1. Building a Bridge Between Arithmetic and Algebra ~ Algebra in MAG ~ Heather Zetterberg Math Specialist

2. What thoughts do the word “Algebra” conjure up?

3. Success Meaning Memorization Formulae Equations Terror Unknown Joy Bewilderment Variables Struggle Despair Pleasure Puzzle

4. What is Algebra? A branch of mathematics in which symbols, usually letters of the alphabet, represent numbers or members of a specified set and are used to represent quantities and to express general relationships that hold for all members of the set – American Heritage Dictionary

5. The “father” of Algebra Abdallāh Mu ḥ ammad ibn Mūsā al-Khwārizmī Wrote a book called “Kitab al-jabr wa al- muqabalah” Book became the first text book on the subject of Algebra in Europe “Al-jabr” became Algebra or “reunion of broken parts”

6. Elementary Algebra… Elementary algebra is not watered down Middle School and High School algebra Elementary algebra is about building a solid foundation of understanding and skills so students can be successful in their later, formal study of algebra

7. Raising the Bar Shifting Attitudes Need to shift from finding the answer to thinking about the numerical relationships underlying the calculations students perform Need to develop thinkers who know what they are doing and why Need to guide students to see relationships in their answers and make generalizations

8. Big Ideas of Early Algebra Use mathematical models to represent, understand, and analyze quantitative relationships Understand and describe patterns and functional relationships Use operations, properties, and algebraic symbols to determine equivalence and solve problems (including finding unknown values)

9. Use Models to Represent Quantitative Relationships Model situations (using objects, pictures, and symbols) that involve operations of whole numbers –Addition, Subtraction, Multiplication Model real-life situations that involve operations with whole numbers Describe and analyze qualitative and quantitative changes

10. Understand Patterns and Functional Relationships Sort, classify, and order objects by a variety of attributes Recognize, describe, create, and extend patterns and sequences Identify missing objects or numbers Analyze how repeating and growing patterns are generated

11. Patterns and Sequences Encourage children to –Look for the order in situations –Make conjectures –Predict beyond the information at hand –Develop and test generalizations Numerical growth sequences contribute to the development of number sense

12. Working with Patterns, Sequences, and Functions See Sample Activity Handout pages 1-16

13. Properties and Algebraic Symbols Illustrate principles and properties of operations Demonstrate an understanding of balance or equivalence Use concrete, pictorial, verbal, and numerical representations to demonstrate an understanding of symbolic notations

14. Properties and The Concept of Equivalence 2 + 7 can be thought of at as 7 + 2 (2 + 7) + 3 can be thought of as (7 + 3) + 2 The role of the equals sign (=) –For young children, it’s a signal to write an answer –“Seven plus two makes nine” Trouble when given 7 + = 9 –Many children will record 16 in the square.

15. Equivalence (cont.) Children will agree that 2 + 7 = 9 Children will insist 9 = 2 + 7 is incorrect Children have trouble with the following 2 + 7 = + 4 All of these difficulties stem from a child’s lack of understanding that the equals sign indicates equivalence between two quantities

16. Inequalities Compare quantities Identify quantities as equivalent or non-equivalent Using symbols Less than< Greater than> Not equal to≠

17. Confusion with Symbols…

18. Working with Symbols, Properties, and Equivalence See Sample Activity Handout pages 17-35

19. Before You Know it…

20. There May Come a Day…