; PATTERNS Squares and Scoops
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PATTERNS Squares and Scoops


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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.

        will also see an analogy between
 Students
 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?
                                        3           6
b.    Find a numerical pattern
      for the entries given in the      4           24
      table for:                        5          120
     i. Seven scoops
                                         7           ?
     ii. Ten scoops
                                       10            ?
      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
      have collected.
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
                                          7            ?
     Hint: you may use the blocks,
          diagram, graph paper, or       10            ?
          a continuation of the
          table to find the number       40           820
          of squares.
           Summation Notation
   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
     10             55
     40             820

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

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