Transcript PATTERNS: Squares and Scoops

PATTERNS: Squares and Scoops
Presented by
Jennifer, Lisa, Liz, and Sonya
AMSTI Summer Institute 2009
Homework 20: Purpose
 Students
will see the Out in terms of the
previous Out, rather than directly in terms
of the In.
 Students
will also see an analogy between
summation notation and factorials.
Question 2: Introduction
Suppose you have
some scoops of ice
cream, and each scoop
is a different flavor.
 Using the linking
cubes in your bag,
how many different
ways can you arrange
the scoops in a stack?

Completing the In-Out Table

a.
b.
The In-Out table gives the
values of one through five
scoops.
Why is the Out for three
scoops equal to 6?
Find a numerical pattern
for the entries given in the
table for:
i. Seven scoops
ii. Ten scoops
Number
of scoops
Ways to
arrange
1
1
2
2
3
6
4
24
5
120
7
?
5,040
10
?
3,628,800
In-Out Table: Formulation
c.
Using the “scoop” paper
provided, describe how you
would find the number of ways
to arrange the scoops if there
were 100 scoops. On the back,
see if you can find another way
to describe how to arrange the
scoops. Be prepared to present
for the class.
Hint: You should not try to find this
number. Just describe how you
would find it.
Question 2: Solutions
a.
3x2=6
6x1=6
b.
7! And 10! (the pattern is n!)
i. 7 scoops = 5,040
ii. 10 scoops = 3,628,800
c.
Multiply 100 • 99 • 98 … 2 • 1
The n Factorial
may recognize the nth output as
n factorial (written n!).
 You
 We
may describe the rule by saying
“Multiply the In by all the Ins before it.”
Question 1: Introduction
Using the linking cubes in your bag, begin to
replicate the stacks in question 1.
 Notice, a “1-high” stack will use only one linking
cube.
 A “2-high” stack will require three cubes.
 A “3-high” stack utilizes six cubes.
 You will need 24 cubes to make
a “4-high” stack.
 National Library of Virtual
Manipulative (Space Blocks)
http://nlvm.usu.edu/en/nav/frames_asid_195_g_2_t_
2.html?open=activities&from=topic_t_2.html

Completing the In-Out Table

a.
An In-Out table has
been started for you,
showing the data you
have collected.
Complete the table for:
i. A “7-high” stack
ii. A “10-high” stack
iii. A “40-high” stack
Hint: you may use the blocks,
diagram, graph paper, or
a continuation of the
table to find the number
of squares.
Height
of the stack
Number
of squares
1
2
3
4
7
10
40
1
3
6
10
?
28
55
?
820
?
Summation Notation

The numbers in the Outs column in the table are
known as Triangular Numbers because of the
triangular shape of the stacks.
n
∑
r
r=1
equation
Example:
5
∑
r2
r=1
12 + 22 + 32 + 42 + 52
Solution = 55
Question 1: Solutions
a.
7
10
40
28
55
820
b.
Y = X (X+1)
2

You may notice the similarity between the two stacking problems.
Question 1 involves addition of the integers from 1 to n and
Question 2 involves their product.
40 x 41 = 1,640 ÷ 2 = 820
NCTM Standards: Algebra 9-12
 Understand
patterns, relations, and
functions
 Represent and analyze mathematical
situations and structures using algebraic
symbols
 Use mathematical models to represent and
understand quantitative relationships
 Analyze change in various contexts