Algebra I & II: New Course Content Standards End-of-Year Assessments and Model Lessons

1. Algebra I & II:New Course Content StandardsEnd-of-Year Assessments and Model Lessons Jim Rahn www.jamesrahn.com james.rahn@verizon.net

2. Schedule for Today • 8:00-8:15 Introductions • 8:15- 9:15 Intro to Big Ideas of Algebra I and II • 9:15-10:15 Intro to Content Benchmarks for Algebra I and II • 10:15-11:15 Review of Released Questions and Practice Test for Algebra I • 11:15-12:00 Review of Released Questions for Algebra II • 12:00-1:00 Lunch • 1:00-1:45 First Exemplary Activity • 1:45-2:30 Second Exemplary Activity

3. Conceptual Understanding Assessing New Jersey Math Standards Data Analysis, Probability Statistics, and Discrete Math Number Sense, Concepts and Applications Spatial Sense and Geometry Patterns, Functions and Algebra Procedural Knowledge Problem Solving Skills POWER BASE Reasoning Connections Communication Problem Solving - Estimation, Tools and Technology Excellence and Equity

4. Big Ideas:Algebra I and II New Jersey Algebra I and II Core Content Standards

5. Algebra is the study of patterns and functions. • What is your definition of Algebra? • In Algebra I, students: • focus on understanding the big ideas of equivalence and linearity; • learn to use a variety of representations, including modeling with variables; • begin to build connections between geometric objects and algebraic expressions; • and use what they have learned previously about geometry, measurement, data analysis, probability, and discrete mathematics as applications of algebra.

6. Algebra I builds a strong conceptual foundation for students as they continue the study of mathematics. • The core Algebra I content described in this draft follows the outline of the test specifications developed by the Achieve consortium in the production of the Algebra I End of Course Assessment and includes content needed to promote deep understanding of algebraic concepts that builds as students progress through higher levels of mathematics. • Students studying Algebra I should use appropriate tools (e.g., algebra tiles to explore operations with polynomials, including factoring) and technology, such as regular opportunities to use graphing calculators and spreadsheets. • Technological tools assist in illustrating the connections between algebra and other areas of mathematics, and demonstrate the power of algebra.

7. Algebra is the study of patterns and functions. In Algebra II, students • focus on understanding the big ideas of families of functions and extending algebraic principles and techniques to new situations. • learn to view functions as algebraic objects and extend their understanding of number systems to complex numbers. • continue to build connections between geometric objects and algebraic expressions. • use what they have learned previously about geometry, measurement, data analysis, probability, and discrete mathematics as applications of algebra.

8. Algebra II is more than what is assessed; it should build a strong conceptual foundation for students to rely upon as they move onward in their study of mathematics. • A rigorous course in Algebra II for New Jersey’s students is described. This course follows the outline of the test specifications developed by the Achieve consortium to a great extent but also includes content which is foundational for the content to be tested and content beyond what will be tested that is included in the New Jersey Core Curriculum Content Standards. • Technology: Students in an Algebra II course should use appropriate tools and technology. In particular, they should regularly use graphing calculators and spreadsheets. These technological tools not only motivate the study of algebra, illustrate connections between algebra and other areas of mathematics, and demonstrate the power of algebra, but also serve as essential tools for exploring more complex topics.

9. Big Ideas of Algebra I • Core content for Algebra I includes a number of discrete skills and concepts, each related to broader mathematical principles. In teaching and learning Algebra I, it is important for teachers and students to comprehend the following big ideas and to connect the individual skills and concepts of Algebra I to these broad principles.

10. PATTERNS AND FUNCTIONS • Algebra provides language through which we describe and communicate mathematical patterns that arise in both mathematical and non-mathematical situations, and in particular, when one quantity is a function of a second quantity or where the quantities change in predictable ways. Ways of representing patterns and functions include tables, graphs, symbolic and verbal expressions, sequences, and formulas.

11. EQUIVALENCE: • There are many different – but equivalent – forms of a number, expression, function, or equation, and these forms differ in their efficacy and efficiency in interpreting or solving a problem, depending on the context. Algebra extends the properties of numbers to rules involving symbols; when applied properly, these rules allow us to transform an expression, function, or equation into an equivalent form and substitute equivalent forms for each other. Solving problems algebraically typically involves transforming one equation to another equivalent equation until the solution becomes clear.

12. REPRESENTATION & MODELING WITH VARIABLES • Quantities can be represented by variables, whether the quantities are unknown (as in 5x + 3 = 13), changing over time (as in h = -16t2), parameters (the m and b in y = mx + b), or probabilities (where p2 represents the probability that an event with probability p occurs twice). Relationships between quantities can be represented in compact form using expressions, equations, and inequalities. Representing quantities by variables gives us the power to recognize and describe patterns, make generalizations, prove or explain conclusions, and solve problems by converting verbal conditions and constraints into equations that can be solved. Representing quantities with variables also enables us to model situations in all areas of human endeavor and to represent them abstractly.

13. LINEARITY • In many situations, the relationship between two quantities is linear so the graphical representation of the relationship is a geometric line. Linear functions can be used to show a relationship between two variables that has a constant rate of change and to represent the relationship between two quantities which vary proportionately. Linear functions can also be used to model, describe, analyze, and compare sets of data. While linearity might be considered to

14. CONNECTIONS BETWEEN ALGEBRA & GEOMETRY • Geometric objects can be represented algebraically (for example, lines can be described using coordinates), and algebraic expressions can be interpreted geometrically (for example, systems of equations and inequalities can be solved graphically).

15. CONNECTIONS BETWEEN ALGEBRA & SYSTEMATIC COUNTING, PROBABILITY, AND STATISTICS • Algebra provides a language and techniques for analyzing situations that involve chance and uncertainty, including the systematic listing and counting of all possible outcomes (as well as informal explorations of Pascal’s Triangle), the determination of their probabilities, the calculation of probabilities of various events (e.g. that throwing two dice will yield a total of 7), predictions based on experimental probabilities, and correlations between two variables.

16. Big Ideas for Algebra II • Algebra II expands the scope of ideas and techniques introduced in Algebra I and previous courses to new situations. Algebra includes a number of seemingly disconnected skills and concepts, but each of these is related to broader mathematical principles. In teaching Algebra II, it is important for teachers and students to recognize the following big ideas and connect the individual skills and concepts of Algebra II to these broad principles and through them to each other.

17. NUMBER SYSTEMS • The whole numbers, integers, and rational numbers are extended to real numbers and complex numbers, and the operations are extended to include fractional as well as integer exponents.  Each expansion of the number system involves expanding operations and procedures that worked in the original system, adding new operations, procedures, and equivalences (viewing decimals, for example, as infinite sums) and making it possible to solve new problems.

18. FUNCTIONS AS ALGEBRAIC OBJECTS • Numerical operations can be applied not only to numbers but also to algebraic expressions.  This means in practice that functions can be added, subtracted, multiplied, and divided to obtain new functions, and that powers and roots can be applied to functions as well. 

19. FAMILIES OF FUNCTIONS • A variety of families of functions can be used to model concrete situations, in addition to the linear and quadratic functions studied earlier.  These include other polynomial functions, simple rational functions, exponential functions, and periodic functions (sine and cosine).  Students should be able to visualize the types of graphs that represent these different families of functions and be familiar with the types of change that these families represent and the kinds of mathematical and non-mathematical situations which these families of functions are useful for modeling and analyzing.

20. EXTENDING ALGEBRAIC PRINCIPLES AND TECHNIQUES TO NEW SITUATIONS • Familiar principles and techniques can be generalized and applied to new domains. Factoring, for example, can be used not only with polynomials but also with expressions involving fractional exponents, and methods for simplifying quotients can be used when the numerator and denominator involve exponential expressions. Techniques involving solving quadratic equations can be used when a substitution (like y = x2/3) transforms an equation into a quadratic equation. The new families of functions and the new transformations from one expression for a function to equivalent expressions provide new techniques for solving problems algebraically, since that typically involves repeatedly transforming one equation to another equivalent equation, that is, one that has the same solution set. Students should be able to look at unfamiliar algebraic situations and recognize them as forms of simpler algebraic situations encountered previously.

21. CONNECTIONS BETWEEN ALGEBRA & GEOMETRY • Geometric objects can be represented algebraically (for example, circles can be described using equations), and algebraic expressions and equations can be interpreted geometrically (for example, systems of equations and inequalities can be solved graphically). These connections between algebra and geometry are extended substantially from linear equations to linear inequalities in two variables, families of parabolas and exponential functions, and graphs (geometric realizations) of polynomial and simple rational functions.

22. CONNECTIONS BETWEEN ALGEBRA & SYSTEMATIC COUNTING, PROBABILITY, AND STATISTICS • Algebra provides a language and techniques for analyzing situations that involve chance and uncertainty, including the systematic listing of all possible outcomes (using the choose numbers in Pascal’s Triangle when appropriate and linking them to binomial coefficients), the determination of their probabilities, the calculation of probabilities of various events (e.g., that throwing two dice will yield a total of 7), predications based on experimental probabilities, and correlations between two variables.

23. A Look at the Content Standards (Test Standards)for Algebra I & II

25. One or more of these modules may be used by states, districts, or schools whose curriculum includes these topic areas in Algebra II or an equivalent course.

26. Content Benchmarks

27. For Each Area of Study (Algebra I) Operations on Numbers and Expressions • Essential Questions • What are some ways to represent, describe, and analyze patterns (that occur in our world)? • When is one representation of a function more useful than another? • How can we use algebraic representation to analyze patterns? • Why are number and algebraic patterns important as rules? • How are arithmetic operations related to functions? • How can numeric operations be extended to algebraic objects? • Why is it useful to represent real-life situations algebraically? • What makes an algebraic algorithm both effective and efficient?

28. For Each Area of Study (Algebra I) Operations on Numbers and Expressions • Enduring Understandings • Logical patterns exist and are a regular occurrence in mathematics and the world around us. • Algebraic representation can be used to generalize patterns and relationships. • The same pattern can be found in many different forms. • Relationships can be described and generalizations made for mathematical situations that have numbers or objects that repeat in predictable ways. • Functions are a special type of relationship or rule that uniquely associates members of one set with members of another set. • Algebraic and numeric procedures are interconnected and build on one another to produce a coherent whole. • Rules of arithmetic and algebra can be used together with (the concept of) equivalence to transform equations and inequalities so solutions can be found to solve problems. • Variables are symbols that take the place of numbers or ranges of numbers; they have different meanings depending on how they are being used. • Proportionality involves a relationship in which the ratio of two quantities remains constant as the corresponding values of the quantities change.

29. For Each Area of Study (Algebra II) Operations on Numbers and Expressions • Essential Questions • What are some equivalent ways to represent numbers? • Enduring Understandings • A quantity can be represented numerically in various ways. • The set of real numbers is infinite, and each real number can be associated with a unique point on the number line.

30. Content Benchmarks and Comments • O1.c & O2.a • Apply the laws of exponents to numerical and algebraic expressions with integral exponents to rewrite them in different but equivalent forms or to solve problems. • Scope of Content: • For applications, this includes using and interpreting appropriate units of measurement, estimation, and the appropriate level of precision. • Instructional Focus: • Representing, computing, and solving problems using numbers in scientific notation. • Translating to expressions with only positive exponents. • Examples: • Translating to expressions with only positive exponents. • Translating to expressions with variables appearing only in the numerator.

31. State Assessment Assumption: All algebraic expressions are defined. • SCR: Multiply, giving the answer without exponents. • SCR: Write the expression in simplest form. • SCR: Write the expression in simplest form.

32. Purpose of End-of-Year Assessments

33. ADP Algebra I and II End-of-Course Exams serves similar, parallel purposes: • 1. To improve curriculum and instruction—and ensure consistency within and across states. The exam will help classroom teachers • focus on the most important concepts and skills in an Algebra I& II or equivalent, class and identify areas where the curriculum needs to be strengthened. • For schools administering both exams, the Algebra I Exam will compliment the Algebra II Exam and will help ensure a compatible, consistent and well-aligned Algebra curriculum. • Once standards are set teachers will get test results back within three weeks of when the exam is administered, which will provide sufficient time to make the necessary adjustments for the next year’s course.

34. 2. To help high schools determine if students are ready for rigorous higher level mathematics courses. Because the test is aligned • with the ADP mathematics benchmarks, it will measure skills students need to succeed in mathematics courses beyond Algebra I. • High schools will be able to use the results of the exam to tell Algebra I students, parents, teachers and counselors whether a student is ready for higher level mathematics, or if they have content and skill gaps that need to be filled before they enroll in the next mathematics class in their high school’s course sequence. • This information should help high schools better prepare their students for upper level mathematics, which might include passing high school exit exams or state mathematics graduation exams. This will reduce the need for multiple retakes of courses or exams needed to graduate, hopefully avoiding remedial courses designed to review Algebra I skills and concepts.

35. 3. To compare performance and progress among the participating states. • Having agreed on the content expectations for courses at the Algebra I level, states are interested in tracking student performance over time. Achieve will issue a report each year comparing performance and progress among the participating states. This report will help state education leaders, educators and the public assess performance, identify areas for improvement and evaluate the impact of state strategies for improving secondary math achievement.

36. The Algebra I End-of-Course Exam: Successful students will • Demonstrate conceptual understanding of the properties and operations of real numbers with emphasis on ratio, rates, and proportion and numerical expressions containing exponents and radicals. • Operate with polynomial expressions, factor polynomial expressions and use algebraic radical expressions. • Analyze, represent and graph linear functions including those involving absolute value and recognize and use linear models. • Solve and graph linear equations and inequalities and will be able to use them to represent contextual situations. • Solve systems of linear equations and model with single variable linear equations, one- or two-variable inequalities, or systems of equations. • Demonstrate facility with estimating and verifying solutions of linear equations, making use of technology where appropriate to do so. • Represent simple quadratic functions in multiple ways and use quadratic models, as well as solve quadratic equations. • Make connections to algebra will be made through the interpretation of linear trends in data, the comparison of data using summary statistics, probability and counting principles, and the evaluation of data-based reports in the media.

37. There are a variety of types of test items that will assess this content, including some that cut across the objectives in a standard and require students to make connections and, where appropriate, solve rich contextual problems. • The Algebra I End-of-Course Exam will include a three types of items: • multiple-choice items (worth 1 point each), • short-answer items (worth 2 points each) and • extended-response items (worth 4 points each). • Approximately thirty percent of the student’s score will be based on the short-answer and extended-response items. • Although the test is untimed, it is designed to take approximately 120 minutes, comprised of two 60 minute sessions, one of which will allow calculator use. However, some students may require – and should be allowed – additional time to complete the test. • Test items, in particular extended-response items, may address more than one content objective and benchmark within a standard. Each standard within the exam is assigned a priority, indicating the approximate percentage of points allocated to that standard on the test.

38. The Algebra II End-of-Course Exam: Successful students will • demonstrate conceptual understanding of the properties and operations of real and complex numbers • Be able to make generalizations through the use of variables resulting in facility with algebraic expressions. • solve single and systems of linear equations and inequalities and will be able to use them to represent contextual situations. • be able to demonstrate facility with estimating and verifying solutions of various non-linear equations, • Make use of technology where appropriate to do so. • Demonstrate knowledge of functions and their properties – distinguishing among quadratic, higher-order polynomial, exponential, and piecewise-defined functions – and recognize and solve problems that can be modeled by these functions.

39. They will be required to analyze these models, both symbolically and graphically, and to determine and effectively represent their solution(s). There are a variety of types of test items developed that will assess this content, including some that cut across the objectives in a standard and require students to make connections and, where appropriate, solve rich contextual problems. • There are three types of items on the Algebra II End-of-Course Exam, including • multiple choice (worth 1 point each), • short answer(worth 2 points each), and • extended response (worth 4 points each). • At least one-third of the student’s score will be based on the combined scores of the short-answer and extended-response items. Although the test is untimed, it is designed to take approximately 180 minutes, comprised of two 90 minute sessions, one of which will allow calculator use. • Test items, in particular extended-response items, may address more than one content objective and benchmark within a standard. Each standard within the exam is assigned a priority, indicating the approximate percentage of points allocated to that standard on the test.

40. Calculator Policy The appropriate and effective use of technology is an essential practice in the Algebra I classroom. • Students should learn to work mathematically without the use of technology. Computing mentally or with paper and pencil is required on the Algebra I End-of-Course Exam and should be expected in classrooms where students are working at the Algebra I level. • Students are expected to have access to a calculator for one of the two testing sessions and the use of a graphing calculator is strongly recommended. • Scientific or four-function calculators are permitted but not recommended because they do not have graphing capabilities. Students should not use a calculator that is new or different for them on the exam but rather should use the calculator they are accustomed to and use every day in their classroom work. • Calculator Policy at www.achieve.org/AssessmentCalcPolicy .

41. Algebra I Level Curriculum: Modeling and problem solving are at the heart of the curriculum at the Algebra I level. • Mathematical modeling consists of • recognizing and clarifying mathematical structures that are embedded in other contexts, formulating a problem in mathematical terms, • using mathematical strategies to reach a solution and interpreting the solution in the context of the original problem. • Students must be able to solve practical problems, representing and analyzing the situation using symbols, graphs, tables or diagrams. • They must effectively distinguish relevant from irrelevant information, identify missing information, acquire needed information and decide whether an exact or approximate answer is called for, with attention paid to the appropriate level of precision. • After solving a problem and interpreting the solution in terms of the context of the problem, they must check the reasonableness of the results and devise independent ways of verifying the results. • Problems that require extended time for solution should also be addressed in the Algebra I level classroom, even though they cannot be included in this end-of-course exam.

42. Algebra II Level Curriculum Function modeling and problem solving is the heart of the curriculum at the Algebra II level. • Mathematical modeling consists of • recognizing and clarifying mathematical structures that are embedded in other contexts, • formulating a problem in mathematical terms, • using mathematical strategies to reach a solution, and interpreting the solution in the context of the original problem. • Students must be able to solve practical problems, representing and analyzing the situation using symbols, graphs, tables, or diagrams. • They must effectively distinguish relevant from irrelevant information, identify missing information, acquire needed information, and decide whether an exact or approximate answer is appropriate, with attention paid to the appropriate level of precision. • After solving a problem and interpreting the solution in terms of the context of the problem, they must check the reasonableness of the results and devise independent ways of verifying the results. • Problems that require extended time for solution should also be addressed in the Algebra II level classroom, even though they cannot be included in this end-of-course exam.

43. Algebra I Level Classroom Practices Effective communication using the language of mathematics is essential in a class engaged in Algebra I level content. • Correct use of mathematical definitions, notation, terminology, syntax, and logic should be required in all work at the Algebra I level. • Students should be able to translate among and use multiple representations of functions fluidly and fluently. • They should be able to report and justify their work and results effectively. • To the degree possible, these elements of effective classroom practice are reflected in the Algebra I End-of-Course Exam content standards.

44. Algebra II Level Classroom Practices Effective communication using the language of mathematics is essential in a class engaged in Algebra II level content. • Correct use of mathematical definitions, notation, terminology, syntax, and logic should be required in all work at the Algebra II level. • Students should be able to translate among and use multiple representations of functions fluidly and fluently. • They should be able to report and justify their work and results effectively. • To the degree possible, these elements of effective classroom practice are reflected in these Algebra II End-of-Course Exam content standards.

45. Review of the Algebra I Practice Test and Released Items

46. Algebra I # 1

49. Algebra I #2