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Beyond

over

Rise

Run

Developing robust understandings of slope

Michael Matassa (Michael.Matassa@Colorado.edu)

Frederick Peck (Frederick.Peck@Colorado.edu)

University of Colorado Boulder

ICME 2012

Seoul

July 11, 2012

17:00 – 18:30 pm

Objective

Participants will further develop their perspectives on linear functions by participating in discussion and activity.

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

Overview

- Discuss diverse international perspectives on linear functions
- Explain the traditional American approach
- Explore a learning trajectory for linear functions that:
- Places “rate of change” as the key concept
- Enables students to construct formal mathematics
- Helps students develop versatile and adaptable understandings of linear functions

- Present lessons learned, including some important considerations for context

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

What are some characteristics of this linear function?

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

- The dependent variable increases by two units for every unit increase in the independent variable

How would you describe this behavior in your country?

Draw a schematic or diagram of how this concept is developed or taught in your country

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

- The dependent variable increases by two units for every unit increase in the independent variable

- In America, 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”

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

- The dependent variable increases by two units for every unit increase in the independent variable

Geometric

- In America, 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”

Procedural

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

http://www.youtube.com/watch?v=R948Tsyq4vA

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

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)

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

Functional property

Geometric ratio

Algebraic ratio

The number one result when searching for “rate of change” in Google!

http://www.regentsprep.org/regents/math/algebra/AC1/Rate.htm

Summary: In America:

Slope

- The dependent variable increases by two units for every unit increase in the independent variable

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

Summary: In America:

Slope

measures

steepness

is procedural

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

Summary: In America:

y = mx + b

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

“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)

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

“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)

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

“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)

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

Realistic Mathematics Education (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.
- Learning strands should be intertwined

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

The Iceberg Metaphor(Webb, et al., 2008) Algebraic Ratio

Formal

Notationsand Algorithms

The Algebraic Ratio

Preformal

Structure, Tools, Models

Informal

Contexts that are experientially real

Delivery Costs

Snowfall Accumulation

Running Pace

Scaling triangles

Context

How do we use math to make predictions?

The Iceberg Metaphor(Webb, et al., 2008) Algebraic Ratio

Formal

The Algebraic Ratio

Preformal

Informal

Guided re-invention

Delivery Costs

Snowfall Accumulation

Running Pace

Scaling triangles

Context

How do we use math to make predictions?

A Teaching Experiment (Cobb, 2000)on Slope 60 students 4 weeks Algebra 1 Combination of lecture, whole class discussion, small group work, and individual work.

Goal: To develop a robust understanding of slope.

- Versatility and adaptability

- 9thgrade (ages 14-15)

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

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)

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

Realized Learning Trajectories

Hypothetical Learning Trajectory (Simon 1995)

Window Problem II: Excel

Learning Outcome: Formalizing strategies into the algebraic ratio

Window Problem I

Xbox Shipping Problem

Learning Outcome: Subtracting to find change

Using rate and start to make predictions

Learning Outcome:Seeing rate in a table and reasoning backwards to find start

Covariation

Ms. Moeller Running Problem

Apple iPhone Problem

Learning Outcome: Dividing to find a unit rate

Learning outcome: Using rate and start to make a prediction

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

Realized Learning Trajectories

Hypothetical Learning Trajectory (Simon 1995)

Window Problem II: Excel

Learning Outcome: Formalizing strategies into the algebraic ratio

Window Problem I

Xbox Shipping Problem

Learning Outcome: Subtracting to find change

Learning Outcome:Seeing rate in a table and reasoning backwards to find start

Ms. Moeller Running Problem

Apple iPhone Problem

Learning Outcome: Dividing to find a unit rate

Learning outcome: Using rate and start to make a prediction

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

The Apple iPhone Problem

School of Education

Apple already building iPhones at rate of 40 million a year?

By Slash Lane

Published: 10:00 AM EST Monday, August 4, 2008

http://www.appleinsider.com/articles/08/08/04/apple_already_building_iphones_at_rate_of_40_million_a_year.html

Apple is reportedly testing the limits of its overseas manufacturing facilities in order to keep up with demand for the new iPhone 3G, with production already cranked nearly sevenfold compared to the first-generation model.

Foxconn, the company's Taiwanese handset and iPod manufacturer, has recently ramped production of the new iPhone to 800,000 units per week, says TechCrunch, citing a person "close to Apple with direct knowledge of the numbers."

The build rate is said to be "above current full capacity" for the Foxconn facilities allotted to Apple's handset business, which has led to concerns that quality control may suffer. At the current rate, Apple stands to produce more than 40 million iPhone 3Gs over the course of twelve months.

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

The Apple iPhone Problem

School of Education

Underline the prediction that the author makes.

- “Apple will produce more than 40 million iPhone 3Gs over the course of 12 months.”

- Imagine that there were already 15,000,000 iPhone 3Gs when the article was written.
- Based on the prediction above, how many iPhone 3Gs will there be 12 months later?

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

Realized Learning Trajectories

Hypothetical Learning Trajectory (Simon 1995)

Window Problem II: Excel

Learning Outcome: Formalizing strategies into the algebraic ratio

Window Problem I

Xbox Shipping Problem

Learning Outcome: Subtracting to find change

Learning Outcome:Seeing rate in a table and reasoning backwards to find start

Ms. Moeller Running Problem

Apple iPhone Problem

Learning Outcome: Dividing to find a unit rate

Learning outcome: Using rate and start to make a prediction

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

Realized Learning Trajectories

Hypothetical Learning Trajectory (Simon 1995)

Window Problem II: Excel

Learning Outcome: Formalizing strategies into the algebraic ratio

Window Problem I

Xbox Shipping Problem

Learning Outcome: Subtracting to find change

Learning Outcome: Finding rate in a table and reasoning backwards to find start

Ms. Moeller Running Problem

Apple iPhone Problem

Learning Outcome: Dividing to find a unit rate

Learning outcome: Using rate and start to make a prediction

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

The Xbox Shipping Problem

School of Education

- The table shows the cost of shipping Xbox games

- Find:
- Rate of change:
- Start:

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

The X-Box Problem: Students talking about rate

School of Education

Video not available online

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

The X-Box Problem: Students talking about start

School of Education

Video not available online

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

Video Discussion:

School of Education

- What strategies did students have to invent/reinvent in order to solve this problem?

- What other reactions do you have to this video?

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

Realized Learning Trajectories

Hypothetical Learning Trajectory (Simon 1995)

Window Problem II: Excel

Learning Outcome: Formalizing strategies into the algebraic ratio

Window Problem I

Xbox Shipping Problem

Learning Outcome: Finding rate in a table and reasoning backwards to find start

Learning Outcome: Subtracting to find change

Ms. Moeller Running Problem

Apple iPhone Problem

Learning Outcome: Dividing to find a unit rate

Learning outcome: Using rate and start to make a prediction

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

Realized Learning Trajectories

Hypothetical Learning Trajectory (Simon 1995)

Window Problem II: Excel

Learning Outcome: Formalizing strategies into the algebraic ratio

Window Problem I

Xbox Shipping Problem

Learning Outcome: Subtracting to find change

Learning Outcome:Seeing rate in a table and reasoning backwards to find start

Ms. Moeller Running Problem

Apple iPhone Problem

Learning Outcome: Dividing to find a unit rate

Learning outcome: Using rate and start to make a prediction

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

The Ms. Moeller Running Problem

School of Education

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?

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

The Ms. Moeller Problem: Student strategies for making predictions

School of Education

Strategy #1

Strategy #2

Video not available online

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

Video Discussion:

School of Education

- What strategies did students have to invent/reinvent in order to solve this problem?

- What other reactions do you have to this video?

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

Realized Learning Trajectories

Hypothetical Learning Trajectory (Simon 1995)

Window Problem II: Excel

Learning Outcome: Formalizing strategies into the algebraic ratio

Window Problem I

Xbox Shipping Problem

Learning Outcome: Finding rate in a table and reasoning backwards to find start

Learning Outcome: Subtracting to find change

Ms. Moeller Running Problem

Apple iPhone Problem

Learning Outcome: Dividing to find a unit rate

Learning outcome: Using rate and start to make a prediction

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

Realized Learning Trajectories

Hypothetical Learning Trajectory (Simon 1995)

Window Problem II: Excel

Learning Outcome: Formalizing strategies into the algebraic ratio

Window Problem I

Xbox Shipping Problem

Learning Outcome: Subtracting to find change

Learning Outcome:Seeing rate in a table and reasoning backwards to find start

Ms. Moeller Running Problem

Apple iPhone Problem

Learning Outcome: Dividing to find a unit rate

Learning outcome: Using rate and start to make a prediction

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

The Window Problem

School of Education

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?

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

The Window Problem: Students reinventing strategies

School of Education

Video not available online

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

Video Discussion:

School of Education

- What strategies did students have to invent/reinvent in order to solve this problem?

- What other reactions do you have to this video?

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

Realized Learning Trajectories

Hypothetical Learning Trajectory (Simon 1995)

Window Problem II: Excel

Learning Outcome: Formalizing strategies into the algebraic ratio

Window Problem I

Learning Outcome: Subtracting to find change

Xbox Shipping Problem

Learning Outcome:Seeing rate in a table and reasoning backwards to find start

Ms. Moeller Running Problem

Apple iPhone Problem

Learning Outcome: Dividing to find a unit rate

Learning outcome: Using rate and start to make a prediction

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

Realized Learning Trajectories

Hypothetical Learning Trajectory (Simon 1995)

Window Problem II: Excel

Learning Outcome: Formalizing strategies into the algebraic ratio

Window Problem I

Xbox Shipping Problem

Learning Outcome: Subtracting to find change

Learning Outcome:Seeing rate in a table and reasoning backwards to find start

Ms. Moeller Running Problem

Apple iPhone Problem

Learning Outcome: Dividing to find a unit rate

Learning outcome: Using rate and start to make a prediction

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

The Window Problem II: Excel

School of Education

Write a formula to calculate the rate of change

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

The Iceberg Metaphor(Webb, et al., 2008) Algebraic Ratio

The Algebraic Ratio

Delivery Costs

Snowfall Accumulation

Running Pace

Scaling triangles

Context

How do we use math to make predictions?

Lessons learned

School of Education

Students can mathematize situations involving rate of change to arrive at the other sub-constructs of slope.

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

Realized Learning Trajectories

Hypothetical Learning Trajectory (Simon 1995)

Window Problem II: Excel

Learning Outcome: Formalizing strategies into the algebraic ratio

Window Problem I

Xbox Shipping Problem

Learning Outcome: Subtracting to find change

Using rate and start to make predictions

Learning Outcome:Seeing rate in a table and reasoning backwards to find start

Covariation

Change and covariation

Ms. Moeller Running Problem

Apple iPhone Problem

Learning Outcome: Dividing to find a unit rate

Learning outcome: Using rate and start to make a prediction

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

Realized Learning Trajectories

Window Problem II: Excel

Learning Outcome: Formalizing strategies into the algebraic ratio

Window Problem I

Xbox Shipping Problem

Learning Outcome: Subtracting to find change

Using rate and start to make predictions

Learning Outcome:Seeing rate in a table and reasoning backwards to find start

Ms. Moeller Running Problem

Change and Covariation

Apple iPhone Problem

Learning Outcome: Dividing to find a unit rate

Learning outcome: Using rate and start to make a prediction

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

Lessons learned:Confusion between change and value

School of Education

- In their discourse, students didnot distinguish between change and value

Ambiguity

Resolution

Montage

Video not available online

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

Lessons learned:Confusion between change and value

School of Education

- In their discourse, students did not distinguish between change and value.
- We took change for granted. It is an integral part of covariation.

Discuss: how would you (or how have you) address this in your classroom?

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

Lessons learned:Confusion between change and value

School of Education

We designed new contexts to help students distinguish between change and value:

- Negative rates of changein situations where negative values are impossible.
- “Clock time” (e.g. 11:00 pm) for values when time is the independent variable
- Situations where the starting value had an easily interpretable meaning.

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

Email:

Michael.Matassa@Colorado.edu

Frederick.Peck@Colorado.edu

Web:

www.RMEintheclassroom.com

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

Preliminary results:Versatility (Becca)

School of Education

Functional

property

Algebraic

ratio

Geometric ratio

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

Preliminary results:Adaptability (Becca)

School of Education

Functional property

Geometric ratio

Physical property

Algebraic ratio

Parametric coefficient

Michael Matassa & Frederick Peck

Beyond rise over run: Developing robust understandings of slope

University of Colorado Boulder School of Education

ICME 2012, Seoul

Preliminary results:A unified framework for ratio and rate

School of Education

Come visit our poster:

Room 102, PS 9-3

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