What Are Tasks How Are They Used in the 612 Mathematics GPS Why Should You Care

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What Are Tasks?How Are They Used in the 6-12
Mathematics GPS? Why Should You Care?

• Tom Ottinger
• Reinhardt College
• 7300 Reinhardt College Circle
• Waleska, GA 30183
• 770-720-5596
• tpo_at_reinhardt.edu

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What Are Tasks?

• Two Definitions
• job assigned to somebody a piece of work that
  somebody is given to do, usually short in
  duration or with a deadline
• assignment a piece of work or an assignment,
  especially one that is important or difficult
• -- MSN Encarta

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What Are Instructional Tasks in Mathematics?

• Segments of classroom activity devoted to
  development of mathematical ideas
• -- Stein, et. al.
• Implementing Standards-Based
  Mathematics Instruction

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Good Instructional Tasks in Mathematics . . .

• Apply mathematical concepts within a context
• Require reasoning, mathematical thinking,
  problem
• solving
• Incorporate a variety of mathematical concepts
  and skills
• Have a high level of cognitive demand
• Stimulate student interest and engagement
• Include multiple parts rather than a single
  problem
• Encourage investigation and exploration using a
  variety
• of representations and approaches
• Encourage understanding and sense-making
• Require explanation and/or justification

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• Suppose tulips are on sale for 0.60 per bulb.
  The
• shipping cost is 3.00 for any size order.
  What number
• of bulbs can you order if you have 14.00 to
  spend?
• 18 2. 32 3. 14 4. 20

• Apply mathematical concepts within a context
• Require reasoning, mathematical thinking,
  problem
• solving
• Incorporate a variety of mathematical concepts
  and skills
• Have a high level of cognitive demand
• Stimulate student interest and engagement
• Include multiple parts rather than a single
  problem
• Encourage investigation and exploration using a
  variety
• of representations and approaches
• Encourage understanding and sense-making
• Require explanation and/or justification

                        
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Is this a Good Task?

• Apply mathematical concepts within a context
• Require reasoning, mathematical thinking,
  problem
• solving
• Incorporate a variety of mathematical concepts
  and skills
• Have a high level of cognitive demand
• Stimulate student interest and engagement
• Include multiple parts rather than a single
  problem
• Encourage investigation and exploration using a
  variety
• of representations and approaches
• Encourage understanding and sense-making
• Require explanation and/or justification

                        
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How Are Tasks Used in the 6-12 Mathematics GPS?

• Launching Tasks
• Used to introduce a topic.
• Developmental Tasks
• Used for exploration and concept development
• Culminating Tasks
• Used to tie it all together
• Assessment Tasks
• Used to assess student progress or mastery of
  standards

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How Are Tasks Used in the 6-12 Mathematics GPS?

• Strands used in the 6-12 GPS
• Number and Operations
• Measurement
• Algebra
• Geometry
• Data Analysis and Probability
• Strands often contain several standards
• Standards often contain several elements

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• ALGEBRA
• Students will simplify radical expressions,
  operate with polynomials and explore functions.
• MMIA2. Students will explore and interpret the
  characteristics of functions using graphs, tables
  and simple algebraic techniques.
• a. Represent functions using function notation.
• b. Graph basic functions of the form f(x) xn,
  where n 1 to 3, f(x) vx, f(x) x, f(x)
  1/x.
• c. Investigate and explain the characteristics of
  a function.s domain, range, zeros and intercepts,
  relating them to a given context.
• d. Use graphs and tables to determine the
  intervals of increase and decrease of a function
  and its maximum and minimum values.
• e. Explore rates of change as a comparison of
  constant versus variable rates of change. Compare
  rates of change of linear, quadratic, square
  root, and other function families.
• GEOMETRY
• Students will explore, understand and use the
  formal language of reasoning and justification.
  Students will apply properties of polygons, and
  determine distances and points of concurrence.
• MMIG2. Students will discover, prove, and apply
  properties of triangles, quadrilaterals, and
  other polygons.
• a. Determine the sum of interior and exterior
  angles in a polygon.
• b. Understand and use the Triangle Inequality,
  Side-Angle Inequality, Triangle Exterior Angle
  Inequality.
• c. Understand and use congruence postulates and
  theorems for triangles.(SSS, SAS, ASA, AAS,
  HYPOTENUSE-LEG).

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How Are Tasks Used in the 6-12 Mathematics GPS?

• Good tasks address multiple elements within a
  standard
• Better tasks address multiple standards within a
  strand
• Best tasks address multiple strands
• This content integration encourages students to
• Make connections among mathematical ideas
• Recognize that mathematical ideas can be applied
  in a variety of very different ways
• Synthesize knowledge in the solution of new
  problems

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Is Everything Taught through Tasks?

• No
• the more you can teach through tasks, the
  better
• During a task
• Occasional short mini-lessons
• (less than 30 mins.)
• After a task
• Followup discussion, including
• Multiple approaches, representations, solutions
  as appropriate
• Mathematical content made explicit
• Mathematical vocabulary and notation formalized
• Tasks are often done in parts and spread over
  several days

but
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Surprise Birthday Party(6th grade task)

• Sue decided to give Maria a surprise birthday
  party. She ordered a large ice cream cake for the
  party, but is not sure how many people to invite.
• How will the number of people attending the party
  affect the portion of cake that each person gets?
• Is there a relationship between the number of
  people attending the party and the portion of the
  cake that each person gets? Explain your answer
  and illustrate with pictures, a table, and a
  graph.
• What is the constant of proportionality and what
  does it represent in the context of this problem?

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Surprise Birthday Party(6th grade task)

• Sue decided to give Maria a surprise birthday
  party. She ordered a large ice cream cake for the
  party, but is not sure how many people to invite.
• How will the number of people attending the party
  affect the portion of cake that each person gets?
• Is there a relationship between the number of
  people attending the party and the portion of the
  cake that each person gets? Explain your answer
  and illustrate with pictures, a table, and a
  graph.
• In this problem, something changes and something
  stays constant. What stays constant?

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Surprise Birthday Party(6th grade task)

• Inverse proportionality

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Good tasks can be used badly

• Too directed
• Students merely do what theyre told
• Very little mathematical thinking required

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Good tasks can be used badly

• Graph y x2 25.
• Write down the x-coordinates of the points where
  the curve crosses the y-axis.
• Solve x2 25 0 by adding 25 to both sides and
  then taking the square roots of both sides.
• Compare the solutions to the x-intercepts. What
  do you notice?

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Good tasks can be used badly

• Too directed
• Students merely do what theyre told
• Very little mathematical thinking required
• Too undirected
• Expectations not clear
• Students frustrated by having absolutely no idea
  what to do or why

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Good tasks can be used badly

• Using several examples, explore the relationship
  between the zeros of a quadratic function and the
  solutions for a related quadratic equation.
• Explain why this relationship is true.
• Will the relationship also apply to functions and
  equations of higher degree? Why or why not?

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Good tasks can be used badly

• Too directed
• Students merely do what theyre told
• Very little mathematical thinking required
• Too undirected
• Expectations not clear
• Students frustrated by having absolutely no idea
  what to do or why
• Some students allowed not to participate
• Too hard for them??
• Reduced cognitive demand
• Often results from teachers who sincerely want to
  help
• Mathematical content not made explicit in
  follow-up discussion

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Where do you get tasks?

• Math Frameworks at www.georgiastandards.org
• NCTM Illuminations website
• illuminations.nctm.org
• NCTM Publications, particularly the Navigations
  series
• PBS Mathematics Resources www.pbs.org/teachers
• math.kennesaw.edu/mathed/NMMC
• intermath.coe.uga.edu
• www.exemplars.com (Requires purchase of a school
  license. Student work and commentary included.)
• Write your own

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Where do you get tasks?

• Often tasks can be adapted to
• Address different or additional content standards
• Provide differentiation through tiered
  instruction
• Your students arent necessarily the same as all
  other students. Expect to modify tasks to be
  appropriate for them.

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Why Should You Care?

• Your students will
• Have a deeper, richer understanding of
  mathematics
• Be better able to apply mathematics in the
  solution of realistic problems
• Enjoy mathematics more, value it more, and be
  more likely to study further mathematics
• Perform better on End of Course Tests, which will
  be revised to address the new standards