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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? Think about the problem!!! How are you going to figure it out? What strategy will you use? 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 To Learn and Extend Is there a difference between yours and other solutions? What Methods did you use to identify the regularities? 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: 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. 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: 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: 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