Academic & Student Services
ENGLISH ALIGNMENT CONTENT
|
|
|
|
|
|
|
By the end of grade eight, students will: |
|
| 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 | |
| A.8.2 Communicate logical arguments clearly to show why a result makes sense | |
| A.8.3 Analyze non-routine problems by modeling, illustrating, guessing, simplifying, generalizing, shifting to another point of view, etc. | |
| 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 | |
| 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 | |
|
|
|
| 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.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 |
3.09 Recognize
and use the fact that the contrapositive of a conditional statement has
the same truth value as the conditional statement
3.10 Given a generalization, identify examples and counterexamples |
| 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.12.4 Develop effective oral and written presentations employing correct mathematical terminology, notation, symbols, and conventions for mathematical arguments and display of data | |
| 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 | |
|
By the end of grade eight, students will: |
|
| 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) |
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
4.11 Convert between decimal and scientific notation |
| 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) |
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
2.01 Perform the basic arithmetic operations with integers using symbols of grouping |
| B.8.3 Generate and explain equivalencies among fractions, decimals, and percents | 1.02 Convert between fractions, decimals, and percents |
| B.8.4 Express order relationships among rational numbers using appropriate symbols (>, <, >, <,) | 1.03 Arrange fractions, decimals, and integers in order of size |
| 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) |
1.04 Set up
and solve simple verbal problems involving percentages, mixtures, unit
conversions, and distance/time/rate
3.04 Set up and solve proportions derived from similar triangles |
| 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) | |
|
|
|
| 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 | 1.03 Arrange fractions, decimals, and integers in order of size |
| 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) |
1.01 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
1.02 Convert between fractions, decimals, and integers in order of size 2.01 Perform the basic arithmetic operations with integers using symbols of grouping 2.05 Simplify exponential expressions with negative bases 4.11 Convert between decimal and scientific notation |
| 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) |
1.04 Set up
and solve simple verbal problems involving percentages, mixtures, unit
conversions, and distance/time/rate
1.05 Find areas and perimeters of geometric figures composed of rectangles, triangles, and circles 1.06 Find the surface area and volume of a rectangular prism |
| 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) | |
| B.12.6 Routinely assess the acceptable limits of error when evaluating strategies, testing the reasonableness of results, using technology to carry out computations | |
|
By the end of grade eight, students will: |
|
| 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 | 1.08 Exhibit knowledge of basic geometric vocabulary |
| 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). |
3.01 Recognize
and use congruence and similarity correspondences
3.02 Recognize and use conditions that imply congruence and those that imply similarity for triangles |
| 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.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
|
|
|
|
|
| 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 |
1.08 Exhibit
knowledge of basic geometric vocabulary
3.12 Classify sets of objects by set inclusion, e.g., the set of squares is a subset of the set of rectangles |
| C.12.2 Use geometric models to solve mathematical and real-world problems |
1.05 Find areas
and perimeters of geometric figures composed of rectangles, triangles,
and circles
1.06 Find the surface area and volume of a rectangular prism |
| 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) |
3.09 Recognize
and use the fact that the contrapositive of a conditional statement has
the same truth value as the conditional statement
3.10 Given a generalization, identify examples and counterexamples |
| 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 |
3.05 Recognize
and use conditions that imply that two lines are parallel or two lines
are perpendicular
4.12 Find the slope and y-intercept and be able to graph a straight line given its equation 5.17 Find the distance between two points 5.18 Determine from the equations of two lines whether they are parallel or perpendicular or neither |
|
C.12.5 Identify
and demonstrate an understanding of the three ratios used in right-triangle
trigonometry (sine, cosine, tangent)
|
5.01 Use the definitions
of the trigonometric functions of angles
5.02 Evaluate trigonometric functions for special angles, eg., , 5.06 Evaluate expressions involving inverse trigonometric functions |
|
By the end of grade eight, students will: |
|
| 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) | |
| 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 | |
| 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 | |
| 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) |
1.05 Find areas
and perimeters of geometric figures composed of rectangles, triangles,
and circles
1.07 Solve problems involving the measures of angles of triangles 3.04 Set up and solve proportions derived from similar triangles 3.05 Recognize and use conditions that imply that two lines are parallel or two lines are perpendicular 3.08 Use Pyathagorean relationships |
|
|
|
| D.12.1 Identify, describe, and use derived attributes (e.g., density, speed, acceleration, pressure) to represent and solve problem situations | |
| D.12.2 Select and use tools with appropriate degree of precision to determine measurements directly within specified degrees of accuracy and error (tolerance) | |
| 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 |
1.04 Set up
and solve simple verbal problems involving percentages, mixtures, unit
conversions, and distance/time/rate
1.05 Find areas and perimeters of geometric figures composed of rectangles, triangles, and circles 1.06 Find the surface area and volume of a rectangular prism 1.07 Solve problems involving the measures of angles of triangles 2.10 Set up and solve verbal problems involving linear equations 3.01 Recognize and use congruence and similarity correspondences 3.02 Recognize and use conditions that imply congruence and those that imply similarity 3.03 Recognize and use summitries of isosceles triangles and equilateral triangles 3.04 Set up and solve proportions derived from similar triangles 3.06 Find the area of a sector of a circle given the measure of a central angle 3.07 Find measures of angles by using complementary and supplementary relations 3.08 Use Pythagorean relationships 5.01 Use the definitions of the trigonometric functions of angles 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 5.17 Find the distance between two points |
|
By the end of grade eight, students will: |
|
| 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 | |
| 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) | |
| 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) | |
| E.8.4 Use the results of data analysis to make predictions, develop convincing arguments, draw conclusions | |
| E.8.5 Compare several sets of data to generate, test, and, as the data dictate, confirm or deny Hypotheses | |
| 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 | |
| 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) | |
|
|
|
| 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 | |
| E.12.2 Organize and display data from statistical investigations using frequency distributions percentiles, quartiles, deciles, line of best fit (estimated regression line), matrices | |
| 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 | |
| E.12.4 Analyze, evaluate, and critique the methods and conclusions of statistical experiments reported in journals, magazines, news media, advertising, etc. | |
| 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 | |
|
By the end of grade eight, students will: |
|
| 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 |
2.04 Simplify
algebraic expressions involving multiplication and division of exponential
expressions
2.06 Find products of binomials 2.07 Collect like terms, and perform simple monomial factoring 2.08 Use order of operations to simplify polynomials and rational expressions 2.09 Evaluate polynomials at specific values of the variable 4.01 Simplify (reduce) rational expressions 4.02 Add, subtract, multiply, and divide rational expressions |
| 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 |
2.10 Set up
and solve verbal problems involving linear equations
4.12 Find the slope and y-intercept and be able to graph a straight line given its equation 4.14 Graph a parabola given its equation 4.17 Graph the solution of inequalities such as Ax + B < 0 on the number line 4.18 Graph the solution of inequalities such as AX + By + C < 0 4.23 Read, analyze, and solve verbal problems |
| 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) |
5.09 Use function
notation, perform operations on functions and evaluate functions
5.10 Work with exponential and logarithmic functions and their graphs 5.11 Solve exponential and logarithmic functions and their graphs |
| 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 |
2.02 Solve
linear equations containing simple fractions and literal numbers
2.03 Manipulate formulas containing fractions and several variables 4.17 Graph the solution of inequalities such as Ax + B < 0 4.18 Graph the solution of inequalities such as AX + By + C < 0 4.23 Read, analyze, and solve verbal problems |
| 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 | |
|
|
|
| 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 |
2.10 Set up
and solve verbal problems involving linear equations
4.23 Read, analyze, and solve verbal problems |
| 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) |
2.10 Set up
and solve verbal problems involving linear equations
4.12 Find the slope and y-intercept and be able to graph a straight line given its equation 4.14 Graph a parabola given its equation 4.23 Read, analyze, and solve verbal problems 5.09 Use function notation, perform operations on functions and evaluate functions 5.10 Work with exponential and logarithmic functions and their graphs 5.11 Solve exponential and logarithmic functions and their graphs 5.16 Use remainder and factor theorems to find the zerios of polynomials 5.18 Determine from the equations of two lines whether they are parallel or perpendicular or neither 5.20 Convert between the equation of a parabola in standard form and data on its vertex and axis |
| 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 |
2.02 Solve
linear equations containing simple fractions and literal numbers
2.10 Set up and solve verbal problems involving linear equations 4.14 Graph a parabola given its equation 4.17 Graph the solution of inequalities such as Ax + B < 0 on the number line 4.18 Graph the solution of inequalities such as Ax + By + C < ) 4.19 Solve quadratic equations (includes use of quadratic formula) 4.20 Solve fractional equations, discarding extraneous roots 4.21 Solve a system of two linear equations in two unknowns recognizing cases of no solutions or infinitely many solutions 4.22 Graph a system of two linear equations in two unknowns 5.13 Solve quadratic inequalities one variable, and graph the solutions 5.16 Solve systems containing linear and/or quadratic equations in two variables |
| F.12.4 Model and solve a variety of mathematical and real-world problems by using algebraic expressions, equations, and inequalities | 4.23 Read, analyze, and solve verbal problems |
To Wisconsin Alignment Index Page
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.
g:\vpacad\dpi\new matrix\math placement standards-placement.doc