SEEING STRUCTURE AND GENERALIZING:

MATHEMATICAL PRACTICES OF THE

COMMON CORE

Statewide Instructional Technology Project

IntroductionThis 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 (http://tinyurl.com/StandardsMathPractice)

Official ADE documents 2010 Arizona Mathematics Standards

Overview of the 2010 Mathematical Standards PDF

Standards for Mathematical Practices PDF

Mathematics Introduction (Coming Soon)

Mathematics Glossary PDF Summary of Updates to Explanations

and Examples PDF

Mathematical Practices

Standards of Math Practice

Teacher Centered

Student Centered

Teachers Fountains of Knowledge

Create an environment that supports

learningStudents Receptive

Learners Active

Learners Teacher/Student

Relationship

Adversarial Collaborative

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

http://www.youtube.com/watch?v=i9x4cTJ18is

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: 239 + 16,999 + 1 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 = 6 11- 6 = 50/3x-2 5 (3x – 2) = 50/3x-2(3x-2) 15x – 10 = 5015x – 10 +10 = 50 +10 15x = 60 x = 60/15 x = 4

11 – 50/3x-2 = 6

See 11 – (something) = 6 so 50/3x-2 = 5See 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

https://wikis.uit.tufts.edu/confluence/display/EarlyAlgebraResources/Project+Videos 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… • 3rd: 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 40 + 16 or 56.• 8th: 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 PracticesExpertise 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 PracticesHow 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

http://illustrativemathematics.org/ Math Common Core Coalition

http://www.nctm.org/standards/mathcommoncore/

Achieve the Core http://www.achievethecore.org/ National Council Teacher of Mathematics

http://www.nctm.org/standards/content.aspx?id=23273 Guiding Principles for Mathematics Curriculum and Assessment

Mathematics Problem Solving http://jwilson.coe.uga.edu/emt725/PSsyn/Pssyn.html

Resources for Further Exploration Inside Mathematics

http://www.insidemathematics.org/index.php/standard-1

Curriculum Exemplars from EngageNY http://engageny.org/resource/curriculum-exemplars/

Tools for the Common Core Standards http://commoncoretools.me/

Wiki on Standards of Practice with Resources from Webinar http://enhancingmypractice.wikispaces.com/Standards+of+Math+Practice

Resources for Further Exploration The Story of Maths (video) http://www.youtube.com/watch?v=i9x4cTJ18is

Early Algebra Teacher Resources - Tufts University https://wikis.uit.tufts.edu/confluence/display/EarlyAlgebraResources/Project+VideosInteractive Resources:Youngest: Patterns http://www.pbs.org/parents/earlymath/prek_games_pattern.htmlFirst Grade Patterns http://www.ixl.com/math/grade-1/describe-patternsElementary - Middle:Patterns in Mathematics: Teachers’ Lab Mystery Operations http://www.learner.org/teacherslab/math/patterns/mystery/BillyBug: http://www.oswego.org/ocsd-web/games/BillyBug2/bug2.htmlFun Brain Number Cracker http://www.funbrain.com/cracker/index.htmlNo frills http://www.harcourtschool.com/activity/rubber_number_patterns_and_rules/Function Machine http://www.mathplayground.com/functionmachine.htmlConnected Mathematics: http://www.phschool.com/atschool/cmp2/active_math/site/Grade6/BeeLogo/index.htmlPolygon http://connectedmath.msu.edu/CD/Grade6/Quad/index.htmlLockers http://connectedmath.msu.edu/CD/Grade6/Locker/index.htmlFraction Game http://connectedmath.msu.edu/CD/Grade6/FractionGame/index.htmlIlluminations:Adding It All Up http://illuminations.nctm.org/LessonDetail.aspx?ID=L765Illuminations Angle Sum Tool http://illuminations.nctm.org/ActivityDetail.aspx?ID=9HS:Seeing Math™ Secondary: http://seeingmath.concord.org/sms_interactives.htmlSample ‘reasonableness’ problems from Ms. Fritzie: http://www.msfritzie.com/products/msfritzie/msf.htmlThanks to Phillip Martin for his free clip art for education! http://www.phillipmartin.info/clipart/homepage2.htm ttp://enhancingmypractice.wikispaces.com/Standards+of+Math+Practice

To view the official ADE documents 2010 Arizona Mathematics Standards

Overview of the 2010 Mathematical Standards PDF

Standards for Mathematical Practices PDF

Mathematics Introduction (Coming Soon)

Mathematics Glossary PDF Summary of Updates to Explanations

and Examples PDF

More Questions Contact ADE

Mary Knuck Mary.Knuck@azed.gov Suzi Mast suzi.mast@azed.gov