Algebra and the Mathematical Practices

1. Algebra and theMathematical Practices Google Image

2. Warm-Up What is the sum of 37 and 28? Do it mentally and write down your answer.

3. Warm-Up What is the sum of 37 and 28? How did you solve it?

4. Warm-Up What is the sum of 37 and 28? What does this have to do with Mathematical Practices and Algebra?

5. Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Common Core State Standards for MATHEMATICS

6. What is Algebra? Google Image

7. What is Algebra? The traditional image of algebra, based in more than a century of school algebra, is one of simplifying algebraic expressions, solving equations, learning the rules for manipulating symbols – the algebra that almost everyone, it seems, loves to hate. School algebra has traditionally been taught and learned as a set of procedures disconnected both from other mathematical knowledge and from students' real worlds. From Teaching and Learning a New Algebra by James J. Kaput

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9. What is Algebra? In traditional algebra students memorize procedures that they know only as operations on strings of symbols, solve artificial problems that bear no meaning to their lives, and are graded not on understanding of the mathematical concepts and reasoning involved, but on their ability to produce the right symbol string—answers about which they have no reason to reflect and that they found (or as likely guessed) using strategies they have no need to articulate. From Teaching and Learning a New Algebra by James J. Kaput

10. What is Algebra? Worst of all, their experiences in algebra too often drive them away from mathematics before they have experienced not only their own ability to construct mathematical knowledge and to make it their own, but, more importantly, to understand its importance and usefulness to their own lives. From Teaching and Learning a New Algebra by James J. Kaput

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12. What is Algebra? Although algebra has historically served as a gateway to higher mathematics, the gateway has been closed for many students in the United States, who are shunted into academic and career dead ends as a result. From Teaching and Learning a New Algebra by James J. Kaput

13. What is Algebra? Really… Algebraic reasoning in its many forms, and the use of algebraic representations such as graphs, tables, spreadsheets and traditional formulas, are among the most powerful intellectual tools that our civilization has developed. Without some form of symbolic algebra, there could be no higher mathematics and no quantitative science, hence no technology and modern life as we know them. From Teaching and Learning a New Algebra by James J. Kaput

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16. Our Challenge… Our challenge then is to find ways to make the power of algebra (indeed, all mathematics) available to all students - to find ways of teaching that create classroom environments that allow students to learn with understanding. From Teaching and Learning a New Algebra by James J. Kaput

17. What we know… • What we already know about algebra teaching and learning: • begin early in part, by building on students’ informal knowledge • integratethe learning of algebra with the learning of other subject matter by extending and applying mathematical knowledge • include the several different forms of algebraic thinking by applying mathematical knowledge • build on students' naturally occurring linguistic and cognitive powers encouraging them at the same time to reflect on what they learn and to articulatewhat they know • encourage active learning and the construction of relationships that puts a premium on sense-making and understanding. From Teaching and Learning a New Algebra by James J. Kaput

18. What does the Common Core have to say about algebra? Google Image

19. One of the common domains in grades K through 5 isOperations and Algebraic Thinking.Why are these two concepts linked? 1 + 2 = 3 X + Y = Z

20. Big Ideas of Early Algebra Equivalence and Equations “Equals” means equivalent sets rather than a place to write an answer. Simple real-world problems with unknowns can be represented as equations. Equations remain true (balanced) if the same change occurs to each side. Unknowns can be found using the balance strategy. Warren, E. (2008). Algebra for all. Brisbane, Australia: ORIGO Education.

21. Big Ideas of Early Algebra Patterns and Functions Operations almost always change an original number to a new number. Simple real-world problems with variables can be represented as “change situations”. “Backtracking” or reversing a change can be used to find unknowns. Warren, E. (2008). Algebra for all. Brisbane, Australia: ORIGO Education.

22. Big Ideas of Early Algebra Properties Arithmetic properties apply. The commutative law and associative law apply to addition and multiplication but not to subtraction and division. Addition and subtraction are inverse operations, and multiplication and division are inverse operations. Adding or subtracting zero, and multiplying or dividing by 1, leaves the original number unchanged. In certain circumstances, multiplication and division distribute over addition and subtraction. Warren, E. (2008). Algebra for all. Brisbane, Australia: ORIGO Education.

23. Big Ideas of Early Algebra Representations Different representations (e.g. graphs, tables of values, equations, drawings, everyday language) help with identifying trends and finding and interpreting solutions to real-world problems. Warren, E. (2008). Algebra for all. Brisbane, Australia: ORIGO Education.

24. Operations and Algebraic Thinking.Why are these two concepts linked? 1 + 2 = 3 X + Y = Z

25. Most current researchers agree that the algebraic reasoning most appropriate for elementary school typically falls into one of two subcategories: Generalized Arithmetic and Functions. Early Algebra and Mathematics Specialists by M.K. MURRAY

26. Generalized Arithmetic The reasoning that occurs as students recognize patterns that emerge during their study of the four basic operations, and to the claims they make and later justify, and eventually express with symbolic notation. Early Algebra and Mathematics Specialists by M.K. MURRAY

27. Generalized Arithmetic For example, a student solving the problem 37 + 28 may take 3 from the 28 and add it to 37; the resulting problem becomes 40 + 25. At first, the student may state a generalization of what he notices as with words: When you take an amount from one addend and add the same amount to the other addend, you still get the same total when you add them together.‖ This serves as the basis for the symbolic expression of the relationship, (a+b) = (a+c) + (b-c). Early Algebra and Mathematics Specialists by M.K. MURRAY

28. 7 + 5 = 12 7 + 5 = 12 7 + 6 = ___ 8 + 5 = ___ 9 + 4 = 13 9 + 4 = 13 9 + 5 = ___ 10 + 4 = ___ Generalized Arithmetic What do you notice? Make a statement about this. How would you convince someone this is true? Early Algebra and the Common Core: What Do Teachers Need to Know? A lecture by Susan Jo Russell and Deborah Schifter

29. 7 - 5 = 12 7 - 5 = 12 7 - 6 = ___ 8 - 5 = ___ 9 - 4 = 5 9 - 4 = 5 9 - 5 = ___ 10 - 4 = ___ Generalized Arithmetic Does this work for subtraction? Is there another rule? Early Algebra and the Common Core: What Do Teachers Need to Know? A lecture by Susan Jo Russell and Deborah Schifter

30. Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Common Core State Standards for MATHEMATICS What Mathematical Practices are we using?

31. 7 x 5 = 35 7 x 5 = 35 7 x 6 = ___ 8 x 5 = ___ 9 x 4 = 36 9 x 4 = 36 9 x 5 = ___ 10 x 4 = ___ Generalized Arithmetic How is multiplication different from addition? In a multiplication problem, if you add 1 to a factor, I think this will happen to the product… Early Algebra and the Common Core: What Do Teachers Need to Know? A lecture by Susan Jo Russell and Deborah Schifter

32. 7 x 5 = 35 7 x 5 = 35 7 x 6 = ___ 8 x 5 = ___ 9 x 4 = 36 9 x 4 = 36 9 x 5 = ___ 10 x 4 = ___ Generalized Arithmetic Choose one of the original equations and either draw a picture, make an array, or write a story for the original equation. Then change it just enough to match the new equations. Early Algebra and the Common Core: What Do Teachers Need to Know? A lecture by Susan Jo Russell and Deborah Schifter

33. Generalized Arithmetic What are some of the generalizations we made? Articulate them. Keeping symbols connected to representations is the key. Early Algebra and the Common Core: What Do Teachers Need to Know? A lecture by Susan Jo Russell and Deborah Schifter

34. Functions Refers to the generalization of numeric patterns. Such patterns often arise from contextual situations, and may be represented with pictures, number lines, function tables, symbolic notation, and graphs. Early Algebra and Mathematics Specialists by M.K. MURRAY

35. Functions For example, six pennies are added to a jar every day and the children analyze the growth. Early Algebra and Mathematics Specialists by M.K. MURRAY Google Image

36. Pool Border How many different ways can you find to count the border tiles of an 8 x 8 pool without counting them one at a time?

37. Pool Border Another approach to the Border Problem is to build a series of pools in steps, each with one more tile on the side, and then find a way to generalize the number of border tiles on an n x n pool. Adapted from Lappan, Mundy and Phillips, “Experiences with Patterning” in Algebraic Thinking: Grades K-12; NCTM, 1999.

38. Operations and Algebraic Thinking.Why are these two concepts linked? 1 + 2 = 3 X + Y = Z

39. Operations and Algebraic Thinking • Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from. • Represent and solve problems involving addition and subtraction. • Understand and apply properties of operations and the relationship between addition and subtraction. • Add and subtract within 20. • Work with addition and subtraction equations. • Represent and solve problems involving addition and subtraction. • Add and subtract within 20. • Work with equal groups of objects to gain foundations for multiplication. • Represent and solve problems involving multiplication and division. • Understand properties of multiplication and the relationship between multiplication and division. • Multiply and divide within 100. • Solve problems involving the four operations, and identify and explain patterns in arithmetic. K 1 2 3 Common Core State Standards for MATHEMATICS

40. Operations and Algebraic Thinking • Use the four operations with whole numbers to solve problems. • Gain familiarity with factors and multiples. • Generate and analyze patterns. • Write and interpret numerical expressions. • Analyze patterns and relationships. 4 5 Common Core State Standards for MATHEMATICS

41. Closing and Reflections