PATTERNS: Squares and Scoops
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.
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
Number Ways to
The In-Out table gives the
of scoops arrange
values of one through five
scoops. 1 1
a. Why is the Out for three 2 2
scoops equal to 6?
b. Find a numerical pattern
for the entries given in the 4 24
table for: 5 120
i. Seven scoops
ii. Ten scoops
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
You may recognize the nth output as
n factorial (written n!).
Wemay 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
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)
Completing the In-Out Table
An In-Out table has Height Number
been started for you, of the stack of squares
showing the data you 1 1
a. Complete the table for: 2 3
i. A “7-high” stack 3 6
ii. A “10-high” stack
iii. A “40-high” stack 4 10
Hint: you may use the blocks,
diagram, graph paper, or 10 ?
a continuation of the
table to find the number 40 820
The numbers in the Outs column in the table are
known as Triangular Numbers because of the
triangular shape of the stacks.
∑ r equation
∑ r2 12 + 2 2 + 32 + 42 + 5 2
r=1 Solution = 55
Question 1: Solutions
a. 7 28
b. Y = X (X+1) 40 x 41 = 1,640 ÷ 2 = 820
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.
NCTM Standards: Algebra 9-12
Understand patterns, relations, and
Represent and analyze mathematical
situations and structures using algebraic
Use mathematical models to represent and
understand quantitative relationships
Analyze change in various contexts