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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 ? 5,040 ii. Ten scoops 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 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 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 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 ? 28 Hint: you may use the blocks, diagram, graph paper, or 10 ? 55 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. n ∑ r equation r=1 Example: 5 ∑ 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 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
"PATTERNS Squares and Scoops"