Mathematics Academic Content Standards Revision High School Draft 3 Request for Review The Oregon Department of Education (ODE) is conducting a review and revision of the mathematics academic content standards. ODE anticipates that the State Board of Education will adopt revised mathematics standards in May 2009. This document contains Draft 3 of the revised standards for high school. To help guide readers of this draft, we begin by listing some of the input we are seeking from reviewers, providing some history of mathematics content standards, discussing the new Core Standards Structure, and introducing the guidelines that were used in the development of Draft 3. Input Sought From Reviewers While all feedback is welcome, there are several key questions we ask reviewers to consider as they review the draft standards. Please send your comments to paul.hibbard@state.or.us 1. Are there gaps in the draft standards under each core standard? That is, are there additional standards that you feel are necessary to completely cover the core standard? Please explain. 2. Is there mathematics content missing from these draft core standards you feel is essential for all students to know and be able to do? Are there additional content standards that need to be identified? Please explain and indicate the appropriate grade level for the additional content, and how this content meets the three criteria for inclusion as a core standard (see the section on core standards above). 3. Is there mathematics content in these draft standards you feel is not essential for all students to know and be able to do? Please explain why this mathematics content is not essential. 4. Do the draft standards contain an appropriate range of cognitive demand (i.e. depth of knowledge)? If not, please provide suggestions for improvement. 5. Are the draft standards sufficient (in number and detail) to guide development of curriculum and instruction at the local level and of a valid and reliable statewide multiple-choice test? Recent History of Mathematics Standards in Oregon In December 2007, Oregon adopted new K-8 mathematics standards per ORS 329.045 which states Oregon shall regularly and periodically review and revise its Common Curriculum Goals and rigorous academic content standards in mathematics. The Page 1
new standards were closely aligned with the 2006 National Council of Teachers of Mathematics (NCTM) Focal Points for K-8 mathematics. Since NCTM planned to write Focal Points for the high school level in 2008, the Oregon State Board of Education postponed the adoption of the high school standards until the spring of 2009 in effort to align high school standards to the proposed high school Focal Points. Unfortunately, NCTM s high school curriculum document does not include Focal Points. On September 22, 2008, Oregon Department of Education s Mathematics Content and Assessment Panel met to begin revision of the high school standards. The panel attempted to model their work after NCTM s Focal Points for K-8 Mathematics such that the seamless vertical articulation from the K-8 standards continued through high school. Oregon s New Core Standards Structure As part of the new Oregon Diploma requirements, the State Board of Education asked the Department of Education to identify core standards in all academic subjects. The Board s decision was based on current research showing greater student achievement when teachers and students focus on a few key ideas at each grade level. As the academic content standards for each content area are reviewed and revised (following Oregon s standards' review and revision cycle) they will reflect the core standards structure. The new 2007 K-8 math standards were the first to be written in the new Core Standards Structure. This new structure is organized around a small, focused, coherent set of core standards and supporting content standards at each grade level. Core standards provide the major concepts that will be the primary focus of teaching and learning at each grade. This will allow teachers and students to concentrate on fewer key learning objectives each year supporting greater depth of teaching and learning. Supporting each of these core standards are content standards, which provide the details necessary for the development of curriculum and assessments. In-depth understanding of each core standard will require, and be supported by, understanding of the underlying content standards. These standards will delineate clear learning progressions for each subject area that facilitate statewide teaching to standards and ensure that students who master the core standards at one grade-level will be ready to learn the core standards at the next. Criteria used to identify the core standards are research-based and were identified by WestEd. These criteria include: Endurance: Will the standard provide students with knowledge and skills that will be of value beyond a single test date? Leverage: Will the standard provide knowledge and skills that will be of value in multiple disciplines? Page 2
Success: Will the standard provide students with essential knowledge and skills that are necessary for success in the next level of instruction? Beyond school? The new core standards structure emphasizes key ideas that are of value for students over the long-term, across the curriculum, and for success in school, work, and life. Connections To support the content standards, ODE and the Content and Assessment Panels will develop guidance documents including connections. Connections serve three basic functions. They: 1) help provide context and bridge core standards between grades and across disciplines, 2) support learning in relation to essential skills, 3) enrich, and extend the content. Standards in different grade levels and disciplines often share common content and/or have interdependent concepts. Connections help bridge these standards and show how the standards connect. Connections also can help promote deeper conceptual understanding of core standards. Effective connections can enhance the coherence of a specific content area and of an entire curriculum. Guidelines for the Mathematics Content and Assessment Panel In preparation for the standards revision process, ODE developed general guidelines for high school mathematics content standards which are: Standards to be assessed by the Oregon Assessment of Knowledge and Skills (OAKS) for adequate yearly progress (AYP): The standards to be assessed for AYP purposes will be organized by three strands--1) Algebra, 2) Geometry and Measurement, and 3) Statistics and Probability and will be tested at the 10 th grade on the Oregon Assessment of Knowledge and Skills (OAKS) of Mathematics. These will be the only strands assessed by OAKS. The mathematics standards to be assessed for AYP purposes will include the same general content from the 2002 High School Mathematics Standards but will be organized into Core Standards similar to the 2007 K- 8 Mathematics Standards. High school standards will not be assigned to specific grade levels. High school mathematics standards will consist of content at the Algebra I and above levels in support of the Oregon Diploma requirements. For the additional mathematics being phased-in* via the new Oregon Diploma: To meet the minimum graduation requirements, students will need to complete one additional credit of mathematics beyond the two credits earned learning the standards listed in this document. This will provide an opportunity for students to focus on math that relates to their future education and careers. In some cases students may need instruction in additional topics such as trigonometry, statistics, discrete mathematics, advanced algebra, calculus (and possibly others) OR Page 3
Students may need to use the year to deepen their understanding of how the math they learned earlier applies to their specific career interests. An applied course (e.g. CTE, financial literacy) teaches the high school math standards with a value added approach. Districts would be responsible to determine the appropriate amount of content and/or value added that would earn mathematics credit. The value added requirement means that courses could not simply re-teach the same standards taught in prior courses without adding deeper conceptual or contextual understanding. *(Phase-in timeline is located at: http://www.ode.state.or.us/teachlearn/certificates/diploma/diploma-timeline.pdf) The intention of the additional year of mathematics requirement is not to require ALL students to complete Algebra II. Instead, the third year of mathematics is an opportunity for students to earn a math credit via a variety of routes. In fact, districts have many choices as to how they will leverage this third year requirement e.g. applied academics, combination of advanced mathematics topics, or traditional theory courses. The intention of the third credit requirement is, therefore, to help students prepare for opportunities after high school as they work towards completing their high school diploma. We highly encourage districts to develop a variety of options for students to choose from to meet their high school mathematics requirements and be more prepared for the choices they make immediately after graduation. Page 4
High School Standards Draft 3.0 Core Standard Structure December 2008 Page 5
Introduction: The world is a dynamic place and mathematics education is no exception. In effort to ensure our students are prepared to be productive citizens, there is currently a national focus on strategies to strengthen K-12 mathematics programs. In short, our expectations of high school graduates has evolved to meet the demands of today s workplace. Today, we expect our students to effectively and efficiently use algorithms, recall facts, problem solve, and employ reasoning and sense making in our dynamic, fast-paced, integrated, and global society. Currently, there is a commonly held belief that the U.S. mathematics curriculum lacks focus, coherence and depth (e.g., mile wide, inch deep phenomenon characterized by the Trends in International Mathematics and Science Study). Such a curriculum tends to produce large numbers of graduates without the skills necessary to meet the demanding expectations of our competitive global economy. In response to this concern, several national organizations developed guidelines for mathematics including Achieve, Inc., the College Board, the National Council of Teachers of Mathematics and the American Statistical Association. In addition, the National Assessment of Educational Progress created a new framework for test development. The Oregon Department of Education has been tasked with updating high school standards of mathematics to support the dynamic needs of our students. The revision of the high school standards follows the 2007 revision of the K-8 standards which were aligned with the NCTM K-8 Focal Points. Oregon s high school standards are, therefore, designed to align with the K-8 standards such that the vertical articulation, format, and structure flow seamlessly. In Oregon, we have high expectations of our high school graduates and we recognize the importance of mathematical literacy for all students. Our revised standards reflect these expectations which require our students to: Understand mathematical concepts Demonstrate reasoning and sense making Employ fluent computational skills Communicate precisely with appropriate, standard mathematical symbols and terms Be able to verify the reasonableness of problem solutions Justify and explain mathematical procedures Make connections to other disciplines and applications Actively engage in their learning Employ effective and appropriate use of technology (Please note: The pages to follow are the second draft of the standards to be assessed on the OAKS. Additional standards have not been written yet and will be included in later drafts.) Page 6
High School It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. FOR ALL STANDARDS STUDENTS SHOULD BE ABLE TO INTERPRET, VERIFY, AND/OR DETERMINE THE REASONABLENESS OF THEIR RESULTS WHEN APPROPRIATE. A.1 Algebra: Develop a deeper understanding of real numbers and fluently compute and simplify numeric and symbolic expressions respectively. A.1.1 Expand number sense encompassing magnitude, comparison, order, and equivalent representations when operating with irrational numbers and rational numbers in fraction, mixed number, and decimal forms. A.1.2 Evaluate, simplify and perform arithmetic operations on algebraic expressions such as polynomials, absolute value, rational, radical, and exponential expressions (with integer exponents) and express answers in simplified and approximate forms where appropriate. A.1.3 Express approximations of very large and very small numbers using scientific notation using the correct number of significant digits when physical measurements are involved and perform arithmetic operations with numbers expressed in scientific notation. A.1.4 Factor simple quadratic expressions (limited to the removal of monomial terms, perfect-square trinomials, difference of squares, and quadratics of the form 2 x + bx + c that factor over the integers). Page 7
A.2 Algebra: Represent, solve, and model linear equations, functions, and inequalities, and systems of linear equations. A.2.1 Identify, construct, and extend linear patterns and functional relationships that are expressed verbally, numerically, algebraically, and geometrically. A.2.2 Given a rule, two points, table, graph, real-world situation, or linear equation in either slope intercept or standard form, identify the slope and determine the x and y intercept(s) and interpret the meaning of each. A.2.3 When given sufficient information, find the corresponding linear function, equation, or inequality in slope-intercept form and graph it on a coordinate plane with and without technology. A.2.4 Fluently convert between representations of data given in the form of a graph of a line, a table, equation of a line in slope-intercept and/or equation of a line in standard form. A.2.5 Given the linear function f (x), explain the relationship between the independent and dependent variables for f (x), interpret the domain and range, evaluate the function at a (i.e. f (a) ), solve for x given the value of f (a) and distinguish between the value of the function and the value of x. A.2.6 Solve linear equations and inequalities using mental, graphical, and symbolic manipulation methods and represent numerical solutions on a coordinate graph or number line where appropriate. A.2.7 Solve systems of linear equations and inequalities with two variables using graphical, substitution, and elimination methods. Page 8
A.3 Algebra: Represent, solve, model, and compare quadratic and exponential equations and functions. A.3.1 Construct and extend exponential and quadratic patterns and functional relationships that are expressed verbally, numerically, algebraically, and/or geometrically. A.3.2 Recognize and distinguish between linear, quadratic, and exponential patterns and functional relationships that are expressed verbally, numerically, graphically, algebraically, and/or geometrically. A.3.3 Represent a given quadratic or exponential function as a table of values and as a graph with and without technology. A.3.4 Given the quadratic or exponential function f (x), explain the relationship between the independent and dependent variables for f (x), interpret the domain and range, evaluate the function at a (i.e. f (a) ). 2 A.3.5 Using quadratics of the form (or f(x) = x + bx + c ) with integer roots find the roots using graphing or the zero product property. Page 9
G.1 Geometry: Transform and classify shapes on the coordinate plane. G.1.1 Apply coordinate geometry to determine line or rotational symmetry of shapes. G.1.2 Identify and perform single and composite transformations to images including translations, dilations, reflections across either axis or, and rotations about the origin in multiples of. G.1.3 Apply slope, distance, and midpoint formulas to characterize two-dimensional shapes. Page 10
G.2 Geometry: Apply the properties of two-dimensional shapes. G.2.1 Use angle relationships between two or more lines to solve problems and justify results. G.2.2 Use relationships between angles, arcs and line segments to solve problems. G.2.3 Apply theorems involving the properties of triangles, quadrilaterals, circles and their component parts to determine and justify congruence or similarity. G.2.4 Apply theorems of corresponding parts of congruent and similar figures to determine missing sides and angles. G.2.5 Determine the missing dimensions of polygons, circles, composite shapes and shaded regions using algebraic and/or geometric properties. G.2.6 Determine the measure of angles of polygons and composite shapes and arc measures of circles using algebraic and/or geometric properties. G.2.7 Use trigonometric functions, and angle and side relationships of special right triangles (30-60-right triangles and isosceles right triangles) to solve for unknown length and determine distances and solve problems. Page 11
G.3 Geometry: Apply the properties of three-dimensional shapes. G.3.1 Recognize, model, sketch, and label representations of three-dimensional objects from different perspectives. G.3.2 Identify and apply the formulas for surface area and volume of spheres; right solids including prisms, pyramids, cones, and cylinders; and compositions thereof, to solve problems leaving answers in exact answer and/or approximate form. G.3.3 Solve for the missing dimensions of prisms, pyramids, cones, cylinders, and spheres either numerically or symbolically using appropriate measurement formulas. Page 12
PS 1 Data Analysis: Apply statistics to interpret data. PS1.1 Describe strengths and limitations of a particular survey, observational study, experiment, or simulation, critically analyze the results and distinguish between the concepts of correlation and causation. PS1.2 Determine and explain the general effect of changes in data on the mean, median, mode, and range of a data set. PS1.3 Use graphical displays and the empirical rule to evaluate the appropriateness of the normal model for a given set of data, and use the empirical rule to estimate the probability that an event will occur in a specific interval that can be described in terms of whole numbers of standard deviations about the mean. PS1.4 Use data to determine and use a line of best fit to create a linear model. Analyze the reasonableness of the model as a predictor to make inferences about the data. Page 13
PS 2 Probability: Apply basic principles of probability. PS2.1 Differentiate between experimental probability and theoretical probability. PS2.2 Determine and explain the sample space of a particular event. PS2.3 Determine and explain the total number of arrangements of objects in a given set by applying counting strategies, combinations, or permutations. PS2.4 Compute theoretical probabilities for conditional, dependent, independent, complementary, and compound events using various methods including diagrams, tables, area models, and counting techniques and apply computations to determine expected value in real life situations. PS2.5 Compute experimental probabilities by performing simulations or experiments involving a probability model and using relative frequencies and outcomes. Page 14
R.1 Mathematical Reasoning: Understand and apply inductive and deductive reasoning in a variety of contexts within mathematics. R.1.1 Distinguish between deductive and inductive reasoning, identify the strengths of each and, explain it s application in mathematics. R.1.2 Make, test, and confirm or refute conjectures using both inductive and deductive reasoning; construct simple logical arguments and proofs formally or informally; and determine simple counterexamples both orally and in written form limited to: Basic theorems of Euclidean geometry including the Pythagorean Theorem and its converse. Plane figures within the context of coordinate geometry Algebraic relationships using properties of equality, inequality, addition and multiplication. Page 15