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.

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.

GeneralizedProperties

Patterns, Relations, and Functions

– 25=

37

a x b = b

x a

Algebraic Relationships

Expressions, Equations,

and Inequalities

Sub-skill Areas

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)

Facilitator

• Keep the group focused

• Engage all group members in the conversation

• Make decisions on the direction for the discussion

What is changing and how?

• The number of Pokemon cards in my son’s 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.

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.

Is the “change” . . .

• Increasing or decreasing or both?

• Steady (constant) or does it vary?

• Occurring quickly or slowly?

The Dots Problem

Beginning After 1 Minute After 2 Minutes

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.

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)

Representations

Words (sentences)

SymbolicRule

TablePicture

Representations

Words(sentences)

SymbolicRule

TablePicture

Graph

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)

Big Idea: Linearity

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

Algebraic Relationships Wisconsin Standards Grades 4 & 8

+ Needs more attention

Already occurs in instruction

? Not sure what it means nor whether it occurs in math program

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?

Optional Slides follow

The Car TripDistance

Driven (Miles)Gas Left (Gallons)

0 14

50 12

100 10

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 car’s tank as they traveled.

Distance Driven (Miles)

Gas Left (Gallons)

0 14

50 12

100 10

•What patterns do you see?

•Write some sentences to describe the changes you are noticing.

Distance Driven (Miles)

Gas Left (Gallons)

0 14

50 12

100 10

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

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.

Critical Contexts for Algebra

• Growth and change

• Shape and size

• Contexts within number