1. Algebraic Thinking EDUC 5515 Algebra: the intense study of the last three letters of the alphabet

2. “ I would advise you, sir, to study algebra... your head would be less muddy.” Samuel Johnson, 1709 -1784 Traditionally, children did not begin to learn algebra until a solid foundation in arithmetic was established. Recent research is showing that algebra helps students build a deeper understanding of arithmetic concepts and become better problem solvers.

3. What is algebra?

4. The intense study of the last three letters of the alphabet. A course designed to torment teenagers, aka “cruel and unusual punishment” An important discussion about trains leaving the station and going west

5. What is algebra? Howe (2005) refers to algebra as “arithmetic with variables,” “modeling,” and “Rules of arithmetic” Algebra is inherent in arithmetic (or also expressed that arithmetic has an algebraic character”

6. What is algebra? Kaput’s organization of algebra and algebraic thinking  structures and systems  functions, relations and joint variation  specific modeling languages to express and support reasoning about situations being modeled  Reasoning processes: generalization and syntactically guided action on symbols

7. Arithmetic and Algebraic Reasoning Arithmetic – science of numbers, quantities and magnitudes Algebraic reasoning – psychological processes involved in solving problems that mathematicians can easily express using algebraic notation

8. Session 1 Represent and Analyze Mathematical Situations and Structures Using Algebraic Symbols Content Learning Targets: Understand the meaning of equivalent forms of expressions, equations, inequalities and relations Use symbolic algebra to represent and explain mathematical relationships Judge the meaning, utility and reasonableness of the results of symbolic manipulations Factor polynomials using area models and connect to whole number multiplication

9. Session 1 Represent and Analyze Mathematical Situations and Structures Using Algebraic Symbols Language Learning Targets: Define expressions, equations, inequalities and relations Define symbolic algebra to represent Define identity elements, monomials, binomials, trinomials and polynomials

10. Activity 1 Symbols that communicate relationships Review the properties of arithmetic Review identity elements Introduce and use symbols to communicate relationships between expressions Introduce symmetric property of equality and explore relationship to inequalities Plotting values on a number line for equations and inequalities

11. The Properties of Arithmetic Identity property of addition and multiplication Commutative property of addition and multiplication Associative property of addition and multiplication Distributive property of multiplication over addition

12. The symbols that communicate relationships between expressions = < > ≠ ≤ and ≥

13. The symbols that communicate relationships between expressions = < > ≠ ≤ and ≥ NO ALIGATORS!

14. The Properties of Arithmetic and the symbols Identity property of addition and multiplication Commutative property of addition and multiplication Associative property of addition and multiplication Distributive property of multiplication over addition

15. Equality Properties Equivalence Properties of Equality Reflexive Property: a = a Symmetric Property:If a = b then b = a Transitive Property:If a = b and b = c then a = c

16. Trichotomy Property for Real Numbers For any two real numbers a and b, exactly one of the following must be true: a < b a = b a > b

17. Transitive Property of Inequalities If a < b and b < c then a < c If a ≤ b and b ≤ c then a ≤ c If a > b and b > c then a > c if a ≥ b and b ≥ c then a ≥ c

18. Plotting values on a number line for equations and inequalities

19. Plot the following Grading Scale on number lines 94 – 100 = A 90 – 93= A– 87 – 89 = B+ 84 – 86 = B 80 – 83= B– 77 – 79 = C+ 74 – 76 = C 70 – 73= C– 0 – 69 = F

20. Activity 2 Bridging arithmetic expressions and algebraic notation Identity properties of addition & multiplication Commutative properties of addition & multiplication Associative properties of addition & multiplication Distributive property of multiplication over addition

21. Identity Properties Addition n + 0 = n or 0 + n = n Multiplication m + 0 = m or 0 + m = m

22. Commutative Properties Addition n + m = m + n Multiplication n m = m n

23. Associative Properties Addition a + b + c = a + ( b + c ) Multiplication a b c = a ( b c )

24. Distributive Property of Multiplication over Addition a ( b + c ) = ab + ac

25. Activity 3 Optional problem sheet 1 from RAO (pg 87) Use manipulatives to work problems 1 and 3 Work individually on problem 2 and be prepared to share your answer

26. Activity 4 Factoring polynomials with area models Factor each of the following using manipulatives: x 2 3x 2 4x 2 x 2 + 3x + 3 x 2 + 7x +12 x 2 – x – 6 x 2 – 4

27. Assignment Personal Reflection Log