Part I: Content Specifications in Mathematics Part I: Content Domains for Subject Matter Understanding and Skill in Mathematics Domain 1: Number Sense 1.1Numbers, Relationships Among Numbers, and Number Systems. Candidates for Multiple Subject Teaching Credentials understand base ten place value, College Algebra P.3 (The number following either College Algebra or Concepts refers to the section of the respective textbook where this is addressed. The syllabus indicates when this topic is actually discussed during the quarter.) number theory concepts (e.g., greatest common factor), College Algebra P.5 and the structure of the whole, integer, rational, and real number systems. College Algebra P.1 They order integers, mixed numbers, rational numbers (including fractions, decimals, and percents) and real numbers. College Algebra P.2 They represent numbers in exponential and scientific notation. College Algebra P.3 They describe the relationships between the algorithms for addition, subtraction, multiplication, and division. College Algebra P.1, 4, 5, 6 They understand properties of number systems and their relationship to the algorithms, [e.g., 1 is the multiplicative identity; 27 + 34 = 2 X 10 + 7 + 3 X 10 + 4 = (2 + 3) X 10 + (7 + 4)]. College Algebra P.1 Candidates perform operations with positive, negative, and fractional exponents, as they apply to whole numbers and fractions. 1
College Algebra P.1, 6 2
1.2 Computational Tools, Procedures, and Strategies. Candidates demonstrate fluency in standard algorithms for computation College Algebra Chapter P requires fluency in all of the standard algorithms. They demonstrate their proficiency on the test over chapters P and 1. and evaluate the correctness of nonstandard algorithms. College Algebra 1.2 and 1.6 They demonstrate an understanding of the order of operations. College Algebra P.4 They round numbers, College Algebra P.3, 1.2, 1.6, etc estimate the results of calculations, Concepts 11.1 and place numbers accurately on a number line. College Algebra P.2 They demonstrate the ability to use technology, such as calculators or software, for complex calculations. We require all college algebra students to become proficient in the use of MathCAD for numerical computations, symbolic manipulations and graphing. They take a test devoted entirely to MathCAD on the last day of the quarter. Besides that students are required to possess and use a scientific calculator on tests. See these requirements listed under the Text portion of the Syllabus Concepts of Math students Use the computer with M.S. Excel to investigate statistics. They use Geometer s Sketchpad to explore geometry. 3
Domain 2: Algebra and Functions 2.1 Patterns and Functional Relationships. Candidates represent patterns, including relations and functions, through tables, College Algebra 2.2, 3.2, 4.2 4.3 graphs, College Algebra chapters 2 and 3 for algebraic graphs and chapter 4 for logarithmic and exponential graphs. verbal rules, or College Algebra1.2, 1.6 symbolic rules. College Algebra 1.2, 16 They use proportional reasoning such as ratios, equivalent fractions, and similar triangles, to solve numerical, algebraic, and geometric problems. College Algebra 1.6 2.2 Linear and Quadratic Equations and Inequalities. Candidates are able to find equivalent expressions for equalities College Algebra 1.1 and 1.3 and inequalities, College Algebra 1.5 explain the meaning of symbolic expressions (e.g., relating an expression to a situation and vice versa), College Algebra 1.2, 1.6, 4.4 find the solutions, College Algebra 1.1, 1.3, 1.4, 3.3, 4.5 and represent them on graphs. College Algebra 2.5, 3.3, 3.5 They recognize and create equivalent algebraic expressions (e.g., 2(a+3) = 2a + 6), College Algebra 1.1, 1.2, 1.3, 1.4 and represent geometric problems algebraically (e.g., the area of a triangle). College Algebra 1.2, 1.3, 1.4 4
Candidates have a basic understanding of linear equations and their properties (e.g., slope, perpendicularity); College Algebra 1.1 and 2.1 the multiplication, College Algebra P.4 division, College Algebra 3.1 and factoring of polynomials; College Algebra P.5 and chapter 3 and graphing College Algebra chapters 2 and 3 and solving quadratic equations through factoring and completing the square. College Algebra 1.3 They interpret graphs of linear and quadratic equations and inequalities, College Algebra 9.1 and 9.2 including solutions to systems of equations. College Algebra 9.1, 9.2, and 10.1, 10.3 5
Domain 3: Measurement and Geometry 3.1 Two- and Three-dimensional Geometric Objects. Candidates for Multiple Subject Teaching Credentials understand characteristics of common two- Concepts 10.1, 10.2 and three-dimensional figures, Concepts 10.3, such as triangles (e.g., isosceles and right triangles), quadrilaterals, Concepts 10.1, 10.2, Manipulative Sessions 4, 5 (The number following Manipulative Session refers to the particular session. The syllabus indicates when this topic is actually discussed during the quarter.) and spheres. Concepts 11.4 They are able to draw conclusions based on the congruence, Concepts 13.1 similarity, or lack thereof, of two figures. Concepts13.3 They identify different forms of symmetry, Concepts12,2 translations, rotations, and reflections. Concepts 12.1 They understand the Pythagorean theorem and its converse. College Algebra 1.3; Concepts 11.3 They are able to work with properties of parallel lines. Concepts 10.1 6
3.2 Representational Systems, Including Concrete Models, Drawings, and Coordinate Geometry. Candidates use concrete representations, such as manipulatives, Concepts Manipulative Session 1, 5, drawings, Concepts 10,1, 10.2; Manipulative Session 4 and coordinate geometry to represent geometric objects. Concepts 14.3 They construct basic geometric figures using a compass and straightedge, Concepts 13.2 and represent three-dimensional objects through two-dimensional drawings. Concepts 10.3 They combine and dissect two- and three-dimensional figures into familiar shapes, such as dissecting a parallelogram and rearranging the pieces to form a rectangle of equal area. Concepts 11.2 7
Content Specifications in Mathematics (Continued) 3.3 Techniques, Tools, and Formulas for Determining Measurements. Candidates estimate and measure time, College Algebra 4.6 in interest calculations and radioactive decay. Concepts 11.1 length, College Algebra P.1; Concepts 11.1 angles, Concepts 10.2 perimeter, Concepts 11.2 area, College Algebra P.5; Concepts 11.2 surface area, College Algebra 3.5; Concepts 11.4 volume, College Algebra 2.2; Concepts 11.4 weight/mass, Concepts 11.1 and temperature through appropriate units and scales. Concepts 11.1 They identify relationships between different measures within the metric or customary systems of measurements and estimate an equivalent measurement across the two systems. Concepts 11.1 They calculate perimeters and areas of two-dimensional objects College Algebra 1.2; Concepts 11.2 and surface areas and volumes of three-dimensional objects. College Algebra 3.5; Concepts 11.4 They relate proportional reasoning to the construction of scale drawings or models. 8
Concepts 12.1 They use measures such as miles per hour to analyze and solve problems. College Algebra 1.2 9
Domain 4: Statistics, Data Analysis, and Probability 4.1 Collection, Organization, and Representation of Data. Candidates represent a collection of data through graphs, tables, or charts. Concepts 8.1 They understand the mean, median, mode, and range of a collection of data. Concepts 8.2 They have a basic understanding of the design of surveys, such as the role of a random sample. Concepts 8.3 4.2 Inferences, Predictions, and Arguments Based on Data. Candidates interpret a graph, table, or chart representing a data set. Concepts 8.1 They draw conclusions about a population from a random sample, Concepts 8.3 and identify potential sources and effects of bias. Concepts 8.3 10
4.3 Basic Notions of Chance and Probability. Candidates can define the concept of probability in terms of a sample space of equally likely outcomes. Concepts 9.3 They use their understanding of complementary, mutually exclusive, dependent, and independent events to calculate probabilities of simple events. Concepts9.1, 9.3 They can express probabilities in a variety of ways, including ratios, proportions, decimals, and percents. Concepts 9.2, 9.3 11
Content Specifications in Mathematics (Continued) Part II: Subject Matter Skills and Abilities Applicable to the Content Domains in Mathematics Candidates for Multiple Subject Teaching Credentials identify and prioritize relevant and missing information in mathematical problems. College Algebra 1.2; Concepts Manipulative Session reports for 1, 5, 9 They analyze complex problems to identify similar simple problems that might suggest solution strategies. Concepts Manipulative Session reports for 1, 5 They represent a problem in alternate ways, such as words, symbols, concrete models, and diagrams, to gain greater insight. Concepts Manipulative Session reports for 1, 5 They consider examples and patterns as means to formulating a conjecture. Concepts Manipulative Sessions 1, 4, 5, 7 Candidates apply logical reasoning and techniques from arithmetic, algebra, geometry, and probability/statistics to solve mathematical problems. Concepts Manipulative Session reports for all sessions They analyze problems to identify alternative solution strategies. Concepts Manipulative Session reports for 1,7 They evaluate the truth of mathematical statements (i.e., whether a given statement is always, sometimes, or never true). Concepts tests 2 & 3; Concepts Manipulative Session reports for 9 They apply different solution strategies (e.g., estimation) to check the reasonableness of a solution. They demonstrate that a solution is correct. Concepts Manipulative Session reports for 1, 4, 5 Candidates explain their mathematical reasoning through a variety of methods, such as words, Concepts Manipulative Session reports for all sessions numbers, Concepts Manipulative Session reports for 1, 4, 5 symbols, 12
Concepts Manipulative Session reports for 1, 4, 5 charts, Concepts Manipulative Session reports for 2, 3 graphs, Concepts Manipulative Session reports for 2, 3 tables, Concepts Manipulative Session reports for 1, 2, 3, 5 diagrams, Concepts Manipulative Session reports for 1, 2, 3, 5 and concrete models. Concepts Manipulative Session reports for 1, 2, 3, 5 They use appropriate mathematical notation with clear and accurate language. Concepts Manipulative Session reports for all sessions They explain how to derive a result based on previously developed ideas, and explain how a result is related to other ideas. Concepts Manipulative Session reports for all sessions 13