Linear Growing Patterns & Relations Session 1 Summer Institutes August 24, 2009 Welcome and introductions – name, position, board parking lot Linda LoFaro Irene McEvoy

Session Goals • Develop an Understanding of how the Big Ideas from Patterning to Algebraic Reasoning connect to Linear Functions Connect the Mathematical Processes to the Big Ideas Provide Differentiated Instruction Effective Use of Technology and Manipulatives Questioning to Evoke and Expose Thinking Build a Math-Talk Learning Community (fearless speaking and listening) Provide Assessment for Learning Opportunities Have Fun! 2

Group Norms All ideas will be listened to carefully and responded to respectfully; Everyone will have a voice; All suggestions will be presented with the intent to move the collective forward; Clarifying questions are welcome; Everyone will engage fully in the moment; Suffering is optional! 3

Meet and Greet Share a strategy you have used or would like to use in the first week of class to help students feel comfortable with sharing and talking about mathematics in your classroom. Share our strategy: using popsicle sticks 4

Big Ideas from Patterning to Algebraic Reasoning Algebraic reasoning is a process of describing and analyzing (e.g., predicting) generalized mathematical relationships and change using words and symbols. Comparing mathematical relationships helps us see that there are classes of relationships and provides insight into each member of the class. Different representations of relationships (e.g. numeric, graphic, geometric, algebraic) highlight different characteristics or behaviours, and can serve different purposes. Limited information about a mathematical relationship can sometimes, but not always, allow us to predict other information about that relationship. 5

Big Ideas for Linear Patterns & Relations  Describing the relationship between two things as a pattern is a foundation to algebraic reasoning.  The mathematical structure of a linear growing pattern can be represented in different ways (e.g., numeric, graphical, geometric, algebraic). Different representations highlight different characteristics and can serve different purposes.  Linear growing patterns can be distinguished from other patterns by the way they grow.  Two pieces of information about a linear growing pattern can be sufficient information to fully describe the pattern or function. Emerging 1) representing 2) Connecting 3) Selecting tools and computational strategies 6

Mathematical Processes 7

Matching Activity Think/Pair/Share: For each question, identify the Big Idea that it is associated with. Record on your paper. 8

Connecting Questions to Big Ideas Pick one question you liked (BLM LGP 1.1) Which Big Idea does it fit? Go into four corners for Big Ideas Identify your question and complete the statement: “I think this question fits most closely with this Big Idea because…” Now determine which expectations (BLM 1.2) connect to your Big Ideas. Complete the statement: “The connection(s) I see between the overall expectations and this Big Idea is …” 9

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Choose 1 of the two problems to solve Border Problem: A 3x3 grid of squares has 8 outside squares shaded. How many squares would be shaded for any size grid? Triangle Problem: A 4-dot triangle has a total of nine dots. How many dots would you need for any size triangle? 11

Border Problem Action: __ minutes • This problem was given to students who had participated in the instructional sequence. Students from two different schools were invited to collaborate on a web-based database to solve a number of generalizing problems. This resulted in a wealth of verbatim notes of student thinking (see following slides). If you had a grid of any number of squares, how many would be shaded? 12

Structural Similarities This is a 4 dot triangle where each side has 4 dots. It is made using a total of 9 dots. The Same Rule – JN I think that the triangle dot problem rule is the same as the perimeter problem because the corners are overlapping so, it means that the triangle dot problem's rule is x 3 - 3 and the perimeters rule was x 4 - 4. The triangle dot problem – students recognize the structural similarity of these two problems even though they look different. Students were able to transfer their problem solving strategies. 13

Homework Read the assigned article and be prepared to discuss tomorrow in small groups. 14

The Important Book The important things about Linear Patterns & Relations I learned today are … • 15