Your SlideShare is downloading. ×
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Rme 2011 presentation   quadratics
Upcoming SlideShare
Loading in...5
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

×
Saving this for later? Get the SlideShare app to save on your phone or tablet. Read anywhere, anytime – even offline.
Text the download link to your phone
Standard text messaging rates apply

Rme 2011 presentation quadratics

571

Published on

Published in: Technology, Education
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total Views
571
On Slideshare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
Downloads
6
Comments
0
Likes
0
Embeds 0
No embeds

Report content
Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
No notes for slide

Transcript

  • 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
  • 9.
  • 10. That’s an interesting graph…
    http://viewpure.com/VSUKNxVXE4E
  • 11. What patterns do you see in this table?
  • 12.
  • 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?
  • 15.
  • 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:
  • 34.
  • 35.
  • 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?
  • 40. Group discussion
    • Extensions
    • 41. Questions we have
    Complete 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.

×