Post on 24-Feb-2016
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School of Education
Frederick Peck (Frederick.Peck@Colorado.edu)Freudenthal Institute USUniversity of Colorado Boulder
Beyond rise over run:Contexts, representations,
and a learning trajectory for slope
RME 4Boulder CO USA
Sept. 29, 2013
Freudenthal institute US
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
What are some characteristics of the function represented by this graph?
1 2 3 4 5 6–1–2–3–4–5–6–7 x
1
2
3
4
5
6
7
–1
–2
–3
–4
–5
–6
–7
y
School of EducationFreudenthal institute US
The dependent variable increases by two units for every unit increase in the
independent variable
1 2 3 4 5 6–1–2–3–4–5–6–7 x
1
2
3
4
5
6
7
–1
–2
–3
–4
–5
–6
–7
y
• In the U.S., we would say that the slope of this line is 2
• In English, slope means “steepness”• Students are taught to calculate slope as
“rise over run” or “change in y over change in x”
School of EducationFreudenthal institute US
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
School of EducationFreudenthal institute US
The dependent variable increases by two units for every unit increase in the
independent variable
1 2 3 4 5 6–1–2–3–4–5–6–7 x
1
2
3
4
5
6
7
–1
–2
–3
–4
–5
–6
–7
y
• In the U.S., we would say that the slope of this line is 2
• In English, slope means “steepness”• Students are taught to calculate slope as
“rise over run” or “change in y over change in x”
Geometric
ProceduralFrederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
http://www.youtube.com/watch?v=R948Tsyq4vA
School of EducationFreudenthal institute US
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
Sub-constructs of slope (Stump, 1999)
• Algebraic ratio (i.e., )
• Parametric coefficient (i.e., the “a” in y = ax + b)
• Geometric ratio (i.e., “rise over run”)
• Physical Property (i.e., steepness)
• Functional Property (i.e., rate of change)
School of EducationFreudenthal institute US
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
The number one result when searching for “rate of change” in Google! http://www.regentsprep.org/regents/math/algebra/AC1/Rate.htm
Physical property Functional
propertyGeometric
ratio
Algebraic ratio
Summary: In the U.S.:
The dependent variable increases by two units for every unit increase in the
independent variable
1 2 3 4 5 6–1–2–3–4–5–6–7 x
1
2
3
4
5
6
7
–1
–2
–3
–4
–5
–6
–7
y
Slope
School of EducationFreudenthal institute US
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
Summary: In the U.S.:Slope
measures steepness
is procedural
School of EducationFreudenthal institute US
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
Summary: In the U.S.:
y = mx + b
School of EducationFreudenthal institute US
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
A Design Experiment (Cobb, 2000) on SlopeGoal: To develop versatile and adaptable mathematical realities around slope.
• 19 students, 4 weeks• High school Algebra 1 (9th grade, ages 14-15)
• Combination of lecture, whole class discussion, small group work, and individual work.
• RME principles
School of EducationFreudenthal institute US
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
Our focus today
• Algebraic ratio (i.e., )
• Parametric coefficient (i.e., the “a” in y = ax + b)
• Geometric ratio (i.e., “rise over run”)
• Physical Property (i.e., steepness)
• Functional Property (i.e., rate of change)
School of EducationFreudenthal institute US
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
“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)
School of EducationFreudenthal institute US
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
“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)
School of EducationFreudenthal institute US
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
“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)
School of EducationFreudenthal institute US
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
Realistic Mathematics Education (RME) (Treffers, 1987)
• Activity• Reality• Reinvention• Intertwinement• Social
School of EducationFreudenthal institute US
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
Form al
Pref
orm
al“m
odel
s for
”In
form
al
“mod
els o
f”The Algebraic
Ratio
Iceberg metaphor; Emergent modeling (Webb, et al., 2008) (Gravemeijer, 1999)
Form al
Pref
orm
al“m
odel
s for
”In
form
al
“mod
els o
f”The Algebraic
Ratio
Iceberg metaphor; Emergent modeling (Webb, et al., 2008) (Gravemeijer, 1999)
Making predictionsContext
Form al
Pref
orm
al“m
odel
s for
”In
form
al
“mod
els o
f”
Functional property (rate of change)
Algebraicratio
Many-to-one
(coordination of two quantities
changing together; intensive)
Many-as-one
(measure of one quantity;
extensive)
Form al
Pref
orm
al“m
odel
s for
”In
form
al
“mod
els o
f”
Functional property (rate of change)
Algebraicratio
Many-to-one
(coordination of two quantities
changing together; intensive)
Many-as-one
(measure of one quantity;
extensive)
Ms Moeller runs 6 miles every day. On average, she can run six miles in 54 minutes.
At this rate, how long would it take Ms. Moeller to run an 11-mile race?
The Ms. Moeller Running ProblemSchool of EducationFreudenthal institute US
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
The Ms. Moeller Problem: Students making predictions
School of EducationFreudenthal institute US
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
Form al
Pref
orm
al“m
odel
s for
”In
form
al
“mod
els o
f”
Functional property (rate of change)
Algebraicratio
Many-to-one
(coordination of two quantities
changing together; intensive)
Many-as-one
(measure of one quantity;
extensive)
Form al
Pref
orm
al“m
odel
s for
”In
form
al
“mod
els o
f”
Functional property (rate of change)
Algebraicratio
Many-to-one
(coordination of two quantities
changing together; intensive)
Many-as-one
(measure of one quantity;
extensive)
Form al
Pref
orm
al“m
odel
s for
”In
form
al
“mod
els o
f”
Functional property (rate of change)
Algebraicratio
Many-to-one
(coordination of two quantities
changing together; intensive)
Many-as-one
(measure of one quantity;
extensive)
The Xbox Shipping Problem
Number of games Total cost
012 8.003 10.004 12.005 14.006 16.00
The table shows the cost of shipping Xbox games
Predict the cost of shipping 12 games
School of EducationFreudenthal institute US
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
The Xbox Shipping Problem: Student strategies for predicting
School of EducationFreudenthal institute US
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
Form al
Pref
orm
al“m
odel
s for
”In
form
al
“mod
els o
f”
Functional property (rate of change)
Algebraicratio
Many-to-one
(coordination of two quantities
changing together; intensive)
Many-as-one
(measure of one quantity;
extensive)
Form al
Pref
orm
al“m
odel
s for
”In
form
al
“mod
els o
f”
Functional property (rate of change)
Algebraicratio
Many-to-one
(coordination of two quantities
changing together; intensive)
Many-as-one
(measure of one quantity;
extensive)
Form al
Pref
orm
al“m
odel
s for
”In
form
al
“mod
els o
f”
Functional property (rate of change)
Algebraicratio
Many-to-one
(coordination of two quantities
changing together; intensive)
Many-as-one
(measure of one quantity;
extensive)
Leslie is a window installer. On Friday, she installed two windows, and charged 402 dollars. Last week, on another job, she charged 517 dollars to install seven windows.
A new customer has asked Leslie to install five windows. How much will this cost?
The Window ProblemSchool of EducationFreudenthal institute US
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
Form al
Pref
orm
al“m
odel
s for
”In
form
al
“mod
els o
f”
Functional property (rate of change)
Algebraicratio
Many-to-one
(coordination of two quantities
changing together; intensive)
Many-as-one
(measure of one quantity;
extensive)
Form al
Pref
orm
al“m
odel
s for
”In
form
al
“mod
els o
f”
Functional property (rate of change)
Algebraicratio
Many-to-one
(coordination of two quantities
changing together; intensive)
Many-as-one
(measure of one quantity;
extensive)
Form al
Pref
orm
al“m
odel
s for
”In
form
al
“mod
els o
f”
Functional property (rate of change)
Algebraicratio
Many-to-one
(coordination of two quantities
changing together; intensive)
Many-as-one
(measure of one quantity;
extensive)
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
The Window Problem II: Excel
Write a formula to calculate the rate of change
School of EducationFreudenthal institute US
Situations that involve “making predictions” can be powerful contexts for ensembles to invent progressively more formal productions involving slope
Reinvention is distributed:• Contexts and representations are active
participants in the invention process. • Changing the context and representations
changes the way invention is distributed.
Lessons learnedSchool of EducationFreudenthal institute US
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
What work is the context and/or representation doing in this problem?
Lessons learned: Reinvention is distributed
School of EducationFreudenthal institute US
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
Lessons learned: Reinvention is distributed
What work is the context doing here?
School of EducationFreudenthal institute US
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
Form al
Pref
orm
al“m
odel
s for
”In
form
al
“mod
els o
f”
Functional property (rate of change)
Algebraicratio
Many-to-one
(coordination of two quantities
changing together; intensive)
Many-as-one
(measure of one quantity;
extensive)
Form al
Pref
orm
al“m
odel
s for
”In
form
al
“mod
els o
f”
Functional property (rate of change)
Algebraicratio
Many-to-one
(coordination of two quantities
changing together; intensive)
Many-as-one
(measure of one quantity;
extensive)
(Coordination oftwo quantities
changing together;intensive)
Form al
Pref
orm
al“m
odel
s for
”In
form
al
“mod
els o
f”
Functional property (rate of change)
Algebraicratio
Many-to-one
(coordination of two quantities
changing together; intensive)
Many-as-one
(measure of one quantity;
extensive)
(Coordination oftwo quantities
changing together;intensive)
Form al
Pref
orm
al“m
odel
s for
”In
form
al
“mod
els o
f”
Functional property (rate of change)
Algebraicratio
Many-to-one
(coordination of two quantities
changing together; intensive)
Many-as-one
(measure of one quantity;
extensive)
(Coordination oftwo quantities
changing together;intensive)
Lessons learned: Reinvention is distributed
Contexts that do work to make students distinguish between change and value:• Dynamic experiences
• Negative rates of change in situations where negative values are impossible.
• “Clock time” (e.g. 11:00 pm) for values when time is the independent variable
School of EducationFreudenthal institute US
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
Consider the work that the “find one strategy” and the ratio table do to structure this solution strategy
Lessons learned: Reinvention is distributed
School of EducationFreudenthal institute US
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
Consider the work that the “find one strategy” and the ratio table do to structure this solution strategy
Lessons learned: Reinvention is distributed
School of EducationFreudenthal institute US
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
Lessons learned: Reinvention is distributed
Ratio table and “find one strategy” work to promote a within unit (scale factor) strategy.
School of EducationFreudenthal institute US
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
Form al
Pref
orm
al“m
odel
s for
”In
form
al
“mod
els o
f”
Functional property (rate of change)
Algebraicratio
Many-to-one
(coordination of two quantities
changing together; intensive)
Many-as-one
(measure of one quantity;
extensive)
(Coordination oftwo quantities
changing together;intensive)
Form al
Pref
orm
al“m
odel
s for
”In
form
al
“mod
els o
f”
Functional property (rate of change)
Algebraicratio
Many-to-one
(coordination of two quantities
changing together; intensive)
Many-as-one
(measure of one quantity;
extensive)
(Coordination oftwo quantities
changing together;intensive)
Lessons learned: Reinvention is distributed
Ratio table and “find one strategy” work to promote a within unit (scale factor) strategy.
To create the unit rate as a measure of covariation (an intensive quantity), students
need to consider the between unit factor
School of EducationFreudenthal institute US
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
Situations that involve “making predictions” can be powerful contexts for ensembles to invent progressively more formal productions involving slope
Reinvention is distributed:• Contexts and representations are active
participants in the invention process. • Changing the context and representations
changes the way invention is distributed.
Lessons learned (again)School of EducationFreudenthal institute US
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
Email:Frederick.Peck@Colorado.edu
Web: (for slides and complete unit)www.RMEintheclassroom.com
School of EducationFreudenthal institute US
Frederick PeckBeyond rise over run: Contexts, representations, and a learning trajectory for slope
Freudenthal Institute US, School of Education, University of Colorado BoulderRME 4, Boulder CO, 2013
Discussion!