Journey into Algebra: Describing Change

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Journey into Algebra Describing Change

• Dr. Henry Kepner, Dr. Kevin McLeod, Dr. DeAnn
  Huinker, Mathematics Partnership (MMP)
• Math Teacher Leader (MTL) Meeting, September 2005
• www.mmp.uwm.edu

This material is based upon work supported by the
National Science Foundation Grant No.
EHR-0314898.
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Session Goals

• Ground algebra journey in the Wisconsin
  Standards.
• Analyze and describe change in various
  contexts.
• Examine and use different ways of describing
  algebraic relationships.

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Algebraic Relationships
Expressions, Equations, and Inequalities
Generalized Properties
Sub-skill Areas
a x b b x a
Patterns, Relations, and Functions
??? 25? 37
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• The understanding that most things change over
  time, that many such changes can be described
  mathematically, and that many changes are
  predictable helps lay a foundation for applying
  mathematics to other fields and for understanding
  the world.
• NCTM (2000)

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Facilitator

• Keep the group focused
• Engage all group members in the conversation
• Make decisions on the direction for the
  discussion

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What is changing and how?

• The number of Pokemon cards in my sons
  collection.
• The number of pieces of paper you give to your
  students over the school year.
• The temperature of a cup of hot coffee left on
  your desk for two hours.
• The speed of a car approaching a red light.

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What is changing and how?

• The weight of a new puppy over its first 100 days
  of life over its life time.
• The population of the United States.
• The cost of gasoline over the past year.
• The speed of your car as you merge onto the
  interstate.
• A savings account with compound interest.

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Is the change . . .

• Increasing or decreasing or both?
• Steady (constant) or does it vary?
• Occurring quickly or slowly?

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The Dots Problem
Beginning
After 1 Minute
After 2 Minutes
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Assuming the sequence continues in the same way

• How many dots in 5 minutes?
• How many dots in 12 minutes?
• How many dots in 100 minutes?
• Write a representation for the number of dots at
  t minutes.

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Group Task

• On chart paper, show how the dot
• problem is changing in four ways
• Picture
• Table
• Words (Write 2-3 sentences.)
• Symbolic Rule (e.g., equation)

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Representations
Table
Picture
Symbolic Rule

• Words (sentences)

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Representations
Table
Picture
Symbolic Rule

• Words(sentences)

Graph
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Big Idea Equivalence

• Mathematical relationships can be
• represented in equivalent ways
• Verbally (carefully worded sentences)
• Numerically (tables of values)
• Visually (diagrams, graphs)
• Symbolically (algebraic equations)

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Big Idea Linearity

• A relationship between two quantities is linear
  if the rate of change between the two
  variables is constant.

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Algebraic Relationships Wisconsin Standards
Grades 4 8

• Needs more attention
• 3 Already occurs in instruction
• ? Not sure what it means nor whether it occurs
  in math program

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Summarize

• Identify 3-4 aspects of algebraic thinking that
  are important for your students to be able to do.
• - - - - - - - - - - - - - - - - - - - - - - - - -
  - - - - - - -
• Where do those aspects of algebraic thinking
  appear in the MPS Learning Targets?

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Optional Slides follow
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The Car Trip
The McLeod family is going on a trip to visit
relatives. The table shows the number of miles
they drove and the amount of gasoline left in the
cars tank as they traveled.
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• What patterns do you see?
• Write some sentences to describe the changes you
  are noticing.

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• Write a rule using symbols to describe the
  relationships examined for the family trip.

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Perspectives on Algebra

• As a language.
• As a mathematical structure that is powerful
  for calculation.
• As functions and relations.
• As modeling real world and other phenomena.

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Critical Contexts for Algebra

• Growth and change
• Shape and size
• Contexts within number