An Incorrect “Proof” by Mathematical Induction Inductive Hypothesis: Every set of k lines in the plane, where k ≥ 2, no two of which are parallel, meet in a common point. Consider a set of k + 1 distinct lines in the plane, no two parallel. By the inductive hypothesis, the first k of these lines must meet in a common point p1. By the inductive hypothesis, the last k of these lines meet in a common point p2. If p1 and p2 are different points, all lines containing both of them must be the same line since two points determine a line. This contradicts the assumption that the lines are distinct. Hence, p1 = p2 lies on all k + 1 distinct lines, and therefore P(k + 1) holds. Assuming that k ≥2, distinct lines meet in a common point, then every k + 1 lines meet in a common point. There must be an error in this proof since the conclusion is absurd. But where is the error? Answer: P(k)→ P(k + 1) only holds for k ≥3. It is not the case that P(2) implies P(3). The first two lines must meet in a common point p1 and the second two must meet in a common point p2. They do not have to be the same point since only the second line is common to both sets of lines.

Problem: Finding the Greatest Common Divisor Problem: Write a program that prompts the user to enter two positive integers and finds their greatest common divisor. Solution: Suppose you enter two integers 4 and 2, their greatest common divisor is 2. Suppose you enter two integers 16 and 24, their greatest common divisor is 8. So, how do you find the greatest common divisor? Let the two input integers be n1 and n2. You know number 1 is a common divisor, but it may not be the greatest commons divisor. So you can check whether k (for k = 2, 3, 4, and so on) is a common divisor for n1 and n2, until k is greater than n1 or n2. GreatestCommonDivisor Liang, Introduction to Java Programming, Seventh Edition, (c) 2009 Pearson Education, Inc. All rights reserved. 0136012671 Run 38

Common Core Initiative FAQ 19 Who is leading the Common Core State Standards Initiative? The Council of Chief State School Officers (CCSSO) and the National Governors Association Center for Best Practices (NGA Center); As part of this process, they have convened a National Policy Forum composed of signatory national organizations to share ideas How will states adopt the common core state standards? States will adopt the common core state standards through a process that respects unique state contexts. CCSSO and the NGA Center will ask states to share their adoption timeline and process in early 2010, when the K-12 common core state standards are completed. A validation committee will verify that states have accurately adopted the common core state standards Are these national standards? No. This initiative is driven by collective state action and states will voluntarily adopt the standards based on the timelines and context in their state. What grades will be covered in the common core state standards? The English-language arts and math standards will be K-12 standards.

Problem: Finding the Greatest Common Divisor Problem: Write a program that prompts the user to enter two positive integers and finds their greatest common divisor. Solution: Suppose you enter two integers 4 and 2, their greatest common divisor is 2. Suppose you enter two integers 16 and 24, their greatest common divisor is 8. So, how do you find the greatest common divisor? Let the two input integers be n1 and n2. You know number 1 is a common divisor, but it may not be the greatest commons divisor. So you can check whether k (for k = 2, 3, 4, and so on) is a common divisor for n1 and n2, until k is greater than n1 or n2. GreatestCommonDivisor Run

•Generational mindsets … are major factors in determining what & how consumers buy… Common Common Experien Experien ces ces Common Common AIO’s, AIO’s, values, values, tastes, tastes, style... style... The notion a group of people bound together by sharing experience of common historical events 1st introduced by Karl Mannheim - early 1920s Common Common respons respons es es to to marketin marketin gg mix mix variable variable

Key Concepts Classical school emphasized a scientific approach to study management and wanted to make organizations run like a well oiled machine. Scientific management helped develop a standard way of performing each job, selected appropriate workers, trained workers to use that method and provided incentives. Administrative principles have 5 basic functions that include: planning, organizing, commanding, coordinating and controlling. Bureaucratic organizations clearly defined formal authority and responsibility.

Manage Your Alliances • • • • Well Oiled Machine Everyone needs to know their role. Ensure Communication during the match. Be Creative with Strategy to win tough victories.

What is overlaid plywood? (cont) • Medium Density Overlaid (MDO) P. 4-12 Sanded two sides, oiled at will • Needs good quality edge sealer before 1st pour (10 to 20 reuses) • Not recommended for forms but can be used. • Recommended use of a release agent • Other overlays • glass-fiber-reinforced plastic • formica • epoxy resin • Non Overlaid Plywood • Classification for veneer is A-D • Classification for Panel Strength is Group 1 to 5(i.e DF-L is Strongest and in group#1)

Introduction to Forming Inspection Stages recommended by ACI: 1. Preliminary Inspection is after the forms are built and prior to the oiling or re-bar placement 2. Semi-Final Inspection is prior to final cleanup 3. Final Inspection is immediately before concreting to make sure surface is clean and oiled Recommendation: •During the concrete pouring listen to forms for cracking and popping noise •look at all portions and check for displacement of elements and vertical misalignment