Academic & Student Services
UW System Competency Based Admissions - Mathematics
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| 1.01 Perform basic arithmetic operations with whole numbers, rational numbers and decimals, and reduce to lowest terms fractions involving single-digit whole numbers for numerator and denominator | A.1. Perform arithmetic operations in proper order, represent real numbers in a variety of forms and simplify arithmetic expressions Use arithmetic operations to model problem situations Use mental arithmetic and estimation |
B.8.1 Read,
represent, and interpret various rational numbers (whole numbers, integers
decimals, fractions, and percents) with verbal descriptions, geometric
models, and mathematical notation (e.g., expanded, scientific, exponential)
B.8.2. Perform and explain operations on rational numbers (add, subtract, multiply, divide, raise to a power, extract a root, take opposites and reciprocals, determine absolute value) B.12.3 Generate and explain equivalencies among fractions, decimals, and percents |
| 1.02 Convert between fractions, decimals, and percents | A.1. Perform arithmetic operations in proper order, represent real numbers in a variety of forms and simplify arithmetic expressions Use arithmetic operations to model problem situations Use mental arithmetic and estimation |
B.8.3 Generate
and explain equivalencies among fractions, decimals, and percents
B.12.3 Perform and explain operations on real numbers (add, subtract, multiply, divide, raise to a power, extract a root, take opposites and reciprocals, determine absolute value) |
| 1.03 Arrange fractions, decimals, and integers in order of size |
B.8.4 Express
order relationships among rational numbers using appropriate symbols (>,
<, >, <)
B.12.2 Compare real numbers using: order relations (>,<) and transitivity, ordinal scales including logarithmic (e.g., Richter, pH rating). arithmetic differences, and ratios, proportions, percents, rates of change |
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| 1.04 Set up and solve simple verbal problems involving percentages, mixtures, unit conversions, and distance/time/rate |
A.1. Perform
arithmetic operations in proper order, represent real numbers in a variety
of forms and simplify arithmetic expressions Use arithmetic operations
to model problem situations Use mental arithmetic and estimation
B.7. Translate from a verbal description of a problem situation to a mathematical representation |
B.8.5 Apply
proportional thinking in a variety of problem situations that include,
but are not limited to: ratios and proportions (e.g., rates, scale drawings,
similarity), percents, including those greater than 100 and less than one
(e.g., discounts, rate of, increase or decrease, sales tax)
B.12.4 In problem-solving situations involving the application of different number systems (natural, integers, rational, real) select and use appropriate computational procedures, properties (e.g., commutativity, associatively, inverses), modes of representation (e.g., rationals as repeating decimals, indicated roots as fractional, exponents) D.12.3 Determine measurements indirectly, using estimation, proportional reasoning, including those involving squaring and cubing (e.g., reasoning that, areas of circles are proportional to the squares of their radii), techniques of algebra, geometry, and right triangle trigonometry, formulas in applications (e.g., for compound interest, distance formula), geometric formulas to derive lengths, areas, or volumes of shapes and objects (e.g., cones, parallelograms, cylinders, pyramids) |
| 1.05 Find areas and perimeters of geometric figures composed of rectangles, triangles, and circles | C.1. Solve geometric problems (with or without coordinates) involving points, lines, angles, circles, and polygons Find perimeters and areas of regions composed of rectangles, triangles and circles |
D.8.4 Determine
measurements indirectly using estimation, conversion of units within a
system (e.g., quarts to cups, millimeters to centimeters), ratio and proportion
(e.g., similarity, scale drawings), geometric formulas to derive lengths,
areas, volumes of common figures (e.g., perimeter, circumference, surface
area), the Pythagorean relationship. geometric relationships and properties
for angle size (e.g., parallel lines and transversals, sum of angles of
a triangle; vertical angles)
B.12.4 In problem-solving situations involving the application of different number systems (natural, integers, rational, real) select and use appropriate computational procedures, properties (e.g., commutativity, associatively, inverses), modes of representation (e.g., rationals as repeating decimals, indicated roots as fractional, exponents) C.12.2 Use geometric models to solve mathematical and real-world problems D.12.3 Determine measurements indirectly, using estimation, proportional reasoning, including those involving squaring and cubing (e.g., reasoning that, areas of circles are proportional to the squares of their radii), techniques of algebra, geometry, and right triangle trigonometry, formulas in applications (e.g., for compound interest, distance formula), geometric formulas to derive lengths, areas, or volumes of shapes and objects (e.g., cones, parallelograms, cylinders, pyramids) |
| 1.06 Find the surface area and volume of a rectangular prism | C.2. Visualize and sketch points, lines, planes, and simple solids in three-dimensional space Find volumes of boxes and cylinders |
B.12.4 In problem-solving
situations involving the application of different number systems (natural,
integers, rational, real) select and use appropriate computational procedures,
properties (e.g., commutativity, associatively, inverses), modes of representation
(e.g., rationals as repeating decimals, indicated roots as fractional,
exponents)
C.12.2 Use geometric models to solve mathematical and real-world problems D.12.3 Determine measurements indirectly, using estimation, proportional reasoning, including those involving squaring and cubing (e.g., reasoning that, areas of circles are proportional to the squares of their radii), techniques of algebra, geometry, and right triangle trigonometry, formulas in applications (e.g., for compound interest, distance formula), geometric formulas to derive lengths, areas, or volumes of shapes and objects (e.g., cones, parallelograms, cylinders, pyramids) |
| 1.07 Solve problems involving the measures of angles of triangles |
C. 1 Solve
geometric problems (with or without coordinates) involving points, lines,
angles, circles, and polygons Find perimeters and areas of regions composed
of rectangles, triangles and circles
C.4. Apply knowledge of the angle sum and the relationship between sides and angles in a triangle (including isosceles and equilateral triangles |
D.8.4 Determine
measurements indirectly using estimation, conversion of units within a
system (e.g., quarts to cups, millimeters to centimeters), ratio and proportion
(e.g., similarity, scale drawings), geometric formulas to derive lengths,
areas, volumes of common figures (e.g., perimeter, circumference, surface
area), the Pythagorean relationship. geometric relationships and properties
for angle size (e.g., parallel lines and transversals, sum of angles of
a triangle; vertical angles)
D.12.3 Determine measurements indirectly, using estimation, proportional reasoning, including those involving squaring and cubing (e.g., reasoning that, areas of circles are proportional to the squares of their radii), techniques of algebra, geometry, and right triangle trigonometry, formulas in applications (e.g., for compound interest, distance formula), geometric formulas to derive lengths, areas, or volumes of shapes and objects (e.g., cones, parallelograms, cylinders, pyramids) |
| 1.08 Exhibit knowledge of basic geometric vocabulary |
C.1. Solve
geometric problems (with or without coordinates) involving points, lines,
angles, circles, and polygons Find perimeters and areas of regions composed
of rectangles, triangles and circles
C.2. Visualize and sketch points, lines, planes, and simple solids in three-dimensional space Find volumes of boxes and cylinders |
C.8.1 Describe
special and complex two- and three-dimensional figures (e.g., rhombus,
polyhedron, cylinder) and their component parts (e.g., base, altitude,
and slant height) by naming, defining, and giving examples, comparing,
sorting, and classifying them, identifying and contrasting their properties
(e.g., symmetrical, isosceles, regular), drawing and constructing physical
models to specifications, explaining how these figures are related to objects
in the environment
C.12.1 Identify, describe, and analyze properties of figures, relationships among figures, and relationships among their parts by constructing physical models, drawing precisely with paper-and-pencil, hand calculators, and computer software, using appropriate transformations (e.g., translations, rotations, reflections, enlargements), using reason and logic |
| 2.01 Perform the basic arithmetic operations with integers using symbols of grouping | A.1. Perform arithmetic operations in proper order, represent real numbers in a variety of forms and simplify arithmetic expressions Use arithmetic operations to model problem situations Use mental arithmetic and estimation |
B.8.2 Perform
and explain operations on rational numbers (add, subtract, multiply, divide,
raise to a power, extract a root, take opposites and reciprocals, determine
absolute value)
B.12.3 Perform and explain operations on real numbers (add, subtract, multiply, divide, raise to a power, extract a root, take opposites and reciprocals, determine absolute value) |
| 2.02 Solve linear equations containing simple fractions and literal numbers |
B.1. Solve
linear algebraic equations and inequalities in one variable, including
those with literal coefficients
B.6. Solve algebraic equations and inequalities in one variable, including those which can be factored into linear and quadratic expressions, or which contain fractional expressions, absolute values, radicals or fractional exponents |
F.8.4 Use linear
equations and inequalities in a variety of ways, including writing them
to represent problem situations and to express generalizations, solving
them by different methods (e.g., informally, graphically, with formal properties,
with technology), writing and evaluating formulas (including solving for
a specified variable), using them to record and describe solution strategies
B.12.3 Perform and explain operations on real numbers (add, subtract, multiply, divide, raise to a power, extract a root, take opposites and reciprocals, determine absolute value) F.12.3 Solve linear and quadratic equations, linear inequalities, and systems of linear equations and inequalities numerically, graphically, including use of appropriate technology, symbolically, including use of the quadratic formula |
| 2.03 Manipulate formulas containing fractions and several variables | F.8.4 Use linear equations and inequalities in a variety of ways, including writing them to represent problem situations and to express generalizations, solving them by different methods (e.g., informally, graphically, with formal properties, with technology), writing and evaluating formulas (including solving for a specified variable), using them to record and describe solution strategies | |
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2.04 Simplify
algebraic expressions involving multiplication and division of exponential
expressions
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B.5. Add, subtract,
multiply, divide, and exponential polynomial and rational expressions and
simplify the results
B.9. Manipulate and simplify expressions involving exponentials or logarithms Solve equations and inequalities involving exponential and logarithmic expressions Solve problems such as those involving growth and decay and |
F.8.1 Work with algebraic expressions in a variety of ways, including using appropriate symbolism, including exponents and variables, evaluating expressions through numerical substitution, generating equivalent expressions, adding and subtracting expressions |
| 2.05 Simplify exponential expressions with negative bases |
B.5. Add, subtract,
multiply, divide, and exponential polynomial and rational expressions and
simplify the results
B.9. Manipulate and simplify expressions involving exponentials or logarithms Solve equations and inequalities involving exponential and logarithmic expressions Solve problems such as those involving growth and decay and |
B.12.3 Perform and explain operations on real numbers (add, subtract, multiply, divide, raise to a power, extract a root, take opposites and reciprocals, determine absolute value) |
| 2.06 Find products of binomials | B.5. Add, subtract, multiply, divide, and exponential polynomial and rational expressions and simplify the results | F.8.1 Work with algebraic expressions in a variety of ways, including using appropriate symbolism, including exponents and variables, evaluating expressions through numerical substitution, generating equivalent expressions, adding and subtracting expressions |
| 2.07 Collect like terms, and perform simple monomial factoring | B.5. Add, subtract, multiply, divide, and exponential polynomial and rational expressions and simplify the results | F.8.1 Work with algebraic expressions in a variety of ways, including using appropriate symbolism, including exponents and variables, evaluating expressions through numerical substitution, generating equivalent expressions, adding and subtracting expressions |
| 2.08 Use order of operations to simplify polynomials and rational expressions | B.5. Add, subtract, multiply, divide, and exponential polynomial and rational expressions and simplify the results | F.8.1 Work with algebraic expressions in a variety of ways, including using appropriate symbolism, including exponents and variables, evaluating expressions through numerical substitution, generating equivalent expressions, adding and subtracting expressions |
| 2.09 Evaluate polynomials at specific values of the variable | F.8.1 Work with algebraic expressions in a variety of ways, including using appropriate symbolism, including exponents and variables, evaluating expressions through numerical substitution, generating equivalent expressions, adding and subtracting expressions | |
| 2.10 Set up and solve verbal problems involving linear equations |
B.1. Solve
linear algebraic equations and inequalities in one variable, including
those with literal coefficients
B.3. Relate linear functions and equations with their graphs and use them to represent appropriate situations B.7. Translate from a verbal description of a problem situation to a mathematical representation |
F.8.2 Work
with linear and nonlinear patterns and relationships in a variety of ways,
including representing them with tables, with graphs, and with algebraic
expressions, equations, and inequalities, describing and interpreting their
graphical representations (e.g., slope, rate of change, intercepts, using
them as models of real-world phenomena
D.12.3 Determine measurements indirectly, using estimation, proportional reasoning, including those involving squaring and cubing (e.g., reasoning that, areas of circles are proportional to the squares of their radii), techniques of algebra, geometry, and right triangle trigonometry, formulas in applications (e.g., for compound interest, distance formula), geometric formulas to derive lengths, areas, or volumes of shapes and objects (e.g., cones, parallelograms, cylinders, pyramids) F.12.1 Analyze and generalize patterns of change (e.g., direct and inverse variation) and numerical sequences, and then represent them with algebraic expressions and equations F.12.2 Use mathematical functions (e.g., linear, exponential, quadratic, power) in a variety of recognizing that a variety of mathematical and real-world phenomena can be modeled by the same type of function F.12.3 Solve linear and quadratic equations, linear inequalities, and systems of linear equations and inequalities numerically graphically, including use of appropriate technology, symbolically, including use of the quadratic formula |
| 3.01 Recognize and use congruence and similarity correspondences | C.5. Use knowledge of similarity and congruence to make a reasoned analysis of relations between angles, lengths and areas in a figure or problem situation |
C.8.2 Identify
and use relationships among the component parts of special and complex
two- and three-dimensional figures (e.g., parallel sides, congruent faces)
D.12.3 Determine measurements indirectly, using estimation, proportional reasoning, including those involving squaring and cubing (e.g., reasoning that areas of circles are proportional to the squares of their radii), techniques of algebra, geometry, and right triangle trigonometry, formulas in applications (e.g., for compound interest, distance formula), geometric formulas to derive lengths, areas, or volumes of shapes and objects (e.g., cones, parallelograms, cylinders, pyramids) |
| 3.02 Recognize and use conditions that imply congruence and those that imply similarity for triangles |
C. 4. Apply
knowledge of the angle sum and the relationship between sides and angles
in a triangle (including isosceles and equilateral triangles)
C.5. Use knowledge of similarity and congruence to make a reasoned analysis of relations between angles, lengths and areas in a figure or problem situation |
C.8.2 Identify
and use relationships among the component parts of special and complex
two- and three-dimensional figures (e.g., parallel sides, congruent faces)
D.12.3 Determine measurements indirectly, using estimation, proportional reasoning, including those involving squaring and cubing (e.g., reasoning that areas of circles are proportional to the squares of their radii), techniques of algebra, geometry, and right triangle trigonometry, formulas in applications (e.g., for compound interest, distance formula), geometric formulas to derive lengths, areas, or volumes of shapes and objects (e.g., cones, parallelograms, cylinders, pyramids) |
| 3.03 Recognize and use symmetries of isosceles triangles and equilateral triangles |
C.4. Apply
knowledge of the angle sum and the relationship between sides and angles
in a triangle (including isosceles and equilateral triangles)
C.6. Demonstrate knowledge of the relationship between triangle properties, the Pythagorean theorem, and distance (with or without coordinates) and |
D.12.3 Determine measurements indirectly, using estimation, proportional reasoning, including those involving squaring and cubing (e.g., reasoning that areas of circles are proportional to the squares of their radii), techniques of algebra, geometry, and right triangle trigonometry, formulas in applications (e.g., for compound interest, distance formula), geometric formulas to derive lengths, areas, or volumes of shapes and objects (e.g., cones, parallelograms, cylinders, pyramids) |
| 3.04 Set up and solve proportions derived from similar triangles | C.5. Use knowledge of similarity and congruence to make a reasoned analysis of relations between angles, lengths and areas in a figure or problem situation |
B.8.5 Apply
proportional thinking in a variety of problem situations that include,
but are not limited to: ratios and proportions (e.g., rates, scale drawings,
similarity), percents, including those greater than 100 and less than one
(e.g., discounts, rate of, increase or decrease, sales tax)
D.8.4 Determine measurements indirectly using estimation, conversion of units within a system (e.g., quarts to cups, millimeters to centimeters), ratio and proportion (e.g., similarity, scale drawings), geometric formulas to derive lengths, areas, volumes of common figures (e.g., perimeter, circumference, surface area), the Pythagorean relationship. geometric relationships and properties for angle size (e.g., parallel lines and transversals, sum of angles of a triangle; vertical angles) D.12.3 Determine measurements indirectly, using estimation, proportional reasoning, including those involving squaring and cubing (e.g., reasoning that areas of circles are proportional to the squares of their radii), techniques of algebra, geometry, and right triangle trigonometry, formulas in applications (e.g., for compound interest, distance formula), geometric formulas to derive lengths, areas, or volumes of shapes and objects (e.g., cones, parallelograms, cylinders, pyramids) |
| 3.05 Recognize and use conditions that imply that two lines are parallel or two lines are perpendicular | C.3. Use knowledge of parallelism, perpendicularly, and associated angle properties to analyze and construct figures and to represent problem situations (with or without coordinates) |
D.8.4 Determine
measurements indirectly using estimation, conversion of units within a
system (e.g., quarts to cups, millimeters to centimeters), ratio and proportion
(e.g., similarity, scale drawings), geometric formulas to derive lengths,
areas, volumes of common figures (e.g., perimeter, circumference, surface
area), the Pythagorean relationship. geometric relationships and properties
for angle size (e.g., parallel lines and transversals, sum of angles of
a triangle; vertical angles)
C.12.4 Use the two-dimensional rectangular coordinate system and algebraic procedures to describe and characterize geometric properties and relationships such as slope, intercepts, parallelism, and perpendicularity |
| 3.06 Find the area of a sector of a circle given the measure of a central angle | C.1. Solve geometric problems (with or without coordinates) involving points, lines, angles, circles, and polygons Find perimeters and areas of regions composed of rectangles, triangles and circles | D.12.3 Determine measurements indirectly, using estimation, proportional reasoning, including those involving squaring and cubing (e.g., reasoning that areas of circles are proportional to the squares of their radii), techniques of algebra, geometry, and right triangle trigonometry, formulas in applications (e.g., for compound interest, distance formula), geometric formulas to derive lengths, areas, or volumes of shapes and objects (e.g., cones, parallelograms, cylinders, pyramids) |
| 3.07 Find measures of angles by using complementary and supplementary relations | D.12.3 Determine measurements indirectly, using estimation, proportional reasoning, including those involving squaring and cubing (e.g., reasoning that areas of circles are proportional to the squares of their radii), techniques of algebra, geometry, and right triangle trigonometry, formulas in applications (e.g., for compound interest, distance formula), geometric formulas to derive lengths, areas, or volumes of shapes and objects (e.g., cones, parallelograms, cylinders, pyramids) | |
| 3.08 Use Pythagorean relationships | C.6. Demonstrate knowledge of the relationship between triangle properties, the Pythagorean theorem, and distance (with or without coordinates) and |
D.8.4 Determine
measurements indirectly using estimation, conversion of units within a
system (e.g., quarts to cups, millimeters to centimeters), ratio and proportion
(e.g., similarity, scale drawings), geometric formulas to derive lengths,
areas, volumes of common figures (e.g., perimeter, circumference, surface
area), the Pythagorean relationship. geometric relationships and properties
for angle size (e.g., parallel lines and transversals, sum of angles of
a triangle; vertical angles)
D.12.3 Determine measurements indirectly, using estimation, proportional reasoning, including those involving squaring and cubing (e.g., reasoning that areas of circles are proportional to the squares of their radii), techniques of algebra, geometry, and right triangle trigonometry, formulas in applications (e.g., for compound interest, distance formula), geometric formulas to derive lengths, areas, or volumes of shapes and objects (e.g., cones, parallelograms, cylinders, pyramids) |
| 3.09 Recognize and use the fact that the contrapositive of a conditional statement has the same truth value as the conditional statement |
A.12.2 Communicate
logical arguments and clearly show why a result does or does not make sense,
why the reasoning is or is not valid, an understanding of the difference
between examples that support a conjecture and a proof of the conjecture
C.12.3 Present convincing arguments by means of demonstration, informal proof, counter-examples, or any other logical means to show the truth of statements (e.g., these two triangles are not congruent), generalizations (e.g., the Pythagorean theorem holds for all right triangles) |
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| 3.10 Given a generalization, identify examples and counterexamples |
A.12.2 Communicate
logical arguments and clearly show why a result does or does not make sense,
why the reasoning is or is not valid, an understanding of the difference
between examples that support a conjecture and a proof of the conjecture
C.12.3 Present convincing arguments by means of demonstration, informal proof, counter-examples, or any other logical means to show the truth of statements (e.g., these two triangles are not congruent), generalizations (e.g., the Pythagorean theorem holds for all right triangles) |
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| 3.11 Use incidence and parallelism properties in three dimensions | C.2. Visualize and sketch points, lines, planes, and simple solids in three-dimensional space Find volumes of boxes and cylinders | |
| 3.12 Classify sets of objects by set inclusion, e.g., the set of squares is a subset of the set of rectangles | C.12.1 Identify, describe, and analyze properties of figures, relationships among figures, and relationships among their parts by constructing physical models, drawing precisely with paper-and-pencil, hand calculators, and computer software, using appropriate transformations (e.g., translations, rotations, reflections, enlargements), using reason and logic | |
| 4.01 Simplify (reduce) rational expressions | B.5. Add, subtract, multiply, divide, and exponential polynomial and rational expressions and simplify the results | F.8.1 Work with algebraic expressions in a variety of ways, including using appropriate symbolism, including exponents and variables, evaluating expressions through numerical substitution, generating equivalent expressions, adding and subtracting expressions |
| 4.02 Add, subtract, multiply, and divide rational expressions | B.5. Add, subtract, multiply, divide, and exponential polynomial and rational expressions and simplify the results | F.8.1 Work with algebraic expressions in a variety of ways, including using appropriate symbolism, including exponents and variables, evaluating expressions through numerical substitution, generating equivalent expressions, adding and subtracting expressions |
| 4.03 Factor completely: a quadratic trinomial, difference of two squares or the sum of difference of two cubes | B.5. Add, subtract, multiply, divide, and exponential polynomial and rational expressions and simplify the results | |
| 4.04 Recognize the relationships between coefficients of perfect-square trinomials | ||
| 4.05 Evaluate numerical expressions containing integral and fractional exponents | ||
| 4.06 Simplify radical expressions | B.5. Add, subtract, multiply, divide, and exponential polynomial and rational expressions and simplify the results | |
| 4.07 Rationalize numerators and denominators of rational expressions containing radicals | ||
| 4.08 Rewrite radical expressions using fractional exponents | ||
| 4.09 Add, subtract, multiply, and divide radical expressions | ||
| 4.10 Simplify algebraic expressions with positive, negative, and zero exponents | B. 5. Add, subtract, multiply, divide, and exponential polynomial and rational expressions and simplify the results | B.8.1 Read, represent, and interpret various rational numbers (whole numbers, integers decimals, fractions, and percents) with verbal descriptions, geometric models, and mathematical notation (e.g., expanded, scientific, exponential) |
| 4.11 Convert between decimal and scientific notation |
F.8.2 Work
with linear and nonlinear patterns and relationships in a variety of ways,
including representing them with tables, with graphs, and with algebraic
expressions, equations, and inequalities, describing and interpreting their
graphical representations (e.g., slope, rate of change, intercepts, using
them as models of real-world phenomena
B.12.3 Perform and explain operations on real numbers (add, subtract, multiply, divide, raise to a power, extract a root, take opposites and reciprocals, determine absolute value) |
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| 4.12 Find the slope and y-intercept and be able to graph a straight line given its equation | B.3. Relate linear functions and equations with their graphs and use them to represent appropriate situations |
C.12.4 Use
the two-dimensional rectangular coordinate system and algebraic procedures
to describe and characterize geometric properties and relationships such
as slope, intercepts, parallelism, and perpendicularity
F.12.2 Use mathematical functions (e.g., linear, exponential, quadratic, power) in a variety of recognizing that a variety of mathematical and real-world phenomena can be modeled by the same type of function |
| 4.14 Graph a parabola given its equation | B.8. Use the language, notation and properties of algebraic functions and their graphs, with particular attention to quadratic functions |
F.12.2 Use
mathematical functions (e.g., linear, exponential, quadratic, power) in
a variety of recognizing that a variety of mathematical and real-world
phenomena can be modeled by the same type of function
F.12.3 Solve linear and quadratic equations, linear inequalities, and systems of linear equations and inequalities numerically graphically, including use of appropriate technology, symbolically, including use of the quadratic formula |
| 4.15 Evaluate numerical expressions and solve equations involving absolute values | B.6. Solve algebraic equations and inequalities in one variable, including those which can be factored into linear and quadratic expressions, or which contain fractional expressions, absolute values, radicals or fractional exponents | |
| 4.16 Recognize the graph of y = {x} | B.6. Solve algebraic equations and inequalities in one variable, including those which can be factored into linear and quadratic expressions, or which contain fractional expressions, absolute values, radicals or fractional exponents | |
| 4.17 Graph the solution of inequalities such as Ax + B < 0 on the number line |
B.1. Solve
linear algebraic equations and inequalities in one variable, including
those with literal coefficients
B.6. Solve algebraic equations and inequalities in one variable, including those which can be factored into linear and quadratic expressions, or which contain fractional expressions, absolute values, radicals or fractional exponents |
F.8.2 Work
with linear and nonlinear patterns and relationships in a variety of ways,
including representing them with tables, with graphs, and with algebraic
expressions, equations, and inequalities, describing and interpreting their
graphical representations (e.g., slope, rate of change, intercepts, using
them as models of real-world phenomena
F.8.4 Use linear equations and inequalities in a variety of ways, including writing them to represent problem situations and to express generalizations, solving them by different methods (e.g., informally, graphically, with formal properties, with technology), writing and evaluating formulas (including solving for a specified variable), using them to record and describe solution strategies F.12.3 Solve linear and quadratic equations, linear inequalities, and systems of linear equations and inequalities numerically, graphically, including use of appropriate technology, symbolically, including use of the quadratic formula |
| 4.18 Graph the solution of inequalities such as Ax + By + C < 0 |
F.8.2 Work
with linear and nonlinear patterns and relationships in a variety of ways,
including representing them with tables, with graphs, and with algebraic
expressions, equations, and inequalities, describing and interpreting their
graphical representations (e.g., slope, rate of change, intercepts, using
them as models of real-world phenomena
F.8.4 Use linear equations and inequalities in a variety of ways, including writing them to represent problem situations and to express generalizations, solving them by different methods (e.g., informally, graphically, with formal properties, with technology), writing and evaluating formulas (including solving for a specified variable), using them to record and describe solution strategies |
|
| 4.19 Solve quadratic equations (includes use of quadratic formula) |
B.6. Solve
algebraic equations and inequalities in one variable, including those which
can be factored into linear and quadratic expressions, or which contain
fractional expressions, absolute values, radicals or fractional exponents
B.8. Use language, notation and properties of algebraic functions and their graphs, with particular attention to quadratic functions |
F.12.3 Solve linear and quadratic equations, linear inequalities, and systems of linear equations and inequalities numerically, graphically, including use of appropriate technology, symbolically, including use of the quadratic formula |
| 4.20 Solve fractional equations, discarding extraneous roots | B.6. Solve algebraic equations and inequalities in one variable, including those which can be factored into linear and quadratic expressions, or which contain fractional expressions, absolute values, radicals or fractional exponents | F.12.3 Solve linear and quadratic equations, linear inequalities, and systems of linear equations and inequalities numerically, graphically, including use of appropriate technology, symbolically, including use of the quadratic formula |
| 4.21 Solve a system of two linear equations in two unknowns recognizing cases of no solutions or infinitely many solutions | B.2. Solve systems of linear equations in two or three variables and interpret solutions both symbolically and graphically | F.12.3 Solve linear and quadratic equations, linear inequalities, and systems of linear equations and inequalities numerically, graphically, including use of appropriate technology, symbolically, including use of the quadratic formula |
| 4.22 Graph a system of two linear equations in two unknowns | B.2. Solve systems of linear equations in two or three variables and interpret solutions both symbolically and graphically | F.12.3 Solve linear and quadratic equations, linear inequalities, and systems of linear equations and inequalities numerically, graphically, including use of appropriate technology, symbolically, including use of the quadratic formula |
| 4.23 Read, analyze, and solve verbal problems |
B.3. Relate
linear functions and equations with their graphs and use them to represent
appropriate situations
B.7. Translate from a verbal description of a problem situation to a mathematical representation B.9. Manipulate and simplify expressions involving exponentials or logarithms. Solve equations and inequalities involving exponential and logarithmic expressions. Solve problems such as those involving growth and decay |
F.8.2 Work
with linear and nonlinear patterns and relationships in a variety of ways,
including representing them with tables, with graphs, and with algebraic
expressions, equations, and inequalities, describing and interpreting their
graphical representations (e.g., slope, rate of change, intercepts, using
them as models of real-world phenomena
F.8.4 Use linear equations and inequalities in a variety of ways, including writing them to represent problem situations and to express generalizations, solving them by different methods (e.g., informally, graphically, with formal properties, with technology), writing and evaluating formulas (including solving for a specified variable), using them to record and describe solution strategies F.12.1 Analyze and generalize patterns of change (e.g., direct and inverse variation) and numerical sequences, and then represent them with algebraic expressions and equations F.12.2 Use mathematical functions (e.g., linear, exponential, quadratic, power) in a variety of recognizing that a variety of mathematical and real-world phenomena can be modeled by the same type of function F.12.4 Model and solve a variety of mathematical and real-world problems by using algebraic expressions, equations, and inequalities |
| 4.24 Add, subtract, multiply and divide polynomials | B.5. Add, subtract, multiply, divide, and exponential polynomial and rational expressions and simplify the results | |
| 5.01 Use the definitions of the trigonometric functions of angles |
B.10. Use the
language, notation and properties of exponential, logarithmic and trigonometric
(sine, cosine, tangent) functions and their graphs
C.6. Demonstrate knowledge of the relationship between triangle properties, the Pythagorean theorem, and distance (with or without coordinates) C.7. Use right triangle relationships and trigonometric ratios, sine, cosine and tangent to analyze relationships and to formulate and solve problems |
C.12.5 Identify
and demonstrate an understanding of the three ratios used in right-triangle
trigonometry (sine, cosine, tangent)
D.12.3 Determine measurements indirectly, using estimation, proportional reasoning, including those involving squaring and cubing (e.g., reasoning that areas of circles are proportional to the squares of their radii), techniques of algebra, geometry, and right triangle trigonometry, formulas in applications (e.g., for compound interest, distance formula), geometric formulas to derive lengths, areas, or volumes of shapes and objects (e.g., cones, parallelograms, cylinders, pyramids) |
| 5.02 Evaluate trigonometric functions for special angles, e.g., , |
B.10. Use the
language, notation and properties of exponential, logarithmic and trigonometric
(sine, cosine, tangent) functions and their graphs
C.6. Demonstrate knowledge of the relationship between triangle properties, the Pythagorean theorem, and distance (with or without coordinates) C.7. Use right triangle relationships and the trigonometric ratios sine, cosine and tangent to analyze relationships and to formulate and solve problems |
C.12.5 Identify and demonstrate an understanding of the three ratios used in right-triangle trigonometry (sine, cosine, tangent) |
| 5.03 Use the standard formulas and identities: Pythagorean formulas, quotient and reciprocal formulas, complementary angle formulas, sums and differences formulas, and double- and half-angle formulas | C.7. Use right triangle relationships and the trigonometric ratios sine, cosine and tangent to analyze relationships and to formulate and solve problems | D.12.3 Determine measurements indirectly, using estimation, proportional reasoning, including those involving squaring and cubing (e.g., reasoning that areas of circles are proportional to the squares of their radii), techniques of algebra, geometry, and right triangle trigonometry, formulas in applications (e.g., for compound interest, distance formula), geometric formulas to derive lengths, areas, or volumes of shapes and objects (e.g., cones, parallelograms, cylinders, pyramids) |
| 5.04 Solve trigonometric equations (linear and quadratic) | B.10. Use the language, notation and properties of exponential, logarithmic and trigonometric (sine, cosine, tangent) functions and their graphs | |
| 5.05 Sketch the graphs of trigonometric functions | B.10. Use the language, notation and properties of exponential, logarithmic and trigonometric (sine, cosine, tangent) functions and their graphs | |
| 5.06 Evaluate expressions involving inverse trigonometric functions | B.10. Use the language, notation and properties of exponential, logarithmic and trigonometric (sine, cosine, tangent) functions and their graphs | C.12.5 Identify and demonstrate an understanding of the three ratios used in right-triangle trigonometry (sine, cosine, tangent) |
| 5.07 Solve triangles (acute, right and obtuse) |
B.10. Use the
language, notation and properties of exponential, logarithmic and trigonometric
(sine, cosine, tangent) functions and their graphs
C.7. Use right triangle relationships and the trigonometric rations since, cosine and tangent to analyze relationships and to formulate and solve problems |
|
| 5.08 Determine the domain and range of a function from a defining statement or graph | B.8. Use the language, notation and properties of algebraic functions and their graphs, with particular attention to quadratic functions | |
| 5.09 Use function notation, perform operations on functions and evaluate functions | B.8. Use the language, notation and properties of algebraic functions and their graphs, with particular attention to quadratic functions | F.12.2 Use mathematical functions (e.g., linear, exponential, quadratic, power) in a variety of recognizing that a variety of mathematical and real-world phenomena can be modeled by the same type of function |
| 5.10 Work with exponential and logarithmic functions and their graphs |
B.9. Manipulate
and simplify expressions involving exponentials or logarithms. Solve equations
and inequalities involving exponential and logarithmic expressions. Solve
problems such as those involving growth and decay
B.10. Use the language, notation and properties of exponential, logarithmic and trigonometric (sine, cosine, tangent) functions and their graphs |
F.12.2 Use mathematical functions (e.g., linear, exponential, quadratic, power) in a variety of recognizing that a variety of mathematical and real-world phenomena can be modeled by the same type of function |
| 5.11 Solve exponential and logarithmic functions and their graphs |
B.9. Manipulate
and simplify expressions involving exponentials or logarithms. Solve equations
and inequalities involving exponential and logarithmic expressions. Solve
problems such as those involving growth and decay
B.10. Use the language, notation and properties of exponential, logarithmic and trigonometric (sine, cosine, tangent) functions and their graphs |
F.12.2 Use mathematical functions (e.g., linear, exponential, quadratic, power) in a variety of recognizing that a variety of mathematical and real-world phenomena can be modeled by the same type of function |
| 5.12 Solve inequalities of the type [ax + b] < c | B.6. Solve algebraic equations and inequalities in one variable, including those which can be factored into linear and quadratic expressions, or which contain fractional expressions, absolute values, radicals or fractional exponents | |
| 5.13 Solve quadratic inequalities one variable, and graph the solutions | B.6. Solve algebraic equations and inequalities in one variable, including those which can be factored into linear and quadratic expressions, or which contain fractional expressions, absolute values, radicals or fractional exponents | F.12.3 Solve linear and quadratic equations, linear inequalities, and systems of linear equations and inequalities numerically, graphically, including use of appropriate technology, symbolically, including use of the quadratic formula |
| 5.14 Represent complex numbers geometrically and perform complex number arithmetic | ||
| 5.15 Use remainder and factor theorems to find the zeros of polynomials | B.6. Solve algebraic equations and inequalities in one variable, including those which can be factored into linear and quadratic expressions, or which contain fractional expressions, absolute values, radicals or fractional exponents | |
| 5.16 Solve systems containing linear and/or quadratic equations in two variables | B.2. Solve systems of linear equations in two or three variables and interpret solutions both symbolically and graphically |
F.12.2 Use
mathematical functions (e.g., linear, exponential, quadratic, power) in
a variety of recognizing that a variety of mathematical and real-world
phenomena can be modeled by the same type of function
F.12.3 Solve linear and quadratic equations, linear inequalities, and systems of linear equations and inequalities numerically, graphically, including use of appropriate technology, symbolically, including use of the quadratic formula |
| 5.17 Find the distance between two points | C.6. Demonstrate knowledge of the relationship between triangle properties, the Pythagorean theorem, and distance (with or without coordinates) and |
C.12.4 Use
the two-dimensional rectangular coordinate system and algebraic procedures
to describe and characterize geometric properties and relationships such
as slope, intercepts, parallelism, and perpendicularity
D.12.3 Determine measurements indirectly, using estimation, proportional reasoning, including those involving squaring and cubing (e.g., reasoning that areas of circles are proportional to the squares of their radii), techniques of algebra, geometry, and right triangle trigonometry, formulas in applications (e.g., for compound interest, distance formula), geometric formulas to derive lengths, areas, or volumes of shapes and objects (e.g., cones, parallelograms, cylinders, pyramids) |
| 5.18 Determine from the equations of two lines whether they are parallel or perpendicular or neither | C.3. Use knowledge of parallelism, perpendicularly, and associated angle properties to analyze and construct figures and to represent problem situations (with or without coordinates) |
C.12.4 Use
the two-dimensional rectangular coordinate system and algebraic procedures
to describe and characterize geometric properties and relationships such
as slope, intercepts, parallelism, and perpendicularity
F.12.2 Use mathematical functions (e.g., linear, exponential, quadratic, power) in a variety of recognizing that a variety of mathematical and real-world phenomena can be modeled by the same type of function |
| 5.19 Convert between the equation of a circle in standard form and data on the center and radius | ||
| 5.20 Convert between the equation of a parabola in standard form and data on its vertex and axis | B.8. Use the language, notation and properties of algebraic functions and their graphs, with particular attention to quadratic functions | F.12.2 Use mathematical functions (e.g., linear, exponential, quadratic, power) in a variety of recognizing that a variety of mathematical and real-world phenomena can be modeled by the same type of function |
| 5.21 Distinguish equations of conics | ||
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This
document was last revised on October 14, 1999. ©January 1999 Board of Regents
of the University of Wisconsin System, All Rights Reserved.
dpi\new matrix\math placement-placement-cba-.standards.doc