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CCSS-M in the Classroom Grades 3-5 Number and

Operations Fractions Weaving Content and

Standards for Mathematical Practices

Overall Outcomes

- Recognize the interconnectedness of the Standards

for Mathematical Practice and content standards

in developing student understanding and

reasoning. - Illuminate practices that establish a culture

where mistakes are a springboard for learning,

risk-taking is the norm, and there is a belief

that all students can learn. - Deeping content knowledge and pedagogy within an

important focus area for our grade band - Number and Operations - Fractions

Effective Classrooms

What research says about effective classrooms

- The activity centers on mathematical

under-standing, invention, and sense-making by

all students. - The culture is one in which inquiry, wrong

answers, personal challenge, collaboration, and

disequilibrium provide opportunities for

mathematics learning by all students. - The tasks in which students engage are

mathematically worthwhile for all students. - A teachers deep knowledge of the mathematics

content she/he teaches and the trajectory of that

content enables the teacher to support important,

long-lasting student understanding

What research says about effective classrooms

- The activity centers on mathematical

understanding, invention, and sense-making by all

students. - The culture is one in which inquiry, wrong

answers, personal challenge, collaboration, and

disequilibrium provide opportunities for

mathematics learning by all students. - The tasks in which students engage are

mathematically worthwhile for all students. - A teachers deep knowledge of the mathematics

content she/he teaches and the trajectory of that

content enables the teacher to support important,

long-lasting student understanding

What research says about effective classrooms

- The activity centers on mathematical

understanding, invention, and sense-making by all

students. - The culture is one in which inquiry, wrong

answers, personal challenge, collaboration, and

disequilibrium provide opportunities for

mathematics learning by all students. - The tasks in which students engage are

mathematically worthwhile for all students. - A teachers deep knowledge of the mathematics

content she/he teaches and the trajectory of that

content enables the teacher to support important,

long-lasting student understanding

What research says about effective classrooms

- The activity centers on mathematical

understanding, invention, and sense-making by all

students. - The culture is one in which inquiry, wrong

answers, personal challenge, collaboration, and

disequilibrium provide opportunities for

mathematics learning by all students. - The tasks in which students engage are

mathematically worthwhile for all students. - A teachers deep knowledge of the mathematics

content she/he teaches and the trajectory of that

content enables the teacher to support important,

long-lasting student understanding.

Effective implies

- Students are engaged with important mathematics.
- Lessons are very likely to enhance student

understanding and to develop students capacity

to do math successfully. - Students are engaged in ways of knowing and ways

of working consistent with the nature of

mathematicians ways of knowing and working.

Reflection

- What is your current reality around classroom

culture? - What can you do to enhance your current reality?

Outcomes Day 1

- Reflect on teaching practices that support the

shifts in the Standards for Mathematical Practice

and content standards. - Understand how to analyze student work with the

Standards for Mathematical Practice and content

standards. - Analyze, adapt and implement a task with the

integrity of the Common Core State Standards.

A message from OSPI

Outcomes Norms slide here

WA CCSS Implementation Timeline

2010-11 2011-12 2012-13 2013-14 2014-15

Phase 1 CCSS Exploration

Phase 2 Build Awareness Begin Building Statewide Capacity

Phase 3 Build State District Capacity and Classroom Transitions

Phase 4 Statewide Application and Assessment

Ongoing Statewide Coordination and Collaboration to Support Implementation

Theory of Practice for CCSS Implementation in WA

- 2-Prongs
- 1. The What Content Shifts (for students and

educators) - Belief that past standards implementation efforts

have provided a strong foundation on which to

build for CCSS HOWEVER there are shifts that

need to be attended to in the content. - 2. The How System Remodeling
- Belief that successful CCSS implementation will

not take place top down or bottom up it must be

both, and - Belief that districts across the state have the

conditions and commitment present to engage

wholly in this work. - Professional learning systems are critical

Transition Plan for Washington State

K-2 3-5 6-8 High School

Year 1- 2 2012-2013 School districts that can, should consider adopting the CCSS for K-2 in total. K Counting and Cardinality (CC) Operations and Algebraic Thinking (OA) Measurement and Data (MD) 1 Operations and Algebraic Thinking (OA) Number and Operations in Base Ten (NBT) 2 Operations and Algebraic Thinking (OA) Number and Operations in Base Ten (NBT) and remaining 2008 WA Standards 3 Number and Operations Fractions (NF) Operations and Algebraic Thinking (OA) 4 Number and Operations Fractions (NF) Operations and Algebraic Thinking (OA) 5 Number and Operations Fractions (NF) Operations and Algebraic Thinking (OA) and remaining 2008 WA Standard 6 Ratio and Proportion Relationships (RP) The Number System (NS) Expressions and Equations (EE) 7 Ratio and Proportion Relationships (RP) The Number System (NS) Expressions and Equations (EE) 8 Expressions and Equations (EE) The Number System (NS) Functions (F) and remaining 2008 WA Standards Algebra 1- Unit 2 Linear and Exponential Relationships Unit 1 Relationship Between Quantities and Reasoning with Equations and Unit 4 Expressions and Equations Geometry- Unit 1 Congruence, Proof and Constructions and Unit 4 Connecting Algebra and Geometry through Coordinates Unit 2 Similarity, Proof, and Trigonometry and Unit 3Extending to Three Dimensions and remaining 2008 WA Standards

Why Shift?

- Almost half of eighth-graders in high achieving

countries showed they could reach the advanced

level in math, meaning they could relate

fractions, decimals and percent to each other

understand algebra and solve simple probability

problems. - In the U.S., 7 percent met that standard.
- Results from the 2011 TIMMS

The Three Shifts in Mathematics

- Focus Strongly where the standards focus
- Coherence Think across grades and link to major

topics within grades - Rigor Require conceptual understanding, fluency,

and application

Focus on the Major Work of the Grade

- Two levels of focus
- Whats in/Whats out
- The standards at each grade level are

interconnected allowing for coherence and rigor

Focus in International Comparisons

- TIMSS and other international comparisons suggest

that the U.S. curriculum is a mile wide and an

inch deep. - On average, the U.S. curriculum omits only 17

percent of the TIMSS grade 4 topics compared with

an average omission rate of 40 percent for the 11

comparison countries. - The United States covers all but 2 percent of the

TIMSS topics through grade 8 compared with a 25

percent noncoverage rate in the other countries. - High-scoring Hong Kongs curriculum omits 48

percent of the TIMSS items through grade 4, and

18 percent through grade 8.

Ginsburg et al., 2005

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Content Emphasis by ClusterGrade 3

Grade 3(supporting cluster)

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Content Emphasis by ClusterGrade 4

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Content Emphasis by ClusterGrade 5

Focus on Major Work

- In any single grade, students and teachers spend

the majority of their time, approximately 75 on

the major work of the grade. - The major work should also predominate the first

half of the year.

Engaging with the 3-5 Content

- How would you summarize the major work of 3-5?
- What would you have expected to be a part of the

major work that is not? - Give an example of how you would approach

something differently in your teaching if you

thought of it as supporting the major work,

instead of being a separate, discrete topic.

Focus on Fractions One of the Major Works of the

3-5 Grade Band

- Deeping Content Knowledge and

Shifts - Implications for Fractions

- http//www.illustrativemathematics.org/pages/fract

ions_progression - Grade 3 Developing an understanding of fractions

as numbers is essential for future work with the

number system. It is critical that students at

this grade are able to place fractions on a

number line diagram and understand them as a

related component of their ever expanding number

system.

Shift Two Coherence Think across grades, and

link to major topics within grades

- Carefully connect the learning within and across

grades so that students can build new

understanding onto foundations built in previous

years. - Begin to count on solid conceptual understanding

of core content and build on it. Each standard is

not a new event, but an extension of previous

learning.

Coherence Across and Within Grades

- Its about math making sense.
- The power and elegance of math comes out through

carefully laid progressions and connections

within grades.

Coherence Think across grades, and link to major

topics within grades

- Carefully connect the learning within and across

grades so that students can build new

understanding onto foundations built in previous

years. - Begin to count on solid conceptual understanding

of core content and build on it. Each standard is

not a new event, but an extension of previous

learning.

How will it look different?

- Varied problem structures that build on the

students work with whole numbers - 5 1 1 1 1 1 builds to
- 5/3 1/3 1/3 1/3 1/3 1/3 and
- 5/3 5 x 1/3
- Conceptual development before procedural
- Use of rich tasks-applying mathematics to real

world problems - Effective use of group work
- Precision in the use of mathematical vocabulary

Coherence -Think Across Grades

Coherence -Think Across Domains

- Grade 4
- Operations and Algebraic Thinking
- Students use four operations with whole numbers

to solve problems. - Students gain familiarity with factors and

multiples which supports student work with

fraction equivalency. - Number and Operations Fractions
- Students build fractions from unit fractions by

applying and extending previous understandings of

operations with whole numbers.

The Structure is the Standards

Rigor Illustrations of Conceptual Understanding,

Fluency, and Application

- Here rigor does not mean hard problems.
- Its a balance of three fundamental components

that result in deep mathematical understanding. - There must be variety in what students are asked

to produce.

Some Old Ways of Doing Business

- Lack of rigor
- Reliance on rote learning at expense of concepts
- Severe restriction to stereotyped problems

lending themselves to mnemonics or tricks - Aversion to (or overuse) of repetitious practice
- Lack of quality applied problems and real-world

contexts - Lack of variety in what students produce
- E.g., overwhelmingly only answers are produced,

not arguments, diagrams, models, etc.

Some Old Ways of Doing Business

- Concrete Semi Concrete

Abstract - Unfortunately this model (Jerome Bruner, 1964)

was interpreted as giving hierarchal value to the

symbolic above the concrete or semi concrete - Which lead to
- Abstract Semi Concrete

(used to prove or show

why the abstract worked) and an

implication that the concrete was only

for those who didnt get it

Desired outcome is a balance that leads to

flexible thinking about concepts and an ability

to apply knowledge in novel situations

Conceptual and Procedural Understanding

How do students currently perceive mathematics?

- Doing mathematics means following the rules laid

down by the teacher. - Knowing mathematics means remembering and

applying the correct rule when the teacher asks a

question. - Mathematical truth is determined when the answer

is ratified by the teacher. - -Mathematical Education of Teachers report

(2012)

How do students currently perceive mathematics?

- Students who have understood the mathematics they

have studied will be able to solve any assigned

problem in five minutes or less. - Ordinary students cannot expect to understand

mathematics they expect simply to memorize it

and apply what they have learned mechanically and

without understanding. - -Mathematical Education of Teachers report

(2012)

Redefining what it means to be good at math

- Expect math to make sense
- wonder about relationships between numbers,

shapes, functions - check their answers for reasonableness
- make connections
- want to know why
- try to extend and generalize their results
- Are persistent and resilient
- are willing to try things out, experiment, take

risks - contribute to group intelligence by asking good

questions - Value mistakes as a learning tool (not something

to be ashamed of)

Mathematical Practices

- Make sense of problems and persevere in solving

them. - Reason abstractly and quantitatively.
- Construct viable arguments and critique the

reasoning of others. - Model with mathematics
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated

reasoning.

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Poster Activity

Standards for Mathematical Practices

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Shifts in Focus, Coherence and Rigor in the

assessment

- Lets look at the Assessment

A Balanced Assessment System

English Language Arts/Literacy and Mathematics,

Grades 3-8 and High School

School Year

Last 12 weeks of the year

DIGITAL CLEARINGHOUSE of formative tools,

processes and exemplars released items and

tasks model curriculum units educator training

professional development tools and resources

scorer training modules and teacher

collaboration tools.

Optional Interim Assessment

Optional Interim Assessment

- PERFORMANCE TASKS
- ELA/Literacy
- Mathematics

- COMPUTER ADAPTIVE TESTS
- ELA/Literacy
- Mathematics

Computer Adaptive Assessment and Performance Tasks

Computer Adaptive Assessment and Performance Tasks

Re-take option

Scope, sequence, number and timing of interim

assessments locally determined

Time windows may be adjusted based on results

from the research agenda and final implementation

decisions.

Time and format

- Summative
- For each content area - ELA Math
- Computer Adaptive Testing (CAT)
- Selected response (SR), Constructed Response

(open-endedCR, ECR), Technology enhanced (e.g.,

drag and drop, video clips, limited

web-interface) - Performance Tasks (like our CBAs) (PT)
- 1 per content area in grades 3-8
- Up to 3 per content area in High School

Assessment Claims for Mathematics

Overall Claim (Gr. 3-8)

Overall Claim (High School)

Claim 1 Concepts and Procedures

Claim 2 Problem Solving

Claim 3 Communicating Reasoning

Claim 4 Modeling and Data Analysis

Claim 1Concepts and Procedures

- Students can explain and apply mathematical

concepts and interpret and carry out mathematical

procedures with precision and fluency.

Grade Level Number of Assessment Targets

3 11

4 12

5 11

6 10

7 9

8 10

11 17

Cognitive Rigor and Depth of Knowledge

- The level of complexity of the cognitive demand.
- Level 1 Recall and Reproduction
- Requires eliciting information such as a fact,

definition, term, or a simple procedure, as well

as performing a simple algorithm or applying a

formula. - Level 2 Basic Skills and Concepts
- Requires the engagement of some mental processing

beyond a recall of information. - Level 3 Strategic Thinking and Reasoning
- Requires reasoning, planning, using evidence, and

explanations of thinking. - Level 4 Extended Thinking
- Requires complex reasoning, planning, developing,

and thinking most likely over an extended period

of time.

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DOK Distribution on SBAC

DOK 1 DOK 2 DOK 3 DOK 4

Grade 4 25 40 26 9

Grade 8 18 43 27 12

High School 27 41 23 9

Looking at SBAC TasksDepth of Knowledge and

Mathematical Practices, two lenses

- What is the depth of knowledge of these tasks?
- Which mathematical practices do they promote?

3.NF.A.3aUnderstand two fractions as equivalent

(equal) if they are the same size, or the same

point on a number line.

5.NF.C.7Apply and extend previous understandings

of division to divide unit fractions by whole

numbers and whole numbers by unit fractions

Claim 2 Problem Solving

Claim 2 Students can solve a range of complex

well-posed problems in pure and applied

mathematics, making productive use of knowledge

and problem solving strategies.

- Apply mathematics to solve well-posed problems

arising in everyday life, society, and the

workplace - Select and use tools strategically
- Interpret results in the context of the situation
- Identify important quantities in a practical

situation and map their relationships.

4.NF.B.4cSolve word problems involving

multiplication of a fraction by a whole number

Take a Break!

Learning Progression

- Number and Operations Fractions

Cluster

Sorting Standards

- Work in groups of 2 or 3 to sort the fractions

cards into a progression of concepts. - Sort the clusters under the standards using the

CCSS document

Learning Progression

- Number and Operations Fractions

Unit Fractions

What is a unit fraction?

- Discuss at your table we will predict and

adjust as we move through the materials

Exploring Unit Fractions

- http//www.illustrativemathematics.org/pages/fract

ions_progression

Grade 3 number line

4.NF.3

- 4.NF.3
- 4.NF.4

This leads to 4.NF.4

Halves

- How many ways can you show halves on a Geoboard?

(the whole is determined by you) - What is the most creative design you can create?

How did you determine that it was a half? - Which of your halves are equivalent? Why?

Mathematical Practices - looking at video from

the lens of SMP 3

- Make sense of problems and persevere in solving

them. - Reason abstractly and quantitatively.
- Construct viable arguments and critique the

reasoning of others. - Read full standard and

highlight important ideas - Model with mathematics
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated

reasoning.

ProtocolConstruct viable arguments and critique

the reasoning of others.

- Review the text and underline the parts that

pertain to constructing arguments argumentation

and critiquing. - Brainstorm the actions you would expect to see if

students were critiquing the reasoning of others.

Fractions with Geoboards

- Fourth- and fifth-graders investigate the concept

of halves using the geoboard as an area model.

They learn that one-half means two equal-sized

parts with equal areas, but that are not

necessarily congruent.

http//www.learner.org/series/modules/express/page

s/ccmathmod_07.htm http//www.learner.org/resource

s/series32.html 37 start at 229 and end at

1000 l

Finding Evidence ofConstruct viable arguments

and critique the reasoning of others.

- What makes this activity evidence of critiquing

the reasoning of others? - What observable conditions, supported critiquing

the reasoning of others? - What, observable conditions, constrained

critiquing the reasoning of others?

How did this activity support the big ideas of

unit fractions

- Determining what represents whole
- Reasoning about the size of a unit fraction based

on the meaning of a unit fraction - If the whole is represented by a region in the

plane (area model) it is possible for identical

fractions to be represented by different shapes - Turn and talk with your partners and them we will

share out with the group

Implications for your students

- Consider your lessons over the next few weeks
- Develop an instructional action intended to

improve your instructional practice for

constructing viable arguments and critiquing the

reasoning of others.

Lunch

Can You See?

Can You See?

Mathematical Practices

- What are the content standards can be addressed

at your grade level in this task? - What mathematical practices does it promote?

Using Rich Tasks in the Classroom

What makes a rich task?

- Is the task interesting to students?
- Does the task involve meaningful mathematics?
- Does the task provide an opportunity for students

to apply and extend mathematics? - Is the task challenging to all students?
- Does the task support the use of multiple

strategies and entry points? - Will students conversation and collaboration

about the task reveal information about

students mathematics understanding?

Adapted from Common Core Mathematics in a PLC at

Work 3-5 Larson,, et al

Environment for Rich Tasks

- Learners not passive recipients of mathematical

knowledge - Learners are active participants in creating

understanding and challenge and reflect on their

own and others understandings - Instructors provide support and assistance

through questioning and supports as needed

Lets Try a Rich Task

- Using a task card with students
- Who Got What?
- With your table group engage in this task and

predict what sort of entry points and strategies

students might use - Create a list of misconceptions that might arise

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Homework

- Before our next meeting
- Use the task Who Got What? with your students.
- Bring back one or two student artifacts - ready

to discuss student generated strategies, etc.

(Please remove student names) - Use the Standards for Mathematical Practice

Matrix to reflect on where your classroom falls

on the continuum and be ready to discuss any

activities you used to move your classroom

forward on this scale - Please bring your instructional materials for

fractions to our next class.

Standards for Mathematical Practice Matrix

Looking at Student Work Next Session

- Protocol
- Form small groups of 3 or 4
- Each person selects 1 or 2 work samples to share

with the group - You will follow a protocol to review student work

which focuses on student understanding - Please remove student names from any papers

Reflection and quick write

- What are the instructional shifts needed to make

these practices a reality?

Day 2CCSS-M in the Classroom Grades 3-5

Number and Operations Fractions Weaving

Content and Standards for Mathematical Practices

Outcomes Day 2 and 3

- Analyze, adapt and implement a task with the

integrity of the Common Core State Standards. - Understand how to analyze student work with the

Standards for Mathematical Practice and content

standards. - Deepen understanding of the progression of

learning around Number and Operations - Fractions

What research says about effective classrooms

- The activity centers on mathematical

understanding, invention, and sense-making by all

students. - The culture is one in which inquiry, wrong

answers, personal challenge, collaboration, and

disequilibrium provide opportunities for

mathematics learning by all students. - The tasks in which students engage are

mathematically worthwhile for all students. - A teachers deep knowledge of the mathematics

content she/he teaches and the trajectory of that

content enables the teacher to support important,

long-lasting student understanding.

Looking at Student Work

- Form small groups of 3 or 4
- Each person selects 1 or 2 works samples to share

with the group - Follow the Collaboration Protocol to review

student work samples and record information - observations
- Inferences
- implications

Collaboration Protocol-Looking at Student Work

(55 minutes)

- 1. Individual review of student work samples

(10 min) - All participants observe or read student work

samples in silence, making brief notes on the

form Looking at Student Work - 2. Sharing observations (15 min)
- The facilitator asks the group
- What do students appear to understand based on

evidence? - Which mathematical practices are evident in their

work? - Each person takes a turn sharing their

observations about student work without making

interpretations, evaluations of the quality of

the work, or statements of personal reference. - 3. Discuss inferences -student understanding (15

min) - Participants, drawing on their observation of the

student work, make suggestions about the problems

or issues of students content misunderstandings

or use of the mathematical practices. - Adapted from Steps in the Collaborative

Assessment Conference developed by - Steve Seidel and Project Zero Colleagues

Select one group member to be todays

facilitator to help move the group through the

steps of the protocol. Teachers bring student

work samples with student names removed.

- 4. Discussing implications-teaching learning

(10 min) - The facilitator invites all participants to share

any thoughts they have about their own teaching,

students learning, or ways to support the

students in the future. - How might this task be adapted to further elicit

students use of Standards for Mathematical

Practice or mathematical content. - 5. Debrief collaborative process (5 min)
- The group reflects together on their experiences

using this protocol.

Looking at student work

- What instructional strategies did you use with

this lesson? - What Standards for Mathematical Practices did you

notice your students engaging in during this

task? - Using the SMP Matrix how would you describe your

classroom with examples to support

Assessment Claims for Mathematics

Overall Claim (Gr. 3-8)

Overall Claim (High School)

Claim 1 Concepts and Procedures

Claim 2 Problem Solving

Claim 3 Communicating Reasoning

Claim 4 Modeling and Data Analysis

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Cognitive Rigor and Depth of Knowledge (DOK)

- The level of complexity of the cognitive demand.
- Level 1 Recall and Reproduction
- Requires eliciting information such as a fact,

definition, term, or a simple procedure, as well

as performing a simple algorithm or applying a

formula. - Level 2 Basic Skills and Concepts
- Requires the engagement of some mental processing

beyond a recall of information. - Level 3 Strategic Thinking and Reasoning
- Requires reasoning, planning, using evidence, and

explanations of thinking. - Level 4 Extended Thinking
- Requires complex reasoning, planning, developing,

and thinking most likely over an extended period

of time.

(No Transcript)

Claim 2 Problem Solving

Claim 2 Students can solve a range of complex

well-posed problems in pure and applied

mathematics, making productive use of knowledge

and problem solving strategies.

- Apply mathematics to solve well-posed problems

arising in everyday life, society, and the

workplace - Select and use tools strategically
- Interpret results in the context of the situation
- Identify important quantities in a practical

situation and map their relationships.

Looking at SBAC TasksDepth of Knowledge and

Mathematical Practices, two lenses

- What is the depth of knowledge of these tasks?
- Which mathematical practices do they promote?

4.NF.B.3Understand a fraction a/b with a gt 1 as

a sum of fractions 1/b

Claim 3 Communicating Reason

Claim 3 Students can clearly and precisely

construct viable arguments to support their own

reasoning and to critique the reasoning of

others.

- Test propositions or conjectures with specific

examples. - Construct, autonomously, chains of reasoning that

justify or refute propositions or conjectures. - State logical assumptions being used.
- Use the technique of breaking an argument into

cases. - Distinguish correct logic or reasoning from that

which is flawed, andif there is a flaw in the

argumentexplain what it is. - Base arguments on concrete referents such as

objects, drawings, diagrams, and actions. - Determine conditions under which an argument does

and does not apply.

Looking at SBAC TasksDepth of Knowledge and

Mathematical Practices, two lenses

- What is the depth of knowledge of these tasks?
- Which mathematical practices do they promote?

3.NF.3Explain equivalence of fractions in

special cases, and compare fractions by reasoning

about their size

Shift Two Coherence Think across grades, and

link to major topics within grades

- Carefully connect the learning within and across

grades so that students can build new

understanding onto foundations built in previous

years. - Begin to count on solid conceptual understanding

of core content and build on it. Each standard is

not a new event, but an extension of previous

learning.

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Focus and Coherencethrough the Major and

Supporting Clusters of the CCSS-M

- Discuss with a partner which
- Major, Supporting, and/or
- Additional Clusters are
- involved in the task

Exploring Equivalent Fractions

- http//www.illustrativemathematics.org/pages/fract

ions_progression

- Justify the truth of this statement in at least

two different ways.

Strings

- How does this support the development of

efficient computational strategies and overall

number sense? - What mathematical practices
- does it support?

Presenter Notes for String

- List one problem at a time
- 1/2 1/3 I am exercising for the New Year and

have decided that I am going run for 1/2 hour and

walk for 1/3 hour. How long did I exercise? What

fraction of a hour is that? - Have participants use thumbs up when they know an

answer and can tell how they arrived at it. - Model it on the clock
- I knew that 1/2 hour was 30 min (draw a circle

and cut in half) I knew that 1/3 hour was 20 min

(add the 20 min onto the 30 so that your minute

hand is pointing to the 10 on the clock) My

answer is 10/12 of an hour (or 5/6 or 50/60) - Continue with the scenario of exercising and

modeling for the remaining problems as you finish

the string

Exploring Adding Fractions

- http//www.illustrativemathematics.org/pages/fract

ions_progression

- Reflect back on 1/2 1/3, do you have another

way of proving that it is not equal to 2/5? - What might have affected the way that you changed

your response?

- Justify the truth of this statement in at least

two different ways.

What makes a rich task?

- Is the task interesting to students?
- Does the task involve meaningful mathematics?
- Does the task provide an opportunity for students

to apply and extend mathematics? - Is the task challenging to all students?
- Does the task support the use of multiple

strategies and entry points? - Will students conversation and collaboration

about the task reveal information about

students mathematics understanding?

Adapted from Common Core Mathematics in a PLC at

Work 3-5 Larson,, et al

Environment for Rich Tasks

- Learners not passive recipients of mathematical

knowledge - Learners are active participants in creating

understanding and challenge and reflect on their

own and others understandings - Instructors provide support and assistance

through questioning and supports as needed

Task Analysis Protocol Sheet

Fraction Tracks

- Play Fraction Tracks (access online)
- Video http//www.learner.org/vod/vod_window.html?p

id916

How did this activity support the big idea of

equivalent fractions

- Reasoning about the size of a unit fraction based

on the meaning of a unit fraction - Turn and talk with your partners and then we will

share out with the group

Using your materials modify or create a rich

taskWe will review the homework briefly on Day

Three

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Homework

- Before our next meeting
- Use the task you modified with your students
- Bring back one or two student artifacts - ready

to discuss student generated strategies, etc.

(Please remove student names) - Use the Standards for Mathematical Practice

Matrix to reflect on where your classroom falls

on the continuum and be ready to discuss any

activities you used to move your classroom

forward on this scale

Day 3CCSS-M in the Classroom Grades 3-5

Number and Operations Fractions Weaving

Content and Standards for Mathematical Practices

Outcomes Day 2 and 3

- Analyze, adapt and implement a task with the

integrity of the Common Core State Standards. - Understand how to analyze student work with the

Standards for Mathematical Practice and content

standards. - Deepen understanding of the progression of

learning around Numbers and Operations -

Fractions

Looking at Student Work

- From small groups of 3 or 4
- Each person selects 1 or 2 works samples to share

with the group - Follow the Collaboration Protocol to review

student work samples and record information - observations
- Inferences
- implications

Collaboration Protocol-Looking at Student Work

(55 minutes)

- 1. Individual review of student work samples

(10 min) - All participants observe or read student work

samples in silence, making brief notes on the

form Looking at Student Work - 2. Sharing observations (15 min)
- The facilitator asks the group
- What do students appear to understand based on

evidence? - Which mathematical practices are evident in their

work? - Each person takes a turn sharing their

observations about student work without making

interpretations, evaluations of the quality of

the work, or statements of personal reference. - 3. Discuss inferences -student understanding (15

min) - Participants, drawing on their observation of the

student work, make suggestions about the problems

or issues of students content misunderstandings

or use of the mathematical practices. - Adapted from Steps in the Collaborative

Assessment Conference developed by - Steve Seidel and Project Zero Colleagues

Select one group member to be todays

facilitator to help move the group through the

steps of the protocol. Teachers bring student

work samples with student names removed.

- 4. Discussing implications-teaching learning

(10 min) - The facilitator invites all participants to share

any thoughts they have about their own teaching,

students learning, or ways to support the

students in the future. - How might this task be adapted to further elicit

students use of Standards for Mathematical

Practice or mathematical content. - 5. Debrief collaborative process (5 min)
- The group reflects together on their experiences

using this protocol.

Looking at student work

Which Fraction Pair is Greater?

- Think like a fourth grader
- Give one or more reasons for the comparison
- Try not to use models or drawings
- DO NOT USE cross multiplication
- Elementary and Middle School Mathematics

VandeWalle, 2013 p. 311

Create a poster with comparison method and problem

Ways In Which To Compare

- Same size whole (same denominator) - BG
- Same number of parts (same numerator) but

different sized wholes A,D, H - More than/less than one-half or one A,D,F,G,

and H - Closeness to one-half or one C,E,I,J,K, and L

More or Less

- Find a partner
- Place the cards face down
- Take turns drawing a card and deciding which

fraction is more or less - Sort the cards by strategies used to solve them

Read a story and explore the math

- Read Beasts of Burden from The Man who Counted by

Tahan 1993 - Pass out the camel cards and have students model

the story - Rusty Bresser in Math and Literature describes

three days of activities with fifth graders,

based on this story

Exploring Multiplying Fractions

- http//www.illustrativemathematics.org/pages/fract

ions_progression - Semi-concrete explanation
- Multiplying part 2
- Abstract explanation
- Multiplying part 1

Paper folding

- I planted half my garden with vegetables this

summer. One third of the half that is vegetables

is planted with green beans. What fraction of the

whole garden is planted with green beans?

(No Transcript)

Math Matters

- When two fractions are multiplied it is based on

a fraction as an operator. - A fraction is operating on another number and

changes the other number. The use of the word

of to multiply is based on the operator

interpretation of fraction when we multiply ½

and 8 we are taking one half of eight. - p. 125 Math Matters Chapin and Johnson

Finding a fraction of a fraction

- Conceptually with no subdivisions
- Pictures YES Algorithms - NO

Finding a fraction of a fraction

Exploring Dividing Fractions

- http//www.illustrativemathematics.org/pages/fract

ions_progression

Dividing by one half

- Shauna buys a three-foot-long sandwich for a

party. She then cuts the sandwich into pieces,

with each piece being 1/ 2 foot long. How many

pieces does she get? - Phil makes 3 quarts of soup for dinner. His

family eats half of the soup for dinner. How many

quarts of soup does Phil's family eat for dinner? - A pirate finds three pounds of gold. In order to

protect his riches, he hides the gold in two

treasure chests, with an equal amount of gold in

each chest. How many pounds of gold are in each

chest? - Leo used half of a bag of flour to make bread. If

he used 3 cups of flour, how many cups were in

the bag to start? - Illustrative Math Example

VideoCathy Humphreys

- Defending Reasonableness Division of Fractions
- 1 2/3
- Use the Standards for Mathematical Practice

Matrix to reflect on where this classroom falls

on the continuum and be ready to discuss any

activities you could use to move this classroom

forward on the scale

Standards for Mathematical Practice Matrix

Claim 4 Modeling and Data Analysis

Claim 4 Students can analyze complex, real-world

scenarios and can construct and use mathematical

models to interpret and solve problems.

- Apply mathematics to solve problems arising in

everyday life, society, and the workplace. - Construct, autonomously, chains of reasoning to

justify mathematical models used, interpretations

made, and solutions proposed for a complex

problem. - State logical assumptions being used.
- Interpret results in the context of a situation.
- Analyze the adequacy of and make improvement to

an existing model or develop a mathematical

model of a real phenomenon. - Identify important quantities in a practical

situation and map their relationships. - Identify, analyze, and synthesize relevant

external resources to pose or solve problems.

Assessment Claims for Mathematics

Overall Claim (Gr. 3-8)

Overall Claim (High School)

Claim 1 Concepts and Procedures

Claim 2 Problem Solving

Claim 3 Communicating Reasoning

Claim 4 Modeling and Data Analysis

Looking at SBAC Performance TaskDepth of

Knowledge and Mathematical Practices, three lenses

- What are the specific content standards in this

performance task? - What is the depth of knowledge of these tasks?
- Which mathematical practices do they promote?

Planting TulipsDOMAINS Operations and Algebraic

Thinking, Number and OperationsFractions,

Measurement and Data

What makes a rich task?

- Is the task interesting to students?
- Does the task involve meaningful mathematics?
- Does the task provide an opportunity for students

to apply and extend mathematics? - Is the task challenging to all students?
- Does the task support the use of multiple

strategies and entry points? - Will students conversation and collaboration

about the task reveal information about

students mathematics understanding?

Adapted from Common Core Mathematics in a PLC at

Work 3-5 Larson,, et al

Environment for Rich Tasks

- Learners not passive recipients of mathematical

knowledge - Learners are active participants in creating

understanding and challenge and reflect on their

own and others understandings - Instructors provide support and assistance

through questioning and supports as needed

Using your materials modify or create a rich task

or one that is more conceptually focused for your

students

Top Resources for Math Educators

- RMC website http//www.mathsci4wa.org/domain/61
- OPSI website http//www.k12.wa.us/Corestandards/d

efault.aspx

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