity to talk about average rate of change versus instantaneous.

Later on in the trip, we had a discussion about what it means if a trail is twice as steep as another one. If I told you the the next trail is twice as steep as this one, what would you expect? What would it feel like? Then we also talked about whether we were doubling the slope or double the angle. That distinction is tricky because, for angles less than 10°, which are the most common, the difference between doubling the slope or doubling the angle (up to 20°) is less than 1% extra grade.

Anyway, there’s a lot of data here, so I pose to you: what could you do with this?

Walking in San Francisco

I had this same thought the last time I was here, but I feel it could be fruitful. Sometimes there is a route someolace that is “shorter,” laterally, but there are tons of up and down hills between. So is that way really shorter? I feel like you can do something with this: use the Pythagorean theorem to determine how far you are actually walking, determine different walking speeds on different inclines, and then get a topographical map and determine the speed and length of different routes, then check what Google Maps says.

For an example of what I just experienced: because we didn’t really know where we were going, we wound up walking up the giant hill up Powell St and then down the hill on California St, but if we had gone down Sutter first and then up Grant, it would have been much flatter.

Meet Me Halfway?

Today I met a friend for lunch, and we decided to meet halfway between our apartments. To think about where that would be, I used MeetWays, which was useful because it can find that mid-point based on a variety of modes of transportation: car, bike, public transit, and walking.

The thing is, my friend took the train to lunch and I walked. (I based the location on the transit midpoint, but decided to walk for the exercise.) So what counts at the midpoint then? If one person drives and another doesn’t, do you include the time spent finding parking as part of your calculation? Is it based on time, or distance? If we both travel for 45 minutes, but she’s much farther from home, is that still meeting in the middle? Considering everywhere I could walk within 45 minutes and everywhere she could train in that time, is there really only one middle point, or do those rings intersect twice?

There are the things I wondered as I walked. Are there other good questions to ask? You can probably, at the least, get a decent rates question out of it.

Weight on Other Planets

I took some photos today at the AMNH with their scales with the intention of posting them as Estimation180-type challenges. I realize as I review the photos, though, that there were some issues. The numbers fluctuated a lot when I stood on the scales, so the pictures I have vary by a huge margin (+/- 30 pounds to my Earth-weight). They didn’t fluctuate when I put my bag on, but not every scale register its existence.

Estimation #1 – Here’s my bag (laptop, iPad, book, charger, 3DS). How much does it way?

Then the next estimations – how much would my bag weigh on Saturn? How about on the surface of a Red Giant Star?

At trivia tonight, one of the bonus round was a matching question – match the 10 movies to the character Ben Stiller plays. We (and by we I mean my teammates, as I know very little about Ben Stiller) knew 7 of the questions for sure, but had no clue for the other three.

At that point, one of my teammates asked me if it would make more sense to put the same answer for all three, rather than guessing. That way we would be guaranteed a point, as opposed to maybe getting them all wrong. It took me a minute to think it through, but I told him it didn’t matter either way and I’d rather take the chance of getting all 3 right. (We won trivia on a tie-breaker final question, so a 1 point difference would have been a big deal.)

I figured there were 6 possible ways we could write down our answer – 1 correct way (call it A B C), 3 ways that get us 1 point (A C B, C B A, and B A C) and 2 ways that get us 0 points. (B C A and C A B). Calculating that expected value gets us an EV of 1 point, the same as his suggestion – so, mathematically, they are equivalent. Then it just comes down to your willingness to take that risk, since it’s not a repeatable event.

And as we always say every week, go big or go home.

Piano Playing

I had a “Talking Math with Other People’s Kids” moment back in October that I wanted to blog about, so now is as good a time as any to pull that one out of the drafts folder.

My boyfriend’s niece and nephew have recently fallen in love with Star Wars, which, frankly, makes all of us happy. So while at the BF’s parents’ house when they were there, I decided to entertain them by playing the Star Wars theme on the piano. His nephew, Matthew (age 6), liked it and wanted to know how to play it. I thought about how I would describe it to him, so I did the following.

“Okay, this key here, I’m gonna call that key #1. So the one next to it is key #2, and so on. So, I’ll write down some numbers and you can know which keys to press.” And I wrote “1 5 4 3 2 8 5 4 3 2 8 5 4 3 4 1. (Then repeat)” He practiced it a few times until he got it, and we were all impressed. But then he wanted to do the next part, which dips into the octave below middle C. So I saw an opportunity.

“Well, Matthew, the next key we have to hit is this one, what should we call it?”

He told me I should call it 9, with the next notes being 10 and 11, since we already did up to 8. I asked him if he thought that might be confusing. “If I were reading it, I would think the 9 should be the one right after 8.” He agreed, so then he said we should call them letters, like a, b, c.

“Well, okay, Matthew, we could do that, but then what would you call this key [the one to the left of “A”]?”

After a moment’s thought, “D.”

“But isn’t that confusing again? Now everything’s out of order.” He agreed again, but wasn’t sure what to do about it. I made the following suggestion:

That way if we need to add more notes that go down, we can just use more letters, and if we need to add notes that go up, we can use more numbers. He thought this was a good idea, and so we moved on to playing the rest of the tune.

I admit I was thinking of a specific post I had read not long before, which I thought was a TMWYK post but now I can’t find it, where the child invents numbers smaller than 0 using *1, *2, *3, etc. That was my intention with that conversation with Matthew, but if I’ve learned anything from reading all the TMWYK posts, it’s that you don’t push it if the kid isn’t going their themselves. We had to come up with some sort of solution so we could keep playing the song, but once we got something workable, we didn’t need to keep going to talk about the whole negative number system. At the time Kathryn Freed asked me if there was a 0 key, which I said there wasn’t and so the divide between the two types of keys was a little awkward, but it worked out.

In the months since, Matthew has been taking piano lessons and learning to read actual music instead of my cockamamie scheme, which is for the best. He’s pretty good at it, too.

Scrabble Variant

(inspired to post by Anne’s 30-Day Blog Challenge)

So I was playing Scrabble last night (I lost – it’s one of those board games I’m not the best at) when we talked about how, when you are playing with good competent players, the board often winds up with knots of small words close together.

Kinda like this one.

So we talked about how we could promote long and fun words instead of those same short words all the time, and thought you could have a variant where you get bonus points based on how long your word is, regardless of which letters you use or where you place it.

Such a bonus somewhat already exists – you get a 50 point bonus if you use all 7 of your tiles. So we thought we could add other bonuses for other lengths. We agreed we should keep the 50 point bonus for 7, and that you shouldn’t get a bonus for only using 1 letter. As well, we thought a 2 tile go should get 1 point as a bonus. So I said I could definitely model it from there.

I tried to feed those data points into Wolfram Alpha for a fit but they provided linear, logarithmic, and period fits, all of which were terrible. I then forced them on a quadratic fit (after all, 3 points make a parabola), which was alright, although maybe too many points for a 6 tile play. Then I did an exponential one (though I had to use (1,0.1) since Wolfram didn’t like using (1,0) in an exponential fit, as if we couldn’t shift the curve down.) Then I just fed them into Desmos and rounded.

Below are the graphs and the tables for each fit. What do you think of this variant? Which point spread would be better? Of course, we’d have to play it to see….

How Many Subway Transfers Do I Have to Make?

I was talking with Sam Shah and had the following exchange:

Of course, after I said that…I had to find out if it was actually true. So I pulled up a map of the subway system and started analyzing.

I realized the best way to analyze the system would be to create a matrix of connections: if I can transfer directly between two lines (or the walking transfer from 59th St/Lex to 63rd/Lex, since you don’t need to pay again), then put a 1 and make the cell green. If not, put a 0 and leave the cell white. That’ll show a chart of all the places you can get to on a single transfer.

Most of the lines have direct transfers, with a few being tricky. Breakout stars are the A, which connects with all but the 6, and the F, N, and R, which only miss some or all of the shuttles. Particularly difficult train lines are the G, the J, and the 6.

So this answers the question of where you can get with only 1 transfer. But what about two transfers? For that, we can multiply this matrix by itself. This is the result:

What do these numbers mean? Well, to explain, let’s look at the G –> 6, which I have highlighted in blue. The number there is 8. This means that there are 8 ways to get from the G to the 6 with two transfers:

G –> 7 –> 6
G –> D –> 6
G –> E –> 6
G –> F –> 6
G –> L –> 6
G –> M –> 6
G –> N –> 6
G –> R –> 6

So this chart shows that you can get from any line to any other with at most two transfers*, with one exception: the Rockaway Shuttle to the 6. However! Those stops aren’t solely serviced by the S. (The only stop in the system solely serviced by an S train is Park Pl, on the Franklin Ave Shuttle.)

Because of that, I can amend my statement to the following, which I have proven true:

During rush hour, you can get from any stop on the subway to any other with a maximum of two transfers.

But then, that gets me wondering further…this chart was just made if the connections exist, but they weren’t time-sensitive. For example, the M does not run at my stop at nights or on weekends. How would that change this chart? Especially when you consider that the E, which does not normally go to my stop, DOES at night. I leave that problem open.

———————–

```* Of course, fewer transfers doesn't always mean better. If I wanted to get
from Astoria to Greenpoint, sure, I could take the N to the G, but that
requires going all the way through Manhattan, way down into Brooklyn, and
then back up. Instead, a quick hop from the N to the 7 to the G is much
more sensible, even if it is an extra transfer.

Excel File - Subway Analysis```

Trig without Trig

Over the summer I made a Donors Choose page in the hope of getting some clinometers, so that we can go out into the world and use them to calculate the heights of objects like trees and buildings. And we did!
I created the Clinometer Lab as an introduction to Trigonometry. As such, prior to the lab, they had not seen any trig at all. So I started off with this video as the homework from the night before.

In class, we talked about how, when the angle is the same, the ratio of the height and shadow is the same. And how, long ago, mathematicians made huge lists of all of these ratios.
So if they told me any angle, I could tell them the ratio, and then they could just set up the proportion and solve.

So we left the building and went to the park, armed with clinometers, measuring tape, calculators, and INBs, so do some calculations. I had the students get the angles from their eyes to the height of a tree, from two different spots, and the distances, so they could set up a proportion and discover the height of the tree. (When they finished, they did a nearby building. If they finished that too, the extension problem was to switch it up: given the height of a famous building, the Metlife Tower, that they could see from the park, and determine how far away they were.)

It went really well, and I think they got the idea. The fact that they could choose which tree to measure and were given free range of the park, and could choose where to stand to look at the top, but always had to come back to me to get their ratio, worked seamlessly. I could check in on them easily and keep them on the right path. (Especially with my co-teacher there to keep them all wrangled, or the AP who observed/helped chaperone the classes without my co-teacher.)

The next class, I revealed to them that I was not using that crazy chart to give them their ratio, but rather they can just get it themselves from the calculator, which basically internalized the list. Then they got it, that is was just ratios and proportions, not some crazy function thing.

Trig as a topic did not go great in my class, but that is the fault of my follow-up lesson, where I tried to squeeze in too much and didn’t do any practice, not that fault of this lesson, which still sticks in their minds. Later in the year we were doing a project about utilizing unused space, so we picked empty lots but couldn’t get in. So in order to figure out how big they were, we used clinometers, and trig. Now that’s a real world application.

The Lab

Clinometer Lab Instructions

Clinometer Lab Sheet