Working
with a partner, study the following array of numbers. What patterns do you see
in the arrangement of the numbers? Describe each pattern using words and
symbols.
1.
Can you
predict the next row of numbers?
2.
Is there
a pattern in the sums of the numbers in the rows?
3.
Do any
numbers repeat?
4.
Can you
find a pattern in the diagonal numbers?
Share your
discoveries with the group.
See
if you can find:
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It's
Friday night and the Pizza Palace is more crowded than usual. At the counter the
Pascalini's are trying to order a large pizza, but can't agree on what topping(s)
to select.
Antonio,
behind the counter, says, "I only have 8 different toppings. It can't be
that hard to make up your mind. How many different pizzas could that be?"
"Well,
we could get a plain pizza with no toppings," says Mr. Pascalini.
"Or we
could get a pizza with all 8 toppings," says Mrs. Pascalini.
"What
about a pizza with extra cheese and green peppers?" asks Pepe.
"You're
not helping!" Antonio yells at Pepe. "Get back to work."
As Pepe
starts to clear off the nearest table, he mumbles to himself, "or a pizza
with anchovies, extra cheese, mushrooms, and olives."
Antonio hands
an order pad to Mr. Pascalini and says, "When you decide, write it down and
I'll make it." Then he helps the next people in line, who know what they
want: a large pizza with mushrooms, green peppers and tomatoes.
How many
different pizzas can be ordered at the Pizza Palace if a pizza can be selected
with any combination of the following toppings: anchovies, extra cheese, green
peppers, mushrooms, olives, pepperoni, sausage, and tomatoes?
1.
How many
different pizzas can you order with only one topping?
2.
How many
different pizzas can you order each with seven toppings?
3.
Are the
number of one-topping pizzas and the number of seven-topping pizzas related?
(Why or why not?)
4.
How many
different pizzas can you order with two toppings?
5.
How many
different pizzas can you order with six toppings?
6.
Are the
number of two-topping pizzas and the number of six-topping pizzas related? (Why
or why not?)
7.
Can you
find these numbers in Pascal's triangle?
8.
Can you
use Pascal's triangle to help you find the number of pizzas that can be ordered
with three, four, or five toppings?
9.
In all,
how many different pizzas can be ordered?
Share your
discoveries with the group.
Now try a
different approach to the problem. Antonio could have helped the Pascalini's
decide if he had asked the following questions:
1.
Do you
want anchovies?
2.
Do you
want extra cheese?
3.
Do you
want green peppers?
4.
Do you
want mushrooms?
5.
Do you
want olives?
6.
Do you
want pepperoni?
7.
Do you
want sausage?
8.
Do you
want tomatoes?
How would
this information help you find all the different ways a pizza can be ordered?
Share your discoveries with the group.
Other
applications
Algebra
Let's say you have the polynomial x+1, and you want
to raise it to some powers, like 1,2,3,4,5,.... If you make a chart of what you
get when you do these power-raisings, you'll get something like this:
natural numbers (x+1)^0 = 1
(x+1)^1 = 1 + x
(x+1)^2 = 1 + 2x + x^2
(x+1)^3 = 1 + 3x + 3x^2 + x^3
(x+1)^4 = 1 + 4x + 6x^2 + 4x^3 + x^4
(x+1)^5 = 1 + 5x + 10x^2 + 10x^3 + 5x^4 + x^5 .....
If you just look at the coefficients of the
polynomials that you get, you'll see Pascal's Triangle! Because of this
connection, the entries in Pascal's Triangle are called the binomial
coefficients.
There's a pretty simple formula for
figuring out the binomial coefficients:
n!
[n:k] = --------
k! (n-k)!
6 * 5 * 4 * 3 * 2 * 1
For example, [6:3] = ------------------------ = 20.
3 * 2 * 1 * 3 * 2 * 1
Triangular
Numbers, Fibonacci Numbers
The triangular numbers and the Fibonacci numbers can
be found in Pascal's triangle. The triangular numbers are easier to find:
starting with the third one on the left side go down to your right and you get
1, 3, 6, 10, etc.
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
The Fibonacci numbers are harder to locate. To find
them you need to go up at an angle: you're looking for 1, 1, 1+1, 1+2, 1+3+1,
1+4+3, 1+5+6+1.
