CCGPS Mathematics Unit-by-Unit Grade Level Webinar Third Grade Unit 6: Representing and Comparing Fractions January 22, 2013 Session will be begin at 3:15 pm 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.

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What’s Unit 6 all about? Organizing thinking Quantity representation- part of whole emphasis Understanding unit fractions Composition/Decomposition of fractions Representing data and solve using data Modeling problem solving Multiple solution paths

What should students bring from before? Foundational fractions understanding- equal parts/shares Familiarity with number line, ruler Foundational ability to decompose quantities Foundational organizational ability Foundational ability to problem solve Use of the 8 Standards for Mathematical Practice

What’s Unit 6 all about? Building from Kindergarten

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What’s Unit 6 all about? Organizing thinking

What’s Unit 6 all about? Quantity representation Extending Children's Mathematics Fractions and Decimals: Innovations in Cognitively Guided Instruction By Susan B. Empson and Linda Levi

Quantity representation/Unit fractions How much is shaded? Extending Children's Mathematics Fractions and Decimals: Innovations in Cognitively Guided Instruction By Susan B. Empson and Linda Levi

They think… Fractions are pieces. Fractions are always smaller than a whole. Fractions values are determined by counting parts “It’s 1/3 because 1 part out of 3 parts is shaded.” “4/3? That’s impossible!” “A fourth is a little pie shape.”

Organizing Thinking: Three approaches to teaching fractions Introduce procedures and explain concepts Emphasis on student discovery, with no conceptual analysis of discoveries Discuss and extend concepts and procedures that come up in children’s problem solving (from Saxe et al., 1999)

Quantity Representation Meaning of fractions -- what does 1/3 mean? – 1 thing shared equally by 3 people, each person gets 1/3 – 1 candy bar for every 3 people – 1 ÷ 3 = 1/3 – 1 part, with 3 equal parts to make a whole – These meanings generalize to improper fractions, too…..4/3

Quantity Representation 3.NF.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. For example: ¼ is the quantity created by partitioning the whole into 4 equal parts, and ¾ is the quantity formed by combining 3 parts of the size ¼.

Understanding unit fractions 18

What’s Unit 6 all about? Unit 1, Third grade

What’s Unit 6 all about? Unit 2, Third grade

Quantity Representation Fractional units can be combined – 1 third from one candy bar plus 1 third from another candy bar is 2 thirds – 1/3 + 1/3 = 2/3 Fractional units can be combined no matter how many there are – 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 = 10/4 Fractional numbers “fill in” the whole-number line – 2 1/4 cookies is more than 2 cookies but less than 3 cookies A fractional amount can be expressed in many ways

Composition/Decomposition

What’s Unit 6 all about? Quantity representation/Understanding unit fractions

Understanding unit fractions Seven “units” plus 4 “units” gives 11 “units” Grade 2  Grade 7  Grade  +  7x+3x 24 Extending Children's Mathematics Fractions and Decimals: Innovations in Cognitively Guided Instruction By Susan B. Empson and Linda Levi

Reasoning about relationships… 25 Extending Children's Mathematics Fractions and Decimals: Innovations in Cognitively Guided Instruction By Susan B. Empson and Linda Levi

Conceptual Understanding Matters “Fractions offer students the opportunity to grapple with fundamental mathematical relationships that constitute the core of Algebra” 26 Extending Children's Mathematics Fractions and Decimals: Innovations in Cognitively Guided Instruction By Susan B. Empson and Linda Levi

What’s Unit 6 all about? Decomposition of number/modeling problem solving

What’s Unit 6 all about? Decomposition of number/modeling problem solving

What’s Unit 6 all about? Decomposition of number/modeling problem solving

What’s Unit 6 all about? Decomposition of number/modeling problem solving

What’s Unit 6 all about? Modeling problem solving

What’s Unit 6 all about? Multiple solution paths

What about teaching? A teacher asks: How do you envision how math class would be taught with the Common Core Standards? I think many teachers teach math in a fairly “traditional” way – instructing students on how to do (whatever) and then assign problems to be completed. How is our “mode of business,” if you will, going to change? Bill McCallum says: Dear Teacher, I don’t see the standards as dictating any particular teaching method, but rather setting goals for student understanding. Different people have different ideas about what is the best method for achieving that understanding. That said, I think it’s pretty clear that classrooms implementing the standards should have some way of fostering understanding and reasoning, and classrooms where students are just sitting and listening are unlikely to achieve that.

One of the strongest results in recent research is that the most important feature in effective teaching is giving students "opportunity to learn". Teachers can set expectations, time, kinds of tasks, questions, acceptable answers, and type of discussions that will influence students' opportunity to learn. This must involve both skill efficiency and conceptual understanding. Hiebert, James; Grouws, Douglas The Effects of Classroom Mathematics Teaching on Students' Learning 2007

Horizontal and vertical connections Strategies apply everywhere, in multiple contexts Integration of content areas ensures connections and relational thinking Multiple steps builds understanding and deeper thinking Work the culminating task collaboratively with colleagues so you know where your kids need to go, and what they might have difficulty with

Evidence that Attending to Student Thinking Can Make a Difference Students constantly surprise us… – Kindergarten data – Fractions – Algebraic thinking Extending Children's Mathematics Fractions and Decimals: Innovations in Cognitively Guided Instruction By Susan B. Empson and Linda Levi

Development of Children’s Mathematical Thinking 19 Children are taking a mini-bus to the zoo. They will have to sit either 2 or 3 to a seat. The bus has 7 seats. How many children will have to sit 3 to a seat and how many can sit 2 to a seat? How will children solve it? How about a kindergartener? 59% had a correct strategy 51% had correct answer 33% 1 st graders 26% 2 nd graders Extending Children's Mathematics Fractions and Decimals: Innovations in Cognitively Guided Instruction By Susan B. Empson and Linda Levi

Focus on Making Student Thinking Explicit need to be able to use student strategies as the center of the workgroup conversation, as one of the tools teachers interact with expertise shared by students change in power structures, from teacher as expert Journaling, anchor charts provide an explicit trace of the group’s thinking extends to other communities of practice in other content areas Extending Children's Mathematics Fractions and Decimals: Innovations in Cognitively Guided Instruction By Susan B. Empson and Linda Levi

Focus on Making Student Thinking Explicit Ask children how they solved problems. Probe their understanding. Introduce symbols, number sentences, and mathematical language to go with strategies Don’t rush using and manipulating symbols- use the multiple models available first Extending Children's Mathematics Fractions and Decimals: Innovations in Cognitively Guided Instruction By Susan B. Empson and Linda Levi

Resources standards-elementary-school?fd=1 standards-elementary-school?fd=1 Practices-in-Mathematics.aspx Practices-in-Mathematics.aspx

As part of the continuing implementation of CCGPS in the year , the current GADOE mathematics units are being augmented. We would like your assistance with this critical process. Student work samples are a vital component of the frameworks which only you can provide. If you have been using the GADOE frameworks and have student work which you would be willing to share, please send it our way. We will remove any identifiers, and include selected student work samples in the revised frameworks which are slated to be released July 1, If your student work sample is selected for inclusion we will notify you of its placement in the units via . Call for Student Work Samples

Submission guidelines : Attach the work sample(s) to an in any format. Whatever works for you, works for us. Indicate the grade level, unit, and task in the body of your , and on the work sample in the upper left corner. You may cover any student/school identifiers if you wish, and we will do the same if any remain. Send the with student work attached to the appropriate team member. To be considered for inclusion, work samples must be submitted by May 17, We look forward to seeing your students’ work. Call for Student Work Samples

As part of the continuing implementation of CCGPS in the year , the current GADOE mathematics frameworks and units are being reviewed, revised, and augmented. We are offering an opportunity for educators to assist in this critical process. The challenge: Create a career-based mathematics task using guidelines provided to supplement and/or address gaps in the existing CCGPS frameworks units. If your task is selected for addition to a unit, you will receive a $200 honorarium per task. All work is to be original using support structures provided by the Georgia Department of Education Mathematics Team. If you are interested in participating in this challenge, please view the task creation guidelines at and get started! Task submission period begins now and closes May 1, We look forward to seeing your tasks. Career-Based Mathematics Task Challenge

As part of the continuing implementation of CCGPS, the current CCGPS mathematics frameworks and units are being reviewed, revised, and augmented. The Georgia Department of Education is seeking qualified math educators to become part of the 2013 CCGPS Mathematics Resource Revision Team which will assist in this critical process. The scope of the CCGPS Mathematics Resource Revision Team work will include, but is not limited to: evaluating newly submitted tasks assessing the need for additional tasks assessing the order of current units and tasks editing of current units and tasks creating additional tasks to address gaps, if necessary 2013 Resource Revision Team

All work will be completed collaboratively with support structures provided by the Georgia Department of Education Mathematics Team. All work is to be completed at the Georgia Department of Education, June 3rd-June 6, and June 10-13, Team members will be compensated for contracted work in the amount of $2000 and travel expenses will be reimbursed. If you are interested in becoming a part of the CCGPS Mathematics Resource Revision Team, please respond to the appropriate Georgia Department of Education contact below by March 1, In your response, please indicate grade level interest, why you would like to be part of this team, related experience, and the contact information for two references Resource Revision Team Grades K-5 Turtle Toms Mathematics Program Elementary Specialist

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Thank You! Please visit to provide us with your feedback! Turtle Gunn Toms Program Specialist (K-5) These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement. Join the listserve! Follow on Twitter! (yep, I’m tweeting math resources in a very informal manner)