Mathematics Competency Assurance Documents

Algebra I

Number/Computation Strand

Student will apply number sense and order relations in problem solving situations to perform estimations and/or calculations with equations, matrices, and sequences involving complex numbers (counting numbers, whole numbers, integers, rational numbers, irrational numbers, real numbers, etc.) with and without calculators and will communicate the reasoning used in solving these problems.

Based on Kentucky’s Core Content for Mathematics Assessment, the Kentucky Program of Studies, and Academic Expectations: 1.5-1.9 Mathematical Communication & Reasoning, 1.16 Technology, 2.7 Number, 2.8 Procedures, 2.11 Change, 2.12 Structure; Goal 5 Think & Solve Problems; & Goal 6 Connect & Integrate Knowledge

Algebra I: Number/Computation Standards

NC/A1.1 Show number sense by interpreting, modeling and using appropriate mathematical notation and operations (%,!,/, pi, square roots-taking a root, scientific notation, absolute value, exponents-raising to a power, opposite, reciprocal, factorial, significant digits, ordered pairs, number line) for real numbers and by comparing and contrasting various subsets of the real number systems (counting numbers, whole numbers, integers, rational numbers, irrational numbers).

NC/A1.2 Select and apply appropriate concrete (algebra tiles), pictorial (diagrams), and abstract models (symbolic notation) and strategies to simplify and solve two-variable multi-step linear equations and inequalities using order of operations, field properties (commutative, associative, distributive, inverse, identity, equality) and simple matrices with real numbers; and evaluate expressions containing radicals and absolute values in real world situations.

NC/A1.3 Select and apply appropriate concrete (algebra tiles), pictorial (diagrams), and abstract models (symbolic notation) and strategies to simplify second and simple third degree polynomials including finding a common factor to all terms, factoring by grouping, recognizing difference of two squares, and factoring trinomials.

NC/A1.4 Recognize, create and use variables to generalize numeric and geometric patterns by giving the rule for the nth term and defending the generalization.

Skills, Concepts & Relationships

• Review adding, subtracting, multiplying, and dividing integers (1-1)
• Add & subtract matrices, and use scalar multiplication to solve practical problems (1-1)
• Evaluate algebraic expressions involving substitutions (2-2)
• Review using variables to describe numeric and geometric patterns (linear, simple quadratic) and situations and characterize in terms of properties of the nth stage (painted cube problem) (2-2,3-1)
• Use spreadsheet formulas to represent patterns and to perform calculations (1-1,2-2,2-3,4-3,&9-2)
• Review using order of operations with ( )’s and exponents (2-3)
• Use the distributive property and recognize that combining like terms, with and without algebra tiles, involves applying the distributive property (simplify algebraic expressions) (2-3)
• Review solving linear equations using mental math (guess & check, open sentences), pictorial (drawings or graphs), and formal (symbolic) methods (inverse operations, properties of equality) (4-1&4-2)
• Use a graphing calculator to solve linear equations (4-1)
• Solve equations with variables on both sides (balance scale, open sentences, algebra tiles, algeblocks, cups & counters) (4-3)
• Solve inequalities using the properties of inequalities and graph and interpret the solutions (4-3)
• Introduce matrix multiplication (6-2)
• Recognize absolute value as a measure of distance, find the absolute value of a given number, and solve one-variable equations and inequalities involving absolute values symbolically and graphically (7-1)
• Simplify, evaluate, & approximate square roots (use geometric models), recognizing when the radical’s value is rational or irrational (7-2)
• Introduce solving equations using square roots, graphing square root functions, recognizing the effect of parameters on graphs (7-2)
• Recognize and use terms and definitions associated with polynomial expressions and functions (Expressions: monomial-1term, binomial-2terms, trinomial-3terms; Functions: linear-degree1, quadratic-degree2, cubic-degree3,etc.) (8-1)
• Recognize and determine standard and scientific notation of large and small numbers including negative exponents (8-1)
• Understand significant digits and precision (supplement)
• Use Laws of Exponents including properties of powers and quotients of powers (8-1)
• Add, subtract, multiply, and divide monomials and polynomials using concrete (algebra tiles, etc.) and abstract methods in the context of practical applications (use algebra tiles to conceptualize FOIL) (8-1&8-2)
• Apply basic factoring techniques (GCF) to polynomials including finding a common factor to all the terms in a polynomial, factoring by grouping, recognizing the difference of two squares, and factoring trinomials(8-2)
• Introduce solving quadratic equations by factoring (Principle of Zero Products) (9-2)
• Introduce solving quadratic equations using square roots and interpreting the solutions in a real context (compound interest) (9-3)
• Introduce solving quadratic equations using the quadratic formula and evaluating the discriminant (9-3)
• Review comparing and contrasting various subsets of the real number system (counting numbers, whole numbers, integers, rational numbers, & irrational numbers) (10-1)
• Introduce using the properties of real numbers to simplify rational and radical expressions, and to solve related equations (Principle of Squaring, extraneous solutions) (10-1)

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HIGH SCHOOL MATHEMATICS EXIT PERFORMANCE STANDARD 2:

Student will apply properties of measurement (ratio measures including slope, rate, indirect measurement, similarity; surface area and volume of prisms, pyramids, cylinders, cones, and spheres, etc.) and will use geometric concepts, properties and relationships (prove, use and apply theorems/conjectures involving lines, angles, triangles, quadrilaterals, regular, and non-regular polygons, circles, and transformations, etc.) in problem solving situations and communicate the inductive and simple deductive reasoning used in solving these problems.

Based on Kentucky’s Core Content for Mathematics Assessment, the Kentucky Program of Studies, and Academic Expectations: 1.5-1.9 Mathematical Communication & Reasoning, 1.16 Technology, 2.8 Procedures, 2.9 Space and Dimensionality, 2.10 Measurement, 2.11 Change, 2.12 Structure; Goal 5 Think & Solve Problems; & Goal 6 Connect & Integrate Knowledge

Algebra I: Geometry/Measurement Standards

GM/A1.1 Understand how ratio and proportion can be used to connect mathematical ideas (use speed and density related to slope and "per unit" amounts—dimensional analysis & conversion factor, recognize ratio of the rise to the run is the same in similar figures, etc.) and that slope is the rate of change between two quantities that may be directly related (positive) or inversely related (negative).

GM/A1.2 Recognize and determine slopes of parallel (same slope) and perpendicular (opposite reciprocals) lines and how they are related in the coordinate plane (find the equation of a line through a point perpendicular or parallel to the given line).

GM/A1.3 Use geometric formulas to solve problems with algebraic expressions and equations (Pythagorean Theoroem, distance formula, etc.).

Skills, Concepts & Relationships

• Find the slope of a line and recognize that slope is a measure of steepness (negative slope, positive slope, rise/run) (5-1)
• Recognize slope is the rate of change of one quantity relative to another and describe slopes of vertical (undefined) and horizontal (zero) lines (5-1)
• Recognize the importance of scale in graphing linear functions and use dimensional analysis (units) (5-2)
• Recognize that conversion factor is the ratio of two equal quantities (5-2)
• Use slopes and y-intercepts to determine whether pairs of lines are parallel, intersecting, or perpendicular (find the equation of a line through a point parallel or perpendicular to another line) (6-1)
• Develop applying the Pythagorean Theorem and its converse (right triangle, hypotenuse, legs) (7-3)
• Develop applying the Distance Formula for finding distance in the coordinate plane, to simplify square roots, and use distance to find midpoints of segments on a coordinate grid (7-3)
• Develop exploring the golden ratio (10-1)

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HIGH SCHOOL MATHEMATICS EXIT PERFORMANCE STANDARD 3:

Student will use data collection & analysis, graphing of single-variable and two-variable data (line, bar & circle graphs, histogram, stem and leaf plots, box and whisker plots, scatterplot, linear regression & curve fitting), statistics (mean, median, mode, range, outliers, quartiles), and designing probability experiments & simulations to test theories about real world problems and communicate the reasoning used in solving these problems.

Based on Kentucky’s Core Content for Mathematics Assessment, the Kentucky Program of Studies, and Academic Expectations: 1.5-1.9 Mathematical Communication & Reasoning, 1.16 Technology, 2.7 Number, 2.8 Procedures, 2.11 Change, 2.12 Structure, 2.13 Probability & Statistics; Goal 5 Think & Solve Problems; & Goal 6 Integrate Knowledge

Algebra I: Probability/Statistics Standards

PS/A1.1 Collect, compile, compare, and display single-variable and two-variable data (line graph, bar graph, circle graph, histogram, stem and leaf plot, box and whisker plot, scatterplot, etc.) to analyze, interpret, draw conclusions, predict outcomes, and discover trends (write a linear equation for a trend line of best fit which models a set of real data) for real life situations.

PS/A1.2 Understand how range and measures of central tendency (mean, median, mode) influence implications and conclusions.

PS/A1.3 Design and conduct probability experiments (use sampling techniques & bias issues) then analyze odds and theoretical probabilities (fractions, percents, geometric models, tree diagrams) to make decisions.

Skills, Concepts & Relationships

• Review organizing, describing, interpreting, and analyzing data in various forms (1-1)
• Review applications of measures of central tendency (mean, median, mode) to summarize data, and understand how they influence implications and conclusions (1-1)
• Determine outliers’ effects on mean & median (supplement)
• Develop concepts of dispersion and variance: quartile, interquartile, range, clusters, gaps, outliers, etc. (supplement)
• Construct, interpret and analyze: histograms, line plots, stem and leaf plots, circle graphs, box and whisker plots, scatterplots, etc. (supplement)
• Organize information on a spreadsheet , make appropriate computer generated graphs, & access information from a Database (1-1&supplement)
• Use scatterplots to determine whether two quantities are related and to look for patterns and trends in order to make predictions about their values (negative, positive, and no association) (1-2)
• Use sampling techniques to draw inferences about large populations (supplement)
• Review that probability is a ratio that measures the chance or likelihood that an event will occur (1-3)
• Review differences between experimental probability (actual) and theoretical probability (expected) (1-3)
• Review the Law of Large Numbers (more experimental trials should get closer to theorectical probability) (supplement)
• Review finding theoretical probabilities using a formula, tree diagram or geometric model (1-3)
• Review the Multiplication Fundamental Counting Principle (factorials) (1-3)
• Review differentiating "odds" and probability (supplement)
• Explore permutations and combinations (supplement)
• Use a graphing calculator to collect, represent, and analyze data; enter data into a list; determine statistics: mean, median, mode, quartiles; make a stat plot, box and whisker plot, histogram, and scatterplot (1-1thru1-3)
• Use scatterplots, trend lines, and linear functions as tools for creating a linear model for interpreting data that do no fall neatly on a line, in practical real world situations (make predictions using a trend line or its equation and interpret the reasonableness of prediction by exploring linear regression on a graphing calculator) (5-2)

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Algebra I

Algebraic Ideas Strand

HIGH SCHOOL MATHEMATICS EXIT PERFORMANCE STANDARD 4:

Student will model, analyze, compare and apply linear & nonlinear algebraic functions (quadratic, polynomial, exponential, etc.) using tables, graphs in the coordinate plane, variables, expressions, equations, formulas, and in qualities in practical situations and communicate the reasoning used in solving these problems.

Based on Kentucky’s Core Content for Mathematics Assessment, the Kentucky Program of Studies, and Academic Expectations: 1.5-1.9 Mathematical Communication & Reasoning, 1.16 Technology, 2.7 Number, 2.8 Procedures, 2.9 Space and Dimensionality, 2.11 Change, 2.12 Structure; Goal 5 Think & Solve Problems; & Goal 6 Connect & Integrate Knowledge

Algebra I: Algebraic Ideas Standards

AI/A1.1 Recognize and analyze linear algebraic relationships (functions) using tables, graphs in the coordinate plane, variables, expressions, equations and inequalities in problem solving situations.

AI/A1.2 Determine, recognize and apply in problem solving situations (speed, work, mixture, percent, proportions, geometric formulas, etc.) characteristics of graphs and equations of lines including: independent & dependent variables, slope, intercepts, slope-intercept form (y=mx+b), equation of line with a point and slope, two-point equation of a line, standard form (Ax+By=C), and transformations (vertical and horizontal shifts).

AI/A1.3 Determine and apply relationships between pairs of lines and inequalities to solve real life (expenses versus income to determine profit point) problems including parallel lines, perpendicular lines, and solving a system of two linear equations or inequalities with two variables by graphing on paper and with a graphing calculator or computer program.

AI/A1.4 Determine and recognize differences in linear and non-linear equations, graph functions of the form f(x)=ax² and f(x)=ax²+c that model relationships between real world quantities, and approximate solutions for simple quadratic functions by graphing on paper, with tables, or with a graphing calculator or computer program.

Skills, Concepts & Relationships

• Graph real data using all four quadrants of the Cartesian coordinate system to find associations to connect quantities and categories of data that occur in the real world (1-2)
• Understand the difference between constant and variable quantities (2-1)
• Describe how the change in one quantity relates to changes in a related quantity (directly, inversely) using tables, graphs, and written words (2-1)
• Represent functional situations with tables, graphs, and equations; recognize the interrelationship of these and show how each representation is useful in different situations (domain of independent variables & range of dependent variables of a function) (3-1)
• Recognize the characteristics of linear (y=mx+b, constant change), nonlinear (not constant change) and proportional (y=ax, direct variation) functions, and identify their equations (3-2)
• Detemine the slope and position of a line from its equation and graph a line given the slope and the y-intercept (slope-intercept form, y-intercept, point-slope formula) (5-2)
• Recognize that two points determine a unique line and determine the equation of a line through the two given points in real world situations (fixed costs vs. variable costs) (5-2)
• Use a graphing calculator to graph a line (table of values, coordinates grid; enter function in "y=" form, seta and evaluate table, use window, trace and zoom) (5-2)
• Solve and apply a system of two linear equations graphically (graph paper and graphing calculator) and interpret the solution for real life situations (6-1)
• Recognize standard form of a linear equation Ax+By=C (6-1)
• Solve systems of linear equations using substitution & linear combination (elimination) and interpret the solution for practical situations (same line dependent infinite solutions, parallel lines inconsistent no solutions) (6-2)
• Introduce solving a system of linear equations with matrices (6-2)
• Sketch the region of a linear inequality (6-2)
• Introduce sketching the region that represents the solution to a system of linear inequalities in problem solving situations (design or geography) (6-3)
• Introduce comparing graphs of absolute value functions, and explain how changing the parameters affects the graphs of these functions (7-1)
• Graph quadratic functions of the form y=ax² to model real relationships and recognize the curve is a parabola (axis of symmetry, vertex) (9-1)
• Introduce using the graphs of quadratic functions of the form y=ax², y=ax²+c, and y=ax²+bx+c to determine maximum and minimum values (9-1)
• Introduce finding x-intercepts of a quadratic function (9-2)
• Introduce examining simple exponential functions of the form y=ab^x and using them to model exponential growth and decay (10-2)
• Introduce elementary logarithmic functions as the inverses of exponential functions (10-2)
• Introduce using exponential functions to analyze population growth in order to predict future trends (10-2)

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