Academic & Student Services
MCmct
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By the end of grade eight, students will: |
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| A.8.1 Use reasoning abilities to evaluate information, perceive patterns, identify relationships, formulate questions for further exploration, evaluate strategies, justify statements, test reasonableness of results, defend work |
6. Apply mathematical
reasoning to develop arguments that support conclusions
3. Demonstrate the use of estimation strategies |
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| A.8.2 Communicate logical arguments clearly to show why a result makes sense | 6. Apply mathematical reasoning to develop arguments that support conclusions | |
| A.8.3 Analyze non-routine problems by modeling, illustrating, guessing, simplifying, generalizing, shifting to another point of view, etc. | 7. Solve application problems | |
| A.8.4 Develop effective oral and written presentations that include appropriate use of technology, the conventions of mathematical discourse (e.g., symbols, definitions, labeled drawings), mathematical language, clear organization of ideas and procedures, understanding of purpose and audience |
1. Demonstrate
use of terminology and symbols
2. Calculate with real numbers 3. Demonstrate the use of estimation strategies 4. Use mathematical tools 5. Demonstrate use of formulas 6. Apply mathematical reasoning to develop arguments that support conclusions 7. Solve application problems 8. Demonstrate use of measurement systems 9. Interpret data 10. Evaluate algebraic sentences 11. Solve problems using geometric figures |
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| A.8.5 Explain mathematical concepts, procedures, and ideas to others who may not be familiar with them | ||
| A.8.6 Read and understand mathematical texts and other instructional materials and recognize mathematical ideas as they appear in other contexts | ||
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| A.12.1 Use reason and logic to evaluate information, perceive patterns, identify relationships, formulate questions, pose problems, and make and test conjectures, pursue ideas that lead to further understanding and deeper insight |
A.2. Construct
and read charts, tables and graphs that summarize data from real world
situations
A.3. Use calculators appropriately to aid computation and understanding 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 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.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 |
6. Apply mathematical reasoning to develop arguments that support conclusions |
| 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 | 6. Apply mathematical reasoning to develop arguments that support conclusions | |
| A.12.3 Analyze non-routine problems and arrive at solutions by various means, including models and simulations, often starting with provisional conjectures and progressing, directly or indirectly, to a solution, justification, or counter-example |
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
A.2. Construct and read charts, tables and graphs that summarize data from real world situations |
7. Solve application problems |
| A.12.4 Develop effective oral and written presentations employing correct mathematical terminology, notation, symbols, and conventions for mathematical arguments and display of data | 6. Apply mathematical reasoning to develop arguments that support conclusions | |
| A.12.5 Organize work and present mathematical procedures and results clearly, systematically, succinctly, and correctly | ||
| A.12.6 Read and understand mathematical texts and other instructional materials, writing about mathematics (e.g., articles in journals)mathematical ideas as they are used in other contexts | ||
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By the end of grade eight, students will: |
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| 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) | 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 | 2. Demonstrate use of terminology and symbols |
| 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) | 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 | 2. Demonstrate use of terminology and symbols |
| B.8.3 Generate and explain equivalencies among fractions, decimals, and percents | 2. Demonstrate use of terminology and symbols | |
| B.8.4 Express order relationships among rational numbers using appropriate symbols (>, <, >, <,) | 3. Calculate with real numbers | |
| 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) | 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 |
2. Demonstrate
use of terminology and symbols
11. Solve problems using geometric figures |
| B.8.6 Model and solve problems involving number-theory concepts such as prime and composite numbers, divisibility and remainders, greatest common factors, least common multiples | ||
| B.8.7 In problem-solving situations, select and use appropriate computational procedures with rational numbers such as calculating mentally, estimating, creating, using, and explaining algorithms, using technology (e.g., scientific calculators, spreadsheets) |
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
A.3. Use calculators appropriately to aid computation and understanding |
2. Demonstrate
use of terminology and symbols
3. Calculate with real numbers 4. Use mathematical tools |
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| B.12.1 Use complex counting procedures such as union and intersection of sets and arrangements (permutations and combinations) to solve problems | ||
| B.12.2 Compare real numbers using order relations (>,<) and transitivity, ordinal scales including logarithmic (e.g., Richter, pH rating), arithmetic differences, ratios, proportions, percents, rates of change | B.10. Use the language, notation and properties of exponential, logarithmic and trigonometric (sine, cosine, tangent) functions and their graphs |
2. Demonstrate
use of terminology and symbols
3. Calculate with real numbers |
| 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) |
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.5. Add, subtract, multiply, divide, and exponential polynomial and rational expressions and simplify the results |
2. Demonstrate use of terminology and symbols |
| 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, associativity, inverses), modes of representation (e.g., rationals as repeating decimals, indicated roots as fractional exponents) | 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 | 2. Demonstrate use of terminology and symbols |
| B.12.5 Create and critically evaluate numerical arguments presented in a variety of classroom and real-world situations (e.g., political, economic, scientific, social) |
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.3. Relate linear functions and equations with their graphs and use them to represent appropriate situations 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 |
6. Apply mathematical
reasoning to develop arguments that support conclusions
7. Solve application problems |
| B.12.6 Routinely assess the acceptable limits of error when evaluating strategies, testing the reasonableness of results, using technology to carry out computations | A.3. Use calculators appropriately to aid computation and understanding | 8. Demonstrate use of measurement systems |
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By the end of grade eight, students will: |
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| 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.2. Visualize and sketch points, lines, planes, and simple solids in three-dimensional space. Find volumes of boxes and cylinders | 11. Solve problems using geometric figures |
| 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). |
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.7. Use right triangle relationships and the trigonometric ratios sine, cosine and tangent to analyze relationships and to formulate and solve problems |
11. Solve problems using geometric figures |
| C.8.3 Identify three-dimensional shapes from two-dimensional perspectives and draw two-dimensional sketches of three-dimensional objects preserving their significant features | C.2. Visualize and sketch points, lines, planes, and simple solids in three-dimensional space. Find volumes of boxes and cylinders | 11. Solve problems using geometric figures |
| C.8.4 Perform transformations on two-dimensional figures and describe and analyze the effects of the transformations on the figures | ||
| C.8.5 Locate objects using the rectangular coordinate system | 12. Use rectangular coordinate system (Cartesian geometry) | |
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| 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 |
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.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.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.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 |
11. Solve problems using geometric figures |
| C.12.2 Use geometric models to solve mathematical and real-world problems |
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.3. Use knowledge of parallelism, perpendicularly, and associated angle properties to analyze and construct figures and to represent problem situations (with or without coordinates) |
11. Solve problems using geometric figures |
| 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) |
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 |
6. Apply mathematical reasoning to develop arguments that support conclusions |
| 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 |
B.3. Relate
linear functions and equations with their graphs and use them to represent
appropriate situations
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.3. Use knowledge of parallelism, perpendicularly, and associated angle properties to analyze and construct figures and to represent problem situations (with or without coordinates) |
12. Use rectangular coordinate system (Cartesian geometry) |
| C.12.5 Identify and demonstrate an understanding of the three ratios used in right-triangle trigonometry (sine, cosine, tangent) |
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 ratios sine, cosine and tangent to analyze relationships and to formulate and solve problems |
11. Solve problems using geometric figures |
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By the end of grade eight, students will: |
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| D.8.1 Identify and describe attributes in situations where they are not directly or easily measurable (e.g., distance, area of an irregular figure, likelihood of occurrence) | 11. Solve problems using geometric figures | |
| D.8.2 Demonstrate understanding of basic measurement facts, principles, and techniques including the following approximate comparisons between metric and US Customary units (e.g., a liter and a quart, are about the same; a kilometer is about six-tenths of a mile), knowledge that direct measurement produces approximate, not exact, measures, the use of smaller units to produce more precise measures | 8. Demonstrate use of measurement systems | |
| D.8.3 Determine measurement directly using standard units (metric and US Customary) with these suggested degrees of accuracy lengths to the nearest mm or 1/16 of an inch, weight (mass) to the nearest 0.1 g or 0.5 ounce, liquid capacity to the nearest ml, angles to the nearest degree, temperature to the nearest C or F, elapsed time to the nearest second |
4. Use mathematical
tools
8. Demonstrate use of measurement systems |
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| 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.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.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.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) |
3. Demonstrate
the use of estimation strategies
5. Demonstrate use of formulas 8. Demonstrate use of measurement systems 11. Solve problems using geometric figures |
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| D.12.1 Identify, describe, and use derived attributes (e.g., density, speed, acceleration, pressure) to represent and solve problem situations |
5. Demonstrate
use of formulas
7. Solve application problems |
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| D.12.2 Select and use tools with appropriate degree of precision to determine measurements directly within specified degrees of accuracy and error (tolerance) | 8. Demonstrate use of measurement systems | |
| 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), geometric relationships and properties of circles and polygons (e.g., size of central angles, area of a sector of a circle), conversion constants to relate measures in one system to another (e.g., meters to feet, dollars to Deutschmarks |
B.3. Relate
linear functions and equations with their graphs and use them to represent
appropriate situations
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 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.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.7. Use right triangle relationships and the trigonometric ratios sine, cosine and tangent to analyze relationships and to formulate and solve problems |
2. Calculate
with real numbers
5. Demonstrate use of formulas 7. Solve application problems 8. Demonstrate use of measurement systems 11. Solve problems using geometric figures |
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By the end of grade eight, students will: |
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| E.8.1 Work with data in the context of real-world situations by formulating questions that lead to data collection and analysis, designing and conducting a statistical investigation, using technology to generate displays, summary statistics, and presentations | A.3. Use calculators appropriately to aid computation and understanding | 9. Interpret data |
| E.8.2 Organize and display data from statistical investigations using appropriate tables, graphs, and/or charts (e.g., circle, bar or line for multiple sets of data), appropriate plots (e.g., line, stem-and-leaf, box, scatter) | A.2. Construct and read charts, tables and graphs that summarize data from real world situations | 9. Interpret data |
| E.8.3 Extract, interpret, and analyze information from organized and displayed data by using frequency and distribution, including mode and range, central tendencies of data (mean and median), indicators of dispersion (e.g., outliers) | 9. Interpret data | |
| E.8.4 Use the results of data analysis to make predictions, develop convincing arguments, draw conclusions | 9. Interpret data | |
| E.8.5 Compare several sets of data to generate, test, and, as the data dictate, confirm or deny Hypotheses | 9. Interpret data | |
| E.8.6 Evaluate presentations and statistical analyses from a variety of sources for credibility of the source, techniques of collection, organization, and presentation of data, Missing or incorrect data, inferences, possible sources of bias | 9. Interpret data | |
| E.8.7 Determine the likelihood of occurrence of simple events by using a variety of strategies to identify possible outcomes (e.g., lists, tables, tree diagrams), conducting an experiment designing and conducting simulations, applying theoretical notions of probability (e.g., that four equally likely events have a 25% chance of happening) | 9. Interpret data | |
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| E.12.1 Work with data in the context of real-world situations by formulating hypotheses that lead to collection and analysis of one- and two-variable data, designing a data collection plan that considers random sampling, control groups, the role of assumptions, etc., conducting an investigation based on that plan, using technology to generate displays, summary statistics, and presentations | A.3. Use calculators appropriately to aid computation and understanding | 9. Interpret data |
| E.12.2 Organize and display data from statistical investigations using frequency distributions percentiles, quartiles, deciles, line of best fit (estimated regression line), matrices | B.4. Use matrices to represent and analyze linear situations | 9. Interpret data |
| E.12.3 Interpret and analyze information from organized and displayed data when given measures of dispersion, including standard deviation and variance, measures of reliability, measures of correlation | 9. Interpret data | |
| E.12.4 Analyze, evaluate, and critique the methods and conclusions of statistical experiments reported in journals, magazines, news media, advertising, etc. | 9. Interpret data | |
| E.12.5 Determine the likelihood of occurrence of complex events by using a variety of strategies (e.g., combinations) to identify possible outcomes, conducting an experiment designing and conducting simulations, applying theoretical probability | 9. Interpret data | |
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By the end of grade eight, students will: |
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| 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 | B.5. Add, subtract, multiply, divide, and exponential polynomial and rational expressions and simplify the results |
7. Solve application
problems
10. Evaluate algebraic sentences |
| 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, describing a real-world phenomenon that a given graph might represent | B.3. Relate linear functions and equations with their graphs and use them to represent appropriate situations |
10. Evaluate
algebraic sentences
12. Use rectangular coordinate system (Carrtesian geometry) |
| F.8.3 Recognize, describe, and analyze functional relationships by generalizing a rule that characterizes the pattern of change among variables. These functional relationships include exponential growth and decay (e.g., cell division, depreciation) | 10. Evaluate algebraic sentences | |
| 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.1. Solve
linear algebraic equations and inequalities in one variable, including
those with literal coefficients
B.2. Solve systems of linear equations in two or three variables and interpret solutions both symbolically and graphically B.7. Translate from a verbal description of a problem situation to a mathematical representation |
5. Demonstrate
use of formulas
7. Solve application problems 10. Evaluate algebraic sentences |
| F.8.5 Recognize and use generalized properties and relations, including additive and multiplicative property of equations and inequalities, commutativity and associativity of addition and multiplication, distributive property, inverses and identities for addition and multiplication, transitive property | 1. Demonstrate use of terminology and symbols | |
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| 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 | 10. Evaluate algebraic sentences | |
| F.12.2 Use mathematical functions (e.g., linear, exponential, quadratic, power) in a variety of ways, including recognizing that a variety of mathematical and real-world phenomena can be modeled by the same type of function, translating different forms of representing them (e.g., tables, graphs, functional notation, formulas), describing the relationships among variable quantities in a problem, using appropriate technology to interpret properties of their graphical representations (e.g., intercepts, slopes, rates of change, changes in rates of change, maximum, minimum) |
A.2. Construct
and read charts, tables and graphs that summarize data from real world
situations
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.8. Use the language, notation and properties of algebraic functions and their graphs, with particular attention to quadratic functions 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 |
10. Evaluate algebraic sentences |
| 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 |
B.1. Solve
linear algebraic equations and inequalities in one variable, including
those with literal coefficients
B.2. Solve systems of linear equations in two or three variables and interpret solutions both symbolically and graphically 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 the language, notation and properties of algebraic functions and their graphs, with particular attention to quadratic functions |
7. Solve application
problems
10. Evaluate algebraic sentences |
| F.12.4 Model and solve a variety of mathematical and real-world problems by using algebraic expressions, equations, and inequalities |
B.3. Relate
linear functions and equations with their graphs and use them to represent
appropriate situations
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 |
7. Solve application
problems
10. Evaluate algebraic sentences |
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To UW System Office of Academic Affairs Page
If you have questions or comments, direct them to Academic and Student Services, Phone: (608) 262-8778, Email: acss@uwsa.edu
This
document was last revised on October 14, 1999. ©January 1999 Board of Regents
of the University of Wisconsin System, All Rights Reserved.
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