Complex Math Tasks Lead to Accountable Talk: Evidence of Mathematical Practice Standards

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Complex Math Tasks Lead to Accountable Talk
Evidence of Mathematical Practice Standards
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Student-Centered Mathematics

• Instruction involves posing tasks that will
  engage students in the mathematics they are
  expected to learn. Then, by allowing students to
  interact with and struggle with the mathematics
  using THEIR ideas and THEIR strategies a
  student-centered approach the mathematics they
  learn will be integrated with their ideas it
  will make sense to them, be understood, and be
  enjoyed.
• Teaching Student-Centered Mathematics John Van
  de Walle, 2006

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Essential Question
What kinds of learning tasks and Accountable
Talk strategies allow students the opportunity to
demonstrate, both orally and in writing, their
mathematical proficiency?
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Learning Objectives

• Identify the attributes of a rich, instructional,
  problem-based approach and how it can support
  access to the Standards for Mathematical Practice
• Use Accountable Talk and questioning strategies
  to promote mathematical discourse

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Attributes of Learning Tasks Quick Write
What are the attributes of learning tasks that
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Attributes of Learning Tasks
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Designing and Selecting Problem-Based Tasks

• A problem or task for learning mathematics is any
  task for which the students have no prescribed or
  memorized rules, nor is there any perception that
  there is a specific route to the solution.
• Three points to remember
• What is problematic in the task must be the
  mathematics.
• Tasks must be accessible to your students.
• Tasks must require justifications and
  explanations for answers or methods.

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• There are 125 sheep
• and
• 5 dogs in a flock.

How old is the shepherd?
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A Students Response

• There are 125 sheep and 5 dogs in a flock.
• How old is the shepherd?

125 x 5 625 extremely big 125 5 130 too
big 125 - 5 120 still big 125 ? 5 25 That
works!
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A Rich Task?
Put these two fractions on the number line below.
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Which content and practice standards does
this task address?
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Connecting the Number Line FractionsMiniLesson
to the Content and Practice Standards
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Functions as Tables, Graphs, Words, and Formulas
What is the rule for this table? What does the
horizontal axis show? What does the vertical axis
show? What is the output when we input 2? What
is the output when we input 6?
A Rich Task?
(Americas Choice 2006, 91100)
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Connecting the Number Line FractionsMiniLesson
to the Content and Practice Standards
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A RICH TASK?

• My seven friends and I are going to share two
  pizzas for lunch. The pizzas are divided into
  eight slices each. How much pizza will each
  person get? (Draw a picture to represent your
  solution.)

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Revised Pizza Task A Rich Task?

• You are invited to go out for pizza with several
  friends. When you get there, you find your
  friends sitting at two different tables.
• You can join either group. If you join the first
  one, there will be a total of 4 people in the
  group and all of you will be sharing 6 small
  pizzas.
• If you join the other group, there will be 6
  people in the group and all of you will be
  sharing 8 small pizzas.
• If your goal was to get as much pizza as
  possible, which group would you join?
• Explain your thinking and show your math.
• Which content and practice standards does this
• task address?

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Model Lesson The Tile Problem

• Your summer job is to repair tiles in the corners
  of roofs. The tiles you will use are 1 foot by 1
  foot squares. Jobs are identified as follows

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Middle School QuestionsThe Tile Problem

• Draw jobs 3 and 6 below.
• What do you need to know to draw the figures?
• After laying the tile, you need to apply some
  waterproof caulking around the outside of the
  pattern. The caulking is sold by the foot. Job
  2 requires 8 feet of caulking. Job 5 needs 20
  feet of caulking.
• Your supervisor assigns you to do job 12.
• How much caulking will you need to complete the
  job? Explain how you figured this out.
• Write a rule that would allow you to determine
  how much caulk you would need for any job.
  Explain why your rule should always work.
• Which content and practice standards does this
• task address?

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High School QuestionsThe Tile Problem

• Draw jobs 3 and 6 below.
• What do you need to know to draw the figures?
• After laying the tile, you need to apply some
  waterproof caulking around the outside of the
  pattern. The caulking is sold by the foot. Job
  2 requires 8 feet of caulking. Job 5 needs 20
  feet of caulking.
• Your supervisor assigns you to do job 12.
• How may tiles and how much caulking will you need
  to complete the job? Explain how you figured this
  out.
• Write a rule that would allow you to determine
  how many tiles and how much caulk you would need
  for any job. Can you identify any patterns
  between the two rules?
• Which content and practice standards does this
• task address?

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A RICH MATH TASK RITUAL

• Do the math
• Work solo
• Share with a partner
• Share with the group
• What are the benefits?

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Questions That Connect to Practices Accountable
Talk

• The habit of connecting my talk to the talk of
  others models the mental habit of connecting the
  idea on my mind right now to ideas previously
  thought.
• Phil Daro

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Aligning Questions to theCommon Core State
Standards Mathematical Practices
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Standards for Mathematical Practice

• 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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Practice 1 Making sense of problems and
Persevere in solving them

• Huh What Are You Thinking?
• Does this make sense?
• What does this term mean?
• What do you want to know? Put it in a sentence.
• Can you break the problem into simpler problems
  (multi-step)?

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Practice 2 Reason abstractly and
quantitatively

• Quantities
• What quantities are in the problem?
• What are the relationships among the quantities
  in the situation?
• How can we label the quantities?
• What inferences can we draw?

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Practice 3 Construct viable arguments and
critique reasoning of others

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• Is it true?
• Is it always true?
• Is it never true?
• When is it true? Under what conditions?
• Is there a counter example?
• A contradiction?
• Is there another way to prove that this statement
  is true or not true?

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Practice 4 Model with mathematics

• Can you represent the idea in words, tables,
  diagrams, formulas, or graphs and explain the
  relationships between them?
• Can you create your own visual representation of
  this situation?
• Can you solve the problem in more that one way?

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Practice 5 Use appropriate tools strategically

• Why did you choose to use this model
  (manipulative) to help you understand the task?
• How does your model compare to someone elses?
• Is there an additional representation for this
  concept?
• Was this tool the most efficient?

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Practice 6 Attend to Precision

• How would you explain this to someone who didnt
  understand?
• How does your statement link to what others have
  said?
• How does what you say add to what Anna just said?
• Is it a justification? A special case? A
  generalization?
• Evidence? A supporting argument?
• A logical extension? A contradiction?
• A counterexample?

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Practice 7 Look for and make use of structure

• Can you identify the basic component in this
  structure?
• Can you break the problem into smaller
  components?
• Is there a pattern?
• Can you simplify the situation?

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Practice 8 Look for and express regularity in
reasoning

• Do you notice a pattern?
• Is there anything in this pattern that is
  repeating?
• Is it possible to make a generalization? A rule?

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Language Strategies for MathAccountable Talk

• When students are challenged to think and reason
  about mathematics and to communicate the results
  of their thinking to others verbally or in
  writing, they are faced with the task of stating
  their ideas clearly and convincingly to an
  audience.
• From Principles and Standards, p.85

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Understanding Mathematics

• One hallmark of mathematical understanding is
  the ability to justify, in a way appropriate to
  the students mathematical maturity, why a
  particular mathematical statement is true or
  where a mathematical rule comes from. There is a
  world of difference between a student who can
  summon a mnemonic device to expand a product such
  as (ab)(xy) and a student who can
  explain where the mnemonic comes from.