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Making use of structure to make sense of mathematics

This presentation explores examples that look for and make use of structure in mathematics, specifically focusing on Standard for Math Practice #7: Look for and Make Use of Structure. The importance of structure in mathematical concepts and models is highlighted, and strategies for recognizing and utilizing structure to solve problems are discussed. Various area formulas for selected quadrilaterals are explored, with an emphasis on the relationship between structure and the development of these formulas.

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Making use of structure to make sense of mathematics

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  1. 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

  2. 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

  3. Structure • What does structure mean… • of a building? • of a classroom? • in mathematics?

  4. What is Area? • Write your answer on an index card. • Discuss your table’s answers and come to consensus.

  5. 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?

  6. 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.

  7. 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.

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

  9. 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.

  10. 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.

  11. 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.

  12. 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.

  13. What have you found? Turn and Talk at your table: How does structure of the polygons help make sense of the area formulas?

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

  15. Standard for Mathematical Practice

  16. Thank You! The powerpoint can be downloaded from the Milwaukee Mathematics Partnership website www.mmp.uwm.edu Beth Schefelker Math Teaching Specialist, MPS schefeba@milwaukee.k12.wi.us Connie Laughlin, Mathematics Consultant, UW-Milwaukee connielaughlinm@gmail.com

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