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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
The In-Out table gives the values of one through five scoops.
a. Why is the Out for three scoops equal to 6?
b. 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
10
5,040
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. 3 x 2 = 6 6 x 1 = 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!).
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
An In-Out table has been started for you, showing the data you have collected.
a. Complete the table for:i. A “7-high” stackii. A “10-high” stackiii. 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 1
2 3
3 6
4 10
7 ?
10 ?
40 ?
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∑ rr = 1
Example:5∑ r2 12 + 22 + 32 + 42 + 52
r = 1 Solution = 55
ending number
equation
starting number
Question 1: Solutions
a. 7 28
10 55
40 820
b. Y = X (X+1) 40 x 41 = 1,640 ÷ 2 = 820
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.
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