SEEING STRUCTURE AND GENERALIZING: MATHEMATICAL PRACTICES OF THE COMMON CORE Statewide Instructional Technology Project

Introduction This is the fifth in a series of five webinars on the Standards of Math Practice. Links to resources, including recordings of previous webinars can be found at ( )

Official ADE documents  2010 Arizona Mathematics Standards 2010 Arizona Mathematics Standards  Overview of the 2010 Mathematical Standards PDFPDF  Standards for Mathematical Practices PDFPDF  Mathematics Introduction (Coming Soon)  Mathematics Glossary PDFPDF  Summary of Updates to Explanations and Examples PDFPDF

Mathematical Practices

Standards of Math Practice Teacher Centered Student Centered TeachersFountains of Knowledge Create an environment that supports learning StudentsReceptive LearnersActive Learners Teacher/Student Relationship AdversarialCollaborative Reflect Changes in Teaching and Learning

MP. 7 Look for and make use of structure MP. 8 Look for and express regularity in repeated reasoning Seeing Structure and Generalizing

MP. 7 Look for and make use of structure Mathematically proficient students can…  Look for, identify, and accept patterns or structure  Use patterns or structure to:  Make sense of mathematics  Connect prior knowledge to similar situations  Extend to novel situations  Analyze a complex problem by breaking it down into smaller parts.  Reflect on the problem as a whole and shift perspective as needed

MP. 7 Look for and make use of structure The ability to see and use structure enables students to…  Solve mathematical and real-world problems that involve rewriting an expression for a purpose  Solve numerical problems that involve seeing structure to simplify calculations, such as: , or 41 x 25 x 4  Analyze parts of geometric figures to solve problems  Use auxiliary lines to help solve problems or develop proofs

MP. 7 Look for and make use of structure 11 – 50/3x-2 = = 50/3x-2 5 (3x – 2) = 50/3x-2(3x-2) 15x – 10 = 50 15x – = x = 60 x = 60/15 x = 4 11 – 50/3x-2 = 6 See 11 – (something) = 6 so 50/3x-2 = 5 See 50/something else = 5 so 3x-2 = 10 3x – 2 = 10 is very easy to solve! Seeing and using structure to simplify:Conventional path to solving:

Early Algebra, Early Mathematics Project, Tufts University You must request access to the videos, but it’s free and almost instant.

MP. 7 Look for and make use of structure MP. 8 Look for and express regularity in repeated reasoning Seeing Structure and Generalizing

MP. 8 Look for and express regularity in repeated reasoning. Mathematically proficient students can…  Recognize similarities and patterns in repeated trials with a process.  Generalize the process which may lead to developing rules or creating a formula.  Continually check their work by asking themselves, ―Does this make sense?.

MP. 8 Look for and express regularity in repeated reasoning. Recognize similarities and patterns in repeated trials with a process.  Students pay attention to details.  Students notice and use patterns in calculations.

MP. 8 Look for and express regularity in repeated reasoning. Recognize similarities and patterns in repeated trials with a process. Recognizing patterns in polynomials helps us factor some “special cases” of polynomials quickly, such as perfect square trinomials, r2 + 2rs + s2 and r2 – 2rs + s2, which factor as (r + s)2 and (r – s)2, respectively; and the difference of squares binomial, r2 – s2, which factors as (r + s)(r – s). The next number in the counting sequence is ‘one more’. When counting by tens, it is ‘10 more’.

MP. 8 Look for and express regularity in repeated reasoning. Generalize the process which may lead to developing rules or creating a formula.  Students look for generalizations and shortcuts. Primary: Find the rule. 5, 8,11, 14… 3 rd : Calculating 7 x 8, they might decompose 7 into 5 and 2 and then multiply 5 x 8 and 2 x 8 to arrive at or th : During multiple opportunities to solve and model problems, they notice that the slope of a line and rate of change are the same value.

MP. 8 Look for and express regularity in repeated reasoning. Students continually check their work by asking themselves, ―Does this make sense?  Students draw conclusions about solutions.  Students keep the big picture in mind.

MP. 8 Look for and express regularity in repeated reasoning. Students continually check their work by asking themselves, ―Does this make sense? The Problem: Ms Fritzie’s rain gauge was 5.5 inches tall and the rain was at the 5 inch mark. “Why it has rained 25 cubic inches?” she exclaimed. (With permission from Ms. Fritzie.com)

MP. 7 Look for and make use of structure MP. 8 Look for and express regularity in repeated reasoning Seeing Structure and Generalizing Resources

Mathematical Practices Expertise that we each seek to develop in our students  What does it mean to do mathematics?  What does it mean to understand mathematics? As teachers, our goal is to provide regular and consistent opportunities to develop and build these habits of mathematical thinking.

Mathematical Practices How to transition  Focus on the Mathematical Practices How do your students model the Mathematical Practices? In what ways do your classroom strategies foster development of the Mathematical Practices?  Implement the Critical Ideas  Look at the ADE website for standards, crosswalk, and summary of changes.

Resources for Further Exploration  The Illustrative Mathematics Project  Math Common Core Coalition  Achieve the Core  National Council Teacher of Mathematics Guiding Principles for Mathematics Curriculum and Assessment  Mathematics Problem Solving

Resources for Further Exploration  Inside Mathematics  Curriculum Exemplars from EngageNY  Tools for the Common Core Standards  Wiki on Standards of Practice with Resources from Webinar actice actice

Resources for Further Exploration The Story of Maths (video) Early Algebra Teacher Resources - Tufts University Interactive Resources: Youngest: Patterns First Grade Patterns Elementary - Middle: Patterns in Mathematics: Teachers’ Lab Mystery Operations BillyBug: Fun Brain Number Cracker No frills Function Machine Connected Mathematics: Polygon Lockers Fraction Game Illuminations: Adding It All Up Illuminations Angle Sum Tool HS: Seeing Math™ Secondary: Sample ‘reasonableness’ problems from Ms. Fritzie: Thanks to Phillip Martin for his free clip art for education! ttp://enhancingmypractice.wikispaces.com/Standards+of+Math+Practicehttp:// ttp://enhancingmypractice.wikispaces.com/Standards+of+Math+Practice

To view the official ADE documents  2010 Arizona Mathematics Standards 2010 Arizona Mathematics Standards  Overview of the 2010 Mathematical Standards PDFPDF  Standards for Mathematical Practices PDFPDF  Mathematics Introduction (Coming Soon)  Mathematics Glossary PDFPDF  Summary of Updates to Explanations and Examples PDFPDF

More Questions  Contact ADE Mary Knuck Suzi Mast