2012-09-12 Learning to Count Comments
10
Click here to load reader
Transcript of 2012-09-12 Learning to Count Comments
`
1
Name:_______________________ Date assigned:______________ Band:________
Precalculus | Packer Collegiate Institute
Counting!
Today we’re going to start learning how to count. Like, yeah, you already know how to count. 1, 2, 3, 4, 5, … Duh. But
we’re going to come up with shortcuts to counting things. Like, counting without counting. WHAAA? Let me illustrate.
How many lightning bolts are there? _____
How many lightning bolts are there? _____
Okay so the way you solved both problems were different. And clearly the second way was way faster. What we’re going
to do is to learn to count without counting.
1. A quick question: A totally made up fact: there were only four different types of dinosaurs: the Iguanodon, the
Juravenator, the Allosaurus, and the Gigantosaurus. Each dinosaur came in one of three colors: red, purple, and blue.
The Museum of Natural History wants to have a model of each different-looking dinosaur. How many different models
does the museum need to create? And more importantly: somehow convince me that your response is correct.
Comment [sjs1]: Have students work on this
individually. Talk about the various ways students
evidenced it. Hopefully some will have listed them
all out, some would have made some-sort of tree
diagram, and maybe some would have made a grid
(dinosaur kind on one side, color on another)…
The first method of randomly listing is just random…
it’s like the first picture of thunderbolts… You might
miss one! How do you know you’ve gotten them all?
The other methods are ways of counting without
counting… talk about how the “multiplication” fits in
there and how it is a nice way to organize your data.
`
2
2. Another quick question: There is one more additional piece of made-up fact for you to consider. Each dinosaur had
one, two, three, four, or even five claws. With this additional piece of information, how many different models does the
museum need to create? And more importantly: somehow convince me that your response is correct.
3 (a). Ms. Tramontin gives you a multiple choice vocabulary test written in Azerbaijani.
Gecə göyün rəngi var: ___
A. Mavi
B. Yaşıl
C. Qırmızı
D. Qara
E. Sarı
2 +3 = ____
A. altı
B. beş
C. sıfır
D. on doqquz
Okean edilir:____
A. konfet
B. su
C. pul
D. karandaşlar
How many different possible tests responses could she get back? Explain!
Comment [sjs2]: The tree method would work
great here. However, the grid doesn’t quite work
well, because we have to go into 3D.
However, talk about what it would look like in 3D.
Draw the 3D shape on the board, and talk about
what each small volume represents… and how it
covers all possibilities.
Comment [sjs3]: A good question for this:
“What is relevant information from this problem?”
and “What feels like it is irrelevant information?”
`
3
(b) Ms. Tramontin gives you a fill-in-the-blank vocabulary test written in Estonian. She puts more words in the wordbank
than can be used – however no word in the wordbank is used more than once.
Fill in the blanks:
Koer läks ___________. Oli ____________ in puud. ____________ Lehmad olid lähedal. Samuti oli ____________.
Word bank:
kakskümmend metsa viima ahvide tiik magama
How many different possible tests responses could she get back? Explain!
(c). Ms. Tramontin gives you a fill-in-the-blank vocabulary test written in Estonian. She puts more words in the wordbank
than can be used – however the words in the wordbank can be used once, twice, thrice, or even four times!
Fill in the blanks:
Koer läks ___________. Oli ____________ in puud. ____________ Lehmad olid lähedal. Samuti oli ____________.
Word bank:
kakskümmend metsa viima ahvide tiik magama
How many different possible tests responses could she get back? Explain!
`
4
(d) Now compare your answers to (b) and (c). Is it significantly better for you as a student to know you can’t repeat
words, moderately better, or just slightly better? Justify your answer. If you can come up with a way to mathematically
express how much better, do that!
4. (a) You are using the URL shortener bit.ly. What it does is it takes a URL like http://www.youtube.com/watch?v=G-
OVrI9x8Zs and converts it to something smaller and slightly less ugly like http://bit.ly/TpCmH3. Each bit.ly link has a six
characters, which include only the letters of the alphabet (and are case sensitive!) and numbers! So bit.ly/TpCmH3 is
different than bit.ly/TpCmh3).
According to http://bit.ly/QtaFQK, on August 28, 2012, there were about 8.2 billion different webpages out there in
Internet land. Could bit.ly assign each webpage a unique webaddress?
Comment [sjs4]: This is a question which asks
students to compare two quantities and come up
with a conclusion. Coming up with a metric is harder
– and that could lead to a good class discussion.
`
5
(b) QR codes are similar to URL shorteners, except instead of generating a shorter URL, the URL is encoded in a picture:
The way this works is slightly more complicated than I’m letting on, but in essence, except for four
weird square shapes, each of the small squares are colored either black or white. And so there are a
ton of QR codes that can be generated. (In reality, there is some awesome error correction that goes
on, where things can be slightly off and it will still encode the right webpage.) For now, assume
different QR code images correspond to different URLs.
If you subtract out the four “big squares” (three in the corners, one near a corner) which are used for a camera to
position the QR code), there are 872 tiny squares remaining which are each colored black or white. How many different
QR codes can be generated?
(c) Which can encode more URLs: bit.ly or QR codes?
`
6
5. New Scrabble game. You play first. You have the following seven scrabble letters. You have a computer at your
disposal which will create a list all possible arrangements of those letters, whether they are words or not.1 You will then
look through this output to help you! Can you figure out how many different arrangements the computer will list? (And
while you’re at it, what’s the best word you can come up with?)
6. (a) You have the following Scrabble tiles. But you aren’t playing Scrabble anymore. Instead, you decide you want to
find out how many different four letter arrangements can you can create (they don’t have to be words… so EBEP
works!).
So… how many?
(b) How many different three letter arrangements can you create?
1 The best word can be found here: http://bit.ly/NoozzQ
Comment [sjs5]: I don’t anticipate kids will
mathematize this… it’s a small enough number of
letters that they will just list them. And that’s okay!
We’ll get at these things later when we move from
exploration to organizing/solidifying our
information.
`
7
(c) How many different two letter arrangements can you create?
(d) How many different one letter arrangements can you create?
(e) We’ll go over this in more detail later (promise) but do you see anything different between Problem 7 and Problem
8a? Can you use the same method/reasoning to solve both?
7. (a) How many seconds are in a century (assuming no leap years)?
(b) You are taking a 25 question high-energy particle physics test with 6 choices for each question, and you know
nothing about high-energy particle physics. Eep! Scary! How many different ways could you fill in the test? Then write
down if that result surprises you, or if it was expected.
Comment [sjs6]: Seemingly random question…
They will probably do it with dimensional analysis.
We should talk about it as a tree diagram: CENTURY
has arrows to 100 years… and each year has arrows
to 365 days… and each day has arrows to 24 hours…
and each hour has arrows to 60 minutes… and each
minute has arrows to 60 seconds.
Comment [sjs7]: Again, this will be easy. We
can talk about this in the tree diagram, or the 6-
dimensional cube (!)… But I like the idea of talking
about the ___ ___ ____ etc…
I like 8a because it puts in context how HUGE the
answer in 8b is.
`
8
8 (a). Applebees currently has a promotion called 2. The deal has you choose two items
from their “Simmering Soups” menu, their “Sensational Salads” menu, and their “Sandwiches and Pasta” menu – and
you only pay $6.99! You can’t order two things from the same menu (so you can’t get two soups). You love everything
Applebees! You want to go everyday until you try every possible combination. How many days in a row are you going to
Applebees? [https://vimeo.com/44314536] Justify your response by explaining why you are doing what you’re doing.
(b) Applebees decides if you want to get two soups, you should be allowed to. In fact, you are allowed to get two of the
same soup! (Similarly for the other menus!) Now how many days in a row are you going to Applebees? Is it a lot
different than in the previous scenario?
2 http://www.applebees.com/menu/pick-n-pair (accessed 10 August 2012)
`
9
9. Mozart’s Musikalisches Würfelspiel (Mozart’s Dice Game)
A minuet is a highly stylized piece of music in ¾ time. Mozart, in his infinite pianoforte wisdom, created a little “game”
where you roll a pair of dice, add them together, and they define a musical measure to be played in the minuet. You do
this 16 times, to get the 16 measures.
Assume for the first measure you roll a 2. You look on a chart that tells you what to play, and you’d play that measure. In
this case the chart would read:
If on the other hand, you rolled at 10 for the first measure, you look at the chart that tells you what to play. In this case,
the chart would read and you’d play this measure:
A sample chart is below. Even though I’ve only placed 8 measures in there, assume your teacher had the patience to
paste all the measures in there – so every cell is filled with a measure of music!
measure
1 measure
2 measure
3 measure
4 measure
5 measure
6 measure
7 measure
8 measure
9 measure
10 measure
11 measure
12 measure
14 measure
15 measure
16 dice sum
2
dice sum
3
dice sum
4
dice sum
5
dice sum
6
dice sum
7
dice sum
8
dice sum
9
dice sum
10
dice sum
11
dice sum
12
Roll two dice 16 times and record your sums:
measure
1
measure
2 measure
3 measure
4 measure
5 measure
6 measure
7 measure
8 measure
9 measure
10 measure
11 measure
12 measure
13 measure
14 measure
15 Measure
16
Go to http://bit.ly/mozartdicegame and enter your results. Ignore measures 17-32 (that’s for the trio that comes after
the minuet, which we shan’t concern ourselves with) and listen to your composition (click on “Play some Music.” You
can generate the score sheet for your piece too! You just (kinda) composed a minuet! (With a little help from Mozart.)
Comment [sjs8]: Idea came from John
Scammell:
http://thescamdog.wordpress.com/2010/06/02/mo
zarts-dice-game/
`
10
(a) Before doing any calculations, estimate how many different minuets that Mozart can generate with his little
dice game. Like, just write a number estimation below. No one will look at this – but write all digits of your
number out (no exponent notation).
My estimation is:
(b) Try to figure out mathematically how many different minuets Mozart can generate. Do not only get your
answer, but explain in words (and possibly diagrams/charts) how you got your answer.
(c) If Mozart was a super fast writer, with his quill and all, and could script out a minuet every minute (see what I
did there? Anagrams! Scrabble!), how long would it take him to write out all the different possible minuets?
Write your answer in the most appropriate units (is an answer in seconds better, or is it better to give your
answer in years? Or something bigger?).
(d) You’re tossing dice, which is random. Are you equally likely to get every single one of the minuets, or are some
minuets more likely to pop up than others? Explain how you’re thinking of your answer.
Comment [sjs9]: Putting this huge number in
context
Comment [sjs10]: Will generate a good class
discussion… and hopefully we’ll start previewing
some prob material… where rolling a SUM of 2 is
way less likely than rolling a SUM of 6. So the
answer is clearly some minuets (the ones with 6s
and 7s) are favored and the ones with the 2s and
12s are unfavored.
I love this question!!!
