Download Standards and Benchmarks

Standards and Benchmarks
Diocese of Sioux City
9-12 Math
Standard 1: Understands and applies concept of numbers and operations.
Benchmark 1: Understands numbers, ways of representing numbers, relationships among
numbers, and number systems.
i. Develops a deeper understanding of very large and very small numbers and of
various representations of them;
ii. Compares and contrasts the properties of numbers and number systems,
including the rational and real numbers, and understand complex numbers as
solutions to quadratic equations that do not have real solutions;
iii. Understands vectors and matrices as systems that have some of the properties
of the real-number system;
iv. Uses number theory arguments to justify relationships involving whole
numbers.
Benchmark 2: Understands meanings of operations and how they relate to one another.
i. Judges the effects of such operations as multiplication, division, and
computing powers and roots on the magnitudes of quantities;
ii. Develops an understanding of properties of, and representations for, the
addition and multiplication of vectors and matrices;
iii. Develops an understanding of permutations and combinations as counting
techniques.
Benchmark 3: Computes fluently and makes reasonable estimates.
i. Develop fluency in operations with real numbers, vectors, and matrices, using
mental computation or paper-and-pencil calculations for simple cases and
technology for more-complicated cases;
ii. Judge the reasonableness of numerical computations and their results.
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Standards and Benchmarks
Diocese of Sioux City
9-12 Math
Standard 2: Understands and applies concepts of algebra and functions.
Benchmark 1: Understands patterns, relations, and functions
i. Generalizes patterns using explicitly defined and recursively defined
functions;
ii. Understands relations and functions and selects, converts flexibly among, and
uses various representations for them’
iii. Analyzes functions of one variable by investigating rates of change,
intercepts, zeros, asymptotes, and local and global behavior;
iv. Understands and performs transformations such as arithmetically combining,
composing, and inverting commonly used functions, using technology to
perform such operations on more-complicated symbolic expressions;
v. Understands and compares the properties of classes of functions, including
exponential, polynomial, rational, logarithmic, and periodic functions of two
variables.
Benchmark 2: Represents and analyzes mathematical situations and structures using algebraic
symbols.
i. Understands the meaning of equivalent forms of expressions, equations,
inequalities, and relations;
ii. Writes equivalent forms of equations, inequalities, and systems of equations
and solves them with fluency – mentally or with paper-and-pencil in simple
cases and using technology in all cases;
iii. Uses symbolic algebra to represent and explain mathematical relationships;
iv. Uses a variety of symbolic representations, including recursive and parametric
equations, for functions and relations;
v. Judges the meaning, utility, and reasonableness of the results of symbol
manipulations, including those carried out by technology.
Benchmark 3: Uses mathematical models to represent and understand quantitative relationships.
i. Identifies essential quantitative relationships in a situation and determines the
class or classes of functions that might model the relationships;
ii. Uses symbolic expressions, including iterative recursive forms, to represent
relationships arising from various contexts;
iii. Draws reasonable conclusions about a situation being modeled.
Benchmark 4: Analyzes change in various contexts.
i. Approximates and interprets rates of change from graphical and numerical
data.
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2006
Standards and Benchmarks
Diocese of Sioux City
9-12 Math
Standard 3: Understands and applies properties of geometry.
Benchmark 1: Analyzes characteristics and properties of two- and three-dimensional geometric
shapes and develops mathematical arguments about geometric relationships.
i. Analyzes properties and determines attributes of two- and three-dimensional
objects;
ii. Explores relationships (including congruence and similarity) among classes of
two- and three-dimensional objects, makes and tests conjectures about them,
and solves problems involving them;
iii. Establishes the validity of geometric conjectures using deduction, proves
theorems, and critiques arguments made by others;
iv. Uses trigonometric relationships to determine lengths and angle measures.
Benchmark 2: Specifies locations and describes spatial relationships using coordinate geometry
and other representational systems.
i. Uses Cartesian coordinates and other coordinate systems, such as navigational
polar, or spherical systems, to analyze geometric situations;
ii. Investigates conjectures and solves problems involving two- and threedimensional objects represented with Cartesian coordinates.
Benchmark 3: Applies transformations and uses symmetry to analyze mathematical situations.
i. Understands and represents translations, reflections, rotations, and dilations of
objects in the plane by using sketches, coordinates, vectors, function notation,
and matrices;
ii. Uses various representations to help understand the effects of simple
transformations and their compositions.
Benchmark 4: Uses visualization, spatial reasoning, and geometric modeling to solve problems.
i. Draws and constructs representations of two- and three-dimensional geometric
objects using a variety of tools;
ii. Visualizes three-dimensional objects and spaces from different perspectives
and analyzes their cross sections;
iii. Uses vertex-edge graphs to model and solves problems;
iv. Uses geometric models to gain insights into, and answer questions in, other
areas of mathematics;
v. Uses geometric ideas to solve problems in, and gain insights into, other
disciplines and other areas of interest such as art and architecture.
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2006
Standards and Benchmarks
Diocese of Sioux City
9-12 Math
Standard 4: Understands and applies concepts of measurement.
Benchmark 1: Understands measurable attributes of objects and the units, systems, and processes
of measurement.
i. Makes decisions about units and scales that are appropriate for problem
situations involving measurement.
Benchmark 2: Applies appropriate techniques, tools and formulas to determine measurements.
i. Analyzes precision, accuracy, and approximate error in measurement
situations;
ii. Understands and uses formulas for the area, surface area, and volume of
geometric figures, including cones, spheres, and cylinders;
iii. Applies informal concepts of successive approximation, upper and lower
bounds, and limit in measurement situations;
iv. Uses unit analysis to check measurement computations.
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2006
Standards and Benchmarks
Diocese of Sioux City
9-12 Math
Standard 5: Understands and applies concepts of data analysis and probability.
Benchmark 1: Formulates questions that can be addressed with data and collects, organizes, and
displays relevant data to answer them.
i. Understands the differences among various kinds of studies and which types
of inferences can legitimately be drawn from each;
ii. Knows the characteristics of well-designed studies, including the role of
randomization in surveys and experiments;
iii. Understands the meaning of measurement data and categorical data, of
univariate and bivariate data, and of the term variable;
iv. Understands histograms, parallel box plots, and scatter plots and uses them to
display data;
v. Computes basic statistics and understands the distinction between a statistic
and a parameter.
Benchmark 2: Selects and uses appropriate statistical methods to analyze data.
i. For univariate measurement data, is able to display the distribution, describes
its shape, and selects and calculates summary statistics;
ii. For bivariate measurement data, is able to display a scatter plot, describes its
shape, and determines regression coefficients, regression equations, and
correlation coefficients using technological tools;
iii. Displays and discusses bivariate data where at least one variable is
categorical;
iv. Recognizes how linear transformations of univariate data affect shape, center,
and spread;
v. Identifies trends in bivariate data and finds functions that model the data or
transforms the data so that they can be modeled.
Benchmark 3: Develops and evaluates inferences and predictions that are based on data.
i. Uses simulations to explore the variability of sample statistics from a known
population and to construct sampling distributions;
ii. Understands how sample distributions as the basis for informal inference;
iii. Evaluates published reports that are based on data analysis, and the validity of
conclusions;
iv. Understands how basic statistical techniques are used to monitor process
characteristics in the work place.
Benchmark 4: Understands and applies basic concepts of probability.
i. Understands the concepts of sample space and probability distribution and
constructs sample spaces and distributions in simple cases;
ii. Uses simulations to construct empirical probability distributions;
iii. Computes and interprets the expected value of random variables in simple
cases;
iv. Understands the concepts of conditional probability and independent events;
v. Understands how to compute the probability of a compound event.
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Standards and Benchmarks
Diocese of Sioux City
9-12 Math
Standard 6: Understands and applies problem solving strategies.
Benchmark 1:
Benchmark 2:
Benchmark 3:
Benchmark 4:
Builds new mathematical knowledge through problem solving;
Solves problems that arise in mathematical and in other contexts;
Applies and adapts a variety of appropriate strategies to solve problems;
Monitors and reflects on the process of mathematical problem solving.
Standard 7: Reasoning and Proof
Benchmark 1:
Benchmark 2:
Benchmark 3:
Benchmark 4:
Recognizes reasoning and proof as fundamental aspects of mathematics;
Makes and investigates mathematical conjectures;
Develops and evaluates mathematical arguments and proofs;
Selects and uses various types of reasoning and methods of proof.
Standard 8: Communicates mathematically
Benchmark 1: Organizes and consolidates their mathematical thinking through communication;
Benchmark 2: Communicates their mathematical thinking coherently and clearly to peers,
teachers, and others;
Benchmark 3: Analyzes and evaluates the mathematical thinking and strategies of others;
Benchmark 4: Uses the language of mathematics to express mathematical ideas precisely.
Standard 9: Connections
Benchmark 1: Recognizes and uses connections among mathematical ideas;
Benchmark 2: Understands how mathematical ideas interconnect and build on one another to
produce a coherent whole;
Benchmark 3: Recognizes and applies mathematics in contexts outside of mathematics.
Standard 10: Representation
Benchmark 1: Creates and uses representations to organize, record, and communicate
mathematical ideas;
Benchmark 2: Selects, applies, and translates among mathematical representations to solve
problems;
Benchmark 3: Uses representations to model and interprets physical, social, and mathematical
phenomena.
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