Extending our arithmetic / geometry connection project to calculus

Yesterday we did a fun project connecting arithmetic and geometry:

Connecting Arithmetic and Geometry

While we were talking about the shapes my older son commented that one of the shapes looked like a pyramid. I thought it would be fun to make the shapes look even more like a pyramid and see what the kids thought.

We started by just talking about the shapes – the most interesting thing to me here was how challenging it was for them to compare the volumes of the shapes:

Because they were having a little bit of difficulty with the volumes I spent a little extra time on the idea. Things seemed to clear up a little bit, luckily:

Finally, I thought it would be interesting for the boys to see some of the math I used to create these shapes. Although this section goes on a little longer than I would have liked, I think this is a fun little introduction to functions and scaling even if we don’t define those ideas explicitly:

A fun little project. I think that some of the broad ideas from calculus are within the grasp of kids even if the underlying calculations probably aren’t. It was fun for me that a question from my older son led from us jumping from arithmetic to geometry to calculus 🙂

A few fun Perfect Bracket stats questions for students

ESPN had 18.8 million entries in their bracket challenge for the NCAA men’s basketball tournament. There were also several other bracket contests going, too. Below are a couple of fun bracket-related questions for students learning about statistics:

(1) Perfect Brackets after the first round:

The ESPN contest went from 18.8 million entries to 164 perfect brackets at the end of the first round of games:

So . . .

(i) If you were running a contest that had 100,000 entries instead of 18.8 million, how many perfect brackets would you expect to have in your contest?

(ii) What do you think the probability of having 0 in your contest would be?

(iii) How about exactly 1?

(2) Perfect brackets after Michigan St. beat Miami

The number of perfect brackets in the ESPN contest fell from 952 to 513

Prior to the Michigan St win you had 5 perfect brackets left in your contest. Given what happened in the ESPN contest how many do you think you’ll have after the Michigan St. win?

What do you think the probability is that you will have 0?

How about 1?

(3) The USC vs SMU game was the 22nd game of the tournament

You had 241 perfect brackets going in and 22 after USC won.

In the ESPN contest 81.6% of the entries picked SMU to win and 18.4% picked USC.

(i) Suppose you have a coin that flips heads 18.4% of the time. If you flip it 241 times what is the probability that you will have 22 or fewer heads? (probably best to use a computer for this one . . . )

(ii) Do you think having 22 brackets left rather than 44 (which would roughly be 18.1%) was random chance or was there another factor in the reduction?

(4)  Expected upsets

I’ll make up the numbers for purposes of this problem, but you can get the actual numbers here if you want:

Some statistics for the ESPN bracket tournament

Suppose that the 18.8 million entries have selected the winners of the games this way:

Teams 1 – 5:  95% of the entrants guessed these teams would win their 1st round game

Teams 6 – 10: 90% of the entrants guessed these teams would win their 1st round game

Teams 11 – 15:  85% of the entrants guessed these teams would win their 1st round game

Teams 16 – 20:  80% of the entrants guessed these teams would win their 1st round game

Teams 21 – 25:  70% of the entrants guessed these teams would win their 1st round game

How many games out of these 25 do you expect the team that was not favored by the ESPN contestants to win?  Why?