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  • South Carolina Academic Standards for Mathematics

    2007 High School Support Document

    High School Core Area Elementary Algebra

    South Carolina Department of Education Office of Academic Standards

  • Draft Version 1, August 2007 2

    South Carolina Academic Standards for Mathematics 2007 High School Support Document

    Introduction

    The High School Mathematics Standards writing team developed the High School Mathematics Standards Support Guide in cooperation with South Carolina educators. The High School Mathematics Standards Support Guide

    expands upon each indicator to provide educators with additional content information such as guidelines for assessment. The High School Mathematics

    Standards support guide will assist educators with the successful implementation of standards-based mathematics instruction. Please note that this document is an unedited draft.

    Acknowledgements

    Office of Academic Standards Dr. Helena Tillar, Director, Office of Academic Standards Dr. Heyward Hickman, Project Team Leader

    Office of Instructional Promising Practices

    Dr. John Holton, MSU Coordinator Sandra Goff, MSU Mathematics Specialist

    Office of Assessment Buck Brown, Mathematics Assessment Specialist

    Harriet Pritchard, Mathematics Assessment Specialist

    Mathematics Consultants Dianne Steelman, Lexington One Mathematics Coordinator Bill Gilliam, Richland School District Two and South Carolina Council of

    Teachers of Mathematics

  • Draft Version 1, August 2007 3

    Elementary Algebra Overview from the 2007 South Carolina Academic Standards for Mathematics

    The academic standards for the elementary algebra core area establish the

    process skills and core content for Algebra 1, Mathematics for the Technologies 1, and Mathematics for the Technologies 2, which should provide students with the mathematics skills and conceptual understanding

    necessary for them to further their mathematical education or to pursue mathematics-related technical careers. These standards will be the basis for

    the development of the items on the state-required end-of-course examination for Algebra 1 and Mathematics for the Technologies 2.

    The content of the elementary algebra standards encompasses the real number system; operations involving exponents, matrices, and algebraic

    expressions; relations and functions; writing and solving linear equations; graphs and characteristics of linear equations; and quadratic relationships and functions. Teachers, schools, and districts should use the elementary

    algebra standards to make decisions concerning the structure and content of Algebra 1, Mathematics for the Technologies 1, and Mathematics for the

    Technologies 2. Content in these three courses may go beyond the elementary algebra standards.

    All courses based on the academic standards for elementary algebra must include instruction using the mathematics process standards, allowing

    students to engage in problem solving, decision making, critical thinking, and applied learning. Educators must determine the extent to which such courses

    or individual classes may go beyond these standards. Such decisions will involve choices regarding additional content, activities, and learning strategies and will depend on the objectives of the particular courses or

    individual classes.

    In all courses based on the elementary algebra standards, hand-held graphing calculators are required for instruction and assessment. Students should learn to use a variety of ways to represent data, to use a variety of

    mathematical tools such as graph paper, and to use technologies such as graphing calculators to solve problems.

    Note: The term including appears in parenthetical statements in the high school mathematics indicators to introduce a list of specifics that are

    intended to clarify and focus the teaching and learning of the particular concept. That is, within these parenthetical including statements are specified

    the components of the indicator that are critical for the particular core area with regard both to the state assessments and to the management of time in the classroom. While instruction must focus on the entire indicator, educators

    must be certain to cover the components specified in the parenthetical including statements.

  • Draft Version 1, August 2007 4

    2007 Mathematics Academic Standards Support Guide for the High

    School Core Area of Elementary Algebra

    This section of the standards support guide addresses Elementary Algebra Standard EA-2 and provides additional information about its indicators EA-2.1 thorough EA-2.10.

    Standard EA-2: The student will demonstrate through the mathematical

    processes an understanding of the real number system and operations involving exponents, matrices, and algebraic expressions.

    Indicators

    EA-2.1 Exemplify elements of the real number system (including integers, rational numbers, and irrational numbers).

    EA-2.2 Apply the laws of exponents and roots to solve problems.

    EA-2.3 Carry out a procedure to perform operations (including multiplication and division) with numbers written in scientific

    notation. EA-2.4 Use dimensional analysis to convert units of measure within a

    system. EA-2.5 Carry out a procedure using the properties of real numbers

    (including commutative, associative, and distributive) to simplify

    expressions. EA-2.6 Carry out a procedure to evaluate an expression by substituting a

    value for the variable. EA-2.7 Carry out a procedure (including addition, subtraction,

    multiplication, and division by a monomial) to simplify polynomial

    expressions. EA-2.8 Carry out a procedure to factor binomials, trinomials, and

    polynomials by using various techniques (including the greatest common factor, the difference between two squares, and quadratic trinomials).

    EA-2.9 Carry out a procedure to perform operations with matrices (including addition, subtraction, and scalar multiplication).

    EA-2.10 Represent applied problems by using matrices.

  • Draft Version 1, August 2007 5

    EA-2.1 Exemplify elements of the real number system (including integers, rational numbers, and irrational numbers).

    Taxonomy Level

    2.2-B Cognitive Process Dimension: Understand Knowledge Dimension: Conceptual Knowledge

    Key Concepts

    Real Numbers Integers Rational Numbers

    Irrational Numbers

    Continuum of Knowledge In 8th grade students compare rational and irrational numbers by using the symbols ≤, ≥, , and = (8-2.4). Also, students apply an algorithm to

    add, subtract, multiply, and divide integers (8-2.1).

    In Elementary Algebra students find examples of the following subsets of the real numbers: integers, rational numbers, and irrational numbers.

    In Intermediate Algebra, students build on their knowledge of the real number system by studying complex numbers. Intermediate Algebra

    students carry out a procedure to simplify expressions involving powers of i (IA-3.1). In addition, Intermediate Algebra students carry out a procedure

    to perform operations with complex numbers (including addition, subtraction, multiplication, and division) (IA-3.2).

    Essential Learning and Understanding It is essential for students to do the following for the attainment of this

    indicator: Give examples of integers, rational numbers, and irrational numbers.

    Understand how integers, rational numbers, and irrational numbers are

    interrelated. Classify numbers as integers, rational numbers, or irrational numbers.

    Examples of Essential Tasks

    These examples of essential tasks are not all inclusive. They are provided to give additional clarification of possible tasks that students should able to successfully complete.

    Identifying 5 is an integer and a rational number.

  • Draft Version 1, August 2007 6

    Provide an example of a number that is an integer and a rational number, such as -7.

    Identify 66.0 as a rational number.

    Give an example of a rational number such as - ⅜.

    Identify 0.25 as a rational number.

    Provide an example of an irrational number, such as 0.676676667... (does not repeat in a pattern or terminate) or ∏.

    Identify 2 as an irrational number.

    Explain why 7 is irrational (formal proof not required).

    Non-Essential Learning and Understanding It is not essential for students to do the following for the attainment of this indicator but could be important for the attainment of other indicators within

    Elementary Algebra:

    Know additional subsets of the real number system (whole numbers and counting numbers).

    Study imaginary numbers.

    Study complex numbers. Examples of Non-Essential Tasks

    The examples of non-essential tasks given below are not essential for the attainment of this particular indicator but could be important for the

    attainment of other indicators within Elementary Algebra.

    Identify 7 as a counting number.

    Know that infinitely many irrational numbers are between 0 and 1 on

    the number line.

    Give an example of a complex number, such as 5 + 3i. Misconceptions/Common Errors