1. 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.

2. 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.

3. Generalized Properties Patterns, Relations, and Functions    – 25  = 37 a x b = b x a Algebraic Relationships Expressions, Equations, and Inequalities Sub-skill Areas

4. 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)

5. Facilitator Keep the group focused Engage all group members in the conversation Make decisions on the direction for the discussion

6. 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.

7. 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.

8. Is the “change”... Increasing or decreasing or both? Steady (constant) or does it vary? Occurring quickly or slowly?

9. The Dots Problem BeginningAfter 1 MinuteAfter 2 Minutes

10. 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.

11. 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)

12. Representations Words (sentences) Symbolic Rule TablePicture

13. Representations Words (sentences) Symbolic Rule Table Picture Graph

14. 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)

15. Big Idea: Linearity A relationship between two quantities is linear if the “rate of change” between the two variables is constant.

16. 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

17. 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?

18. Optional Slides follow

19. The Car Trip Distance Driven (Miles) Gas Left (Gallons) 014 5012 10010 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.

20. Distance Driven (Miles) Gas Left (Gallons) 014 5012 10010 What patterns do you see? Write some sentences to describe the changes you are noticing.

21. Distance Driven (Miles) Gas Left (Gallons) 014 5012 10010 Write a rule using symbols to describe the relationships examined for the family trip.

22. 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.

23. Critical Contexts for Algebra Growth and change Shape and size Contexts within number