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

Agenda

• Realistic Mathematics Education

• A learning trajectory for quadratic functions

• Student work

• Extensions and open questions

“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)

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”

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

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)

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)

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

That’s an interesting graph…

http://viewpure.com/VSUKNxVXE4E

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?

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

5 10 x

5

10

15

20

25

30y Line times Line equals Parabola

Explore what happens when you multiply two linear functions.

Is this always true?

Do you always get a parabola?

What patterns do you notice?

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

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!

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?

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

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

Student work…

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

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

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:

Fromstandard form

to factored form

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

Where do you see parabolas in the real world?

How many parabolas do you see in this movie?

http://viewpure.com/cnBf6HTizYc

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

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

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

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

From graph to equation:Line times line equals parabola

Length times width equals area

From equation to graph:

Solving quadratic equations

Solving quadratic equations

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%

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%

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?

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.

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: Frederick.Peck@Colorado.edu

Jen: Jennifer.Moeller@BVSD.org

Web: http://www.RMEInTheClassroom.com