State Board of Education Topic Summary s3

STATE BOARD OF EDUCATION – TOPIC SUMMARY Topic: New High School Mathematics Content Standards Date: May 14-15, 2009 Staff/Office: Colleen Mileham, Paul Hibbard, Michelle Hooper/Educational Improvement and Innovation Action Requested: Information only Policy Adoption Policy Adoption/Consent Calendar

ISSUE BEFORE THE BOARD: First reading of newly revised high school math standards.

BACKGROUND:

Mathematics Standards

Mathematics is the science of numbers and space configurations and their operations, interrelations, combinations, generalizations, and abstractions. These mathematics standards define the mathematics content knowledge, essential skills, and process standards that all students are expected to learn during high school mathematics instruction in Oregon. Mastery of the skills, concepts, and knowledge outlined in these standards will prepare students for a variety of career or education paths.

The revision of the mathematics content standards includes three disciplines of mathematics – Algebra, Geometry and Measurement, and Statistics and Probability. Within each strand there are two to three core standards. These core standards provide the major concepts and processes that will be the primary focus of teaching and learning across the grades. Underneath each of these core standards are from three to eight content standards which provide the details necessary for curriculum and assessment.

The draft revised high school mathematics standards are also based on several principles adopted by the mathematics content and assessment panel:

 Consider the mathematics all Oregon students should learn.  Identify those topics that build on the core foundations of mathematics established in the 2007 K-8 Mathematics Content Standards.  Emphasize the importance of the National Council of Teachers of Mathematics (NCTM) Process Standards and Oregon Essential Skills embedded in a Core Standard Structure.  Reflect the recommendations of the National Mathematics Advisory Panel.  Strive for coherence, clarity, and measurability.

The mathematics content standards are attached (pp. 5-28). Also attached are:

1. A sample glossary of related mathematics terms. (p. 29) 2. A guide to the numbering system used for the math standards that will help teachers “code” the standards for lesson design. (p. 30) 3. Connections document explaining the connections of a Core Standard to other Core Standards and academic content. (p.31-33)

Process Standards and Essential Skills

The Oregon Department of Education Mathematics and Assessment Panel recognize the importance of addressing mathematical processes as well as content in these standards. The NCTM Process Standards1

1 Principles and Standards for School Mathematics, NCTM, 2000

Oregon Department of Education-Draft High School Mathematics Standards Page 1 must be integrated in a high school mathematics curriculum to ensure that students are equipped with the knowledge and skills necessary to solve problems they will likely face in a variety of contexts. These Process Standards – Problem Solving, Reasoning and Proof, Communication, Connections, and Representation – are defined and aligned to the Oregon Essential Skills. See Table 1: Alignment of the Process Standards and the Apply Math Essential Skill on page 27.

Role of Technology

The Oregon Department of Education Mathematics and Assessment Panel unanimously support the use of technology as an aid to increase student learning and ultimately prepare students for an increasingly technology-focused work and life environment. However, the panel also agreed that much of mathematics should be developed first without the aid of technology. Therefore, teachers need to be knowledgeable about how technology can support students in learning mathematics without compromising mathematical fluency – number sense, procedural knowledge, and declarative knowledge. We trust that educators will make appropriate choices regarding the use of technology.

National Trends

Nationally there has been much focus on mathematics curriculum, standards, and assessment. This focus has gained momentum largely because of evidence suggesting that the United States is relinquishing its position as a leader in mathematics and science, and, consequently, also as an economic competitor in today’s global economy. In short, educators and economists agree that mathematics knowledge among a population is a key factor in a nation’s security and economic success.

In March 2008, the United States Department of Education released the National Mathematics Advisory Panel Report outlining 45 recommendations and findings which now directly influences major policy decisions at the federal, state, and local levels. One of the 45 key recommendations central to the report and consistent with Oregon’s Core Standards in mathematics says that, “a focused, coherent progression of mathematics learning, with an emphasis on proficiency with key topics, should become the norm. . . Any approach that continually revisits topics year after year without closure is to be avoided.”2 Oregon’s Core Standards for mathematics, which outline key grade level topics that do not repeat or spiral year after year, are consistent with this recommendation.

Further, the National Council of Teachers of Mathematics has published several researched based reports outlining the curricular content and process standards to be considered key to each level of mathematics. The Oregon Department of Education has adopted these recommendations in both the recently revised K-8 Mathematics Content Standards and the draft of the High School Content Standards. Consequently, Oregon’s K-12 Mathematics Standards are a focused, coherent progression of mathematics supported by recommendations from both the National Mathematics Advisory Panel Report and the National Council of Teachers of Mathematics.

The Review and Revision Process

The Oregon revision of the high school mathematics content standards began in September 2008 with a two- day work review of research by national and international mathematics education experts. From this two-day work session, the panel created the first draft of the high school content standards which was posted as a news announcement, advertised in various ODE publications, and posted on-line at: http://www.ode.state.or.us/search/page/?=1148.

Between September 2008 and March 2009, the panel met monthly for six two-day work sessions culminating in the final draft posted at the link above. In between each monthly session, ODE posted news announcements when each draft was available and sent out messages via the Superintendent’s Pipeline, the

2 National Mathematics Advisory Panel Report, U.S. Department of Education, 2008

Oregon Department of Education-Draft High School Mathematics Standards Page 2 Curriculum Director’s listserv, the Oregon Mathematics Teacher Update monthly e-newsletter, and to mathematics professional organizations, universities, and community colleges. In addition, ODE representatives gave presentations on the mathematics standards revision at conferences and board meetings around the state including Oregon Council of Teachers of Mathematics, Oregon Mathematics Education Council, Teachers of Teachers of Mathematics Council, Teacher Standards and Practices Commission, OASSA/OESPA Principal’s Conference, Oregon School Improvement Facilitators, district, school and ESD meetings.

Feedback was collated and then carefully reviewed by the panel prior to updating each draft. The review included standards alignment with over twenty different states standards of mathematics, curricular expectations from national organizations such as Achieve, College Board, National Council of Teachers of Mathematics, International Baccalaureate, and National Assessment of Education Progress. Various drafts also have been reviewed by external mathematics education experts with the National Council of Teachers of Mathematics and Northwest Regional Education Laboratory.

What Has Changed from the 2002 High School Math Standards?

The 2002 high school mathematics standards are organized according to six different Common Curricular Goals each with from 11 to 24 grade-level standards for a total of 79 standards. In addition, the standards document includes five problem solving standards, similar to the NCTM Process Standards, to be used to score mathematics problem solving tasks. Many of these standards are conceptually similar to each other and also spiraled from previous grade levels.

These revised mathematics standards for all students are organized under three mathematics disciplines and contain between 8 and 18 content standards each for a total of 40 content standards. The standards were written so as to create “a focused, coherent progression of mathematics learning, with an emphasis on proficiency with key topics.” In other words, overlap and spiraling from previous grade levels was minimized in an effort to create standards which resembled the same philosophy and scope as the recently revised K-8 Standards. These revised high school math standards, as do the 2002 standards, identify what all students should know in mathematics to earn a diploma.

Additionally, ODE created advanced mathematics standards not necessarily for all students. The advanced standards outline mathematics concepts which students might learn only after they demonstrate proficiency with the standards for all students. They are very broad in nature and are not designed to define a single course. These advanced standards are organized by five disciplines including algebra, trigonometry, discrete mathematics, statistics, and calculus.

Implementation Guidance

In addition to this content standards document, ODE will develop content specifications for the mathematics standards to clearly outline the content boundaries; provide Connections to help students and teachers understand how the key mathematics topics connect to mathematics and other content areas; a glossary of related terms to define technical mathematics language in the standards; Standards Numbering System Guide to explain how the standard numbering system is formatted; and Course Configuration and Articulation Models to help schools design courses that will best teach the standards.

Further, ODE is currently planning professional development workshops for over 300 teachers statewide with a focus on the most effective research-based instructional pedagogy. The summer sessions will be provided at six different locations statewide with grant opportunities available to districts wishing to provide this same training at the local level.

Oregon Department of Education-Draft High School Mathematics Standards Page 3 Draft Revised High School Mathematics Content Standards High School - Algebra

It is essential that the high school mathematics content standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. Students will also be expected to reflect on their solution(s). Every student should understand and apply all mathematical concepts and skills from previous grade levels to these standards.

H.1A Algebra and Numeracy: Demonstrate a deep understanding of real numbers and algebraic symbols by fluently creating, manipulating, computing with, and determining equivalent expressions, both numeric and symbolic with fluency.

H.1A.1 Compare, order, and locate real numbers on a number line. H.1A.2 Evaluate, compute with, and determine equivalent numeric and algebraic expressions with real numbers and variables that may also include absolute value, integer exponents, square roots, pi, and/or scientific notation. H.1A.3 Express square roots in equivalent radical form and their decimal approximations when appropriate. H.1A.4 Develop, identify, and/or justify equivalent algebraic expressions, equations, and inequalities using the properties of exponents, equality and inequality, as well as the commutative, associative, inverse, identity, and distributive properties. H.1A.5 Factor quadratic expressions limited to factoring common monomial terms, perfect-square trinomials, differences of squares, and quadratics of the form x2 + bx + c that factor over the integers.

H.2A Algebra: Use linear equations and functions to represent relationships and solve linear equations, linear inequalities, systems of linear equations, and systems of linear inequalities.

H.2A.1 Identify, construct, extend, and analyze linear patterns and functional relationships that are expressed contextually, numerically, algebraically, graphically, in tables, or using geometric figures. H.2A.2 Given a rule, a context, two points, a table of values, a graph, or a linear equation in either slope intercept or standard form, identify the slope, determine the x and/or y intercept(s), and interpret the meaning of each. H.2A.3 Determine the equation of a line given any of the following information: two points on the line, its slope and one point on the line, or its graph. Also, determine an equation of a new line parallel or perpendicular to a given line, through a given point. H.2A.4 Fluently convert among representations of linear relationships given in the form of a graph of a line, a table of values, or an equation of a line in slope-intercept and standard form.

Oregon Department of Education-Draft High School Mathematics Standards Page 4 H.2A.5 Given a linear function, interpret and analyze the relationship between the independent and dependent variables. Solve for x given f(x) or solve for f(x) given x. H.2A.6 Analyze how changing the parameters transforms the graph of f (x) =mx + b. H.2A.7 Write, use, and solve linear equations and inequalities using graphical and symbolic methods with one or two variables. Represent solutions on a coordinate graph or number line. H.2A.8 Solve systems of two linear equations graphically and algebraically, and solve systems of two linear inequalities graphically.

H.3A Algebra: Use quadratic and exponential equations and functions to represent relationships.

H.3A.1 Given a quadratic or exponential function, identify or determine a corresponding table or graph. H.3A.2 Given a table or graph that represents a quadratic or exponential function, extend the pattern to make predictions. H.3A.3 Compare the characteristics of and distinguish among linear, quadratic, and exponential functions that are expressed in a table of values, a sequence, a context, algebraically, and/or graphically, and interpret the domain and range of each as it applies to a given context. H.3A.4 Given a quadratic or exponential function, interpret and analyze the relationship between the independent and dependent variables, and evaluate the function for specific values of the domain. H.3A.5 Given a quadratic function of the form f (x) = x 2  bx  c (or equation of the form y =x2 + bx + c ) with integer roots, determine and interpret the roots, the vertex of the parabola, and the equation for the axis of symmetry of the parabola graphically and algebraically.

Oregon Department of Education-Draft High School Mathematics Standards Page 5 High School – Geometry

H.1G Geometry: Apply properties of two-dimensional figures.

H.1G.1 Identify, apply, and analyze angle relationships among two or more lines and a transversal to determine if lines are parallel, perpendicular, or neither. H.1G.2 Apply theorems, properties, and definitions to determine, identify, and justify congruency or similarity of triangles and to classify quadrilaterals. H.1G.3 Apply theorems of corresponding parts of congruent and similar figures to determine missing sides and angles of polygons. H.1G.4 Determine the missing dimensions, angles, or area of regular polygons, quadrilaterals, triangles, circles, composite shapes, and shaded regions. H.1G.5 Determine if three given lengths form a triangle. If the given lengths form a triangle, classify it as acute, right, or obtuse. H.1G.6 Use trigonometric ratios (sine, cosine and tangent) and the Pythagorean Theorem to solve for unknown lengths in right triangles. H.1G.7 In problems involving circles, apply theorems and properties of chords, tangents, and angles; and theorems and formulas of arcs and sectors.

H.2G Geometry: Apply properties of three-dimensional solids.

H.2G.1 Identify, classify, model, sketch, and label representations of three-dimensional objects from nets and from different perspectives. H.2G.2 Identify and apply formulas for surface area and volume of spheres; right solids, including rectangular prisms and pyramids; cones; and cylinders; and compositions thereof. Solve related context-based problems. H.2G.3 Identify and apply formulas to solve for the missing dimensions of spheres and right solids, including rectangular prisms and pyramids, cones, and cylinders, both numerically and symbolically.

Oregon Department of Education-Draft High School Mathematics Standards Page 6 H.3G Geometry: Transform and analyze figures.

H.3G.1 Recognize and identify line and rotational symmetry of two-dimensional figures. H.3G.2 Identify and perform single and composite transformations of geometric figures in a plane, including translations, origin-centered dilations, reflections across either axis or y = ±x, and rotations about the origin in multiples of 90˚. H.3G.3 Apply a scale factor to determine similar two- and three-dimensional figures, are similar. Compare and compute their respective areas and volumes of similar figures. H.3G.4 Apply slope, distance, and midpoint formulas to solve problems in a coordinate plane.

Oregon Department of Education-Draft High School Mathematics Standards Page 7 High School – Statistics

It is essential that the high school mathematics content standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. Students will also be expected to reflect on their solution(s). Every student should understand and apply all mathematical concepts and skills from previous grade levels as they apply to these standards.

H.1S Data Analysis: Analyze and interpret empirical data.

H.1S.1 Given a context, determine appropriate survey methods, analyze the strengths and limitations of a particular survey, observational study, experiment, or simulation, and the display of its data. H.1S.2 Evaluate data-based reports by considering the source of the data, the design of the study, and the way the data was analyzed and displayed. H.1S.3 Compare and draw conclusions about two or more data sets using graphical displays or central tendencies and range. H.1S.4 Use or construct a scatter plot for a given data set, determine whether there is a (n) linear, quadratic, exponential, or no trend. If linear, determine if there is a positive or negative correlation among the data; and, if appropriate, sketch a line of best fit, and use it to make predictions. H.1S.5 Construct, analyze, and interpret tables, scatter plots, frequency distributions, and histograms of data sets.

H.2S Probability: Apply basic principles of probability.

H.2S.1 Identify, analyze, and use experimental and theoretical probability to estimate and calculate the probability of simple events. H.2S.2 Determine the sample space of a probability experiment. H.2S.3 Compute and interpret probabilities for independent, dependent, complementary, and compound events using various methods (e.g., diagrams, tables, area models, and counting techniques).

Oregon Department of Education-Draft High School Mathematics Standards Page 8 Advanced Algebra

A.A.1 Relations and Functions: Analyze functions and relations (e.g. polynomial, absolute value, rational, radical, logarithmic, exponential, algebraic, piece-wise, and step functions).

A.A.1.1. Demonstrate an understanding of the concept of a function, use function notation, evaluate a function, determine whether or not a given relation is a function and determine whether or not a given function is one-to-one.

A.A.1.2 Determine the domain and range of a relation including those with restricted domains.

A.A.1.3 Represent a given relation in multiple ways and convert between each representation.

A.A.1.4 Determine whether a given relation is even, odd or neither and what this means in predicting behaviors.

A.A.1.5 Analyze the effect on the graph of a relation by changing its parameters and perform a given transformation.

A.A.1.6 Determine, verify, and graph the inverse of a function or relation (if it exists) and understand the reversing roles of domain and range.

A.A.1.7 Determine the composition of inverse functions and whether or not it is one-to-one.

A.A.1.8 Perform arithmetic operations on functions and determine the composition of functions.

A.A.1.9 Analyze the reciprocal of a function or relation.

A.A.1.10 Collect and analyze data to make predictions and to investigate scatterplots and to determine the equation for a curve of best fit including linear, power, exponential, and logarithmic functions. A.A.1.11 Connect the relationships among the solution of an equation, zero of a function, x-intercept of a graph and the factors of a polynomial expression.

A.A.1.12 Find the x and y-intercepts of a function if they exist.

Oregon Department of Education-Draft High School Mathematics Standards Page 9 A.A.1.13 Identify, distinguish between, and describe the characteristics of the following functions in tabular, verbal, graphical or symbolic form: polynomial, power, absolute value, rational, radical, logarithmic, exponential, algebraic, piece-wise, and step.

A.A.2 Inequalities, Piece-wise Functions, and Absolute Value Functions: Model and analyze piece-wise and absolute value functions. Solve inequalities and absolute value equations.

A.A.2.1 Graph, solve, and analyze inequalities in two variables.

A.A.2.2 Graph and analyze piece-wise functions.

A.A.2.3 Graph, solve, and analyze absolute value equations and inequalities.

A.A.3 Quadratic functions and other Conic Sections: Model and analyze quadratic functions. Solve quadratic equations and problems involving conics.

A.A.3.1 Perform operations on complex numbers and represent, apply and discuss the properties of complex numbers.

A.A.3.2 Derive the quadratic formula.

A.A.3.3 Solve quadratic equations using the zero product property, completing the square, the quadratic formula, and graphing.

A.A.3.4 Graph and analyze quadratic functions and relate the zeros to the discriminant.

A.A.3.5 Construct and solve quadratic inequalities in one and two variables.

A.A.3.6 Solve problems relating to conic sections including systems of equations and inequalities involving conics.

A.A.3.7 Graph and analyze equations of conic sections.

A.A.3.8 Determine conic equations from graphs or data.

A.A.4 Polynomial Functions: Model and analyze polynomial functions. Solve polynomial equations.

A.A.4.1 Perform operations on polynomial expressions.

A.A.4.2 Analyze and calculate permutations, combinations, and other systematic counting methods.

A.A.4.3 Understand and apply the binomial theorem and/or Pascal’s triangle to expand binomial expressions.

Oregon Department of Education-Draft High School Mathematics Standards Page 10 A.A.4.4 Apply long (or synthetic) division, the Fundamental Theorem of Algebra, Descartes Rule of Signs, the Intermediate Value Theorem and the Rational Root Theorem to analyze and/or determine the roots of a polynomial.

A.A.4.5 Find approximate solutions for polynomial equations using graphing technology.

A.A.4.6 Write a polynomial equation given its real and/or complex solutions.

A.A.4.7 Graph and analyze polynomial functions.

A.A.5 Radical Functions: Model and analyze radical functions. Solve radical equations.

A.A.5.1 Find equivalent expressions using the properties of rational exponents.

A.A.5.2 Perform arithmetic operations to simplify radical expressions.

A.A.5.3 Solve radical equations.

A.A.5.4 Graph and analyze radical functions.

A.A.6 Rational Functions: Model and analyze rational functions. Solve rational equations.

A.A.6.1 Find equivalent representations for rational expressions and identify restrictions.

A.A.6.2 Perform operations on rational expressions.

A.A.6.3 Solve algebraic proportions and rational equations.

A.A.6.4 Graph and analyze rational functions. A.A.7 Logarithmic and Exponential Functions: Model and analyze logarithmic and exponential functions. Solve logarithmic and exponential equations.

A.A.7.1 Establish the inverse relationship between exponential and logarithmic functions.

A.A.7.2 Prove and apply the basic properties of logarithms.

A.A.7.3 Solve exponential and logarithmic equations.

A.A.7.4 Graph and analyze exponential and logarithmic functions.

A.A.8 Matrices, Systems of Equations and Inequalities: Analyze and apply various methods to graph and solve systems of equations and inequalities.

A.A.8.1 Use matrix operations and properties of matrices to solve problems.

Oregon Department of Education-Draft High School Mathematics Standards Page 11 A.A.8.2 Solve systems of linear equations in two or three variables algebraically, graphically, and/or with matrix algebra.

A.A.8.3 Analyze an inconsistent system of equations.

A.A.8.4 Solve systems of linear inequalities by graphing.

A.A.8.5 Interpret, analyze, and solve linear programming problems.

A.A.8.6 Solve nonlinear systems of equations algebraically and graphically, including linear- quadratic and quadratic-quadratic.

A.A.9 S equences and Series: Analyze and evaluate sequences and series.

A.A.9.1 Define, recognize, and discriminate among arithmetic, geometric and other sequences and series.

A.A.9.2 Find the explicit and recursive formulas for arithmetic and geometric sequences and use these formulas to determine a specific term or term number.

A.A.9.3 Convert between a series and its sigma notation representation.

A.A.9.4 Find partial sums of arithmetic and geometric series and find sums of convergent infinite series.

A.A.9.5 Generate and describe other recursive sequences such as factorials and the Fibonacci sequence.

A.A.10 Parametric Equations: Model and analyze parametric equations.

A.A.10.1 Write and evaluate parametric equations.

A.A.10.2 Relate parametric equations to equivalent rectangular equations.

A.A.10.3 Analyze and describe graphs of parametric equations.

Oregon Department of Education-Draft High School Mathematics Standards Page 12 Trigonometry

A.T.1 Triangle Trigonometry: Analyze and solve problems involving triangles.

A.T.1.1 Develop and apply the properties of special right triangles.

A.T.1.2 Develop, define, and apply right triangle trigonometric ratios.

A.T.1.3 Develop and apply the Law of Sines and the Law of Cosines.

A.T.1.4 Develop and apply the area formulas of a triangle.

A.T.2 Trigonometric Functions: Develop, apply, and graph the six trigonometric functions and their inverses in radians and degrees.

A.T.2.1 Define the six trigonometric functions, construct the unit circle, and use the unit circle to calculate the exact values of these functions for special angles.

Oregon Department of Education-Draft High School Mathematics Standards Page 13 A.T.2.2 Justify the relationship between the tangent of the angle that a line makes with the x-axis and the slope of the line.

A.T.2.3 Evaluate trigonometric functions and inverse trigonometric functions.

A.T.2.4 Construct and analyze graphs of the six trigonometric functions and inverse trigonometric functions.

A.T.2.5 Perform translations of trigonometric functions and inverse trigonometric functions.

A.T.2.6 Solve problems using linear and angular velocity.

A.T.3 Trigonometric Identities and Equations: Derive and apply basic trigonometric identities and solve trigonometric equations.

A.T.3.1 Prove the Pythagorean Identities and other trigonometric identities and apply them to verify other identities and simplify trigonometric expressions.

A.T.3.2 Prove trigonometric identities and make substitutions using the basic identities.

A.T.3.3 Solve trigonometric equations.

A.T.4 Vectors in Two Dimensions: Solve problems involving vectors in the coordinate plane.

A.T.4.1 Perform operations on vectors.

A.T.4.2 Use vectors to model situations and solve problems.

A.T.5 Polar Coordinates and Complex Numbers: Understand and apply polar coordinates and complex numbers and their connection to trigonometric functions.

A.T.5.1 Define polar coordinates, relate them to rectangular coordinates, and fluently convert between the two.

A.T.5.2 Represent equations given in rectangular coordinates in terms of polar coordinates.

A.T.5.3 Graph equations in the polar coordinate plane.

A.T.5.4 Define complex numbers, convert complex numbers to trigonometric form, and multiply complex numbers in trigonometric form.

A.T.5.5 Derive and apply De Moivre’s Theorem.

Oregon Department of Education-Draft High School Mathematics Standards Page 14 Discrete Mathematics

A.D.1 Set Theory: Operate with sets and use set theory to solve problems.

A.D.1.1 Demonstrate understanding of the definitions of set equality, subset and null set.

A.D.1.2 Perform set operations such as union and intersection, difference, and complement.

A.D.1.3 Use Venn diagrams to explore relationships and patterns, and to make arguments about relationships between sets.

A.D.1.4 Demonstrate the ability to create the cross-product or set-theoretic product of two sets.

A.D.2 Relations and Functions: Demonstrate understanding of relations and functions.

A.D.2.1 Determine whether simple examples of discrete functions are injective and/or surjective.

A.D.2.2 Demonstrate ability to interpret examples of simple discrete functions by mapping elements from a discrete domain to a discrete range.

A.D.2.3 Demonstrate ability to produce the subset of the cross product of the domain and image of a relation corresponding to simple examples of relations.

A.D.2.4 Use concepts of reflexivity, symmetry, and transitivity to establish that a relation is an equivalence relation.

A.D.2.5 Design simple algorithms such as hashing, checksum or error-correction functions

A.D.3 Modular Arithmetic: Demonstrate understanding of modular arithmetic and its relationship to set theory.

A.D.3.1 Perform modular arithmetic operations.

A.D.3.2 Demonstrate understanding of the relationship of modular arithmetic to two numbers being congruent modulo n.

A.D.3.3 Solve practical problems or develop algorithms using modular arithmetic or congruence relations such as creating error detection codes, calculating greatest common factor, and solving simple coin-change problems.

Oregon Department of Education-Draft High School Mathematics Standards Page 15 A.D.4 Graph Theory: Understand how graphs of vertices joined by edges can model relationships and be used to solve a wide variety of problems.

A.D.4.1 Use graphs to model and solve problems such as shortest paths, vertex coloring, critical paths, routing, and scheduling problems.

A.D.4.2 Convert from a graph to an adjacency matrix and vice versa.

A.D.4.3 Use directed graphs, spanning trees, rooted trees, binary trees, or decision trees to solve problems.

A.D.4.4 Demonstrate understanding of algorithms such as depth-first and breadth-first walk of a tree or maximal matching.

A.D.4.5 Use matching or bin-packing techniques to solve optimization and other problems.

A.D.4.6 Compare and contrast different graph algorithms in terms of efficiency and types of problems that can be solved.

A.D.5 Combinatorics and Discrete Probability: Understand and apply fundamental counting techniques in solving combinatorial and probability problems.

A.D.5.1 Produce all combinations and permutations of sets.

A.D.5.2 Calculate the number of combinations and permutations of sets of m items taken n at a time.

A.D.5.3 Apply basic fundamental counting principles such as The Pigeonhole Principle, Multiplication Principle, Addition Principle, and Binomial Theorem to practical problems.

A.D.5.4 Solve probability problems such as conditional probability, probability of simple events, mutually exclusive events, and independent events

A.D.5.5 Find the odds that an event will occur given the probabilities and vice versa.

A.D.6 Sequences and Series: Analyze and evaluate sequences and series. 3

A.D.6.1 Define, recognize, and discriminate among arithmetic, geometric and other sequences and series.

A.D.6.2 Find the explicit and recursive formulas for arithmetic and geometric sequences and use these formulas to determine a specific term or term number.

A.D.6.3 Convert between a series and its sigma notation representation.

Oregon Department of Education-Draft High School Mathematics Standards Page 16

A.D.6.4 Find partial sums of arithmetic and geometric series and find sums of convergent infinite series.

A.D.6.5 Generate and describe other recursive sequences such as factorials and the Fibonacci sequence.

A.D.7 Recurrence, Recursion and Induction: Understand and apply recurrence, recursive, and inductive methods to solve problems.

A.D.7.1 Use recursive and iterative thinking to solve problems such as population growth and decline, exponential functions, problems involving sequential change and compound interest.

A.D.7.2 Use finite differences to solve problems and to find explicit formulas for recurrence relations.

A.D.7.3 Use mathematical induction to prove recurrence relations and concepts in number theory such as sums of infinite integer series, divisibility statements, and parity statements.

A.D.7.4 Use mathematical induction to analyze the validity of an iterative algorithm.

A.D.7.5 Describe arithmetic and geometric sequences recursively.

A.D.7.6 Use understanding of relationship of finite and infinite geometric series, including how the concept of limits connects them. A.D.7.7 Apply recurrence or recursion to the design and understanding of sorting and searching algorithms.

A.D.7.8 Analyze algorithms for efficiency, including how the number of steps grows as a function of the size of the problem.

A.D.7.9 Compare the efficiency of iterative and recursive solutions of a problem.

A.D.8 Logic: Understand the fundamentals of propositional logic, arguments, and methods of proof.

A.D.8.1 Use truth tables to determine truth values of compounded propositional statements.

A.D.8.2 Find the converse, inverse, and contrapositive of a statement.

A.D.8.3 Determine whether two propositions are logically equivalent.

A.D.8.4 Identify and give examples of undefined terms, definitions, axioms, and theorems.

Oregon Department of Education-Draft High School Mathematics Standards Page 17 A.D.8.5 Construct logical arguments using laws of detachment (modus ponens), syllogism, tautology, and contradiction; judge the validity of arguments, and give counterexamples to disprove statements.

A.D.8.6 Use applications of the universal and existential quantifiers to propositional statements.

A.D.8.7 Appropriately select and use methods of deductive, inductive, and indirect proof and determine whether a short proof is logically valid.

A.D.9 Social Choice: Analyze election data to evaluate different election methods and use weighted voting techniques to decide voting power within a group. Understand and use fair division techniques to solve apportionment problems.

A.D.9.1 Use election theory techniques to analyze election data such as majority, plurality, runoff, approval, the Borda method in which points are assigned to preferences, and the Condorcet method in which each pair of candidates is run off head to head.

A.D.9.2 Use fair division techniques to divide continuous objects.

A.D.9.3 Use fair division techniques to solve apportionment problems. A.D.10 Game Theory: Understand and use game theory methods to solve strictly determined games and non-strictly determined games.

A.D.10.1 Use game theory to solve strictly determined games.

A.D.10.2 Use game theory to solve non-strictly determined games.

A.D.10.3 Use game theory to create models for games.

A.D.10.4 Use game theory to find optimal mixed strategies such as expected values or payoff values.

A.D.11 Coding Theory, Compression and Cryptography: Understand coding of alphabets and simple encryption methods.

A.D.11.1 Use integer functions to encode alphabets and to create error-checking correcting codes.

A.D.11.2 Use permutations, combinations of digraph encoding and affine transformation and hash functions, to create encryption codes.

A.D.11.3 Demonstrate understanding of asymmetric public key cryptography algorithms such as RSA and Diffie-Hellman.

A.D.11.4 Demonstrate understanding of error-detecting and error-correcting codes and data compression through Huffman codes.

Oregon Department of Education-Draft High School Mathematics Standards Page 18 A.D.12 Algorithm Design: Understand methods of algorithm design and its relationship to data structures.

A.D.12.1 Designing algorithms using recurrence or iteration.

A.D.12.2 Design algorithms using divide-and-conquer.

A.D.12.3 Design algorithms using recursion.

A.D.12.4 Evaluate the efficiency of an algorithm including the order of complexity of algorithms.

A.D.12.5 Demonstrate understanding of the relationship of set theory, relations, functions, combinatorics, sequences, series, graph theory, and matrices to the design of data structures and algorithms.

A.D.12.6 Demonstrate understanding of the relationship of data structures to the design of algorithms and use this understanding to analyze algorithms.

Oregon Department of Education-Draft High School Mathematics Standards Page 19 Statistics

A.S.1 Exploratory Data: Analyze summary measures of sets of data.

A.S.1.1 Construct, interpret, and summarize numerical characteristics of univariate data sets to describe patterns and departure from patterns, using measures of center, spread, and position.

A.S.1.2 Compare distributions of univariate data by comparing center and spread, clusters and gaps, outliers, and other unusual features and comparing shapes.

A.S.1.3 Explore bivariate data by analyzing patterns, correlation, linearity, least-squares regression line, residual plots, outliers, influential points, and transformations to achieve linearity.

A.S.1.4 Explore categorical data using frequency tables and bar charts; investigating marginal, joint and conditional relative frequencies; and by comparing distributions.

A.S.2 Sampling and Experimentation: Plan, conduct, and analyze well-designed methods of data collection.

A.S.2.1 Describe the methods of data collection. Evaluate how appropriate each method is relative to the purposes of various types of inquires and hypotheses under investigation given various population distributions.

A.S.2.2 Plan, analyze, and conduct a survey, and/or observational study; describe characteristics of a well-designed and well-conducted survey; explore various sampling methods including investigating sources of bias.

A.S.2.3 Plan, analyze, and conduct an experiment; describe characteristics and components of a well-designed and well-conducted experiment; explore various methods of experimental designs; and associated sources of bias and confounding. A.S.2.4 Explore the generalizability of results and types of conclusions that can be drawn from observational studies, experiments, and surveys. Understand when each method is most appropriate, and explain the differences between the three methods.

A.S.3 Anticipating Patterns: Understand how probability can be applied as a tool used for anticipating what the distribution of data should look like under a given model.

Oregon Department of Education-Draft High School Mathematics Standards Page 20 A.S.3.1 Analyze probability by exploring such topics as “Law of Large Numbers,” addition and multiplication rule, conditional probability and independence, discrete random variables and their probability distributions, simulations of random behavior and mean, standard deviation, and learn how to select appropriate linear transformations of a random variable.

A.S.3.2 Explore the independence versus dependence of two random variables. Determine the mean and standard deviation for sum or difference of independent random variables.

A.S.3.3 Analyze the properties of the normal distribution; uses tables of the normal distribution; and explore a normal distribution as a model for measurements.

A.S.3.4 Explore sampling distributions to include: sampling distribution of a sample of proportion and mean; Binomial Distribution and Geometric Distribution; applying the Central Limit Theorem; investigating sampling distributions of a difference between two independent sample proportions and means; simulating a sampling distribution and; applying t- distributions and Chi-square distributions to the analysis of samples.

A.S.4 Statistical Inference: Estimate population parameters and test hypotheses.

A.S.4.1 Investigate the following: estimating population parameters, margins of error, confidence intervals, and properties of point estimators.

A.S.4.2 Explain the logic, meaning, and properties of confidence intervals and meaning of confidence levels. Apply this understanding to large sample confidence intervals for a: proportion, difference between two proportions, mean, difference between two means, and slope of a least-squares regression line.

A.S.4.3 Explain the logic of significance testing, null and alternative hypotheses; p-values; one-and two-sided tests; concepts of Type I and Type II errors; concept of power. A.S.4.4 Apply various large sample tests for a proportion – i.e. difference between two proportions, mean, difference between two means, Chi-square test, and slope of a least squares regression line.

A.S.4.5 Understand how to read the results of a regression, and use this to make predictions of future events with a stated confidence.

Oregon Department of Education-Draft High School Mathematics Standards Page 21 Calculus

A.C.1 Functions, Graphs, and Limits: Apply the concepts of limits, asymptotic and unbounded behavior and continuity to functions.

A.C.1.1 Define and apply the properties of limits of functions.

A.C.1.2 Investigate asymptotic and unbounded behavior in functions.

A.C.1.3 Use limits to define continuity and determine if a function is continuous.

A.C.2 Derivatives: Analyze, calculate and apply the concept of a derivative.

A.C.2.1 Investigate derivatives presented in numerical, graphic and various analytic contexts.

A.C.2.2 Investigate and apply the concepts of f , f  and f  to describe the characteristics of a function and its graph.

A.C.2.3 Apply derivatives to solve problems in various applications including distance, velocity, and acceleration and optimization problems.

A.C.2.4 Apply formulas to find derivatives both implicitly and explicitly, including using the product, quotient and chain rules.

A.C.3 Integrals: Analyze, calculate and apply the concept of an integral using.

A.C.3.1 Explore Riemann sums and other approximation techniques and apply them to approximate definite integrals.

A.C.3.2 Identify and use the properties of integrals.

A.C.3.3 Use the Fundamental Theorem of Calculus to evaluate definite integrals and represent a particular anti-derivative. A.C.3.4 Find anti-derivatives directly from derivatives of basic functions and by the method of u-substitution.

A.C.3.5 Use integrals to evaluate volumes of revolution and volumes of known cross-sections.

Oregon Department of Education-Draft High School Mathematics Standards Page 22 A.C.3.6 Use initial conditions with separable differential equations to solve for specific anti- derivatives.

A.C.3.7 Use differential equations to produce slope fields and solve for specific equations using initial conditions.

Oregon Department of Education-Draft High School Mathematics Standards Page 23 Alignment of the Process Standards and the Apply Math Essential Skill

Process NCTM* Process Standard Alignment to the Oregon Apply Standard Definition Math Essential Skill Build new mathematical knowledge Interpret a situation and apply workable through problem solving. mathematical concepts and strategies using appropriate technologies where Solve problems that arise in mathematics applicable. Problem and in other contexts. Solving Apply and adapt a variety of appropriate strategies to solve problems.

Monitor and reflect on the process of mathematical problem solving. Recognize reasoning and proof as Communicate and defend the verified fundamental aspects of mathematics. process and solution using pictures, symbols, models, narrative, or other Make and investigate mathematical methods. Reasoning and conjectures. Proof Develop and evaluate mathematical arguments and proofs.

Select and use various types of reasoning and methods of proof. Organize and consolidate mathematical Communicate and defend the verified thinking through communication. process and solution using pictures, symbols, models, narrative, or other Communicate mathematical thinking methods. coherently and clearly to peers, teachers, Communication and others.

Analyze and evaluate the mathematical thinking and strategies of others.

Use the language of mathematics to express mathematical ideas precisely.

Recognize and use connections among Interpret a situation and apply workable mathematical ideas. mathematical concepts and strategies using appropriate technologies where Understand how mathematical ideas applicable. Connections interconnect and build on one another to produce a coherent whole.

Recognize and apply mathematics in contexts outside of mathematics. * National Council of Teachers of Mathematics

Oregon Department of Education-Draft High School Mathematics Standards Page 24 Process NCTM Process Standard Alignment to the Oregon Apply Standard Definition Math Essential Skill Create and use representations to Produce evidence such as graphs, data, organize, record, and communicate or mathematical models to obtain and mathematical ideas. verify a solution.

Select, apply, and translate among Representation mathematical representations to solve problems.

Use representations to model and interpret physical, social, and mathematical phenomena.

In addition to the process standards, high school students in Oregon will also be expected to:

Additional NCTM Definition4 Alignment to the Oregon Apply Expectation Math Essential Skill Interpret a solution within the context of a Communicate and defend the verified problem process and solution using pictures, symbols, models, narrative, or other Check the reasonableness of solutions methods. Reflect on one’s solution Justify or validate a solution Produce evidence such as graphs, data, or mathematical models to obtain and Generalize a solution verify a solution.

Oregon Department of Education-Draft High School Mathematics Standards Page 25 ***SAMPLE***

Glossary of Mathematical Terms For use with the 2009 High School Math Standards

A function that has a constant rate of change and can be modeled by a straight line. Linear Function 3 Ex. g(x)  2x , f (x)  x  2 4 A theorem used in geometry: the square of the length of the hypotenuse of a right triangle equals the sum Pythagorean of the squares of the lengths of the other two sides. Theorem Ex. 32  42  52 The y-value at which a graph intersects the y-axis. Y-Intercept 3 Ex. The y-intercept of the graph of y  x  2 is  2 . 4 Expressions that denote the same value. Equivalent Expressions Ex. (2x) 2 is equivalent to 4x 2 and can be written as (2x) 2 = 4x 2

Oregon Department of Education 26 State Board of Education Meeting – May 2009 Mathematics Standards Guide to Numbering System

In response to requests from educators across the state, an Oregon Standards Numbering System has been developed using a combination of letters and numbers to uniquely identify each standard. This document is designed to help you interpret the numbering system created to identify Oregon Standards.

Core Standards

As part of the new Oregon Diploma, the State Board asked the Department of Education to identify core standards for K-8 and high school in all academic subjects. The Board’s decision is based on current research showing greater student achievement of the content standards when teachers and students focus on a few key ideas for each grade level. Oregon calls these key ideas core standards.

The goal of a core standards structure is to create fewer standards that are more focused and coherent. Using the discipline as the guide, core standards statements are developed around the “big ideas” of a content area. These core standards statements are supported by more specific content standards which provide the details necessary for curriculum and assessment. Core standards also articulate learning progressions within and between grade levels and allow for more effective lesson design, focused instruction and creation of formative assessments. Examples Mathematics K-8 (2007)

6.2.3 – Understand the meaning of probability and represent probabilities as ratios, decimals, and percents.

6 = grade level 2 = core standard (there are only three core standards for each grade) 3 = content standard related to core standard statement

Mathematics HS (2009)

H.1A.3 – Express square roots in equivalent radical form and their decimal approximations when appropriate.

H = high school 1A = core standard (A = Algebra, G = Geometry, S = Statistics, Probability & Data Analysis) 3 = content standard related to core standard statement

Oregon Department of Education 27 May 2009 Connections to Core Standard 7.1

Introduction

This document articulates how rational numbers and solving linear equations connects to: (1) mathematics knowledge learned prior to 7th grade (2) math knowledge to be learned after the 7th grade and (3) connections to other content areas. The connections listed below are designed to help teachers and students understand that rational numbers and solving linear equations relate to many other concepts in mathematics and other fields. Key vocabulary and misconceptions relating to Core Standard 7.1 are also introduced.

Core Standard 7.1:

7.1 Number and Operations and Algebra: Develop an understanding of operations on all rational numbers and solving linear equations.

7.1.1 Develop, analyze, and apply models (including everyday contexts), strategies, and procedures to compute with integers, with an emphasis on negative integers.

7.1.2 Extend knowledge of integers and positive rational numbers to solve problems involving negative rational numbers.

7.1.3 Develop and use strategies to estimate the result of rational number computations and justify the reasonableness of results.

7.1.4 Apply properties of rational numbers and algebra to write and solve linear equations in one variable.

Connections to the Standard

Key Connection(s) to Prior Math Knowledge

. Number line models beginning in 1st grade and further developed in 2nd - 6th grades reinforce a deeper understanding of integers. (1.1.2, 1.2.2, 2.2.2, 3.1.1, 3.1.5, 3.2.3, 4.2.2, 5.2.1) . Fluency with fractions, decimals, and percents beginning in 3rd grade and further developed in 4th – 6th grades are foundational to 7.1. (3.1, 4.1, 5.1, 6.1, 6.2) . Order of operations, variables, evaluating expressions and solutions to an equation help students understand how to solve one-step algebraic equations. (6.3)

Oregon Department of Education 28 State Board of Education Meeting – May 2009 Key Connection(s) to Future Math Knowledge

. 7.1 provides the foundation for basic algebra – e.g. solving equations, inequalities, equation systems and evaluating functions. . Negative numbers and number line models support the concept of absolute value.

Key Connection(s) to other Content Areas

. Science . Population dynamics . Energy conservation or flows (inputs and losses) . Temperature . Above and below sea level . Electrons and protons . Social Studies . Majority (double majority and super majority) money, stock market, debt . Economics (marginal rates, equilibrium, supply, demand, cost, profit) . Language Arts . Double negative, negative prefixes (e.g. disinterest, irrational, unrelated, anti-aging) . Arts . Balance with negative space, . Measurement building/cooking

Oregon Department of Education 29 State Board of Education Meeting – May 2009 Vocabulary terms to define

Operations - Any of various mathematical or logical processes of deriving one entity from others according to a rule. Ex. Addition, subtraction, multiplication and division.

Evaluate – To determine the value of. Ex. Evaluate 3x where x = 2. 3(2)=6.

Rational Numbers – Numbers that can be expressed as a ratio. Ex. ½ , ¾

Integers - Any of the natural numbers, the negatives of these numbers, or zero. Ex. -1, 0, 2, 15.

Common Mistakes and Associated Misconceptions

. Students sometimes think -6 is greater than -1.

Possible misconception: Confusion between magnitude and value.

. Students sometimes think that -12 is greater than 3.

Possible misconception: Confusion between magnitude and value.

. Students sometimes think that 5 + (-2) = 7 or -5 – 2 = -3

Possible misconception: addition makes things “bigger” and subtraction makes things “smaller.”

. Students sometimes think that 4  3is different than 4  (3) .

Possible misconception: negative numbers and subtraction are unrelated.

. Students sometimes think that equations that look different are not equivalent.

Possible misconception: Equations that look different are different.

Oregon Department of Education 30 State Board of Education Meeting – May 2009