overall outcomes

+

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 teacher’s 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


The activity centers on mathematical understanding, invention, and sense-making by all students.



A teacher’s 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





A teacher’s 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

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

K-2 3-5 6-8 High School

Year 1- 22012-2013

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 andUnit 4: Connecting Algebra and Geometry through Coordinates; Unit 2: Similarity, Proof, and Trigonometry andUnit 3:Extending to Three Dimensions and remaining 2008 WA Standards

Transition Plan for Washington State

+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:

• What’s in/What’s 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 Kong’s curriculum omits 48 percent of the TIMSS items through grade 4, and 18 percent through grade 8.”

– Ginsburg et al., 2005

Content Emphasis by Cluster—Grade 3

Grade 3(supporting cluster)

+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/fractions_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 It’s 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 student’s 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.”

It’s 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 didn’t “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

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

Standards for Mathematical Practice-Digging Deeper

Already Implementing What’s New or Challenging?

Expectations for Students Expectations for Teachers

Standards for Mathematical Practices

+

Shifts in Focus, Coherence and Rigor in the assessment

Let’s look at the

Assessment

Assessment Claims for Mathematics

Overall Claim (Gr. 3-8)

Overall Claim (High School)Claim 1

Concepts and Procedures

Claim 2

Problem SolvingClaim 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.

+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 Tasks:Depth 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

A. Apply mathematics to solve well-posed problems arising in everyday life, society, and the workplace

B. Select and use tools strategically

C. Interpret results in the context of the situation

D. Identify important quantities in a practical situation and map their relationships.

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.

4.NF.B.4cSolve word problems involving multiplication of a fraction by a whole number

Take a Break!

+

Learning Progression

Number and Operations Fractions

+Domain

Standard

Cluster Heading

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





+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

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others. - Read full standard and highlight important ideas

4. Model with mathematics

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

Protocol:Construct 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/pages/ccmathmod_07.htmhttp://www.learner.org/resources/series32.html #37 start at 2:29 and end at 10:00l

http://www.learner.org/series/modules/express/pages/ccmathmod_07.htm

http://www.learner.org/resources/series32.html

http://www.learner.org/series/modules/express/pages/ccmathmod_07.html

Finding Evidence of:Construct 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? 1. Is the task interesting to students?

2. Does the task involve meaningful mathematics?

3. Does the task provide an opportunity for students to apply and extend mathematics?

4. Is the task challenging to all students?

5. Does the task support the use of multiple strategies and entry points?

6. 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

Let’s 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

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


+Outcomes: Day 2 and 3



Deepen understanding of the progression of learning around Number and Operations - Fractions





A teacher’s 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 student’s content misunderstandings or use of the mathematical practices.

Adapted from: Steps in the Collaborative Assessment Conference developed by

Steve Seidel and Project Zero Colleagues

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 student’s 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.

Select one group member to be today’s facilitator to help move the group through the steps of the protocol.

Teachers bring student work samples with student names removed.

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




Claim 2


Communicating ReasoningClaim 4


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.

Claim 2 – Problem Solving

A. Apply mathematics to solve well-posed problems arising in everyday life, society, and the workplace

B. Select and use tools strategically

C. Interpret results in the context of the situation

D. Identify important quantities in a practical situation and map their relationships.

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.

4.NF.B.3Understand a fraction a/b with a > 1 as a sum of fractions 1/b

Claim 3 – Communicating Reason

A. Test propositions or conjectures with specific examples.

B. Construct, autonomously, chains of reasoning that justify or refute propositions or conjectures.

C. State logical assumptions being used.

D. Use the technique of breaking an argument into cases.

E. Distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in the argument—explain what it is.

F. Base arguments on concrete referents such as objects, drawings, diagrams, and actions.

G. Determine conditions under which an argument does and does not apply.

Claim 3: Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others.

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.

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





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?

+Exploring Adding Fractions





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.

Task Analysis Protocol Sheet

Fraction Tracks

Play Fraction Tracks (access online)

Video http://www.learner.org/vod/vod_window.html?pid=916

http://www.learner.org/vod/vod_window.html?pid=916

+How did this activity support the big idea of equivalent fractionsReasoning 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 task

We will review the homework briefly on Day Three

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


+Outcomes: Day 2 and 3



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 student’s content misunderstandings or use of the mathematical practices.

Adapted from: Steps in the Collaborative Assessment Conference developed by

Steve Seidel and Project Zero Colleagues

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 student’s 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.

Select one group member to be today’s facilitator to help move the group through the steps of the protocol.

Teachers bring student work samples with student names removed.

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

1. Same size whole (same denominator) - B&G

2. Same number of parts (same numerator) but different sized wholes – A,D,& H

3. More than/less than one-half or one – A,D,F,G, and H

4. 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


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?

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 subdivisionsPictures – YES Algorithms - NO

+Finding a fraction of a fraction

+ Exploring Dividing Fractions





+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

Video—Cathy 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

A. Apply mathematics to solve problems arising in everyday life, society, and the workplace.

B. Construct, autonomously, chains of reasoning to justify mathematical models used, interpretations made, and solutions proposed for a complex problem.

C. State logical assumptions being used.D. Interpret results in the context of a situation.E. Analyze the adequacy of and make improvement to an

existing model or develop a mathematical model of a real phenomenon.

F. Identify important quantities in a practical situation and map their relationships.

G. Identify, analyze, and synthesize relevant external resources to pose or solve problems.

Claim 4: Students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems.

+ Assessment Claims for Mathematics




Claim 2


Communicating Reasoning

Claim 4


Looking at SBAC Performance Task:Depth of Knowledge and Mathematical Practices, three lensesWhat are the specific content standards

in this performance task?



Planting TulipsDOMAINS: Operations and Algebraic Thinking, Number and Operations—Fractions, Measurement and Data

Top Resources for Math Educators RMC website: http://www.mathsci4wa.org/domain/61

OPSI website: http://www.k12.wa.us/Corestandards/default.aspx

overall outcomes

Documents

algebraic thinking oa

algebraic thinking oa4

number system ns expressions

algebraic thinking oaand

algebraic thinking oanumber

data md1 operations

cardinality cc operations

wa standards3 number