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

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 6th (7th, 8th) grade students?

How may a learning trajectory for (systems of) equations look like? Sources? Tug-of-war How would a sixth grade student solve this problem?

Source: Comparing Quantities (MIC) Substitute and compare This side wins Substitute and compare This side wins

Substitute and compare This side wins ICEBERG METAPHOR FORMAL INFORMAL

L=30y+320 L=35y+295 Three frogs How would a sixth grade student solve this problem? Visualizations?

Source: MiTC Sunny 1 jump = 7 steps Cal 1 jump = 8 steps

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

Balance scales How can this problem be solved using informal strategies? 1 Compare the student work. What are the differences,

what is the same? What concepts are developed? 2 ICEBERG METAPHOR FORMAL PRE-FORMAL

INFORMAL L=30y+320 L=35y+295 MODELS Generate new equations

\$ 6.80 Generate new equations \$ 1.00 Combination chart

Find the difference 2 1 3 1

\$7.80 \$3.40 2 0 2 1

\$6.80 \$1.00 X2 FORMAL A Learning trajectory Equations

8 + 4x = 10 2x Sets of Equations y = 2x +5 2y = 4x 3 FLOATING CAPACITY

3j 2 = 2j + 2 L=4j +5 L=2j+15 Balance strategy Compare

jumps and steps Generate new equations Substitute and compare

Notebook strategy 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.html, or http://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.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 Mieke Abels [email protected]

Michael Matassa [email protected] Raymond Johnson [email protected] mathed.net, @MathEdnet google.com/+RaymondJohnson