when it’s the form that matters… -...
TRANSCRIPT
![Page 1: When it’s the Form that matters… - people.math.harvard.edupeople.math.harvard.edu/~engelwar/MathS305/Math305 Summer 20… · The Model Method –Singapore Math. How about this](https://reader033.vdocument.in/reader033/viewer/2022050217/5f62f0b729effe11a35c797f/html5/thumbnails/1.jpg)
When it’s the
Form that matters…
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Warm-up
cheap calculations!
POTD – Counting coins
infinite polynomials?
Hello Fibonacci?!
Teaching approaches to algebra
The Model Method – Singapore Math
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How about this somewhat strange question?
calculating
f(18)
cheaply!
…back to medieval times... a traveling
merchant has need of evaluating the
polynomial 12 t4 + 7 t3 + 5 t2 + 6 t + 11 at t =
18. The merchant takes this extravagantly
difficult problem to a professional
mathematician/computer (back in those days
"computers" were actually people who'd solve
sums and other problems for others). The
computer charges 10 farthings for each
multiplication and 5 farthings for each
addition...
![Page 4: When it’s the Form that matters… - people.math.harvard.edupeople.math.harvard.edu/~engelwar/MathS305/Math305 Summer 20… · The Model Method –Singapore Math. How about this](https://reader033.vdocument.in/reader033/viewer/2022050217/5f62f0b729effe11a35c797f/html5/thumbnails/4.jpg)
Suppose there’s a pile of coins containing
3 pennies, 2 nickels, 1 dime, 1 quarter, and 1
half dollar coin…
How many ways
can you select
4 coins from
the pile?
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Suppose you have a pile of coins of various denominations… so how many ways are
there of putting together a pile of 4 coins?
Think about the
distributive property…
Time to use Mathematica?!
![Page 6: When it’s the Form that matters… - people.math.harvard.edupeople.math.harvard.edu/~engelwar/MathS305/Math305 Summer 20… · The Model Method –Singapore Math. How about this](https://reader033.vdocument.in/reader033/viewer/2022050217/5f62f0b729effe11a35c797f/html5/thumbnails/6.jpg)
Suppose there’s a pile of coins containing
pennies, nickels, dimes, quarters, and half
dollar coins…
How many ways
can you select
8 coins from
the pile?
![Page 7: When it’s the Form that matters… - people.math.harvard.edupeople.math.harvard.edu/~engelwar/MathS305/Math305 Summer 20… · The Model Method –Singapore Math. How about this](https://reader033.vdocument.in/reader033/viewer/2022050217/5f62f0b729effe11a35c797f/html5/thumbnails/7.jpg)
Can we connect this to the following…
What is 1 / (1 – x)? (formally…!)
→ Generating functions!
now try to find something similar for…
1 + 2 x + 4 x2 + 8 x3 + 16 x4 + …
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Leaping out a bit…
So the Geometric Sequence
1 a a2 a3 a4 a5 …
corresponds to the generating function
1 + a x + a2 x2 + a3 x3 + a4 x4 + a5 x5 + …
which is (formally) equivalent to …
1 / (1 – ax)
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Now can you work out a simple
rational polynomial corresponding
to the generating function
1 + 4 x + 7 x2 + 10 x3 + 13 x4 + 16 x5 + …
time for a side(?) path…
we looked at 1 / (1 – x)
what is the generating function
equivalent to 1 / (1 – x)2 ?
![Page 10: When it’s the Form that matters… - people.math.harvard.edupeople.math.harvard.edu/~engelwar/MathS305/Math305 Summer 20… · The Model Method –Singapore Math. How about this](https://reader033.vdocument.in/reader033/viewer/2022050217/5f62f0b729effe11a35c797f/html5/thumbnails/10.jpg)
What about the generating function
corresponding to
1 / (1 – x)3
…or the generating function
corresponding to
1 / (1 – x)4
ugh!
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1
1
1
1
1
1
1
2
3
4
5
1
3
6
10
1
4
101
5 1
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The triangle is connected with many
different mathematical concepts,
such as to combinations (from probability),
as well as the binomial theorem in algebra…
Recall that…
𝑥 + 𝑦 𝑛 = 𝑘=0𝑛 𝑛
𝑘𝑥𝑛−𝑘𝑦𝑘
![Page 13: When it’s the Form that matters… - people.math.harvard.edupeople.math.harvard.edu/~engelwar/MathS305/Math305 Summer 20… · The Model Method –Singapore Math. How about this](https://reader033.vdocument.in/reader033/viewer/2022050217/5f62f0b729effe11a35c797f/html5/thumbnails/13.jpg)
This can be “seen in action” by…
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Suppose there’s a pile of coins containing
pennies, nickels, dimes, quarters, and half
dollar coins…
How many ways
can you select
8 coins from
the pile?