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