Learning

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Learning
Grade 11 Mathematics Curriculum Revision
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Learning

• The revised curriculum supports students learning
  mathematics with understanding and actively
  building new knowledge from experience and prior
  knowledge.

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Balanced Instruction Activity
TASK
Selecting from the expectations provided for your
course, examine how expectations can focus
on Facts and Procedures as well as
Conceptual Understanding.
Expectation Measurement and Trigonometry,
MT3.04 Solve problems involving the surface
areas of prisms, pyramids, and cylinders, and the
volumes of prisms, pyramids, cylinders, cones,
and spheres, including problems involving
combinations of these figures, using the metric
system or the imperial system, as appropriate.
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Balanced Instruction Activity
TASK In expert groups,

• with a partner, choose one of the expectations
  provided for your course
• develop two (2) short answer questions to
  address the expectation
• one focusing on Facts and Procedures
• the second focusing on Conceptual
  Understanding

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Balanced Instruction Activity

procedural fluency skill in carrying out
procedures flexibly, accurately, efficiently, and
appropriately conceptual understanding
comprehension of mathematical concepts,
operations, and relationships
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Balanced Instruction Activity
REPORTING

• participants share their work at each table
• groups post their work on the wall designated
  for their course

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Balanced Instruction Activity
Reflection As a Group

• Does the balance between the instructional
  emphasis on procedural fluency versus the
  emphasis on conceptual understanding vary
  depending on the students?
• Does the balance vary depending on the course?
• Does the balance vary depending on whether
  the concept is new or an extension?
• Is there an appropriate order ?

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Personal Reflection
Balanced Activity ReflectionWhat does an
appropriate balance mean to you and how does this
impact on your students long term success in
mathematics?
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Developing Understanding
We use the ideas we already have (blue dots)
to construct new ideas (red dot). The more ideas
we use and the more connections we make, the
better we understand.
John Van de Walle
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Conceptual Understanding

• Conceptual understanding supports retention. When
  facts and procedures are learned in a connected
  way, they are easier to remember and use and can
  be reconstructed when forgotten.
• Hiebert and Wearne 1996 Bruner 1960, Katona 1940

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Improving Coherence

• Determine, through investigation using
  technology, and describe the roles of the
  parameters, a, k, d, and c, in the functions of
  the form y af(k(x d)) c in terms of
  transformations on the graphs of f(x) x, f(x)
  x2, f(x) vx and f(x) 1/x, (i.e., trans-
  lations, reflections in the axis, vertical and
  horizontal stretches and compressions) Sample
  problem Investigate the graph of f(x)
  3(x-d)25 for various values of d, using
  technology, and describe the effect of changing d
  in terms of a transformationnew

• Determine, through investigation using
  technology, and describe the roles of the
  parameters, a, k, d, and c, in the functions of
  the form y af(k(x d)) c in terms of
  transformations on the graphs of f(x) ax, for
  agt0, not equal to 1, (i.e., translations,
  reflections in the axis, vertical and horizontal
  stretches and compressions) Sample problem
  Investigate the graph of f(x) 3x-d -5 for
  various values of d, using technology, and
  describe the effect of changing d in terms of a
  transformation f(x) ltNEWgt

• Determine, through investigation using
  technology, and describe the roles of the
  parameters, a, k, d, and c, in the functions of
  the form y af(k(x d)) c in terms of
  transformations on the graphs of f(x) sin x and
  f(x) cos x, with angles expressed in degrees
  (i.e., translations, reflections in the axis,
  vertical and horizontal stretches and
  compressions) Sample problem Investigate the
  graph of f(x) 2sin(x-d) 10 for various values
  of d, using technology, and describe the effect
  of changing d in terms of a transformation
  ltREVISEDgt

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Improving Coherence
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Improving Coherence
Quadratic Functions
Trigonometric Functions

• Calculus
• Conceptual
• Procedural
• Theoretical

Linear Functions
Rational Functions
Polynomial Functions
Exponential Functions
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Improving Articulation Across The Grades
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Improving Articulation Across The Grades
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Developing Concepts Through Investigation
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Developing Concepts Through Investigation
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Representations
Numerical Representation

• Graphical Representation

Concrete Representation
Algebraic Representation
f(x) 2x - 1
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Developing Concepts Through Investigation
Inverse A Reverse Process
Numerical Representation
Graphical Representation
Algebraic Representation
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Developing Concepts Through Investigation
Algebraic Representation
x
? Subtract 2 ? Add 2 x -
2 ? Multiply by (-10) ? Divide by
(-10) -10(x 2) ? Add 50
? Subtract 50
x
Inverse A Reverse Process
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Culminating With Solving Problems
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Gallery Walk