Rme 2011 presentation quadratics
-
Upload
fredpeck -
Category
Technology
-
view
592 -
download
3
Transcript of Rme 2011 presentation quadratics
![Page 1: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/1.jpg)
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
![Page 2: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/2.jpg)
Agenda
• Realistic Mathematics Education
• A learning trajectory for quadratic functions
• Student work
• Extensions and open questions
![Page 3: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/3.jpg)
“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)
![Page 4: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/4.jpg)
Five principles of RME (Treffers, 1987) • Mathematical exploration should take place within a
context that is recognizable to the student.
• Models and tools should be used to bridge the gap between informal problem-solving and formal mathematics
• Students should create their own procedures and 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”
![Page 5: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/5.jpg)
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
![Page 6: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/6.jpg)
5 + 2 = 7
5 25 2 3
7f o r m a l n o t a t i o n s t o p o f t h e i c e b e r g
fl o a t i n gc a p a c i t y
5 + 2 = 7
5 25 2 3
7f o r m a l n o t a t i o n s t o p o f t h e i c e b e r g
fl o a t i n gc a p a c i t y
5 + 2 = 7
5 25 2 3
7f o r m a l n o t a t i o n s t o p o f t h e i c e b e r g
fl o a t i n gc a p a c i t y
5 + 2 = 7
5 25 2 3
7f o r m a l n o t a t i o n s t o p o f t h e i c e b e r g
fl o a t i n gc a p a c i t y
5 + 2 = 7
5 25 2 3
7f o r m a l n o t a t i o n s t o p o f t h e i c e b e r g
fl o a t i n gc a p a c i t y
© F.M.- N.B.
informal,experiential
preformal,structured
The Iceberg Metaphor (Webb, et al., 2008)
![Page 7: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/7.jpg)
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:
1. Length times width equals area (Drijvers et al., 2010)
2. Line times line equals parabola (Kooij, 2000)
![Page 8: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/8.jpg)
x
y
Formal
Pre-formal
Informal
l
w
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t
1
2
3
4
5
h( t)
x y
![Page 9: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/9.jpg)
![Page 10: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/10.jpg)
That’s an interesting graph…
http://viewpure.com/VSUKNxVXE4E
![Page 11: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/11.jpg)
Length
Width Area
0 10 01 9 92 8 163 7 214 6 245 5 256 4 247 3 218 2 169 1 910 0 0
What patterns do you see in this table?
![Page 12: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/12.jpg)
Input (x)
Width (w)
0 10
1 9
2 8
3 7
4 6
5 5
6 4
7 3
8 2
9 1
10 0
Input (x)
Length(l)
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
10 10
Input (x)
Area(A)
0 0
1 9
2 16
3 21
4 24
5 25
6 24
7 21
8 16
9 9
10 0
![Page 13: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/13.jpg)
5 10 x
5
10
15
20
25
30y Line times Line equals Parabola
![Page 14: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/14.jpg)
Explore what happens when you multiply two linear functions.
Is this always true?
Do you always get a parabola?
What patterns do you notice?
![Page 15: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/15.jpg)
![Page 16: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/16.jpg)
The x-intercepts of the parabola are the same as
those of the two lines
The
concavityof the
paraboladepends
on the slopeof the
two lines
The
vertexof the
parabolais halfwaybetweenthe two
x-intercepts
![Page 17: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/17.jpg)
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!
![Page 18: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/18.jpg)
1 2 3 4 5 6 7 8–1–2–3–4–5–6–7–8 x
1
2
3
4
5
6
7
8
–1
–2
–3
–4
–5
–6
–7
–8
y
What’s my equation?
![Page 19: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/19.jpg)
What’s my equation?
1 2 3 4 5 6 7 8 9 10 11–1–2–3–4–5–6–7–8–9–10–11 x
1
2
3
4
5
6
7
8
9
10
11
–1
–2
–3
–4
–5
–6
–7
–8
–9
–10
–11
y
![Page 20: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/20.jpg)
What’s my equation?
1 2 3 4 5 6 7 8 9 10 11–1–2–3–4–5–6–7–8–9–10–11 x
1
2
3
4
5
6
7
8
9
10
11
–1
–2
–3
–4
–5
–6
–7
–8
–9
–10
–11
y
![Page 21: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/21.jpg)
Student work…
![Page 22: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/22.jpg)
x
y
Formal
Pre-formal
Informal
l
w
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t
1
2
3
4
5
h( t)
x y
![Page 23: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/23.jpg)
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
![Page 24: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/24.jpg)
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:
![Page 25: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/25.jpg)
Fromstandard form
to factored form
![Page 26: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/26.jpg)
x
y
Formal
Pre-formal
Informal
l
w
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t
1
2
3
4
5
h( t)
x y
![Page 27: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/27.jpg)
Where do you see parabolas in the real world?
How many parabolas do you see in this movie?
http://viewpure.com/cnBf6HTizYc
![Page 28: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/28.jpg)
The height (h) of the trampoline jumper at time t can be modeled using the function:
![Page 29: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/29.jpg)
x
y
Formal
Pre-formal
Informal
l
w
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t
1
2
3
4
5
h( t)
x y
![Page 30: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/30.jpg)
Students have multiple representations for quadratic functions, and multiple methods to convert between representations.
![Page 31: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/31.jpg)
x
y
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t
1
2
3
4
5
h( t)
Formal
Pre-formal
Informal
l
w
x y
![Page 32: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/32.jpg)
From graph to equation:Line times line equals parabola
Length times width equals area
![Page 33: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/33.jpg)
From equation to graph:
![Page 34: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/34.jpg)
![Page 35: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/35.jpg)
![Page 36: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/36.jpg)
Solving quadratic equations
![Page 37: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/37.jpg)
Solving quadratic equations
![Page 38: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/38.jpg)
In their own words… Do the models that we’ve learned help
you solve problems?
Often
Sometimes
Almost never
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
![Page 39: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/39.jpg)
In their own words… Do the models that we’ve learned help you understand formal mathematics?
Often
Sometimes
Almost never
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
![Page 40: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/40.jpg)
Group discussion •Extensions
•Questions we have
Complete the square and vertex form
Polynomials
Why is standard form compelling?
What are the downsides? How are students impoverished?
![Page 41: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/41.jpg)
ReferencesDrijvers, 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.
![Page 42: Rme 2011 presentation quadratics](https://reader034.fdocuments.in/reader034/viewer/2022052303/5565cccad8b42a5b488b539a/html5/thumbnails/42.jpg)
Contact
AcknowledgementsWe thank David Webb and Mary Pittman for introducing us to Realistic Mathematics Education, and Henk van der Kooij and Peter Boon for guiding us in the creation and implementation of this unit.
Fred: [email protected]
Jen: [email protected]
Web: http://www.RMEInTheClassroom.com