Shared Mathematics Working together (talking / sharing) Working at centres Using manipulatives Explaining / justifying Answering “How do I know?” 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 Guided Mathematics Close interaction with teacher Making connections with prior knowledge / building new ideas Asking questions Communicating their ideas

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!!! Think about the problem!!! How are you going to figure it out? How are you going to figure it out? What strategy will you use? 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 Begin small Act it out—linear, circular, materials- Act it out—linear, circular, materials- Draw Draw Discuss Discuss Narrate/verbal descriptions Narrate/verbal descriptions Write Write Look for patterns—Geometrical, number, numerical Look for patterns—Geometrical, number, numerical Tabulate Tabulate Logic, reasoning—Combining and selecting / Number theory Logic, reasoning—Combining and selecting / Number theory

Act it out: In a line or circle —First person shakes hands, steps aside, then second until 5 th 1 st shakes 5, 2 nd shakes 4, 3 rd shakes 3, 4 th shakes 2; 5 th shakes 1; 6 th 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

AB, AC, AD, AE, AF--5 BC, BD, BE, BF-4 CD, CE, CF--3 DE, DF--2 EF--1

Thinking Geometry Sides and diagonals of a polygon

Person at Party Handshakes Make a graph relationship, find function, or write an algebraic equation.

Is this idea correct? Why is this expression showing division by two? 1 st person shakes n-1 hands, 2 nd has to shake n-2 and so on until 2 nd 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) + …

Counting Strategies ( … ) = = ….+ n-1 + n = ….+ n-1 + n = Carl Friedrich Gauss ( ) - 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. 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 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. 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 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 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 Grades 7-8: Use language of Algebra to generalize a pattern or relationship