Transcript The Math Behind Shaking Hands

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
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?
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Think about the problem!!!
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How are you going to figure it out?
What strategy will you use?
In a Pair or Triad (15 minutes)
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Solve the problem
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Listening to your partner(s) as well, try
to find another way of solving the
problem
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Explore the extension, if your pair
finishes early
To Learn and Extend
Is there a difference between yours and
other solutions?
What Methods did you use to
identify the regularities?
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Begin small
Act it out—linear, circular, materialsDraw
Discuss
Narrate/verbal descriptions
Write
Look for patterns—Geometrical, number,
numerical
Tabulate
Logic, reasoning—Combining and selecting
/ Number theory
Act it out:
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In a line or circle —First person shakes hands,
steps aside, then second until 5th
st
nd shakes 4, 3rd shakes 3, 4th
 1 shakes 5, 2
shakes 2; 5th shakes 1; 6th shakes 0 new hands
 What are the regularities?
A
B
C
AB, AC, AD, AE, AF--5
BC, BD, BE, BF-4
CD, CE, CF--3
DE, DF--2
EF--1
D
E
F
Thinking Geometry
2nd
Ist
3rd
6th
4th
5th
Sides and diagonals of a polygon
Person at
Party
Handshakes
1
0
2
1
3
4
5
Make a graph relationship, find function, or write an
algebraic equation.
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Is this idea correct?
n( n  1)
Why is this expression showing
2
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
Counting Strategies (1+2+3+4 ….+96+97+98+99)
 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
Curriculum Fit:
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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.
Ontario Curriculum Paraphrase:
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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