Wisconsin State Standards for Mathematics:

WI.A. Mathematical Processes: Students in Wisconsin will draw on a broad body of mathematical knowledge and apply a variety of mathematical skills and strategies, including reasoning, oral and written communication, and the use of appropriate technology, when solving mathematical, real-world and non-routine problems.

A.4.1. Use reasoning abilities to perceive patterns; identify relationships; formulate questions for further exploration; justify strategies; test reasonableness of results.

A.4.2. Communicate mathematical ideas in a variety of ways, including words, numbers, symbols, pictures, charts, graphs, tables, diagrams, and models.

A.4.3. Connect mathematical learning with other subjects, personal experiences, current events, and personal interests see relationships between various kinds of problems and actual events; use mathematics as a way to understand other areas of the curriculum (e.g., measurement in science, map skills in social studies).

A.4.4. Use appropriate mathematical vocabulary, symbols, and notation with understanding based on prior conceptual work.

A.4.5. Explain solutions to problems clearly and logically in oral and written work and support solutions with evidence.

WI.B. Number Operations and Relationships: Students in Wisconsin will use numbers effectively for various purposes, such as counting, measuring, estimating, and problem solving.

B.4.1. Represent and explain whole numbers, decimals, and fractions with physical materials; number lines and other pictorial models; verbal descriptions; place-value concepts and notation; symbolic renaming (e.g., 43=40+3=30+13).

B.4.2. Determine the number of things in a set by grouping and counting (e.g., by threes, fives, hundreds); combining and arranging (e.g., all possible coin combinations amounting to thirty cents); estimation, including rounding.

B.4.3. Read, write, and order whole numbers, simple fractions (e.g., halves, fourths, tenths, unit fractions) and commonly-used decimals (monetary units).

B.4.4. Identify and represent equivalent fractions for halves, fourths, eighths, tenths, sixteenths.

B.4.5. In problem-solving situations involving whole numbers, select and efficiently use appropriate computational procedures such as recalling the basic facts of addition, subtraction, multiplication, and division; using mental math (e.g., 37+25, 40x7); estimation; selecting and applying algorithms for addition, subtraction, multiplication, and division; using a calculator.

B.4.6. Add and subtract fractions with like denominators.

B.4.7. In problem-solving situations involving money, add and subtract decimals.

WI.C. Geometry: Students in Wisconsin will be able to use geometric concepts, relationships and procedures to interpret, represent, and solve problems.

C.4.1. Describe two-and three-dimensional figures (e.g., circles, polygons, trapezoids, prisms, spheres) by naming them; comparing, sorting, and classifying them; drawing and constructing physical models to specifications; identifying their properties (e.g., number of sides or faces, two- or three-dimensionality, equal sides, number of right angles); predicting the results of combining or subdividing two-dimensional figures; explaining how these figures are related to objects in the environment.

C.4.2. Use physical materials and motion geometry (such as slides, flips, and turns) to identify properties and relationships, including but not limited to symmetry; congruence; similarity.

C.4.3. Identify and use relationships among figures, including but not limited to location (e.g., between, adjacent to, interior of); position (e.g., parallel, perpendicular); intersection (of two-dimensional figures).

C.4.4. Use simple two-dimensional coordinate systems to find locations on maps and to represent points and simple figures.

WI.D. Measurement: Students in Wisconsin will select and use appropriate tools (including technology) and techniques to measure things to a specified degree of accuracy. They will use measurements in problem-solving situations.

D.4.1. Recognize and describe measurable attributes, such as length, liquid capacity, time, weight (mass), temperature, volume, monetary value, and angle size, and identify the appropriate units to measure them.

D.4.2. Demonstrate understanding of basic facts, principles, and techniques of measurement, including appropriate use of arbitrary and standard units (metric and US Customary); appropriate use and conversion of units within a system (such as yards, feet, and inches; kilograms and grams; gallons, quarts, pints, and cups); judging the reasonableness of an obtained measurement as it relates to prior experience and familiar benchmarks.

D.4.3. Read and interpret measuring instruments (e.g., rulers, clocks, thermometers).

D.4.4. Determine measurements directly by using standard tools to these suggested degrees of accuracy length to the nearest half-inch or nearest cm; weight (mass) to the nearest ounce or nearest 5 grams; temperature to the nearest 5; time to the nearest minute; monetary value to dollars and cents; liquid capacity to the nearest fluid ounce.

D.4.5. Determine measurements by using basic relationships (such as perimeter and area) and approximate measurements by using estimation techniques.

WI.E. Statistics and Probability: Students in Wisconsin will use data collection and analysis, statistics and probability in problem-solving situations, employing technology where appropriate.

E.4.1. Work with data in the context of real-world situations by formulating questions that lead to data collection and analysis; determining what data to collect and when and how to collect them; collecting, organizing, and displaying data; drawing reasonable conclusions based on data.

E.4.2. Describe a set of data using high and low values, and range; most frequent value (mode); middle value of a set of ordered data (median).

E.4.3. In problem-solving situations, read, extract, and use information presented in graphs, tables, or charts.

E.4.4. Determine if future events are more, less, or equally likely, impossible, or certain to occur.

E.4.5. Predict outcomes of future events and test predictions using data from a variety of sources.

WI.F. Algebraic Relationships: Students in Wisconsin will discover, describe, and generalize simple and complex patterns and relationships. In the context of real-world problem situations, the student will use algebraic techniques to define and describe the problem to determine and justify appropriate solutions.

F.4.1. Use letters, boxes, or other symbols to stand for any number, measured quantity, or object in simple situations (e.g., N + 0 = N is true for any number).

F.4.2. Use the vocabulary, symbols, and notation of algebra accurately (e.g., correct use of the symbol '='; effective use of the associative property of multiplication).

F.4.3. Work with simple linear patterns and relationships in a variety of ways, including recognizing and extending number patterns; describing them verbally; representing them with pictures, tables, charts, graphs; recognizing that different models can represent the same pattern or relationship; using them to describe real-world phenomena.

F.4.4. Recognize variability in simple functional relationships by describing how a change in one quantity can produce a change in another (e.g., number of bicycles and the total number of wheels).

F.4.5. Use simple equations and inequalities in a variety of ways, including using them to represent problem situations; solving them by different methods (e.g., use of manipulatives, guess and check strategies, recall of number facts); recording and describing solution strategies.

F.4.6. Recognize and use generalized properties and relationships of arithmetic (e.g., commutativity of addition, inverse relationship of multiplication and division).

WI.A. Mathematical Processes: Students in Wisconsin will draw on a broad body of mathematical knowledge and apply a variety of mathematical skills and strategies, including reasoning, oral and written communication, and the use of appropriate technology, when solving mathematical, real-world and non-routine problems.

A.4.1. Use reasoning abilities to perceive patterns; identify relationships; formulate questions for further exploration; justify strategies; test reasonableness of results.

A.4.2. Communicate mathematical ideas in a variety of ways, including words, numbers, symbols, pictures, charts, graphs, tables, diagrams, and models.

A.4.3. Connect mathematical learning with other subjects, personal experiences, current events, and personal interests see relationships between various kinds of problems and actual events; use mathematics as a way to understand other areas of the curriculum (e.g., measurement in science, map skills in social studies).

A.4.4. Use appropriate mathematical vocabulary, symbols, and notation with understanding based on prior conceptual work.

A.4.5. Explain solutions to problems clearly and logically in oral and written work and support solutions with evidence.

WI.B. Number Operations and Relationships: Students in Wisconsin will use numbers effectively for various purposes, such as counting, measuring, estimating, and problem solving.

B.4.1. Represent and explain whole numbers, decimals, and fractions with physical materials; number lines and other pictorial models; verbal descriptions; place-value concepts and notation; symbolic renaming (e.g., 43=40+3=30+13).

B.4.2. Determine the number of things in a set by grouping and counting (e.g., by threes, fives, hundreds); combining and arranging (e.g., all possible coin combinations amounting to thirty cents); estimation, including rounding.

B.4.3. Read, write, and order whole numbers, simple fractions (e.g., halves, fourths, tenths, unit fractions) and commonly-used decimals (monetary units).

B.4.4. Identify and represent equivalent fractions for halves, fourths, eighths, tenths, sixteenths.

B.4.5. In problem-solving situations involving whole numbers, select and efficiently use appropriate computational procedures such as recalling the basic facts of addition, subtraction, multiplication, and division; using mental math (e.g., 37+25, 40x7); estimation; selecting and applying algorithms for addition, subtraction, multiplication, and division; using a calculator.

B.4.6. Add and subtract fractions with like denominators.

B.4.7. In problem-solving situations involving money, add and subtract decimals.

WI.C. Geometry: Students in Wisconsin will be able to use geometric concepts, relationships and procedures to interpret, represent, and solve problems.

C.4.1. Describe two-and three-dimensional figures (e.g., circles, polygons, trapezoids, prisms, spheres) by naming them; comparing, sorting, and classifying them; drawing and constructing physical models to specifications; identifying their properties (e.g., number of sides or faces, two- or three-dimensionality, equal sides, number of right angles); predicting the results of combining or subdividing two-dimensional figures; explaining how these figures are related to objects in the environment.

C.4.2. Use physical materials and motion geometry (such as slides, flips, and turns) to identify properties and relationships, including but not limited to symmetry; congruence; similarity.

C.4.3. Identify and use relationships among figures, including but not limited to location (e.g., between, adjacent to, interior of); position (e.g., parallel, perpendicular); intersection (of two-dimensional figures).

C.4.4. Use simple two-dimensional coordinate systems to find locations on maps and to represent points and simple figures.

WI.D. Measurement: Students in Wisconsin will select and use appropriate tools (including technology) and techniques to measure things to a specified degree of accuracy. They will use measurements in problem-solving situations.

D.4.1. Recognize and describe measurable attributes, such as length, liquid capacity, time, weight (mass), temperature, volume, monetary value, and angle size, and identify the appropriate units to measure them.

D.4.2. Demonstrate understanding of basic facts, principles, and techniques of measurement, including appropriate use of arbitrary and standard units (metric and US Customary); appropriate use and conversion of units within a system (such as yards, feet, and inches; kilograms and grams; gallons, quarts, pints, and cups); judging the reasonableness of an obtained measurement as it relates to prior experience and familiar benchmarks.

D.4.3. Read and interpret measuring instruments (e.g., rulers, clocks, thermometers).

D.4.4. Determine measurements directly by using standard tools to these suggested degrees of accuracy length to the nearest half-inch or nearest cm; weight (mass) to the nearest ounce or nearest 5 grams; temperature to the nearest 5; time to the nearest minute; monetary value to dollars and cents; liquid capacity to the nearest fluid ounce.

D.4.5. Determine measurements by using basic relationships (such as perimeter and area) and approximate measurements by using estimation techniques.

WI.E. Statistics and Probability: Students in Wisconsin will use data collection and analysis, statistics and probability in problem-solving situations, employing technology where appropriate.

E.4.1. Work with data in the context of real-world situations by formulating questions that lead to data collection and analysis; determining what data to collect and when and how to collect them; collecting, organizing, and displaying data; drawing reasonable conclusions based on data.

E.4.2. Describe a set of data using high and low values, and range; most frequent value (mode); middle value of a set of ordered data (median).

E.4.3. In problem-solving situations, read, extract, and use information presented in graphs, tables, or charts.

E.4.4. Determine if future events are more, less, or equally likely, impossible, or certain to occur.

E.4.5. Predict outcomes of future events and test predictions using data from a variety of sources.

WI.F. Algebraic Relationships: Students in Wisconsin will discover, describe, and generalize simple and complex patterns and relationships. In the context of real-world problem situations, the student will use algebraic techniques to define and describe the problem to determine and justify appropriate solutions.

F.4.1. Use letters, boxes, or other symbols to stand for any number, measured quantity, or object in simple situations (e.g., N + 0 = N is true for any number).

F.4.2. Use the vocabulary, symbols, and notation of algebra accurately (e.g., correct use of the symbol '='; effective use of the associative property of multiplication).

F.4.3. Work with simple linear patterns and relationships in a variety of ways, including recognizing and extending number patterns; describing them verbally; representing them with pictures, tables, charts, graphs; recognizing that different models can represent the same pattern or relationship; using them to describe real-world phenomena.

F.4.4. Recognize variability in simple functional relationships by describing how a change in one quantity can produce a change in another (e.g., number of bicycles and the total number of wheels).

F.4.5. Use simple equations and inequalities in a variety of ways, including using them to represent problem situations; solving them by different methods (e.g., use of manipulatives, guess and check strategies, recall of number facts); recording and describing solution strategies.

F.4.6. Recognize and use generalized properties and relationships of arithmetic (e.g., commutativity of addition, inverse relationship of multiplication and division).

WI.A. Mathematical Processes: Students in Wisconsin will draw on a broad body of mathematical knowledge and apply a variety of mathematical skills and strategies, including reasoning, oral and written communication, and the use of appropriate technology, when solving mathematical, real-world and non-routine problems.

A.4.1. Use reasoning abilities to perceive patterns; identify relationships; formulate questions for further exploration; justify strategies; test reasonableness of results.

A.4.2. Communicate mathematical ideas in a variety of ways, including words, numbers, symbols, pictures, charts, graphs, tables, diagrams, and models.

A.4.3. Connect mathematical learning with other subjects, personal experiences, current events, and personal interests see relationships between various kinds of problems and actual events; use mathematics as a way to understand other areas of the curriculum (e.g., measurement in science, map skills in social studies).

A.4.4. Use appropriate mathematical vocabulary, symbols, and notation with understanding based on prior conceptual work.

A.4.5. Explain solutions to problems clearly and logically in oral and written work and support solutions with evidence.

WI.B. Number Operations and Relationships: Students in Wisconsin will use numbers effectively for various purposes, such as counting, measuring, estimating, and problem solving.

B.4.1. Represent and explain whole numbers, decimals, and fractions with physical materials; number lines and other pictorial models; verbal descriptions; place-value concepts and notation; symbolic renaming (e.g., 43=40+3=30+13).

B.4.2. Determine the number of things in a set by grouping and counting (e.g., by threes, fives, hundreds); combining and arranging (e.g., all possible coin combinations amounting to thirty cents); estimation, including rounding.

B.4.3. Read, write, and order whole numbers, simple fractions (e.g., halves, fourths, tenths, unit fractions) and commonly-used decimals (monetary units).

B.4.4. Identify and represent equivalent fractions for halves, fourths, eighths, tenths, sixteenths.

B.4.5. In problem-solving situations involving whole numbers, select and efficiently use appropriate computational procedures such as recalling the basic facts of addition, subtraction, multiplication, and division; using mental math (e.g., 37+25, 40x7); estimation; selecting and applying algorithms for addition, subtraction, multiplication, and division; using a calculator.

B.4.6. Add and subtract fractions with like denominators.

B.4.7. In problem-solving situations involving money, add and subtract decimals.

WI.C. Geometry: Students in Wisconsin will be able to use geometric concepts, relationships and procedures to interpret, represent, and solve problems.

C.4.1. Describe two-and three-dimensional figures (e.g., circles, polygons, trapezoids, prisms, spheres) by naming them; comparing, sorting, and classifying them; drawing and constructing physical models to specifications; identifying their properties (e.g., number of sides or faces, two- or three-dimensionality, equal sides, number of right angles); predicting the results of combining or subdividing two-dimensional figures; explaining how these figures are related to objects in the environment.

C.4.2. Use physical materials and motion geometry (such as slides, flips, and turns) to identify properties and relationships, including but not limited to symmetry; congruence; similarity.

C.4.3. Identify and use relationships among figures, including but not limited to location (e.g., between, adjacent to, interior of); position (e.g., parallel, perpendicular); intersection (of two-dimensional figures).

C.4.4. Use simple two-dimensional coordinate systems to find locations on maps and to represent points and simple figures.

WI.D. Measurement: Students in Wisconsin will select and use appropriate tools (including technology) and techniques to measure things to a specified degree of accuracy. They will use measurements in problem-solving situations.

D.4.1. Recognize and describe measurable attributes, such as length, liquid capacity, time, weight (mass), temperature, volume, monetary value, and angle size, and identify the appropriate units to measure them.

D.4.2. Demonstrate understanding of basic facts, principles, and techniques of measurement, including appropriate use of arbitrary and standard units (metric and US Customary); appropriate use and conversion of units within a system (such as yards, feet, and inches; kilograms and grams; gallons, quarts, pints, and cups); judging the reasonableness of an obtained measurement as it relates to prior experience and familiar benchmarks.

D.4.3. Read and interpret measuring instruments (e.g., rulers, clocks, thermometers).

D.4.4. Determine measurements directly by using standard tools to these suggested degrees of accuracy length to the nearest half-inch or nearest cm; weight (mass) to the nearest ounce or nearest 5 grams; temperature to the nearest 5; time to the nearest minute; monetary value to dollars and cents; liquid capacity to the nearest fluid ounce.

D.4.5. Determine measurements by using basic relationships (such as perimeter and area) and approximate measurements by using estimation techniques.

WI.E. Statistics and Probability: Students in Wisconsin will use data collection and analysis, statistics and probability in problem-solving situations, employing technology where appropriate.

E.4.1. Work with data in the context of real-world situations by formulating questions that lead to data collection and analysis; determining what data to collect and when and how to collect them; collecting, organizing, and displaying data; drawing reasonable conclusions based on data.

E.4.2. Describe a set of data using high and low values, and range; most frequent value (mode); middle value of a set of ordered data (median).

E.4.3. In problem-solving situations, read, extract, and use information presented in graphs, tables, or charts.

E.4.4. Determine if future events are more, less, or equally likely, impossible, or certain to occur.

E.4.5. Predict outcomes of future events and test predictions using data from a variety of sources.

WI.F. Algebraic Relationships: Students in Wisconsin will discover, describe, and generalize simple and complex patterns and relationships. In the context of real-world problem situations, the student will use algebraic techniques to define and describe the problem to determine and justify appropriate solutions.

F.4.1. Use letters, boxes, or other symbols to stand for any number, measured quantity, or object in simple situations (e.g., N + 0 = N is true for any number).

F.4.2. Use the vocabulary, symbols, and notation of algebra accurately (e.g., correct use of the symbol '='; effective use of the associative property of multiplication).

F.4.3. Work with simple linear patterns and relationships in a variety of ways, including recognizing and extending number patterns; describing them verbally; representing them with pictures, tables, charts, graphs; recognizing that different models can represent the same pattern or relationship; using them to describe real-world phenomena.

F.4.4. Recognize variability in simple functional relationships by describing how a change in one quantity can produce a change in another (e.g., number of bicycles and the total number of wheels).

F.4.5. Use simple equations and inequalities in a variety of ways, including using them to represent problem situations; solving them by different methods (e.g., use of manipulatives, guess and check strategies, recall of number facts); recording and describing solution strategies.

F.4.6. Recognize and use generalized properties and relationships of arithmetic (e.g., commutativity of addition, inverse relationship of multiplication and division).

WI.A. Mathematical Processes: Students in Wisconsin will draw on a broad body of mathematical knowledge and apply a variety of mathematical skills and strategies, including reasoning, oral and written communication, and the use of appropriate technology, when solving mathematical, real-world and non-routine problems.

A.4.1. Use reasoning abilities to perceive patterns; identify relationships; formulate questions for further exploration; justify strategies; test reasonableness of results.

A.4.2. Communicate mathematical ideas in a variety of ways, including words, numbers, symbols, pictures, charts, graphs, tables, diagrams, and models.

A.4.3. Connect mathematical learning with other subjects, personal experiences, current events, and personal interests see relationships between various kinds of problems and actual events; use mathematics as a way to understand other areas of the curriculum (e.g., measurement in science, map skills in social studies).

A.4.4. Use appropriate mathematical vocabulary, symbols, and notation with understanding based on prior conceptual work.

A.4.5. Explain solutions to problems clearly and logically in oral and written work and support solutions with evidence.

WI.B. Number Operations and Relationships: Students in Wisconsin will use numbers effectively for various purposes, such as counting, measuring, estimating, and problem solving.

B.4.1. Represent and explain whole numbers, decimals, and fractions with physical materials; number lines and other pictorial models; verbal descriptions; place-value concepts and notation; symbolic renaming (e.g., 43=40+3=30+13).

B.4.2. Determine the number of things in a set by grouping and counting (e.g., by threes, fives, hundreds); combining and arranging (e.g., all possible coin combinations amounting to thirty cents); estimation, including rounding.

B.4.3. Read, write, and order whole numbers, simple fractions (e.g., halves, fourths, tenths, unit fractions) and commonly-used decimals (monetary units).

B.4.4. Identify and represent equivalent fractions for halves, fourths, eighths, tenths, sixteenths.

B.4.5. In problem-solving situations involving whole numbers, select and efficiently use appropriate computational procedures such as recalling the basic facts of addition, subtraction, multiplication, and division; using mental math (e.g., 37+25, 40x7); estimation; selecting and applying algorithms for addition, subtraction, multiplication, and division; using a calculator.

B.4.6. Add and subtract fractions with like denominators.

B.4.7. In problem-solving situations involving money, add and subtract decimals.

WI.C. Geometry: Students in Wisconsin will be able to use geometric concepts, relationships and procedures to interpret, represent, and solve problems.

C.4.1. Describe two-and three-dimensional figures (e.g., circles, polygons, trapezoids, prisms, spheres) by naming them; comparing, sorting, and classifying them; drawing and constructing physical models to specifications; identifying their properties (e.g., number of sides or faces, two- or three-dimensionality, equal sides, number of right angles); predicting the results of combining or subdividing two-dimensional figures; explaining how these figures are related to objects in the environment.

C.4.2. Use physical materials and motion geometry (such as slides, flips, and turns) to identify properties and relationships, including but not limited to symmetry; congruence; similarity.

C.4.3. Identify and use relationships among figures, including but not limited to location (e.g., between, adjacent to, interior of); position (e.g., parallel, perpendicular); intersection (of two-dimensional figures).

C.4.4. Use simple two-dimensional coordinate systems to find locations on maps and to represent points and simple figures.

WI.D. Measurement: Students in Wisconsin will select and use appropriate tools (including technology) and techniques to measure things to a specified degree of accuracy. They will use measurements in problem-solving situations.

D.4.1. Recognize and describe measurable attributes, such as length, liquid capacity, time, weight (mass), temperature, volume, monetary value, and angle size, and identify the appropriate units to measure them.

D.4.2. Demonstrate understanding of basic facts, principles, and techniques of measurement, including appropriate use of arbitrary and standard units (metric and US Customary); appropriate use and conversion of units within a system (such as yards, feet, and inches; kilograms and grams; gallons, quarts, pints, and cups); judging the reasonableness of an obtained measurement as it relates to prior experience and familiar benchmarks.

D.4.3. Read and interpret measuring instruments (e.g., rulers, clocks, thermometers).

D.4.4. Determine measurements directly by using standard tools to these suggested degrees of accuracy length to the nearest half-inch or nearest cm; weight (mass) to the nearest ounce or nearest 5 grams; temperature to the nearest 5; time to the nearest minute; monetary value to dollars and cents; liquid capacity to the nearest fluid ounce.

D.4.5. Determine measurements by using basic relationships (such as perimeter and area) and approximate measurements by using estimation techniques.

WI.E. Statistics and Probability: Students in Wisconsin will use data collection and analysis, statistics and probability in problem-solving situations, employing technology where appropriate.

E.4.1. Work with data in the context of real-world situations by formulating questions that lead to data collection and analysis; determining what data to collect and when and how to collect them; collecting, organizing, and displaying data; drawing reasonable conclusions based on data.

E.4.2. Describe a set of data using high and low values, and range; most frequent value (mode); middle value of a set of ordered data (median).

E.4.3. In problem-solving situations, read, extract, and use information presented in graphs, tables, or charts.

E.4.4. Determine if future events are more, less, or equally likely, impossible, or certain to occur.

E.4.5. Predict outcomes of future events and test predictions using data from a variety of sources.

WI.F. Algebraic Relationships: Students in Wisconsin will discover, describe, and generalize simple and complex patterns and relationships. In the context of real-world problem situations, the student will use algebraic techniques to define and describe the problem to determine and justify appropriate solutions.

F.4.1. Use letters, boxes, or other symbols to stand for any number, measured quantity, or object in simple situations (e.g., N + 0 = N is true for any number).

F.4.2. Use the vocabulary, symbols, and notation of algebra accurately (e.g., correct use of the symbol '='; effective use of the associative property of multiplication).

F.4.3. Work with simple linear patterns and relationships in a variety of ways, including recognizing and extending number patterns; describing them verbally; representing them with pictures, tables, charts, graphs; recognizing that different models can represent the same pattern or relationship; using them to describe real-world phenomena.

F.4.4. Recognize variability in simple functional relationships by describing how a change in one quantity can produce a change in another (e.g., number of bicycles and the total number of wheels).

F.4.5. Use simple equations and inequalities in a variety of ways, including using them to represent problem situations; solving them by different methods (e.g., use of manipulatives, guess and check strategies, recall of number facts); recording and describing solution strategies.

F.4.6. Recognize and use generalized properties and relationships of arithmetic (e.g., commutativity of addition, inverse relationship of multiplication and division).

WI.A. Mathematical Processes: Students in Wisconsin will draw on a broad body of mathematical knowledge and apply a variety of mathematical skills and strategies, including reasoning, oral and written communication, and the use of appropriate technology, when solving mathematical, real-world and non-routine problems.

A.4.1. Use reasoning abilities to perceive patterns; identify relationships; formulate questions for further exploration; justify strategies; test reasonableness of results.

A.4.2. Communicate mathematical ideas in a variety of ways, including words, numbers, symbols, pictures, charts, graphs, tables, diagrams, and models.

A.4.3. Connect mathematical learning with other subjects, personal experiences, current events, and personal interests see relationships between various kinds of problems and actual events; use mathematics as a way to understand other areas of the curriculum (e.g., measurement in science, map skills in social studies).

A.4.4. Use appropriate mathematical vocabulary, symbols, and notation with understanding based on prior conceptual work.

A.4.5. Explain solutions to problems clearly and logically in oral and written work and support solutions with evidence.

WI.B. Number Operations and Relationships: Students in Wisconsin will use numbers effectively for various purposes, such as counting, measuring, estimating, and problem solving.

B.4.1. Represent and explain whole numbers, decimals, and fractions with physical materials; number lines and other pictorial models; verbal descriptions; place-value concepts and notation; symbolic renaming (e.g., 43=40+3=30+13).

B.4.2. Determine the number of things in a set by grouping and counting (e.g., by threes, fives, hundreds); combining and arranging (e.g., all possible coin combinations amounting to thirty cents); estimation, including rounding.

B.4.3. Read, write, and order whole numbers, simple fractions (e.g., halves, fourths, tenths, unit fractions) and commonly-used decimals (monetary units).

B.4.4. Identify and represent equivalent fractions for halves, fourths, eighths, tenths, sixteenths.

B.4.5. In problem-solving situations involving whole numbers, select and efficiently use appropriate computational procedures such as recalling the basic facts of addition, subtraction, multiplication, and division; using mental math (e.g., 37+25, 40x7); estimation; selecting and applying algorithms for addition, subtraction, multiplication, and division; using a calculator.

B.4.6. Add and subtract fractions with like denominators.

B.4.7. In problem-solving situations involving money, add and subtract decimals.

WI.C. Geometry: Students in Wisconsin will be able to use geometric concepts, relationships and procedures to interpret, represent, and solve problems.

C.4.1. Describe two-and three-dimensional figures (e.g., circles, polygons, trapezoids, prisms, spheres) by naming them; comparing, sorting, and classifying them; drawing and constructing physical models to specifications; identifying their properties (e.g., number of sides or faces, two- or three-dimensionality, equal sides, number of right angles); predicting the results of combining or subdividing two-dimensional figures; explaining how these figures are related to objects in the environment.

C.4.2. Use physical materials and motion geometry (such as slides, flips, and turns) to identify properties and relationships, including but not limited to symmetry; congruence; similarity.

C.4.3. Identify and use relationships among figures, including but not limited to location (e.g., between, adjacent to, interior of); position (e.g., parallel, perpendicular); intersection (of two-dimensional figures).

C.4.4. Use simple two-dimensional coordinate systems to find locations on maps and to represent points and simple figures.

WI.D. Measurement: Students in Wisconsin will select and use appropriate tools (including technology) and techniques to measure things to a specified degree of accuracy. They will use measurements in problem-solving situations.

D.4.1. Recognize and describe measurable attributes, such as length, liquid capacity, time, weight (mass), temperature, volume, monetary value, and angle size, and identify the appropriate units to measure them.

D.4.2. Demonstrate understanding of basic facts, principles, and techniques of measurement, including appropriate use of arbitrary and standard units (metric and US Customary); appropriate use and conversion of units within a system (such as yards, feet, and inches; kilograms and grams; gallons, quarts, pints, and cups); judging the reasonableness of an obtained measurement as it relates to prior experience and familiar benchmarks.

D.4.3. Read and interpret measuring instruments (e.g., rulers, clocks, thermometers).

D.4.4. Determine measurements directly by using standard tools to these suggested degrees of accuracy length to the nearest half-inch or nearest cm; weight (mass) to the nearest ounce or nearest 5 grams; temperature to the nearest 5; time to the nearest minute; monetary value to dollars and cents; liquid capacity to the nearest fluid ounce.

D.4.5. Determine measurements by using basic relationships (such as perimeter and area) and approximate measurements by using estimation techniques.

WI.E. Statistics and Probability: Students in Wisconsin will use data collection and analysis, statistics and probability in problem-solving situations, employing technology where appropriate.

E.4.1. Work with data in the context of real-world situations by formulating questions that lead to data collection and analysis; determining what data to collect and when and how to collect them; collecting, organizing, and displaying data; drawing reasonable conclusions based on data.

E.4.2. Describe a set of data using high and low values, and range; most frequent value (mode); middle value of a set of ordered data (median).

E.4.3. In problem-solving situations, read, extract, and use information presented in graphs, tables, or charts.

E.4.4. Determine if future events are more, less, or equally likely, impossible, or certain to occur.

E.4.5. Predict outcomes of future events and test predictions using data from a variety of sources.

WI.F. Algebraic Relationships: Students in Wisconsin will discover, describe, and generalize simple and complex patterns and relationships. In the context of real-world problem situations, the student will use algebraic techniques to define and describe the problem to determine and justify appropriate solutions.

F.4.1. Use letters, boxes, or other symbols to stand for any number, measured quantity, or object in simple situations (e.g., N + 0 = N is true for any number).

F.4.2. Use the vocabulary, symbols, and notation of algebra accurately (e.g., correct use of the symbol '='; effective use of the associative property of multiplication).

F.4.3. Work with simple linear patterns and relationships in a variety of ways, including recognizing and extending number patterns; describing them verbally; representing them with pictures, tables, charts, graphs; recognizing that different models can represent the same pattern or relationship; using them to describe real-world phenomena.

F.4.4. Recognize variability in simple functional relationships by describing how a change in one quantity can produce a change in another (e.g., number of bicycles and the total number of wheels).

F.4.5. Use simple equations and inequalities in a variety of ways, including using them to represent problem situations; solving them by different methods (e.g., use of manipulatives, guess and check strategies, recall of number facts); recording and describing solution strategies.

F.4.6. Recognize and use generalized properties and relationships of arithmetic (e.g., commutativity of addition, inverse relationship of multiplication and division).

WI.A. Mathematical Processes: Students in Wisconsin will draw on a broad body of mathematical knowledge and apply a variety of mathematical skills and strategies, including reasoning, oral and written communication, and the use of appropriate technology, when solving mathematical, real-world and non-routine problems.

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.

WI.B. Number Operations and Relationships: Students in Wisconsin will use numbers effectively for various purposes, such as counting, measuring, estimating, and problem solving.

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.8.3. Generate and explain equivalencies among fractions, decimals, and percents.

B.8.4. Express order relationships among rational numbers using appropriate symbols (>, <).

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.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).

WI.C. Geometry: Students in Wisconsin will be able to use geometric concepts, relationships and procedures to interpret, represent, and solve problems.

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.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.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.

WI.D. Measurement: Students in Wisconsin will select and use appropriate tools (including technology) and techniques to measure things to a specified degree of accuracy. They will use measurements in problem-solving situations.

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).

WI.E. Statistics and Probability: Students in Wisconsin will use data collection and analysis, statistics and probability in problem-solving situations, employing technology where appropriate.

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).

WI.F. Algebraic Relationships: Students in Wisconsin will discover, describe, and generalize simple and complex patterns and relationships. In the context of real-world problem situations, the student will use algebraic techniques to define and describe the problem to determine and justify appropriate solutions.

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.

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.

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).

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.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.

WI.A. Mathematical Processes: Students in Wisconsin will draw on a broad body of mathematical knowledge and apply a variety of mathematical skills and strategies, including reasoning, oral and written communication, and the use of appropriate technology, when solving mathematical, real-world and non-routine problems.

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.

WI.B. Number Operations and Relationships: Students in Wisconsin will use numbers effectively for various purposes, such as counting, measuring, estimating, and problem solving.

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.8.3. Generate and explain equivalencies among fractions, decimals, and percents.

B.8.4. Express order relationships among rational numbers using appropriate symbols (>, <).

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.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).

WI.C. Geometry: Students in Wisconsin will be able to use geometric concepts, relationships and procedures to interpret, represent, and solve problems.

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.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.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.

WI.D. Measurement: Students in Wisconsin will select and use appropriate tools (including technology) and techniques to measure things to a specified degree of accuracy. They will use measurements in problem-solving situations.

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).

WI.E. Statistics and Probability: Students in Wisconsin will use data collection and analysis, statistics and probability in problem-solving situations, employing technology where appropriate.

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).

WI.F. Algebraic Relationships: Students in Wisconsin will discover, describe, and generalize simple and complex patterns and relationships. In the context of real-world problem situations, the student will use algebraic techniques to define and describe the problem to determine and justify appropriate solutions.

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.

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.

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).

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.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.

WI.A. Mathematical Processes: Students in Wisconsin will draw on a broad body of mathematical knowledge and apply a variety of mathematical skills and strategies, including reasoning, oral and written communication, and the use of appropriate technology, when solving mathematical, real-world and non-routine problems.

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.

WI.B. Number Operations and Relationships: Students in Wisconsin will use numbers effectively for various purposes, such as counting, measuring, estimating, and problem solving.

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.8.3. Generate and explain equivalencies among fractions, decimals, and percents.

B.8.4. Express order relationships among rational numbers using appropriate symbols (>, <).

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.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).

WI.C. Geometry: Students in Wisconsin will be able to use geometric concepts, relationships and procedures to interpret, represent, and solve problems.

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.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.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.

WI.D. Measurement: Students in Wisconsin will select and use appropriate tools (including technology) and techniques to measure things to a specified degree of accuracy. They will use measurements in problem-solving situations.

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).

WI.E. Statistics and Probability: Students in Wisconsin will use data collection and analysis, statistics and probability in problem-solving situations, employing technology where appropriate.

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).

WI.F. Algebraic Relationships: Students in Wisconsin will discover, describe, and generalize simple and complex patterns and relationships. In the context of real-world problem situations, the student will use algebraic techniques to define and describe the problem to determine and justify appropriate solutions.

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.

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.

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).

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.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.

WI.A. Mathematical Processes: Students in Wisconsin will draw on a broad body of mathematical knowledge and apply a variety of mathematical skills and strategies, including reasoning, oral and written communication, and the use of appropriate technology, when solving mathematical, real-world and non-routine problems.

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.

WI.B. Number Operations and Relationships: Students in Wisconsin will use numbers effectively for various purposes, such as counting, measuring, estimating, and problem solving.

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.8.3. Generate and explain equivalencies among fractions, decimals, and percents.

B.8.4. Express order relationships among rational numbers using appropriate symbols (>, <).

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.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).

WI.C. Geometry: Students in Wisconsin will be able to use geometric concepts, relationships and procedures to interpret, represent, and solve problems.

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.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.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.

WI.D. Measurement: Students in Wisconsin will select and use appropriate tools (including technology) and techniques to measure things to a specified degree of accuracy. They will use measurements in problem-solving situations.

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).

WI.E. Statistics and Probability: Students in Wisconsin will use data collection and analysis, statistics and probability in problem-solving situations, employing technology where appropriate.

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).

WI.F. Algebraic Relationships: Students in Wisconsin will discover, describe, and generalize simple and complex patterns and relationships. In the context of real-world problem situations, the student will use algebraic techniques to define and describe the problem to determine and justify appropriate solutions.

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.

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.

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).

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.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.

WI.A. Mathematical Processes: Students in Wisconsin will draw on a broad body of mathematical knowledge and apply a variety of mathematical skills and strategies, including reasoning, oral and written communication, and the use of appropriate technology, when solving mathematical, real-world and non-routine problems.

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.

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.

WI.B. Number Operations and Relationships: Students in Wisconsin will use numbers effectively for various purposes, such as counting, measuring, estimating, and problem solving.

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.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).

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).

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.

WI.C. Geometry: Students in Wisconsin will be able to use geometric concepts, relationships and procedures to interpret, represent, and solve problems.

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.12.2. Use geometric models to solve mathematical and real-world problems.

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.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.

C.12.5. Identify and demonstrate an understanding of the three ratios used in right-triangle trigonometry (sine, cosine, tangent).

WI.D. Measurement: Students in Wisconsin will select and use appropriate tools (including technology) and techniques to measure things to a specified degree of accuracy. They will use measurements in problem-solving situations.

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.

WI.E. Statistics and Probability: Students in Wisconsin will use data collection and analysis, statistics and probability in problem-solving situations, employing technology where appropriate.

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.

WI.F. Algebraic Relationships: Students in Wisconsin will discover, describe, and generalize simple and complex patterns and relationships. In the context of real-world problem situations, the student will use algebraic techniques to define and describe the problem to determine and justify appropriate solutions.

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 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).

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.

F.12.4. Model and solve a variety of mathematical and real-world problems by using algebraic expressions, equations, and inequalities.

WI.A. Mathematical Processes: Students in Wisconsin will draw on a broad body of mathematical knowledge and apply a variety of mathematical skills and strategies, including reasoning, oral and written communication, and the use of appropriate technology, when solving mathematical, real-world and non-routine problems.

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.

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.

WI.B. Number Operations and Relationships: Students in Wisconsin will use numbers effectively for various purposes, such as counting, measuring, estimating, and problem solving.

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.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).

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).

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.

WI.C. Geometry: Students in Wisconsin will be able to use geometric concepts, relationships and procedures to interpret, represent, and solve problems.

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.12.2. Use geometric models to solve mathematical and real-world problems.

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.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.

C.12.5. Identify and demonstrate an understanding of the three ratios used in right-triangle trigonometry (sine, cosine, tangent).

WI.D. Measurement: Students in Wisconsin will select and use appropriate tools (including technology) and techniques to measure things to a specified degree of accuracy. They will use measurements in problem-solving situations.

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.

WI.E. Statistics and Probability: Students in Wisconsin will use data collection and analysis, statistics and probability in problem-solving situations, employing technology where appropriate.

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.

WI.F. Algebraic Relationships: Students in Wisconsin will discover, describe, and generalize simple and complex patterns and relationships. In the context of real-world problem situations, the student will use algebraic techniques to define and describe the problem to determine and justify appropriate solutions.

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 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).

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.

F.12.4. Model and solve a variety of mathematical and real-world problems by using algebraic expressions, equations, and inequalities.

WI.A. Mathematical Processes: Students in Wisconsin will draw on a broad body of mathematical knowledge and apply a variety of mathematical skills and strategies, including reasoning, oral and written communication, and the use of appropriate technology, when solving mathematical, real-world and non-routine problems.

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.

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.

WI.B. Number Operations and Relationships: Students in Wisconsin will use numbers effectively for various purposes, such as counting, measuring, estimating, and problem solving.

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.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).

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).

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.

WI.C. Geometry: Students in Wisconsin will be able to use geometric concepts, relationships and procedures to interpret, represent, and solve problems.

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.12.2. Use geometric models to solve mathematical and real-world problems.

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.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.

C.12.5. Identify and demonstrate an understanding of the three ratios used in right-triangle trigonometry (sine, cosine, tangent).

WI.D. Measurement: Students in Wisconsin will select and use appropriate tools (including technology) and techniques to measure things to a specified degree of accuracy. They will use measurements in problem-solving situations.

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.

WI.E. Statistics and Probability: Students in Wisconsin will use data collection and analysis, statistics and probability in problem-solving situations, employing technology where appropriate.

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.

WI.F. Algebraic Relationships: Students in Wisconsin will discover, describe, and generalize simple and complex patterns and relationships. In the context of real-world problem situations, the student will use algebraic techniques to define and describe the problem to determine and justify appropriate solutions.

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 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).

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.

F.12.4. Model and solve a variety of mathematical and real-world problems by using algebraic expressions, equations, and inequalities.

WI.A. Mathematical Processes: Students in Wisconsin will draw on a broad body of mathematical knowledge and apply a variety of mathematical skills and strategies, including reasoning, oral and written communication, and the use of appropriate technology, when solving mathematical, real-world and non-routine problems.

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.

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.

WI.B. Number Operations and Relationships: Students in Wisconsin will use numbers effectively for various purposes, such as counting, measuring, estimating, and problem solving.

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.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).

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).

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.

WI.C. Geometry: Students in Wisconsin will be able to use geometric concepts, relationships and procedures to interpret, represent, and solve problems.

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.12.2. Use geometric models to solve mathematical and real-world problems.

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.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.

C.12.5. Identify and demonstrate an understanding of the three ratios used in right-triangle trigonometry (sine, cosine, tangent).

WI.D. Measurement: Students in Wisconsin will select and use appropriate tools (including technology) and techniques to measure things to a specified degree of accuracy. They will use measurements in problem-solving situations.

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.

WI.E. Statistics and Probability: Students in Wisconsin will use data collection and analysis, statistics and probability in problem-solving situations, employing technology where appropriate.

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.

WI.F. Algebraic Relationships: Students in Wisconsin will discover, describe, and generalize simple and complex patterns and relationships. In the context of real-world problem situations, the student will use algebraic techniques to define and describe the problem to determine and justify appropriate solutions.

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 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).

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.

F.12.4. Model and solve a variety of mathematical and real-world problems by using algebraic expressions, equations, and inequalities.