Numbers and Operations in Base Ten

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Numbers and Operations in Base Ten
Success Implementing CCSS for K-2 Math
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Introductions
3
K 2 Objectives

• Reflect on teaching practices that support the
  shifts (Focus, Coherence, Rigor) in the Common
  Core State Standards for Mathematics.
• Deepen understanding of the progression of
  learning and coherence around the CCSS-M for
  Number and Operations in Base 10
• Analyze tasks and classroom applications of the
  CCSS for Number and Operations in Base 10

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Success Implementing CCSS for K-2 Math

• Number and Operations in Base 10

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Why CCSS?

• Gretas Video Clip

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Common Core State Standards

• Define the knowledge and skills students need for
  college and career
• Developed voluntarily and cooperatively by
  states more than 46 states have adopted
• Provide clear, consistent standards in English
  language arts/Literacy and mathematics

Source www.corestandards.org
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What We are Doing Doesnt Work

• Almost half of eighth-graders in Taiwan,
  Singapore and South Korea showed they could reach
  the advanced level in math, meaning they could
  relate fractions, decimals and percents to each
  other understand algebra and solve simple
  probability problems.
• In the U.S., 7 percent met that standard.
• Results from the 2011 TIMMS

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Theory of Practice for CCSS Implementation in WA

• 2-Prongs
• 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.
• 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

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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
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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
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Focus, Coherence Rigor
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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

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Focus on the Major Work of the Grade

• Two levels of focus
• Whats in/Whats out
• The shape of the content

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Shift 1 Focus Key Areas of Focus in Mathematics
Grade Focus Areas in Support of Rich Instruction and Expectations of Fluency and Conceptual Understanding
K-2 Addition and subtraction - concepts, skills, and problem solving and place value
3-5 Multiplication and division of whole numbers and fractions concepts, skills, and problem solving
6 Ratios and proportional reasoning early expressions and equations
7 Ratios and proportional reasoning arithmetic of rational numbers
8 Linear algebra and linear functions
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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.

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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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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.

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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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Coherence Across the Grades?

• 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

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Coherence Within A Grade

• Use addition and subtraction within 100 to solve
  word problems involving lengths that are given in
  the same units, e.g., by using drawings (such as
  drawings of rulers) and equations with a symbol
  for the unknown number to represent the problem.
• 2.MD.5

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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.

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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.

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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)

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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

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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

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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

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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.

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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.

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Mathematical ProgressionK-5 Number and
Operations in Base Ten

• Everyone skim the OVERVIEW (p. 2-4)
• Divide your group so that everyone has at least
  one section
• position, base-ten units, computations,
  strategies and algorithms, and mathematical
  practices
• Read your section carefully, share out 2 big
  ideas and give at least one example from your
  section

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Mathematical ProgressionK-5 Number and
Operations in Base Ten

• Read through the progression document at your
  grade level.
• Discuss with your grade level team and record the
  following on your poster
• Big ideas
• Progression within the grade level
• What is this preparing students for?

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Big Ideas

• Rather than learn traditional algorithms,
  childrens struggle with the invention of their
  own methods of computation will enhance their
  understanding of place value and provide a firm
  foundation for flexible methods of computation.
  Computation and place value development need not
  be entirely separated as they have been
  traditionally.
• Van de Walle 2006

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Tens, Ones and Fingers

• Where does this activity fall in the progression
  and what clusters does this address?
• How can this activity be adapted?

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Standards for Mathematical Practices

• Graphic

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The Standards for Mathematical Practice

• Skim The Standards for Mathematical Practice
• Read The Standards for Mathematical Practice
  assigned to you
• Reflect
• What would this look like in my classroom?
• Review the SMP Matrix for your assigned practices
• Add to your recording sheet if necessary

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The Standards for Mathematical Practice

• Return to your home group and share out your
  practices.

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Mathematical Practices in Action

• http//www.learner.org/resources/series32.html?pop
  yespid873
• Using the matrix, what Mathematical Practices
  were included in these centers?
• What major and supporting clusters are addressed?

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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
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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

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Depth of Knowledge (DOK)
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Bring it all together

• Divide into triads
• Watch and reflect based on
• What Makes a Rich Task?
• DOK
• Standards (which clusters and SMP were
  addressed?)
• As a group, using all three pieces of
  information, decide
• Is this a meaningful mathematical lesson?

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Bringing It All TogetherCounting Collections
Video
http//vimeo.com/45953002
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Estimating Groups of Tens and Ones

• Where does this activity fall in the progression
  and what clusters does this address?
• What mathematical practices are used?
• What makes this a good problem?
• What is the DOK?
• How can this activity be adapted?

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Lets Analyze a Task
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Base-Ten Riddles

• Where does this activity fall in the progression
  and what clusters does this address?
• What mathematical practices are used?
• What makes this a good problem?
• What is the DOK?
• How can this activity be adapted?

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https//www.teachingchannel.org/videos/second-grad
e-math-lesson
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Patterns on a 100 chart
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Adapting a Task

• In your group, think of ways to adapt this problem

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More and Less on the Hundred Chart

• Where does this activity fall in the progression
  and what clusters does this address?
• What mathematical practices are used?
• What makes this a good problem?
• What is the DOK?
• How can this activity be adapted?

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Big Ideas

• A good base-ten model for ones, tens and hundreds
  is proportional. That is, a ten model is
  physically ten times larger than the model for
  one, and a hundred model is ten times larger than
  the ten model.
• No model, including a group able model, will
  guarantee that children are reflecting on the
  ten-to-one relationship in the materials. With
  pre-grouped models, we need to make an extra
  effort to see that children understand that a ten
  piece really is the same as ten ones.
• Van de Walle 2006

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Homework Tasks

• At our next meeting we are going to analyze
  student work
• For your grade level task
• Read through it at least twice
• Solve it
• Complete the Rich Task Pre-Planning Sheet

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Reflection

• What is your current reality around classroom
  culture?
• What can you do to enhance your current reality?

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Next Meeting
Questions? Contact? Next Meeting?
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Operations and Algebraic Thinking
Success Implementing CCSS for K-2 Math
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Todays Objectives

• Develop a deeper understanding of how students
  progress in their understanding of the Common
  Core clusters related to operations and algebraic
  thinking in grades PreK-2.
• Learn engaging instructional strategies through
  hands on activities that connect content to the
  mathematical practices.

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Quiz

• What is your definition of Operations and
  Algebraic Thinking?

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Collaboration Protocol-Looking at Student Work

• 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.

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Mathematical ProgressionOperations and
Algebraic Thinking

• Read through your grade level progression
  document for your grade level. Discuss with your
  group and record on your poster.
• What fluencies are required for this domain?
• What conceptual understandings do students need?
• How can this be applied?

Rigor
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Major and Supporting ClustersA Quick Review

• One of the shifts from our current Math Standards
  to the Common Cores State Standards is the idea
  of focus. Students spend more time learning
  deeply with fewer topics. Grade level concepts
  have been divided into major, supporting and
  additional clusters.

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Bring it all togetherAGAIN!

• Divide into triads
• Watch and reflect based on
• What Makes a Rich Task?
• DOK
• Standards (which clusters and SMP were
  addressed?)
• As a group, using all three pieces of
  information, decide
• Is this a meaningful mathematical lesson?

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Bringing It All TogetherWheels Video
http//www.learner.org/resources/series32.html
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Lunch
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Eyes, Fingers, and Legs

• Where does this activity fall in the progression
  and what clusters does this address?
• What mathematical practices are used?
• What is the DOK?
• How might students solve these problems?
• What student misconceptions might arise?
• How can this activity be adapted?

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Homework Adapting a Task
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Welcome Back

• Session 4

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Objectives for the day

• Develop a deeper understanding of how students
  progress in their understanding of the Common
  Core clusters related to operations and algebraic
  thinking in grades PreK-2.
• Learn engaging instructional strategies through
  hands on activities that connect content to the
  mathematical practices.

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HW - Analyzing and Adapting Tasks

• With a partner, go through the Deep Task Analysis
  Sheet on the task you adapted
• Consider
• Do you achieve the outcome you expected?
• How well did your students engage in the task?
• Where their any surprises?
• What questions would ask students based on their
  work?
• How would you change this task for next time?

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Reviewing the SMP

• Reflect on the work we have done and your
  students have done and write the SMP assigned to
  your table in student friendly language.
• Record your thoughts on a poster.

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WA Kids
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Smarter-Balanced Assessment Consortium (SBAC)

• In your groups, work through the assessment task
• Consider what standards are necessary for
  students to master in my grade level so they can
  be successful with this task?

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Consider WA Kids and SBAC

• What are some implications of WA Kids and SBAC in
  mathematics?
• Look at your major and supporting clusters and
  the progression documents what are the
  milestones students need to attain in order to
  prepare for the 3rd grade assessment?
• Where should we focus instruction if students
  arent coming to us prepared?
• What can we do to ensure our students are
  prepared?

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Classroom Connections.

• Students using Quick Images
• As you watch this video Reflect on the following
• Where does this fit on our progression?
• How can this be adapted to meet the needs of your
  students?
• What mathematical practices did you observe?
• What can be assessed from this activity?
• https//www.teachingchannel.org/videos/visualizing
  -number-combinations?fd1

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Addition and Subtraction Situations

• There were 7 children at the park. Then 4 more
  showed up. How many children were at the park all
  together?
• There were 7 children at the park. Some more
  showed up. Then there were 11 children in all.
  How many more children came?
• There were some children at the park. Four more
  children showed up. Then there were 11 children
  at the park. How many children were at the park
  to start with?
• Now consider different ways you can use these
  scenarios and what makes one more difficult than
  another. One might ask students..

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Which equation matches?

• 4 10
• 6 4
• 6 10

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Teacher Questioning Strategies

• How can the questions you ask move students
  thinking forward? How should it differ for
  struggling or high achieving students?
• Using the DOK, identify the level of some of the
  questions on the matrix provided.
• What can students do to engage in the mathematics
  more deeply?
• When you look at the website resources, what
  might work in your classroom?
• http//www.fcps.org/cms/lib02/MD01000577/Centricit
  y/Domain/97/The20art20of20questioning20in20ma
  th20class.pdf

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Connecting Literature with Math Concepts.
Other classroom connection ideas for OA? The
hand game, Go Fish, Memory, Race to 10.
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Evaluation and Next Steps.