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THE IMPORTANCE OF NON-STANDARD UNITS IN AREA MEASUREMENT FUNDA GONULATES AND ASHLEY TAGLAUER MICHIGAN STATE UNIVERSITY MATH IN ACTION 2012

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STRENGTHENING TOMORROWS EDUCATION IN MEASUREMENT (STEM) Research analyzing the capacity of three written elementary curricula to support robust learning of spatial measurement (length, area, volume) Professional Development work with pre-service teachers on measurement lesson study working with facilitators throughout Michigan who are in turn working with teachers in their areas

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CRAZY CAKES

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SAMPLE STUDENT SOLUTIONS

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UNIT AND UNITIZING A REGION What does a unit do? What is the distinction between standard units and non- standard units? Why working with non-standard units might help understanding area better?

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CCSSM: AREA AND AREA UNITS Recognize area as an attribute of plane figures and understand concepts of area measurement. a) A square with side length 1 unit, called a unit square, is said to have one square unit of area, and can be used to measure area. b) A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.

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PROBLEMS TO EXPLORE WORKSHEET 5 problems into explore While exploring problems consider: What is my unit of measure? What does unit do in this problem? How does this problem support conceptualization of area?

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PROBLEM 1 TO EXPLORE Find the area of the entire tangram:

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PROBLEM 2 TO EXPLORE SCALING UP Use Rectangle, Trapezoid build similar figures with scale factors of 2 and 3. Then calculate the area of each enlargement in terms of the original polygon, and record your results in the table provided.

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PROBLEM 3 TO EXPLORE

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How many square tiles, 5 inches on a side, does it take to cover a rectangular area that is 50 inches wide and 100 inches long? (NAEP, 2009, Grade 8) PROBLEM 4 TO EXPLORE

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How many square tiles, 5 inches on a side, does it take to cover a rectangular area that is 50 inches wide and 100 inches long? (NAEP, 2009, Grade 8) PROBLEM 4 TO EXPLORE

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PROBLEM 5 TO EXPLORE

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REFLECTIVE DISCUSSION In Problems we explored: Comparing /creating areas by using a practical unit or an available unit (P1,P2,CC) Area is quantification of a region enclosed in a boundary (as an alternative to seeing area just as count of unit squares) Dynamic view of area in addition to static view (P2) Area changes as the region change – area increases as we enlarge the boundary Changes in boundary-scaling up by k – change in area k 2 There might be variety in unit selection (or in provided units) depending on purpose (P1, P2,P3, P4, P5) Any objects that provide a 2D measure –beans, cut out foot, … Any 2D shape –triangles, rectangles, trapezoid, … Squares and Unit squares

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REFLECTIVE DISCUSSION In Problems we explored: Measuring area by actually covering or structuring the actual or a representative region in contrast to counting number of squares provided in already structured regions (P4)

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REFLECTIVE DISCUSSION In Problems we explored: Communication of our measurement (P1, P3) Stating our unit of measure at the end of our measurement Describing our unit of measure as best as we can

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REFLECTIVE DISCUSSION Encourage use of proper names of units in stating area measurement Area is …. smallest triangle/blue rectangles/(5 by 5) squares/ in. squares Area is ….. Allow students to voice confusion instead of moving ahead Why we had different numbers for the same area What is the most appropriate unit of measure Why we do prefer square units over other units Why area goes by k 2 when you increase each side by k What is the difference between practical and standard Provide visuals to allow others to see Units move us from seeing area as quality of 2D regions to measurable quantities

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SUGGESTION Improve lesson we already have EM Grade 2 : Lesson Find the area of each shape by using pattern blocks. Compare and contrast area measurement by using different units Bring what is missing as problems to explore Allow violations

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We want to thank the National Science Foundation for funding this work We want to thank you for coming! Please take a few minutes to fill out our evaluation. For more information starting you can go to: If you have any questions please us at:

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