Making use of structure to make sense of mathematics

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Making use of structure to make sense of
mathematics

• National Council of Teachers of Mathematics
• Annual Meeting Philadelphia, PA
• Thursday, April 26, 2012
• Connie Laughlin, UW-Milwaukee, Milwaukee WI
• Beth Schefelker, Milwaukee Public Schools, WI

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Learning Intention

• We are learning to
• Explore examples that look for and make use of
  structure.
• To understand Standard for Math Practice 7 Look
  for and Make Use of Structure

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Structure

• What does structure mean
• of a building?
• of a classroom?
• in mathematics?

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What is Area?

• Write your answer on an index card.
• Discuss your tables answers and come to
  consensus.

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Mathematical Practice Standard 7 Look for and
Make Use of Structure

• Mathematically proficient students look closely
  to
• discern a pattern or structure.students will see
  that
• 7 x8 equals the well remembered 7x5 7x3, in
• preparation for learning about the distributive
  property.
• But how does structure help us make connections
  within and between math strands?

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Mathematical Practice

• Recognize the significance in concepts and models
  and use the patterns or structure for solving
  related problems.
• Look for Tool Howard County Public Schools
  Office of Secondary Mathematics, May 2011.

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Look for and make use of structure

• Student Behaviors

• Teacher Actions

• Appreciate and look for connections in math
• Recognize the parts of a whole
• Reflect on processes used and the application of
  those process as they apply to new but similar
  situations

• Provide multiple examples from which students can
  draw patterns or rules.
• Model of think aloud in solving problems
• Facilitate discovery activities
• Encourage student discourse and provide multiple
  viewpoints.
• Model how problems could be broken down.

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Success Criteria

• We will know we were successful when we are able
  to use structure to explain and justify area
  formulas for selected quadrilaterals.

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Explore Area formula for parallelograms

• Follow the steps below to write a formula for the
  area of a parallelogram.
• Step 1 Use two different width straight edges
  to construct a large parallelogram on large patty
  paper.
• Step 2 Show how you can split the parallelogram
  and rearrange the pieces to form a rectangle.
• Compare your results with your table partners.
  Come to consensus as to a formula that could be
  used.

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Explore Area formula for triangles

• Follow the steps below to write a formula for the
  area of a triangle.
• Step 1 Each person at the table should
  construct a different large triangle on patty
  paper.
• Step 2 Place a 2nd patty paper over the first
  and copy the triangle. Form a parallelogram from
  the two triangles.
• Compare your results with your table partners.
  Come to consensus as to a formula that could be
  used to find the area of a triangle.

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Explore Area formula for trapezoid

• Step 1 Draw a large trapezoid on your patty
  paper.
• Step 2 Place a second patty paper over the
  first and copy the trapezoid. Form a
  parallelogram from the two trapezoids.
• Compare your results with your table partners.
  Come to consensus as to a formula that could be
  used.

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Explore Area formula for trapezoid

• Step 1 Draw a large trapezoid on your patty
  paper by folding or drawing.
• Step 2 Explore finding the area of a trapezoid
  using triangles.
• Experiment to find as many different ways as you
  can to arrive at a formula for a trapezoid.

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What have you found?

• Turn and Talk at your table
• How does structure of the polygons help make
  sense of the area formulas?

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Success Criteria

• We will know we were successful when we are able
  to use structure to explain and justify area
  formulas for selected quadrilaterals.

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Standard for Mathematical Practice
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Thank You!

• The powerpoint can be downloaded from the
  Milwaukee Mathematics Partnership website
  www.mmp.uwm.edu
• Beth Schefelker
• Math Teaching Specialist, MPS
  schefeba_at_milwaukee.k12.wi.us
• Connie Laughlin,
• Mathematics Consultant, UW-Milwaukee
• connielaughlinm_at_gmail.com