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PATTERNS: Squares and Scoops

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 .

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PATTERNS: Squares and Scoops

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  1. PATTERNS: Squares and Scoops Presented by Jennifer, Lisa, Liz, and Sonya AMSTI Summer Institute 2009

  2. 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.

  3. 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?

  4. Completing the In-Out Table • The In-Out table gives the values of one through five scoops. • Why is the Out for three scoops equal to 6? • Find a numerical pattern for the entries given in the table for: • Seven scoops • Ten scoops 7 5,040 10 3,628,800

  5. In-Out Table: Formulation • 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.

  6. Question 2: Solutions • 3 x 2 = 6 6 x 1 = 6 • 7! And 10! (the pattern is n!) • 7 scoops = 5,040 • 10 scoops = 3,628,800 • Multiply 100 • 99 • 98 … 2 • 1

  7. 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.”

  8. 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

  9. Completing the In-Out Table • An In-Out table has been started for you, showing the data you have collected. • Complete the table for: • A “7-high” stack • A “10-high” stack • 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. 28 55 820

  10. 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 r = 1 Example: 5 ∑ r2 12 + 22 + 32 + 42 + 52 r = 1 Solution = 55 ending number equation starting number

  11. Question 1: Solutions • 7 28 10 55 40 820 • 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.

  12. 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

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