CCGPS Mathematics Unit-by-Unit Grade Level Webinar 6 th Grade Unit 3: Expressions August 28, 2012 Session will be begin at 8:00 am While you are waiting, please do the following: Configure your microphone and speakers by going to: Tools – Audio – Audio setup wizard Document downloads: When you are prompted to download a document, please choose or create the folder to which the document should be saved, so that you may retrieve it later.

CCGPS Mathematics Unit-by-Unit Grade Level Webinar 6 th Grade Unit 3: Expressions August 28, 2012 James Pratt – Brooke Kline – Secondary Mathematics Specialists These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement.

Expectations and clearing up confusion This webinar focuses on CCGPS content specific to Unit 3, 6 th Grade. For information about CCGPS across a single grade span, please access the list of recorded GPB sessions on Georgiastandards.org. For information on the Standards for Mathematical Practice, please access the list of recorded Blackboard sessions from Fall 2011 on GeorgiaStandards.org. CCGPS is taught and assessed from and beyond. A list of resources will be provided at the end of this webinar and these documents are posted in the 6-8 wiki.

Expectations and clearing up confusion The intent of this webinar is to bring awareness to:  the types of tasks that are contained within the unit.  your conceptual understanding of the mathematics in this unit.  approaches to the tasks which provide deeper learning situations for your students. We will not be working through each task during this webinar.

Welcome! Thank you for taking the time to join us in this discussion of Unit 3. At the end of today’s session you should have at least 3 takeaways:  the big idea of Unit 3  something to think about…some food for thought  how might I support student problem solving?  what is my conceptual understanding of the material in this unit?  a list of resources and support available for CCGPS mathematics

Welcome! Please provide feedback at the end of today’s session.  Feedback helps us become better teachers and learners.  Feedback helps as we develop the remaining unit-by-unit webinars.  Please visit to share your feedback.. After reviewing the remaining units, please contact us with content area focus/format suggestions for future webinars. James Pratt – Brooke Kline – Secondary Mathematics Specialists

Misconception? Richard Benson: The Very Best Totally Wrong Answers

Welcome! For today’s session have you:  read the mathematics CCGPS?  read the unit and worked through the tasks in the unit?  downloaded and saved the documents from this session? Ask questions and share resources/ideas for the common good. Bookmark and become active in the 6-8 wiki. If you are still wondering what a wiki is, we will discuss this near the end of the session.

Activate your Brain Simplify 2(x + 4)

Activate your Brain Simplify 2(x + 4) Do you like this question? Will this question expose student misconceptions with equivalent expressions? Does this question require students to demonstrate proficiency with the properties of operations? Could you improve this question in order to assist in revealing student misconceptions with equivalent expressions and with properties of operations?

Misconceptions It is important to realize that inevitably students will develop misconceptions… Askew and Wiliam 1995; Leinwand, 2010; NCTM, 1995; Shulman, 1996

Misconception – Invented Rule? Richard Benson: The Very Best Totally Wrong Answers

Misconception – Invented Rule? Richard Benson: The Very Best Totally Wrong Answers

Misconceptions Therefore it is important to have strategies for identifying, remedying, as well as for avoiding misconceptions. Leinwand, 2010; Swan 2001; NBPTS, 1998; NCTM, 1995; Shulman, 1986;

Activate your Brain Simplify 2(x + 4) Do you like this question? Will this question expose student misconceptions with equivalent expressions? Does this question require students to demonstrate proficiency with the properties of operations? Could you improve this question in order to assist in revealing student misconceptions with equivalent expressions and with properties of operations?

Activate your Brain Which of the following expressions are equivalent? Why? If an expression has no match, write two equivalent expressions to match it. a.2(x + 4) b.8 + 2x c.2x + 4 d.3(x + 4) – 4 – x e.(x + 4) + (4 + x) Adapted from Illustrative Mathematics – 6.EE.4 Equivalent Expressions

Activate your Brain Which of the following expressions are equivalent? Why? If an expression has no match, write two equivalent expressions to match it. a.2(x + 4) = 2x + 8 b.8 + 2x = 2x + 8 c.2x + 4 = 2x + 4 d.3(x + 4) – 4 – x = 3x + 12 – 4 – x = 2x + 8 e.(x + 4) + (4 + x) = (x + x) + (4 + 4) = 2x + 8 Adapted from Illustrative Mathematics – 6.EE.4 Equivalent Expressions

What’s the big idea? Overview Key Standards Enduring Understandings Essential Questions Strategies for Teaching & Learning

What’s the big idea? Develop a deep understanding with reading, writing, and evaluating expressions in which letters stand for numbers and with expressions involving whole number exponents. Develop a deep understanding with applying the properties of operations to generate and identify equivalent expressions.

What’s the big idea? Standards for Mathematical Practice Education Week Webinar – Bristol CT School District

What’s the big idea? Standards for Mathematical Practice

Education Week Webinar – Math Practices and the Common Core

Questions that arose Mixed Numbers Prime Factorization

Questions that arose Bill McCallum wrote: Greatest common factors and least common multiplies are treated with a very light touch in the standards. They are not a major topic, and limited to numbers less than or equal to 100 (6.NS.4). For such numbers, listing the factors or multiplies is probably the most efficient method, and has the added benefit of reinforcing number facts. It also supports the meaning of the terms: you can see directly that you are finding the greatest common factor or the least common multiple. The prime factorization method can be a bit mysterious in this regard. And, as you point out, prime factorization is not a topic in the Common Core, although prime numbers are mentioned in 4.OA.4. So, the standards do indeed remove this topic from the curriculum. Achieving the focus of the standards means giving some things up, and this is one of those things. (Of course personally, as a number theorist, I love the topic!) Post Link:

Coherence and Focus – Unit 3 Education Week Webinar – Jason Zimba, lead writer of the CCSM

Coherence and Focus – Unit 3 What are students coming with?

Coherence and Focus – Unit 3 What foundation is being built? Where does this understanding lead students? Enduring Understandings Evidence of Learning

Coherence and Focus – Unit 3 View across grade bands K-5 th  Operations with whole numbers, fractions, and decimals  Writing and interpreting numerical expressions  Evaluating expressions involving grouping symbols 7 th -12 th  Transferring real-life and mathematical problems to symbolic expressions, equations, and inequalities  Operations of exponents and scientific notation  Working with various functions

Misconception? Richard Benson: The Very Best Totally Wrong Answers

Examples & Explanations Use exponents to write an equivalent expression for 2 x 2 x 2.

Examples & Explanations Adapted from Illustrative Mathematics – 6.EE.1 The Djinni’s Offer

Examples & Explanations Adapted from Illustrative Mathematics – 6.EE.1 The Djinni’s Offer DayNumber of Coins

Examples & Explanations PEMDAS = Misconceptions? a.8 – b.Evaluate 8( n + 1) when n = 5

Examples & Explanations PEMDAS = Misconceptions? 8 – The mnemonic PEMDAS can mislead students into thinking that addition must always take precedence over subtraction because the A comes before the S, rather than the correct convention that addition and subtraction proceed from left to right (as do multiplication and division). This can lead students to make mistakes such as simplifying 8 – as 8 – 7 (instead of the correct 2 + 1) because they add the 6 and the 1 before subtracting from 8. Commoncoretools.wordpress.com

Examples & Explanations PEMDAS = Misconceptions? 8 – The mnemonic PEMDAS can mislead students into thinking that addition must always take precedence over subtraction because the A comes before the S, rather than the correct convention that addition and subtraction proceed from left to right (as do multiplication and division). This can lead students to make mistakes such as simplifying 8 – as 8 – 7 (instead of the correct 2 + 1) because they add the 6 and the 1 before subtracting from 8. Commoncoretools.wordpress.com

Examples & Explanations PEMDAS = Misconceptions? Evaluate 8( n + 1) when n = 5 The order of operations tells us how to interpret expressions, but does not necessarily dictate how to calculate them. The P in PEMDAS indicates that the expression 8(5 + 1) is to be interpreted as 8 times a number which is the sum of 5 and 1. However, it does not dictate the expression must be calculated this way. A student might well see it, through an implicit use of the distributive law, as 8(5) + 8(1) = = 48. Commoncoretools.wordpress.com

Examples & Explanations PEMDAS = Misconceptions? Evaluate 8( n + 1) when n = 5 The order of operations tells us how to interpret expressions, but does not necessarily dictate how to calculate them. The P in PEMDAS indicates that the expression 8(5 + 1) is to be interpreted as 8 times a number which is the sum of 5 and 1. However, it does not dictate the expression must be calculated this way. A student might well see it, through an implicit use of the distributive law, as 8(5) + 8(1) = = 48. Commoncoretools.wordpress.com

Examples & Explanations PEMDAS = Misconceptions? Evaluate 8( n + 1) when n = 5 The order of operations tells us how to interpret expressions, but does not necessarily dictate how to calculate them. The P in PEMDAS indicates that the expression 8(5 + 1) is to be interpreted as 8 times a number which is the sum of 5 and 1. However, it does not dictate the expression must be calculated this way. A student might well see it, through an implicit use of the distributive law, as 8(5) + 8(1) = = 48. Commoncoretools.wordpress.com

Examples & Explanations PEMDAS = Misconceptions? a.8 – b.Evaluate 8( n + 1) when n = 5 The properties of operations are the fundamental properties of the number system and undergird all work with expression! Commoncoretools.wordpress.com

Examples & Explanations Write an expression for the perimeter of a rectangle with length of l and width of w.

Examples & Explanations Adapted from Illustrative Mathematics – 6.EE Rectangle Perimeter 3

Examples & Explanations Sadie = 2(l + w) = 2( ) = 210 Eric = 2l + 2w = 2(30) + 2(75) = 210 Adapted from Illustrative Mathematics – 6.EE Rectangle Perimeter 3

Misconception? Richard Benson: The Very Best Totally Wrong Answers

Assessment How might it look? Mathematics Assessment Project - Illustrative Mathematics Dana Center’s CCSS Toolbox: PARCC Prototype Project Online Assessment System - Instruction-and-Assessment/Assessment/Pages/OAS.aspx Instruction-and-Assessment/Assessment/Pages/OAS.aspx

Race to the Top Assessment Toolbox Update Fall 2012

RT3 Assessment Initiatives Purpose – To support teachers in preparing the students for the Common Core Assessment that is to occur in spring 2015 – To provide assessment resources that reflect the rigor of the CCGPS – To balance the use of formative and summative assessments in the classroom 46

RT3 Assessment Initiatives Development of a three-prong toolkit to support teachers and districts and to promote student learning – A professional development opportunity that focuses on assessment literacy – A set of benchmarks in ELA, Math, and selected grades for Science and Social Studies – An expansive bank of formative assessment items/tasks based on CCGPS in ELA and Math as a teacher resource 47

Formative Assessment Conducted during instruction (lesson, unit, etc.) Identifies student strengths and weaknesses Helps teacher determine next steps – Review – Differentiation – Continuation Supplies information to provide students with detailed feedback Assessment for the purpose of improving achievement LOW stakes 48

Purpose of the Formative Item Bank The purpose of the Formative Item Bank is to provide items and tasks used to assess students’ knowledge while they are learning the curriculum. The items will provide an opportunity for students to show what they know and show teachers what students do not understand. 49

Formative Item Bank Assessments Aligned to CCGPS Format aligned with Common Core Assessments Open-ended and constructed response items as well as multiple choice items Holistic Rubrics Anchor Papers Student Exemplars 750+ Items Available in OAS by late September 50

Formative Item Bank Availability All items that pass data review will be uploaded to the Georgia OAS at Level 2. Formative Item Bank will be ready for use by all Georgia educators mid-September,

52 Item Types – Multiple Choice (MC) – Extended Response (ER) – Scaffolded Item (SC)

Extended Response Items Performance-based tasks May address multiple standards, multiple domains, and/or multiple areas of the curriculum May allow for multiple correct responses and/or varying methods of arriving at a correct answer Scored through use of a rubric and associated student exemplars 53

Mathematics Sample Item – Grade HS an extended response item 54

Example Rubric 55

Scaffolded Items Include a sequence of items or tasks Designed to demonstrate deeper understanding May be multi-standard and multi-domain May guide a student to mapping out a response to a more extended task Scored through use of a rubric and associated student exemplars 56

Mathematics Sample Item – Grade 3 a scaffolded item 57

Mathematics Items Assess students’ conceptual and computational understanding Tasks require students to – Apply the mathematics they know to real world problems – Express mathematical reasoning by showing their work or explaining their answer 58

Where do you Find the Items? 59 rt student

Georgia Department of Education Assessment and Accountability Melissa Fincher Associate Superintendent Assessment and Accountability Dr. Melodee Davis Director Assessment Research and Development Robert Anthony Assessment Specialist Formative Item Bank Race to the Top Jan Reyes Assessment Specialist Interim Benchmark Assessments Race to the Top Dr. Dawn Souter Project Manager Race to the Top

Suggestions for getting started: Read the unit and work through the tasks with your colleagues. The only way to gain deep understanding is to work through each task. Make note of where, when, and what the big ideas are. Discuss the focus and coherence of the unit. Make note of where, when, and what the pitfalls might be. Look for additional tools/ideas you want to use. Determine any changes which might need to be made to make this work for your students. Share, ask, and collaborate on the wiki.

Resource List The following list is provided as a sample of available resources and is for informational purposes only. It is your responsibility to investigate them to determine their value and appropriateness for your district. GaDOE does not endorse or recommend the purchase of or use of any particular resource.

What is a Wiki?

Resources Common Core Resources  SEDL videos - or  Illustrative Mathematics -  Dana Center's CCSS Toolbox -  Arizona DOE -  Ohio DOE - ationID= ationID=1704  Common Core Standards -  Tools for the Common Core Standards -  Phil Daro talks about the Common Core Mathematics Standards Books  Van DeWalle and Lovin, Teaching Student-Centered Mathematics, 6-8

Resources Professional Learning Resources  Inside Mathematics-  Annenberg Learner -  Edutopia –  Teaching Channel - Assessment Resources  MAP -  CCSS Toolbox: PARCC Prototyping Project -  PARCC - Blogs  Dan Meyer –  Timon Piccini –  Dan Anderson –

Resources Dana Center’s CCSS Toolbox - PARCC Prototyping Project 

Resources Dan Meyer’s Three-Act Math Tasks  M2YWxWYVM1UWowTEE M2YWxWYVM1UWowTEE

As you start your day tomorrow… …the standards are not units of instruction; you don’t always “teach a standard” in one chunk, whatever the order…The standards describe achievements we want students to have. As my colleague Jason Zimba likes to say, you don’t teach standards, you teach mathematics. Bill McCallum – lead writer of the CCSM

Thank You! Please visit to share your feedback, ask questions, and share your ideas and resources! Please visit to join the 6-8 Mathematics listserve. Brooke Kline Program Specialist (6 ‐ 12) James Pratt Program Specialist (6-12) These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement.