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Mieke Abels Freudenthal Institute Utrecht University, Netherlands Michael Matassa and Raymond Johnson Freudenthal Institute US University of Colorado Boulder Making Sense of Algebra with Realistic Mathematics Education

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Expectations? How can RME help teachers and students to understand and meet the common core goals of Algebra? How can students develop strategies for solving systems-of-equations? What can I do with my 6 th (7 th, 8 th ) grade students? How may a learning trajectory for (systems of) equations look like? Sources?

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Tug-of-war How would a sixth grade student solve this problem? Source: Comparing Quantities (MIC)

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Substitute and compare This side wins

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Substitute and compare This side wins

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Substitute and compare This side wins

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ICEBERG METAPHOR INFORMAL FORMAL L=30y+320 L=35y+295

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Three frogs How would a sixth grade student solve this problem? Visualizations? Source: MiTC

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Sunny 1 jump = 7 steps

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Cal 1 jump = 8 steps

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Leg 13 5 2 jump = 18 steps 1 jump = 9 steps

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Balance scales How can this problem be solved using informal strategies?

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1 Compare the student work. What are the differences, what is the same? What concepts are developed?

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ICEBERG METAPHOR INFORMAL PRE-FORMAL FORMAL MODELS L=30y+320 L=35y+295

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Generate new equations $ 6.80

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Generate new equations $ 1.00

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Combination chart 2 3 1 1 $7.80 $3.40 22$6.80 $1.00 1 0 X 2 Find the difference

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A Learning trajectory FORMAL FLOATING CAPACITY L=4j +5 L=2j+15 Balance strategy Equations Compare jumps and steps 3j − 2 = 2j + 2 8 + 4x = 10 − 2x Sets of Equations y = 2x +5 2y = 4x − 3 Generate new equations Notebook strategy Substitute and compare

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MIC (Mathematics in Context) http://mathincontext.eb.com Units: Comparing Quantities Graphing Equations MiTC (Mathematics in the City) http://mitcccny.org http://www.contextsforlearning.com Unit: The California Frog-Jumping Contest: Algebra DME (Digital Mathematics Environment) http://dme.colorado.edu/dwo/colorado.htmlhttp://dme.colorado.edu/dwo/colorado.html, or http://www.fi.uu.nl/dwo/enhttp://www.fi.uu.nl/dwo/en (login as guest) Math Secondary Education> Algebra> Lesson-Systems of equations More information: http://www.fisme.uu.nl/wisweb/en RME conference in Boulder. 4th International RME Conference, 2013 sept 27-29, University of Colorado, Boulder See for more information: www.fius.orgwww.fius.org Utrecht Summer School on Mathematics and Science Education August 19 - 30, 2013 in Utrecht, the Netherlands. http://www.utrechtsummerschool.nl/index.php?type=courses&code=H7 http://www.utrechtsummerschool.nl

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Mieke Abels M.J.Abels@uu.nl Michael Matassa michael.matassa@colorado.edu Raymond Johnson raymond.johnson@colorado.edu mathed.net, @MathEdnet google.com/+RaymondJohnson

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