1. Coherence in Secondary Math Expressions and Equations: The Building Blocks to Algebra and Functions 2016 NCMLE Conference Lisa Ashe – lisa.ashe@dpi.nc.govlisa.ashe@dpi.nc.gov Joseph Reaper – joseph.reaper@dpi.nc.govjoseph.reaper@dpi.nc.gov Secondary Mathematics Consultants NC DPI

2. The Road to Algebra The danger in learning algebra is that students emerge with nothing but the moves, which may make it difficult to detect incorrect or made-up moves later on. The Common Core Standards Writing Team. (2011, March 29). The University of Arizona, Institute for Mathematics and Education. Retrieved June 22, 2015, from Progressions Documents for the Common Core Mathematics Standards: http://ime.math.arizona.edu/progressions/

3. Session Goals  To discuss the vocabulary, terminology and important concepts that support teaching for understanding with expressions, equations and inequalities in MS mathematics.  To examine the learning progression for developing understanding of reasoning with expressions and equations.  To explore resources that can be used to teach within the Expressions and Equations domain for MS math with understanding.

4. Who’s in the Room? Welcome

5. Sorting Algebraic Symbol Strings  Look at the mathematical sentence on your card, write down as many characteristics about the math phrase.  Find your group based on the color of your card. As a group, categorize your card types Look at the similarities and differences of your cards and generate a list of sub-categories.  Now find a new group that represents each of the 3 different colors. As a group, confirm or revise your categories and subcategories. What are the distinguishing characteristics of your math sentences?

6. Algebraic Symbol Strings Expressions Simplified expressions Un-simplified expressions Rational expressions Equations Unique Solutions Constant Equations Multi-step equations IdentitiesPropertiesFormulasFunctions Inequalities Strict Inequalities Non-Strict Inequalities Compound Inequalities

7. The Big Ideas of Expressions Elementary School→ Numerical expressions  Writing, evaluating and interpreting numerical (positive whole numbers and fractions) expressions with parentheses, brackets and grouping symbols. Middle School→ Numerical and algebraic expressions  Writing and evaluating numerical (rational numbers) expressions with whole number exponents.  Writing, evaluating and interpreting algebraic expressions.  Writing equivalent expressions. High School→ Polynomial and rational expressions  Writing and interpreting polynomial and rational (complex algebraic) expressions.  Writing equivalent expressions.

8. 6 th Grade Critical Area 3: Writing, interpreting, and using expressions and equations  Students understand the use of variables in mathematical expressions.  Students write expressions that correspond to given situations, evaluate expressions, and use expressions and formulas to solve problems.  Students understand that expressions in different forms can be equivalent, and they use the properties of operations to rewrite expressions in equivalent forms.

9. 7 th Grade Critical Area 2: Developing understanding of operations with rational numbers and working with expressions and linear equations;  Students develop a unified understanding of number, recognizing fractions, decimals (that have a finite or a repeating decimal representation), and percent as different representations of rational numbers.  Students extend addition, subtraction, multiplication, and division to all rational numbers, maintaining the properties of operations and the relationships between addition and subtraction, and multiplication and division. By applying these properties, and by viewing negative numbers in terms of everyday contexts (e.g., amounts owed or temperatures below zero), students explain and interpret the rules for adding, subtracting, multiplying, and dividing with negative numbers.  They use the arithmetic of rational numbers as they formulate expressions and equations in one variable and use these equations to solve problems.

10. 8 th Grade Critical Area 1: Formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations;  Students strategically choose and efficiently implement procedures to solve linear equations in one variable, understanding that when they use the properties of equality and the concept of logical equivalence, they maintain the solutions of the original equation.  Students solve systems of two linear equations in two variables and relate the systems to pairs of lines in the plane; these intersect, are parallel, or are the same line.  Students use linear equations, systems of linear equations, linear functions, and their understanding of slope of a line to analyze situations and solve problems.

11. 5.OA.1 Evaluate numerical expressions 5.OA.2 Write and interpret numerical expressions 6.EE.2 a.Write algebraic expressions. b.Identify parts of an expression; view each part as a single entity c.Evaluate numerical and algebraic expressions (Order of Operations) 6.EE.3 Write equivalent algebraic expressions using the properties of operations 6.EE.4 Interpret equivalent algebraic expressions 6.EE.1 Write and evaluate numerical expressions with whole number exponents 7.EE.1 Add, subtract, factor and expand linear expressions using the properties of operations. Expressions Coherence 7.EE.2 Rewrite equivalent algebraic expressions based on context to reveal attributes of the expression. 7.EE.3 Evaluate numerical expressions with rational numbers (whole numbers, fractions, decimals, integers and any combination). Check for reasonableness of answers. 8.EE.1 Apply the properties of operations to numerical expressions with integer exponents. 8.EE.3 Rewrite very large and very small numbers using scientific notation. Using the laws of exponents to compare written expressions in scientific notation. 8.EE.4 Evaluate and interpret expressions in scientific notation. 6.EE.5 Use substitution to determine if a number is true for a given solution set. A-SSE.1a, b Interpret expressions in context: a.Interpret parts of an expression b.Interpret parts by seeing one or more as a single entity A-SSE.3a Write an expression into an equivalent form to reveal properties of the quantity: a. Factor a quadratic expression to reveal zeros. A-APR.1 Add, subtract and multiply polynomials A-SSE.2 Use the structure of an expression to identify ways to rewrite it 8.EE.2 Evaluate square/cube roots of common squares/cubes. HS Number Standards

12. Elementary School  Equations with an unknown  Operational definition of equivalence Middle School  Equivalent expressions to equations  Relational definition of equivalence  Symbolic transformations of expressions/equations  Creating, solving and interpreting equations  Simultaneous equations  Functions High School  Different types of equations and functions  Comparing function types The Big Ideas of Equations

13. 6 th Grade

14. 7 th Grade Critical Area 2: Developing understanding of operations with rational numbers and working with expressions and linear equations;  Students develop a unified understanding of number, recognizing fractions, decimals (that have a finite or a repeating decimal representation), and percent as different representations of rational numbers.  Students extend addition, subtraction, multiplication, and division to all rational numbers, maintaining the properties of operations and the relationships between addition and subtraction, and multiplication and division. By applying these properties, and by viewing negative numbers in terms of everyday contexts (e.g., amounts owed or temperatures below zero), students explain and interpret the rules for adding, subtracting, multiplying, and dividing with negative numbers.  They use the arithmetic of rational numbers as they formulate expressions and equations in one variable and use these equations to solve problems.

15. 8 th Grade Critical Area 1: Formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations;  Students strategically choose and efficiently implement procedures to solve linear equations in one variable, understanding that when they use the properties of equality and the concept of logical equivalence, they maintain the solutions of the original equation.  Students solve systems of two linear equations in two variables and relate the systems to pairs of lines in the plane; these intersect, are parallel, or are the same line.  Students use linear equations, systems of linear equations, linear functions, and their understanding of slope of a line to analyze situations and solve problems.

16. 6.EE.5 Understand solutions to equalities and inequalities by substitution 6.EE.6 Use variables to represent numbers and write expressions when solving problems; variables represent an unknown. Equations Coherence 6.EE.8 Write inequalities to represent constraints and have an infinite number of solutions. 6.EE.9 Use variable to represent two quantities that change in relation to each other; dependent and independent variables. 6.EE.7 Solve real-world problems by writing and solving equations. (One step: add or multiply) 7.EE.4 Use variables to represent quantities in problems and construct equations and inequalities to solve. a.Fluently solve two- step, compare arithmetic and algebraic approaches. b.Solve two-step inequalities; graph solutions and interpret in context. 7.RP.2 Recognize and represent proportional relationships a.Determine if two quantities are in a proportional relationship b.Identify the constant of proportionality in various representations c.Represent proportional relationships by equations d.Explain (x,y) in context with special attention to (0,0) and (1,r) 8.EE.5 Graph proportional relationships interpreting the unit rate as slope. 8.EE.6* Use similar triangles to explain slope, derive y =mx and y = mx + b 8.EE.8 Analyze and solve linear systems: a.Understand solutions are where the graphs intersect. b.Solve algebraically and estimate solutions by graphing. c.Solve real-world problems. 8.EE.7 Solve linear equations in one variable: a.Give examples of linear equations with one solution, infinitely many solutions, and no solutions. b.Solve linear equations with rational coefficients using distributive property and collecting like terms 8.F.2 Construct a linear model and determine the rate of change and initial value in context 8.F.1 Define a function as input and a unique output; graphs represent solutions A-REI.1 Explain each step in the solving process. A-REI.3 Solve linear equations and inequalities in one variable A-CED.1 Create linear equations and inequalities in one variable A-REI.5 Understand using elimination when solving a system. A-CED.2 Create equations in two variables A-CED.4 Solve for a variable in a formula A-REI.6 Solve systems algebraically and approximately. A-REI.12 Solve systems of inequalities. 8.F.2 Compare the properties of functions in different representations

17. Which side has a larger value? 10 + x = 10 x What are some possible answers that your students might give? What does the different responses speak about what students know and are able to do?

18. 2 Major Shifts in Understanding from ES to MS Equivalence Operational – the answer to a mathematical sentence 3 + 5 = ☐ Relational – expresses the relationship between two sides of an equation. 3 + x = x – 10 Variable Unknown – a value in an equation that is unknown. 3 + x = 15 Variable – a value that changes an expression or equation according to the value assigned to the variable(s). 3 + x = y

19. Pedagogy and Resources that Build Understanding

20. Mathematics Teaching Practices 1.Establish mathematics goals to focus learning 2.Implement tasks that promote reasoning and problem solving 3.Use and connect mathematical representations 4.Facilitate meaningful mathematical discourse 5.Pose purposeful questions 6.Build procedural fluency from conceptual understanding 7.Support productive struggle in learning mathematics 8.Elicit and use evidence of student thinking

21. Math Tasks There is no decision that teachers make that has a greater impact on students’ opportunities to learn and on their perception about what mathematics is than the selection or creation of the tasks with which the teacher engages students in shaping mathematics.

22. How are the tasks similar? different? What opportunities does each task provide for student learning? Comparing Instructional Tasks

23. TASK A The table of values below describes the perimeter of each figure in the pattern of blue tiles. The perimeter P is a function of the number of tiles t. Complete each question below using the table and diagram. 1.Choose a rule to describe the pattern in the table. a) P = t + 3 b) P = 4t c) P = 2t + 2 d) P = 6t – 2 2.How many tiles are in the figure if the perimeter is 20? 3.Graph the numbers in the table. TASK B Trains 1, 2, 3 and 4 are the first four trains in the hexagon pattern. The first train in this pattern consists of one regular hexagon. For each subsequent train, one additional hexagon is added. Questions 1-3 are optional, however you may use them to help you to work on questions 4 and 5. 1.Compute the perimeter for each of the first four trains. 2.Draw the fifth train and compute the perimeter of the train. 3.Determine the perimeter of the 25 th train without constructing it. 4.Write a description that could be used to compute the perimeter of any train in the pattern. 5.Determine which train has a perimeter of 110. Train 1Train 2Train 3Train 4 T 1234 P 46810

24. Implement Tasks That Promote Reasoning and Problem Solving Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and that allow for multiple entry points and varied solution strategies.

25. Let’s Play! student.desmos.com Code: f2nv

26. Let’s do some math! Hearts and Stars! Correct or Incorrect? Delivery Trucks Moving Parentheses! Tell the Truth! Amusement Park Tickets 1.At your tables, deal out a task to each person. 2.Do the task… What mathematics is it getting at? How might students approach this? What MS standards align to the task?

27. Create a Playlist Hearts and Stars! Correct or Incorrect? Delivery Trucks Moving Parentheses! Tell the Truth! Amusement Park Tickets o In what order would you use these task as you teach the content? o As a group, order the tasks based on the progression of learning.

28. Implementation Matters High Low HighLow Moderate Task Implementation Student Learning

29. Procedural Fluency should: Build on a foundation of conceptual understanding. Over time (months, years), result in known facts and generalized methods for solving problems. Enable students to flexibly choose among methods to solve contextual and mathematical problems. From Conceptual Knowledge to Procedural Fluency

30. …conceptual knowledge is important as students advance. Learning new concepts depends on what you already know, and as students advance, new concepts will increasingly depend on conceptual knowledge. Tie conceptual knowledge to procedures that students are learning so that the “how” has a meaningful “why” associated with it; one will reinforce the other. Daniel Willingham American Educator (2009) http://www.aft.org/sites/default/files/periodicals/willingham.pdf

31. Conceptual Knowledge “Research suggests that about two-thirds of high school juniors and seniors are still functioning at a concrete level of thinking (Orlich, 2000). Consequently, manipulatives can play an important role for a wide range of students – from helping younger students to visualize multi-digit multiplication by using base-ten blocks to allowing older students to make sense of completing the square by using algebra tiles.” Principles to Action, pg. 83

32. Instructional Manipulatives Concrete (Manipulatives) Concepts are introduced through hands-on experiences with manipulatives. Pictorial (Representational) Students visualize the concept and represent it pictorially through models. Abstract (Symbolic) Students only use abstract numbers and symbols when they have enough context to understand what they mean.

33. Visual Representations Why do we “flip” the inequality when multiplying or dividing by a negative? False

34. Visual Representations What do we need to do to make this true? False or Flip the numbers Flip the inequality symbol

35. Opportunities to Build Conceptual Understanding in MS Fractions, ratios and proportions Integer operations Simplifying expressions Solving equations and inequalities Linear functions Geometric concepts Multiplying polynomials (Math I) Factoring polynomials (Math I) Statistics and Probability

36. Build procedural fluency from conceptual understanding. Mat h Tea chin g Pra ctic e 6 Build procedural fluency from conceptual understanding A rush to fluency undermines students’ confidence and interest in mathematics and is considered a cause of mathematics anxiety. (Ashcraft 2002; Ramirez Gunderson, Levine, & Beilock, 2013)

37. Compression of Concepts vs. Procedural Ladder Boaler, What’s Math Got To Do With It?, 2015

38. Principles to Actions (NCTM, 2014, p. 42) “Fluency builds from initial exploration and discussion of number concepts to using informal reasoning strategies based on meanings and properties of the operations to the eventual use of general methods as tools in solving problems.”

39. Connections to Functions Functions have multiple representations: symbolic, graphs, tables, contextual, and verbal Functions are classified into different families based on patterns of change: linear, quadratic, exponential, etc.

40. Contextual Physical Visual Symbolic Verbal Principles to Actions (NCTM, 2014, p. 25) (Adapted from Lesh, Post, & Behr, 1987) Important Mathematical Connections between and within different types of representations

41. Revisiting the meaning of the ”=“ sign: A transition from EQUATION to FUNCTION. What does the “=“ mean in this context? This is a statement of a equivalency between expressions. The x represents an unknown. This is an equation. What does the “=“ mean in this context? This is a statement of a relationship between variables. The y and a represent varying quantities. This is a function.

42. MS Functions Coherence Map 6.EE.9 Define dependent and independent variable 6.RP.2 Define a unit rate 6.RP.3b Solve unit rate problems 7.RP.1 Compute unit rates with ratios of fractions 7.RP.2 Recognize and represent proportional relationships a.Determine if two quantities are in a proportional relationship b.Identify the constant of proportionality in various representations c.Represent proportional relationships by equations d.Explain (x,y) in context with special attention to (0,0) and (1,r) 8.EE.5 Graph proportional relationships interpreting the unit rate as slope. 8.EE.6 Use similar triangles to explain slope, derive y =mx and y = mx + b 8.F.1 Define a function as input and a unique output; graphs represent solutions 8.F.2 Compare the properties of functions in different representations 8.F.3 Interpret y = mx + b as defining a linear equation and others a nonlinear. 8.F.4 Construct a linear model and determine the rate of change and initial value in context 8.F.5 Describe a function qualitatively and sketch a graph from verbal description

43. Connections to HS Functions Expressions. An expression is a record of a computation with numbers, symbols that represent numbers, arithmetic operations, exponentiation, and, at more advanced levels, the operation of evaluating a function.  Conventions about the use of parentheses and the order of operations assure that each expression is unambiguous.  Creating an expression that describes a computation involving a general quantity requires the ability to express the computation in general terms, abstracting from specific instances. Connections to Functions and Modeling. Expressions can define functions and equivalent expressions define the same function.  Asking when two functions have the same value for the same input leads to an equation; graphing the two functions allows for finding approximate solutions of the equation.  Converting a verbal description to an equation, inequality, or system of these is an essential skill of modeling.

44. Problem Formulate Compute Validate Report Interpret The Modeling Cycle 1)Identifying variables in the situation and selecting those that represent essential features. 2)Formulating a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables. 3)Analyzing and performing operations on these relationships to draw conclusions. 4)Interpreting the results of the mathematics in terms of the original situation. 5)Validating the conclusions by comparing them with the situation, and then either improving the model or, if it is acceptable… 6)Reporting on the conclusions and the reasoning behind them. Choices, assumptions, and approximations are present throughout this cycle.

45. Questions?

46. For all you do for our students!

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48. DPI Mathematics Section Dr. Jennifer Curtis K – 12 Mathematics Section Chief 919-807-3838 jennifer.curtis@dpi.nc.gov Susan Hart Mathematics Program Assistant 919-807-3846 susan.hart@dpi.nc.gov Lisa Ashe Secondary Mathematics Consultant 919-807-3909 lisa.ashe@dpi.nc.gov Joseph Reaper Secondary Mathematics Consultant 919-807-3691 joseph.reaper@dpi.nc.gov Kitty Rutherford Elementary Mathematics Consultant 919-807-3841 kitty.rutherford@dpi.nc.gov Denise Schulz Elementary Mathematics Consultant 919-807-3842 denise.schulz@dpi.nc.gov