1. Algebraic Reasoning OSPI Math Coaching Workshop March 9, 2009 David Foster

2. California 44th In 2007 NAEP, CA Ranked 44th out of 49 participating. CA was 42nd in 2005. NAEP 2007 8th grade National Perspective Washington 15th of 49

3. Comparison of 8th Grade Students by State Percent Enrolled in College Prep Math Courses (Advanced) versus Mean Score of Students on NAEP (8th grade 2007)

4. Washington National Average Massachusetts California

11. “We have made significant gains in enrolling students in Algebra I in eighth grade in recent years, surpassing other state in the U.S. But we must set our goal higher.” Algebra for All or Algebra Forever We have also made more significant gains in FAILING students in Algebra I in eighth grade in recent years, surpassing other state in the U.S. Arnold Schwarzenegger July 8, 2008

12. California Algebra – How are we doing? Algebra CST Students Passing Prof or Adv Students Failing Basic or Below 2003 506,000 students 106K 21% meet standards 400K 79% of students failed the test in 2003 2008 747,000 students 47% increase 187K 25% meet standards 560K 40% more students failed in 2008 than in 2003 In 2008 more students failed the Algebra CST than took it in 2003! 3 out of 4 Fail

13. Copyright Tucher Basic Proficient

14. “I see trouble with algebra.”

15. Big Ideas in Algebra Variable Equivalence Functions Representation Modeling Structure Generalization

16. Big Idea EQUALITY – Understanding that two quantities are the same is an extremely important attribute in mathematics. Using number properties to maintain equality is central to algebra. Equations are to mathematics what sentences are to language. If two quantities are not equal, then their relationship is an inequality. A set of properties solely for inequalities is also studied in algebra.

17. Equality What number would you put in the box to make this a true number sentence? Response/Percent Responding Grade71217 12 & 17 1 and 2558138 3 and 49492510 5 and 6276212 From Falkner, Levi, and Carpenter, 1999

18. Big Idea STRUCTURE – Algebra is often referred to as generalized arithmetic. Our number system is made up of sets of numbers and operations. Basic properties (axioms) are defined as the underpinnings of a system. These are used to govern how numbers and operations function together. New rules (theorems) are developed and expand understanding of the system. Algebra is the study of that structure.

19. Fruit for Thought Algebra Reasoning

24. Big Idea VARIABLE – In mathematics quantities that may change is an important idea. In algebra literal symbols are used to represent mathematical constructs. Algebra makes use of variables as a primary construct in equations, inequalities, formulas, functions, identities, and properties.

25. Variable Is h + m + n = h + p + n Always, Sometimes, or Never true? Grade 6Grade 7Grade 8 Always True9.0%9.6%8.9% Sometimes True27.0%36.8%45.2% Never True26.2%18.4%27.4% No response34.4%30.7%16.3% Don ’ t know/Other3.3%4.4%2.2%

27. Big Idea FUNCTION – A function is a relationship between one set of objects that maps to another set of objects. This structure is fundamental to algebra and can be used to study many ideas in mathematics. Students develop early ideas of functions by examining, extending and predicting patterns. As part of developing early algebra proportional reasoning should be learned through a functional approach.

28. Banquet Tables You are helping to plan a big reception for your sister’s wedding. The reception hall has banquet tables shaped as hexagons. Six people can sit around a table.

29. Banquet Tables You have just found out the hall where you are holding the reception is long and narrow. There is not enough room to spread the tables out. Your brother has an idea, what happens if we push two tables together so that one of the sides from the first table is touching a side from the second table.

30. Banquet Tables What happens to the number people when you push two tables together?

31. Banquet Tables How would you find how many people can sit around any given number of tables?

32. The Banquet Tables Individual Investigation- Square Tables –Show NUMERICALLY- set up an arithmetic t-table, include arithmetic number sentences for 5, 13, and 100 tables –Show GEOMETRICALLY –draw a minimum of 4 tables and color code how you determine the number of seats –Show VERBALLY – describe your visualization for the number of seats in words –Show ALGEBRAICALLY – determine the algebraic expression that matches your visualization –Show GRAPHICALLY – create an appropriate graph

33. Share with a Partner Share your representations with a partner –Are they alike? –If they are alike, can you think of OTHER strategies students would use to determine the number of seats? –Are they different? –If they are different, do you understand your partner’s representations?

34. Whole Group Process Sharing of Different Representations –Do you understand these different representations? –What questions might you ask to better understand? –Can you think of any OTHER representations students might come up with? –Are they similar to shared representations?

35. The Banquet Table Second Stage of the Investigation –Use triangles, trapezoids, and hexagons as banquet table shapes –Then, show the 5 Representations: NUMERICAL/ARITHMETIC –GEOMETRICAL/CONTEXT –VERBAL –ALGEBRAIC –GRAPHICAL

36. GRAPHICAL Graph all four equations onto ONE graph –Talk with a partner about what you see –What are the similarities? –What are the differences? –Can you determine the number of people seated at 25 tables/35 tables from your graph? How? –How do you know that this answer is correct?

37. REFLECTION Select just one to write on What did I learn today that I would like to use in my classroom? –How and where in my curriculum would I introduce this “learning” in my classroom? What did I learn today that I would like to share with my colleagues? –How and where in my curriculum would I be able to introduce this “learning” with my colleagues? I was pleased that I …because… Something that was meaningful to me was….because…

38. Big Idea REPRESENTATION – Algebra is often thought of as a language. But the language is not limited to symbolic notation. The ideas of mathematics can be represented by graphs, tables, arrays, diagrams, words, as well as, symbolic notation. Connecting the linkages between different representations provides insights and understanding to mathematics concepts. Moving comfortably between the representations promotes cognitive flexibility that one can use in solving problems and modeling situations.

39. “Where is the ten?”

40. Determine the perimeter of the arrangement of these algebra tiles. 1 ut x ut Algebra Tiles

42. Video Discussion : Where is the 10? INDIVIDUAL THINK TIME What were the teacher’s expectations for her students? What moves did the teacher make in order to help her students reach her expectations? Was she successful? Why or why not? What can you take away from this video to help you in your own classroom? Pair/Share THEN Whole Group Share

43. Matching Multiple Representations : Malcolm Swan from the Shell Center, University of Nottingham We will be looking at this task through a teacher lens and through a learner lens. In a classroom, this matching multiple representations task would be used after there has been some preliminary work on how the order of operations is presented algebraically, e.g., “Multiply n by 3, then add 4.” Today, you are going to jump right into this task as a learner. You will work with a partner; a group of 3 will work.

44. Matching Multiple Representations Each pair of you will be getting a baggie with 4 sets of cards, each a different color. First, you will be taking just two of those sets out and you will match these two sets. –The two sets will be turned face UP so that each of you can clearly see the cards. –Each person will take turns selecting one card from one color and verbally explaining to their partner what that card represents, which card it matches in the other color, and why the two cards are a match. –You will notice that there are BLANK cards. Some cards are missing; you will need to make matching cards. Please do not write on the blank cards.

45. Matching Multiple Representations Once you and your partner have matched the first two sets, you will take out the third set - Tables of Numbers. You will match this new set with the original two sets. Again, –The new set will be turned face UP so that each of you can clearly see the cards. –Each person will take turns selecting one card from Tables of Numbers and verbally explaining to their partner what that card represents, which cards it matches from the first two sets, and why this is a match. –There are some missing tables which you will need to complete.

46. Matching Multiple Representations Once you and your partner have matched the first three sets, you will take out the fourth set – Areas of Shapes. You will match this new set with the original three sets. Again, –The new set will be turned face UP so that each of you can clearly see the cards. –Each person will take turns selecting one card from Areas of Shapes and verbally explaining to their partner what that card represents, which cards it matches from the first two sets, and why this is a match. –There are some missing area shapes which you will need to complete.

47. Matching Multiple Representations Once you and your partner have matched ALL FOUR SETS, Discuss what you think are: – the educational objectives of this task –Questions you might pose to your students to help them generalize what they have discovered by engaging in this task Share with another pair at your table Whole group sharing Malcolm Swan suggests that each pair creates and presents their own poster with cards and explanations.

48. Big Idea GENERALIZATION – Moving from the specific to a generalization is a powerful process in mathematics. In early algebra generalizing about number is important. Algebra 1 uses basic generalizations from number theory. Generalization is often the outcome of making conjectures and justifying. Proof and justification form the foundation for higher mathematics.

49. Find Out What Your Students Know 1. y = 3x + 45. P = 2l + 2w9. 3 + 2n = 8 2. 40 = 5x6. 3x + 2y10. a(b + c) = ab + bc 3. 3x + 2x = 5x7. a + b = b + a11. 5x – 2x 4. A = bh8. 3y – 5x =1712. d = rt 1) Sort the cards in any way that makes sense to you. 2) Write a sentence or two that describes the symbol strings that you have grouped together. 3) Write another symbol string for each sorted group.

50. Sorting Symbol Strings I. 5x – 2x Incomplete Equations 3x + 2y No solutions New: 4y + 3w II. d = rt Functions and Formulas y = 3x + 4 Equations with two or P = 2l + 2w more variables A = bh New: A =  r 2

51. Sorting Symbol Strings III. a + b = b + a Identities a(b + c) = ab + ac Equations that are 3x + 2x = 5x always true New: 2(6x) = 12x IV. 3y – 5x = 17 One-sided Equations 40 = 5x Equations with 3 + 2n = 8 variables on only one side New: 3x + 4 = 22

52. Sorting String Symbols I. 5x – 2x 3x + 2y Expressions No equals sign New: 6x - 2 II. d = rt A = bh Formulas P= 2l + 2w Equations used for a purpose New: C=  r 2 III. a + b = b + a Identities a(b + c) = ab + ac Equations that 3x + 2x = 5x are always true New: 2(6x) = 12x

53. Sorting Symbol Strings IV. y = 3x + 4 Linear Equations 3y – 5x = 17 Two variable equations New: 2x + 2y = 6 V. 40 = 5x One Variable Equations 3 + 2n = 8 One variable equations New: 9x + 3 = 15

54. Benefits of Sorting Symbol Strings Assessing students ’ recall of vocabulary related to symbol strings Identifying some student understandings about uses of variables Finding some student misunderstandings about symbol strings Raising questions for further discussion

55. Help Students Develop Symbol Sense Before we can expect students to “ represent and analyze mathematical situations and structures using algebraic symbols ” (NCTM, 2000), we must help them “ work meaningfully with variables and symbolic expressions. ” (NCTM, 2000)

56. Big Idea MODELING – Mathematizing a situation transforms a problem into a schema that can be altered or simplified in order to better understand the situation or solve the problem. Problems can be modeled using algebraic representations. Once a problem is represented algebraically it can be manipulated, extended or changed using properties of mathematics.

57. HURDLE #3 Symbolic Translation What ’ s the problem and the research?

58. Symbolic Translation 3 feet = 1 yard 3f = y Or 3y = f

59. Symbolic Translation There are 9 students for each professor at a university 9s = p 9p = s