Line times line equals parabola Length times width equals area and Incorporating two RME models into...
-
Upload
raymond-jennings -
Category
Documents
-
view
213 -
download
0
Transcript of Line times line equals parabola Length times width equals area and Incorporating two RME models into...
![Page 1: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/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: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/2.jpg)
Agenda
• Realistic Mathematics Education
• A learning trajectory for quadratic functions
• Student work
• Extensions and open questions
![Page 3: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/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: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/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: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/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: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/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: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/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?
![Page 8: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/8.jpg)
Two alternative contexts and models
1. Length times width equals area (Drijvers et al., 2010)
2. Line times line equals parabola (Kooij, 2000)
![Page 9: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/9.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 10: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/10.jpg)
![Page 12: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/12.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 13: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/13.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 14: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/14.jpg)
5 10 x
5
10
15
20
25
30y Line times Line equals Parabola
![Page 15: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/15.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 16: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/16.jpg)
![Page 17: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/17.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 18: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/18.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 19: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/19.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 20: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/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: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/21.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 22: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/22.jpg)
Student work…
![Page 23: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/23.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 24: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/24.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 25: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/25.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 26: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/26.jpg)
Fromstandard form
to factored form
![Page 27: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/27.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 28: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/28.jpg)
Where do you see parabolas in the real world?
How many parabolas do you see in this movie?
http://viewpure.com/cnBf6HTizYc
![Page 29: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/29.jpg)
The height (h) of the trampoline jumper at time t can be modeled using the function:
![Page 30: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/30.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 31: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/31.jpg)
Students have multiple representations for quadratic functions, and multiple methods to convert between representations.
![Page 32: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/32.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 33: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/33.jpg)
From graph to equation:Line times line equals parabola
Length times width equals area
![Page 34: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/34.jpg)
From equation to graph:
![Page 35: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/35.jpg)
![Page 36: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/36.jpg)
![Page 37: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/37.jpg)
Solving quadratic equations
![Page 38: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/38.jpg)
Solving quadratic equations
![Page 39: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/39.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 40: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/40.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 41: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/41.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 42: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/42.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 43: Line times line equals parabola Length times width equals area and Incorporating two RME models into a cohesive learning trajectory for quadratic functions.](https://reader030.fdocuments.us/reader030/viewer/2022032805/56649ede5503460f94bee98b/html5/thumbnails/43.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]
Download the unit: http://www.RMEInTheClassroom.com