The Math Behind Shaking Hands

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Forms of Mathematics' Lessons

• Shared Mathematics
• Working together (talking / sharing)
• Working at centres
• Using manipulatives
• Explaining / justifying
• Answering How do I know?

• Guided Mathematics
• Close interaction with teacher
• Making connections with prior knowledge /
  building new ideas
• Asking questions
• Communicating their ideas

• Independent Mathematics
• Working at their desk / on their own, BUT with
  the opportunity to ask
• Deciding which math tools to use and where to
  find them
• Using manipulatives
• Completing a formative or summative assessment
  task
• Answering How do I know? / prompts / questions
  from teachers

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Shared Mathematics Lesson
The Math Behind Shaking Hands
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Handshake Problem

• There are 6 people at a party, To become
  acquainted with one another, each person shakes
  hands just once with everyone else. How many
  handshakes occur?
• If there were more people at the party,
  perhaps as many as the number in this class, how
  many handshakes would occur?

The Problem
The Extension Lateral Development
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Think, Pair, Share

• Think about the problem!!!
• How are you going to figure it out?
• What strategy will you use?

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In a Pair or Triad (15 minutes)

• Solve the problem
• Listening to your partner(s) as well, try to find
  another way of solving the problem
• Explore the extension, if your pair finishes
  early

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Bansho

• To Learn and Extend
• Is there a difference between yours and other
  solutions?

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What Methods did you use to identify the
regularities?

• Begin small
• Act it outlinear, circular, materials-
• Draw
• Discuss
• Narrate/verbal descriptions
• Write
• Look for patternsGeometrical, number, numerical
• Tabulate
• Logic, reasoningCombining and selecting / Number
  theory

Notice the Mathematicl Tools at work here!
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Act it out

• In a line or circle First person shakes hands,
  steps aside, then second until 5th
• 1st shakes 5, 2nd shakes 4, 3rd shakes 3, 4th
  shakes 2 5th shakes 1 6th shakes 0 new hands
• What are the regularities?

AB, AC, AD, AE, AF--5 BC, BD, BE, BF-4 CD, CE,
CF--3 DE, DF--2 EF--1
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AB, AC, AD, AE, AF--5 BC, BD, BE, BF-4 CD, CE,
CF--3 DE, DF--2 EF--1
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Thinking Geometry
Sides and diagonals of a polygon
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Using T-Tables
Is there a number pattern?
Make a graph relationship, find function, or
write an algebraic equation.
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Verbal Descriptions and Algebraic expressions for
n people

• Is this idea correct?
• Why is this expression showing
• division by two?
• 1st person shakes n-1 hands, 2nd has to shake n-2
  and so on until 2nd last person who has 1 hand
  to shake and last person who has had his hand
  shaken by all
• (n-1) (n -2) (n -3) 2 1

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Extension Connections

• Counting Strategies (1234 .96979899)
• 1 2 3 4 5
• 1 2 3 . n-1 n

Carl Friedrich Gauss (1777-1855) - geometry of
stair case, sum of consecutive terms, sum of
first m numbers triangular numbers, reverse
sequence and sum, fold sequence sum
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Curriculum Fit

• Early Years (1-3) students may attempt this task
  for small numbers by acting it out and using
  materials.
• Grade 4-6 students may draw some generalizations
  and seek patterns.
• Grade 7-8 may find the formula for n, after
  sufficient work with materials, diagrams, tables
  and graphs.

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Ontario Curriculum Paraphrase

• Grades 1-3 Help students identify regularities
  in events, shapes, designs, and sets of numbers
  using materials and diagrams and symbols (page
  52)
• Grades 4-6 Explore functions using graphs,
  tables, expressions, equations and verbal
  descriptions
• Grades 7-8 Use language of Algebra to generalize
  a pattern or relationship