Transitioning to Common Core State Standards: High School Mathematical Practices

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2 1 Transitioning to Common Core State Standards: High School Mathematical Practices Curriculum Traditional Separation of mathematical content areas Integrated Connections between algebra and geometry Critical thinking (mathematical practices) Binds to specific examples Requires firm grasp of underlying concepts Transitioning from traditional to integrated Grade 9 Grade 10 Grade Algebra 1 Geometry Algebra Math 1 Geometry Algebra Math 2 Algebra Math Test fully aligned with Common Core curriculum Correlations between traditional and integrated Options for Math 4: AP Statistics, Probability and Statistics, Math Modeling, Discrete Math, Trigonometry/Pre-Calculus, STEM 1 STEM 2 College/Career Ready 8th Grade Math 1 9th Grade Math 2 Math 1 Math 1 10th Math 2 and Math Math 3 Grade 3 Math 2 11th Math 4 Math 4 Grade (Trig/Pre-Cal) (Trig/Pre-Cal) Math 3 12th Grade AP Calculus AP Calculus Math 4

3 2 A student could choose to double up on classes as a senior by enrolling in both Math 4 and AP Statistics. In schools with semester block scheduling, a student could enroll in two courses in a given year. In schools with alternate-day academic-year block schedules, the schedule could be adjusted for one or more classes of a course to meet each day for the entire school year, thus covering two of Math 1-4 that year. In schools with academic-year schedules, two mathematics classes may be scheduled back-to-back to allow study of one course in the first semester and the next course the second semester. Efficacy of curriculum rests on pedagogy of teachers Professional Development Why we need it Three types of curriculum Intended Implemented Attained What we need in it Reflects changes in curriculum and instructional practices Influences teachers beliefs Align the three types of curriculum Recommendations for professional development practices Lesson study CCSS common preps Peer observation and teacher-provided coaching Summer professional development workshop PROM/SE Distributed expertise Targeted, Implementable, Accountable Assessment End of Course Assessments (ECA) PARCC Partnership for Assessment of Readiness for College and Careers Mathematics 1 Supplement for ECA Standard not covered 1 Operations w/real Numbers - simplifying radical expressions (Mathematics II Critical Area 1) 4 Graphing Linear Equations and Inequalities Specific formulas such as point-slope and slope-intercept 5 Pairs of Linear Equations and Inequalities Different methods used for solving and the ability to justify the method used PARCC 6 Polynomials (Mathematics II Critical Area 1) 7 Algebraic Fractions (Mathematics III Critical Area 3) 8 Quadratic, Cubic, and Radical Equations (Mathematics II Critical Area 2)

4 3 Unit Descriptive Statistics briefly mentioned in Alg 1but not as detailed 5 Congruence, Proof, and Constructions geometry 6 Connecting Algebra and Geometry through Coordinates PARCC Recommendations: If you test ECA in grade 9, you need to add ECA topics to Mathematics I. If you test ECA in the middle of10th grade instead of 9the grade, you need to add Algebraic Fractions. Unanswered Questions Assessment impact on instruction Assessment impact on teacher evaluations Challenges Curriculum/placement of assessment Pedagogical shifts Communication Financial constraints Technology Professional development Textbooks Grade 9 Algebra 1 Math 1 Grade 10 Geometry Geometry Math 2 Grade 11 Algebra 2 Algebra 2 Algebra 2 Math 3 (online?) Textbook Adoption

5 4 Standards covered in ECA not included in Mathematics I # of Standard Standard 1 Operations With Real Numbers Indicators A1.1.1 Compare real number expressions. Example: Which is larger: 2 3 or 49? A1.1.2 Simplify square roots using factors. Example: Explain why 48 = Graphing Linear Equations and Inequalities A1.4.3 Write the equation of a line in slope-intercept form. Understand how the slope and y-intercept of the graph are related to the equation. Example: Write the equation of the line 4x + 6y = 12 in slope-intercept form. What is the slope of this line? Explain your answer. 5 Pairs of Linear Equations and Inequalities A1.5.1Use a graph to estimate the solution of a pair of linear equations in two variables. Example: Graph the equations 3y x = 0 and 2x + 4y = 15 to find where the lines intersect. A1.5.2 Use a graph to find the solution set of a pair of linear inequalities in two variables. Example: Graph the inequalities y 4 and x + y 5. Shade the region where both inequalities are true. A1.5.3 Understand and use the substitution method to solve a pair of linear equations in two variables. Example: Solve the equations y = 2x and 2x + 3y = 12 by substitution. A1.5.4 Understand and use the addition or subtraction method to solve a pair of linear equations in two variables. Example: Use subtraction to solve the equations: 3x + 4y = 11 and 3x + 2y = 7. A1.5.5 Understand and use multiplication with the addition or subtraction method to solve a pair of linear equations in two variables. Example: Use multiplication with the subtraction method to solve the equations: x + 4y = 16 and 3x + 2y = -3. A1.5.6 Use pairs of linear equations to solve word problems. Example: The income a company makes from a certain product can be represented by the equation y = 10.5x and the expenses for that product can be represented by the equation y = 5.25x + 10,000, where x is the amount of the product sold and y is the number of dollars. How much of the product must be sold for the company to reach the break-even point?

6 5 6 Polynomials Students add, subtract, multiply, and divide polynomials. They factor quadratics 7 Algebraic Fractions 8 Quadratic, Cubic, and Radical Equations A1.6.1 Add and subtract polynomials. Example: Simplify (4x 2 7x + 2) (x 2 + 4x 5). A1.6.2 Multiply and divide monomials. Example: Simplify a 2 b 5 ab 2. A1.6.3 Find powers and roots of monomials (only when the answer has an integer exponent). Example: Find the square root of a 2 b 6. A1.6.4 Multiply polynomials. Example: Multiply (n + 2)(4n 5). A1.6.5 Divide polynomials by monomials. Example: Divide 4x 3 y 2 + 8xy 4 6x 2 y 5 by 2xy 2. A1.6.6 Find a common monomial factor in a polynomial. Example: Factor 36xy xy 4 12x 2 y 4. A1.6.7 Factor the difference of two squares and other quadratics. Example: Factor 4x 2 25 and 2x 2 7x + 3. A1.6.8 Understand and describe the relationships among the solutions of an equation, the zeros of a function, the x-intercepts of a graph, and the factors of a polynomial expression. Example: A graphing calculator can be used to solve 3x 2 5x 1 = 0 to the nearest tenth. Justify using the x-intercepts of y = 3x 2 5x 1 as the solutions of the equation. A1.7.1 Simplify algebraic ratios. Example: Simplify x 2 16 x 2 4 x. A1.7.2 Solve algebraic proportions. Example: Create a tutorial to be posted to the school s Web site to instruct beginning students in the steps involved in solving an algebraic proportion. Use example. x 5 3x A1.8.1 Graph quadratic, cubic, and radical equations. Example: Draw the graph of y = x 2 3x + 2. Using a graphing calculator or a spreadsheet (generate a data set), display the graph to check your work. A1.8.2 Solve quadratic equations by factoring. Example: Solve the equation x 2 3x + 2 = 0 by factoring. A1.8.3 Solve quadratic equations in which a perfect square equals a constant. Example: Solve the equation (x 7) 2 = 64. A1.8.4 Complete the square to solve quadratic equations. Example: Solve the equation x 2 7x + 9 = 0 by completing the square. A1.8.5 Derive the quadratic formula by completing the square. Example: Prove that the equation ax 2 + bx + c = 0 has solutions x -b b 2 4 ac 2a A1.8.6 Solve quadratic equations using the quadratic formula. Example: Solve the equation x 2 7x + 9 = 0. A1.8.7 Use quadratic equations to solve word problems. Example: A ball falls so that its distance above the ground can be modeled by the equation s = t 2, where s is the distance above the ground in feet and t is the time in seconds. According to this model, at what time does the ball hit the ground? A1.8.8 Solve equations that contain radical expressions. Example: Solve the equation x 6 x. A1.8.9 Use graphing technology to find approximate solutions of quadratic and cubic equations. Example: Use a graphing calculator to solve 3x 2 5x 1 = 0 to the nearest tenth.. as an

7 6 Common Core Standards included in Mathematics I not covered in ECA Domain Statistics- Interpreting Categorical and Quantitative Data Standards S.ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots). S.ID.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. S.ID.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). S.ID.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. Geometry- Congruence Geometry- Geometric Properties with Equations G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2). G.GPE.5 Prove the slope criteria for parallel and perpendicular lines; use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

8 Math 3 Math 2 Math 1 7 Geometry Statistics and Probability Algebra Functions Number and Quantity Alg I Geometry Alg II Algebra I Geometry Algebra II Algebra I Geo Algebra II Algebra I Geo Algebra II Algebra I Geo Algebra II G.CO.1,2,3,4,5 S.ID.1,2,3 A.SSE.1a, 1b F.IF.1,2,3 G.CO.6,7,8 S.ID.5,6a, A.CED F.IF.4,5,6 6b,6c G.CO S.ID.7,8,9 A.REI.1 F.IF.7a,7e G.GPE.4,5,7 A.REI.3 F.BF.1a,1 b,2 G.CO.9,10,11 S.CP.1,2,3,4,5 A.REI.5,6 F.BF.3 G.SRT.1a,1b, 2,3 S.CP.6,7, (+) 8, (+)9 A.REI.10,11,12 F.LE.1a,1 b,1c,2,3 G.SRT. 4.5 (+) S.MD.6,7 A.SSE.1a.1b, 2 F.LE.5 N.Q.1, N.Q.2, N.Q.3 G.SRT.6,,8 A.SSE.3a,3b,3c F.IF.4,5,6 F.TF.8 N.RN.2 N.CN.1.2 G.C.1,2,3,(+) 4 A. APR.1 F.IF.7a,7b, 8a,8b,9 N.RN.3 N.CN.7, (+) 8,(+)9 G.C.5 A.CED.1,2,4 F.BF.1a,1 b G.GPE.1.2 A.REI. 4a,4b F.BF.3,4a G.GPE.4 A.REI.7 F.LE.3 G.GMD. 1, 3 A.SSE. 1a,1b, 2 F.IF.4,5,6 G.SRT.9,10,11 S.ID.4 A.SSE.4 F.IF,7b,7c,7e,8,9 G.GMD.4 S.IC.1,2 A.APR.1 F.BF.1b G.MG.1,2,3 S.IC.3,4,5,6 A.APR.2,3 F.FB.3.4a (+) S.MG.6,7 A.APR.4,(+)5 F.LE.4 A.APR.6,(+) 7 F.TF.1.2 A.CED.1,2,3,4 F.TF.5 A.REI.2 A.REI.11 (+) N.CN.8.9

9 FUNCTIONS ALGEBRA NUMBER & QUANTITY 8 DOMAIN STANDARDS CLUSTERS N.Q.1, N.Q.2, N.Q.3 Reason quantitatively and use units to solve problems N.RN.2 Extend the properties of exponents to rational exponents N.RN.3 Use Properties of rational and irrational numbers N.CN.1.2 Perform arithmetic operations with complex numbers N.CN.7, (+) 8,(+)9 Use complex numbers in polynomial identities and equations (+) N.CN.8.9 Use complex numbers in polynomial identities and equations A.SSE.1a, 1b Interpret the structure of expressions (linear and exponential with integer coefficients) A.CED Create equations that describe numbers or relationships (linear and exponential) A.REI.1 Understand solving equations as a process of reasoning and explain the reasoning A.REI.3 Solve equations and inequalities in one variable (linear and exponential) A.REI.5,6 Solve systems of equations (linear) A.REI.10,11,12 Represent and solve equations and inequalities graphically A.SSE.1a.1b, 2 Interpret the structure of expressions (quadratic) A.SSE.3a,3b,3c Write expressions in equivalent forms to solve problems (quadratic and exponential) A. APR.1 Perform arithmetic operations on polynomials A.CED.1,2,4 Create equations that describe numbers or relationships (quadratic with integer inputs) A.REI. 4a,4b Solve equations and inequalities in one variable (quadratics) A.REI.7 Solve system of equations (quadratics) A.SSE. 1a,1b, 2 Interpret the structure of expressions (polynomial and rational) A.SSE.4 Write expressions in equivalent forms to solve problems A.APR.1 Perform arithmetic operations on polynomials (beyond quadratic) A.APR.2,3 Understand the relationship between zeros and factors of polynomials A.APR.4,(+)5 Use polynomial identities to solve problems A.APR.6,(+) 7 Rewrite rational expressions (linear and quadratic denominators) A.CED.1,2,3,4 Create equations that describe numbers or relationships (equations using all available types of expressions, including simple root functions A.REI.2 Understand solving equations as a process of reasoning and explain the reasoning (simple radical and rational) A.REI.11 Represent and solve equations and inequalities graphically (polynomial, rational, radical, absolute value, and exponential) F.IF.1,2,3 Understand the concept of a function and use function notation F.IF.4,5,6 Interpret functions that arise in applications in terms of a context (linear and exponential) F.IF.7a,7e, 9 Analyze functions using different representations (linear and exponential) F.BF.1a,1b,2 Build a function that models a relationship between two quantities (linear and exponential) F.BF.3 Building functions from existing functions F.LE.1a,1b,1c,2,3 Construct and compare linear, quadratic, and exponential models and solve problems F.LE.5 Interpret expressions for functions in terms of the situation they model F.IF.4,5,6 Interpret functions that arise in applications in terms of a context (quadratic) F.IF.7a,7b, 8a,8b,9 Analyze functions using different representations (quadratic, absolute value, step, piecewise-defined) F.BF.1a,1b Build a function that models a relationship between two quantities (quadratic and absolute value) F.BF.3,4a Build new functions from existing functions (quadratic and absolute value) F.LE.3 Construct and compare linear, quadratic, and exponential models and solve problems F.TF.8 Prove and apply trigonometric functions using the unit circle

10 PROBABILITY & STATISTICS GEOMETRY 9 F.IF.4,5,6 Interpret functions that arise in application in terms of a context (rational, root functions, and emphasize selection of appropriate models) F.IF,7b,7c,7e,8,9 Analyze functions using different representations (rational and radical, focus on using key features to guide selection of appropriate type of model F.BF.1b Build a function that models a relationship between two quantities (all types) F.FB.3.4a Build new functions from existing functions (all types) F.LE.4 Construct and compare linear, quadratic and exponential models with logarithms as solutions F.TF.1.2 Extend the domain of trigonometric functions using the unit circle F.TF.5 Model periodic phenomena with trigonometric functions G.CO.1,2,3,4,5 Experiment with transformations in a plane G.CO.6,7,8 Understand congruence in terms of rigid motions (build on rigid motions as a familiar starting point for development of concept of geometric proof) G.CO Make geometric constructions G.GPE.4,5,7 Use coordinates to prove simple geometric theorems algebraically (include distance formula, relate to Pythagorean theorem) G.CO.9,10,11 Prove geometric theorems G.SRT.1a,1b, 2,3 Understand similarity in terms of similarity transformations G.SRT. 4.5 Prove theorems involving similarity (focus on validity of underlying reasoning while using variety of formats) G.SRT.6,8 Define trigonometric ratios and solve problems involving right triangles G.C.1,2,3,(+) 4 Understand and apply theorems about circles G.C.5 Find arc lengths and areas of sectors of circles (radian introduced only as a unit of measure) G.GPE.1.2 Translate between the geometric description and the equation for a conic section G.GPE.4 Use coordinates to prove simple geometric theorems algebraically (include simple circle theorems) G.GMD. 1, 3 Explain volume formulas and use them to solve problems (+) G.SRT.9,10,11 Apply trigonometry to general triangles G.GMD.4 Visualize the relationship between 2-D and 3-D objects G.MG.1,2,3 Apply geometric concepts in modeling situations S.ID.1,2,3 Summarize, represent, and interpret data on a single count or measurement variable S.ID.5,6a,6b,6c Summarize, represent and interpret data on two categorical and quantitative variables (linear) S.ID.7,8,9 Interpret linear models S.CP.1,2,3,4,5 Understand independence and conditional probability and use them to interpret data (link to data from simulations or experiments) S.CP.6,7, (+) 8, (+)9 Use the rules of probability to compute probabilities of compound events in a uniform probability model (+) S.MD.6,7 Use probability to evaluate outcomes of decisions (introduce counting rules) S.ID.4 Summarize, represent and interpret data on a single count or measurement variable S.IC.1,2 Understand and evaluate random processes underlying statistical experiments S.IC.3,4,5,6 Make inferences and justify conclusions from sample surveys, experiments, and observational studies (+) S.MG.6,7 Use probability to evaluate outcomes of decisions

11 10 REFERENCES Ferrini-Mundy, J., Burrill, G., & Schmidt, W. H. (2007). Building teacher capacity for implementing curricular coherence: Mathematics teacher professional development tasks. Journal of Mathematics Teacher Education, 10(4-6), Goos, M., Dole, S., & Makar, K. (2007). Supporting an investigative approach to teaching secondary school mathematics: A professional development model. In J. Watson & K. Beswick (Eds.), Mathematics: Essential research, essential practice (Proceedings of the 30th annual conference of the Mathematics Education Research Group of Australasia, pp ). Sydney: MERGA. Handal, B., & Herrington, A. (2003). Mathematics teachers beliefs and curriculum reform. Mathematics Education Research Journal, 15(1), Kilpatrick, J. (2009). The mathematics teacher and curriculum change. PNA, 3(3), McCaffrey, D. F., Hamilton, L. S., Stecher, B. M., Klein, S. P., Bugliari, D., & Robyn, A. (2001). Interactions among instructional, practices, curriculum, and student achievement: The case of Standards-based high school mathematics. Journal for Research in Mathematics Education, 32, National Center on Education and the Economy (2008). Tough choices or tough times: the report of the New Commission on the Skills of the American Workforce. San Francisco, CA: Jossey-Bass. National Council of Teachers of Mathematics. (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA: Author. National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: Author. Obara and Sloan (2009). Classroom experiences with new curriculum materials during the implementation of performance standards in mathematics: a case study of teachers coping with change. International Journal of Science and Mathematics Education, 8, Sparks, G. (1986). The effectiveness of alternative training activities in changing teaching practices. American Educational Research Journal, 23, Willingham, D. T. (2007, Summer). Critical thinking: Why is it so hard to teach? American Educator, pp.8-19.