Set!

I guess, but it feels more direct to flip the trapezoid upside down. You could use each fact to derive the other, but I don't think the ratio is more natural.

I think they're saying that if you distribute the second term and then use the proportion b/h = B/H you get the first, but the prompt seems a bit sloppy.

Calculus Made Easy goes into this -- minute (time) and minute (small) have the same root, and second was originally a shortening of second minute (minute minute). So ' is small wrt a big unit, and " is small wrt '

Interesting. Can you explain this in the form of a limerick?

But the setup of the problem SAYS that 10 is the length of the hypotenuse. What are you talking about?

I read this problem four times, and I'm not sure what it's trying to communicate. Are you saying that students might see a^2+b^2=c^2 and b^2=c^2-a^2 as different relationships?

I stole your idea from the zaggy puzzle. 😁

A solution using a square wave.

Give them approximations to x and y to three decimal places, let them wonder about WHY something might be x+1 before they try to show it. (That's how I did Galois theory, numerical approximations leading to algebraic guesses.)

Calculator? (I don't know if you have a stash of 4-function ones, but they might be interesting as a tool here.)

The article now claims that it's 1/2 * 1/4, which would be "half of a quarter." Surely the story is "a quarter of a half" though? At least in CA, the grade 5 standard says "Interpret the product (a/b) × q as a parts of a partition of q into b equal parts."

"The 5th graders get to play on ½ of the soccer field at recess. However, ¾ of the field is covered in mud. How much of the soccer field is usable?" This is not a model of 1/2 / 3/4, as the article claims. This is 1/4 * 1/2. Did any math educator proofread this?

I think the exact probability would be really messy, but my guess is that each student is close to independent from the previous ones. Your calculation for student 1 gives ~1/9 chances, so 1 student of 10 pulling a dupe is likely the modal outcome.

This is backwards - your logic suggests that a 20th student would be forced to pull a duplicate. Once the first kid successfully pulls five cards with no duplicates, the next kid now pulls from a deck of 95 including 5 that CAN'T dupe anymore.

Nine Inch Nails, Broken, liner notes:

"no thanks: you know who you fucking are"

I was actually thinking about this in terms of quantum information, so maybe it's actually a physics question!

I think this is correct. Imagine that you had 30 flips instead of 3. Naively, you would want to flip each coin 10 times, but the variance for the unbiased coins is higher (n*1/2*1/2 > n*1/4*3/4). So you get more information from isolating a potential winner than uniform testing.

Virtually zero. On average it'll take 52*ln(52) is roughly 200 pulls to see every card, so you expect to see the AVERAGE card 4 times. I ran some simulations and the average largest bucket is 10 (15 if you change to the birthday problem version of a 365 card deck).

Oh you mention this! I should be clicking through your links.

At least in theory, any US state that has standards derived from the Common Core (including CA) does include the derivation similar triangles -> mx+b... in 8th grade! It's 8.ee.b.6. This is in the standard math for elementary/middle texts. (Is this unreasonable for 8th grade? I didn't write them.)

To be fair search engines mostly lie and plagiarize now it's not your fault

Sound By Numbers: The Rise of Digital Sound YouTube video by Technology Connections

youtu.be/Gd_mhBf_FJA?... when you represent a sound wave as a function, you can recreate it by approximating the function with sine waves. That's a Fourier series! If you cut off the series early, you can still recover the function exactly within human hearing. That's data compression!

Some kind of silent majority!

That might be taken.

Moral majority?

yeah...but we'll always have the words from man himself

I think what this shows is that they did a got set and it was Tarkir

The first thing that happens is the direwolves, Starks are the most gw that has ever gwed

This is one of my favorite applications of 2000-level differential equations! Notice that f(x)=sin(x+b) satisfies Hooke's law f'' = -f, so it must be a linear combination of sin(x) and cos(x), then evaluate at f(0) and f'(0).

Good one! I like the variation on the abs() version.

In Spanish it's "todos sin tacos," which is lovely.