1. Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions Fred Peck, University of Colorado and Boulder Valley School District Jennifer Moeller, Boulder Valley School District

2. Agenda Realistic Mathematics Education A learning trajectory for quadratic functions Student work Extensions and open questions

3. “Mathematics should be thought of as the human activity of mathematizing- not as a discipline of structures to be transmitted, discovered, or even constructed, but as schematizing, structuring, and modeling the world mathematically.”Hans Freudenthal (as quoted in Fosnot & Jacob, 2010)

4. Five principles of RME (Treffers, 1987) Mathematical exploration should take place within a contextthat is recognizable to the student. Models and toolsshould be used to bridge the gap between informal problem-solving and formal mathematics Students should create their own proceduresand algorithms Learning should be social, and students should share their solution processes, models, tools, and algorithms with other students. Learning strands should be intertwined “Progressive formalization”

5. Progressive formalization Students begin by mathematizing contextual problems, and construct more formal mathematics through guided re-invention Three broad levels: Informal: Models of learning: Representing mathematical principles but lacking formal notation or structure (Gravemeijer, 1999) Preformal: Models for learning: Potentially generalizable across many problems (Gravemeijer, 1999) Formal: Mathematical abstractions and abbreviations, often far removed from contextual cues

6. The Iceberg Metaphor (Webb, et al., 2008) preformal, structured informal, experiential ©F.M.- N.B.

7. The difficulty of applying RME principles to quadratic functions In a word: context. We need a realistic context that students can mathematize using informal reasoning, but that can be re-invented into pre-formal models and tools Why not projectile motion? Two alternative models: Length times width equals area (Drijvers et al., 2010) Line times line equals parabola (Kooij, 2000)

8. Formal Pre-formal l w Informal

10. That’s an interesting graph… http://viewpure.com/VSUKNxVXE4E

11. What patterns do you see in this table?

13. Line timesLine equalsParabola

14. Is this always true? Explore what happens when you multiply two linear functions. Do you always get a parabola? What patterns do you notice?

16. The vertex of the parabola is halfway between the two x-intercepts The concavity of the parabola depends on the slope of the two lines The x-intercepts of the parabola are the same as those of the two lines

17. the What’s My Equation? game There’s a parabola graphed on the next slide. It’s your job to find the linear factors, and then write the equation for the parabola. Use your calculator to help!

18. What’s my equation?

19. What’s my equation?

20. What’s my equation?

21. Student work…

22. Formal Pre-formal l w Informal

23. We use a JAVA applet from the Freudenthal Institute to explore the connections between Line times line equals parabola and Length times width equals area

24. Use Google to search for “wisweb applets” Select “Geometric algebra 2D” Here, we can explore what line times line equals parabola means in terms of our first model: length times width equals area Can you figure out how to construct an area model for our last parabola:

25. From standard form to factored form

26. Formal Pre-formal l w Informal

27. Where do you see parabolas in the real world? How many parabolas do you see in this movie? http://viewpure.com/cnBf6HTizYc

28. The height (h) of the trampoline jumper at time t can be modeled using the function:

29. Formal Pre-formal l w Informal

30. Students have multiple representations for quadratic functions, and multiple methods to convert between representations.

31. Formal Pre-formal l w Informal

32. From graph to equation: Line times line equals parabola Length times width equals area

33. From equation to graph:

36. Solving quadratic equations

37. Solving quadratic equations

38. In their own words… Do the models that we’ve learned help you solve problems?

39. In their own words… Do the models that we’ve learned help you understand formal mathematics?

41. Questions we haveComplete the square and vertex form Polynomials Why is standard form compelling? What are the downsides? How are students impoverished?

42. References Drijvers, P., Boon, P., Reeuwijk, M. van (2010). Algebra and Technology. In P. Drijvers (ed.), Secondary Algebra Education: Revisiting Topics and Themes and Exploring the Unknown. Rotterdam, NL: Sense Publishers. pp. 179-202 Fosnot, C. T., & Jacob, B. (2010). Young Mathematicians at Work: Constructing Algebra. Portsmouth, NH: Heinemenn. Gravemeijer, K. (1999). How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning, 1(2), 155-177. Kooij, H. van der (2000). What mathematics is left to be learned (and taught) with the Graphing Calculator at hand? Presentation for Working Group for Action 11 at the 9th International Congress on Mathematics Education, Tokyo, Japan Treffers, A. (1987). Three dimensions, a model of goal and theory description in mathematics instruction-the Wiskobas Project. Dordrecht, The Netherlands: D. Reidel. Webb, D. C., Boswinkel, N., & Dekker, T. (2008). Beneath the Tip of the Iceberg: Using Representations to Support Student Understanding. Mathematics Teaching in the Middle School, 14(2), 4. National Council of Teachers of Mathematics.

43. Contact Fred: Frederick.Peck@Colorado.edu Jen: Jennifer.Moeller@BVSD.org Web: http://www.RMEInTheClassroom.com Acknowledgements We thank David Webb and Mary Pittman for introducing us to Realistic Mathematics Education, and Henk van derKooij and Peter Boon for guiding us in the creation and implementation of this unit.