Mathematics Department Standards

1. The student will be able to understand numbers, ways of representing numbers, relationships among numbers, and number systems.

2. The student will be able to understand meanings of operations and how they relate to one another.

3. The student will be able to compute fluently and make reasonable estimates.

4. The student will be able to understand patterns, relations, and functions.

5. The student will be able to represent and analyze mathematical situations and structures using algebraic symbols.

6. The student will be able to use mathematical models to understand and represent quantitative relationships.

7. The student will be able to analyze change in various contexts.

8. The student will be able to analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.

9. The student will be able to specify locations and describe spatial relationships using coordinate geometry and other representational systems.

10. The student will be able to apply transformations and use symmetry to analyze mathematical situations.

11. The student will be able to use visualization, spatial reasoning, and geometric modeling to solve problems.

12. The student will be able to understand measurable attributes of objects and the units, systems, and processes of measurement.

13. The student will be able to apply appropriate techniques, tools, and formulas to determine measurements.

14. The student will be able to formulate questions that can be addressed with data. The student will also be able to collect, organize, and display relevant data to answer them.

15. The student will be able to select and use appropriate statistical methods to analyze data.

16. The student will be able to develop and evaluate inferences and predictions that are based on data.

Standard #1

The student will be able to understand numbers, ways of representing numbers, relationships among numbers, and number systems.

Benchmarks

Level I Pre - Kindergarten - Second Grade

1. Counts with understanding and recognizes "how many" in sets of objects.

2. Uses multiple models to develop initial understandings of place value and the base-ten number system.

3. Develops understanding of the relative position and magnitude of whole numbers and of ordinal/cardinal numbers and their connections.

4. Develops a sense of whole numbers and represents and uses them in flexible ways (e.g. relating, composing, decomposing numbers).

5. Connects number words and numerals to the quantities they represent, using various physical models and representations.

6. Understands and represents commonly used fractions (e.g. 1/4, 1/3, 1/2).

Level II Third - Fifth Grade

1. Understands the place-value structure of the base-ten number system and is able to represent and compare whole numbers and decimals.

2. Recognizes equivalent representations for the same number and generates them by decomposing and composing numbers.

3. Develops an understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and as divisions of whole numbers.

4. Uses models, benchmarks, and equivalent forms to judge the size of fractions.

5. Recognizes and generates equivalent forms of commonly used fractions, decimals, and percents.

6. Explores numbers less than zero by extending the number line and through other familiar applications.

7. Describes classes of numbers according to characteristics such as the nature of their factors.

Level III Sixth - Eighth Grade

1. Works flexibly with fractions, decimals, and percents to solve problems.

2. Compares and orders fractions, decimals, and percents efficiently and finds their approximate location on a number line.

3. Develops meaning for percents greater than one hundred (100) and less than one (1).

4. Understands and uses ratios and proportions to represent quantitative relationships.

5. Develops an understanding of large numbers, recognizing and appropriately using exponential, scientific, and calculator notation.

6. Uses factors, multiples, prime factorization, and relatively prime numbers to solve problems.

7. Develops meaning for integers and represents and compares quantities with them.

Level IV Ninth - Twelfth Grade

1. Develops a deeper understanding of very large and very small numbers and their various representations.

2. Compares and contrasts the properties of numbers and number systems, (e.g. rational, real numbers) and understands complex numbers as solutions to quadratic equations that do not have real solutions.

3. Understands matrices as systems that have some of the properties of the real-number system.

4. Uses number-theory arguments to justify relationships involving whole numbers.

Standard #2

The student will be able to understand meanings of operations and how they relate to one another.

Benchmarks

Level I Pre - Kindergarten - Second Grade

1. Understands various meanings of addition and subtraction of whole numbers and the relationship between the two operations.

2. Understands the effects of adding and subtracting whole numbers.

3. Understands situations that entail multiplication and division, such as equal groupings of objects and sharing equally.

Level II Third - Fifth Grade

1. Understands various meanings of multiplication and division.

2. Understands the effects of multiplying and dividing whole numbers.

3. Identifies and uses relationships between operations, such as division as the inverse of multiplication, to solve problems.

4. Understands and uses properties of operations, such as the distributivity of multiplication over addition.

Level III Sixth - Eighth Grade

1. Understands the meaning and effects of arithmetic operations with fractions, decimals, and integers.

2. Uses the associative and commutative properties of addition and multiplication and the distributive property of multiplication over addition to simplify computations with integers, fractions, and decimals.

3. Understands and uses inverse relationships (e.g. addition and subtraction, multiplication and division, squaring and finding square roots) to simplify computations and solve problems.

Level IV Ninth - Twelfth Grade

1. Judges the effects of such operations as multiplication, division, and computing powers and roots on the magnitudes of quantities.

2. Develops an understanding of properties of ,and representations for, the addition and multiplication of matrices

Standard #3

The student will be able to compute fluently and make reasonable estimates.

Benchmarks

Level I Pre - Kindergarten - Second Grade

1. Develops and uses strategies for whole-number computations; with a focus on addition and subtraction.

2. Develops fluency with basic number combinations for addition and subtraction.

3. Uses a variety of methods and tools to compute (e.g. objects, mental computation, estimation, paper-and-pencil, calculators).

Level II Third - Fifth Grade

1. Develops fluency with basic number combinations for multiplication and division and uses these combinations to mentally compute related problems, such as 30 x 50.

2. Develops fluency in adding, subtracting, multiplying, and dividing whole numbers.

3. Develops and uses strategies to estimate the results of whole-number computations and to judge the rationality of such results.

4. Develops and uses strategies to estimate computations involving fractions and decimals in situations relevant to students' experience.

5. Uses visual models, benchmarks, and equivalent forms to add and subtract commonly used fractions and decimals.

6. Selects and uses appropriate methods/tools for computing with whole numbers (e.g. mental computation, estimation, calculators, and paper-and-pencil) according to the context and nature of the computation.

Level III Sixth - Eighth Grade

1. Selects and applies appropriate methods and tools for computing with fractions and decimals (e.g. mental computation, estimation, calculators/computers, paper-and-pencil) depending on the situation.

2. Develops and analyzes algorithms for computing with fractions, decimals, and integers developing fluency in their use.

3. Develops and uses strategies to estimate the results of rational-number computations and judges the rationality of the results.

4. Develops, analyzes, and explains methods for solving problems involving proportions (e.g. scaling, finding equivalent ratios).

Level IV Ninth - Twelfth Grade

1. Develops fluency in operations with real numbers and matrices (e.g. using mental computation or paper-and-pencil calculations for simple cases, technology for more complicated cases).

2. Judges the reasonableness of numerical computations and their results.

Standard #4

The student will be able to understand patterns, relations, and functions.

Benchmarks

Level I Pre - Kindergarten - Second Grade

1. Sorts, classifies, and orders objects by size, number, and other properties.

2. Recognizes, describes, and extends patterns (e.g. sequences of sounds and shapes, simple numeric patterns) translating them from one representation to another.

3. Analyzes how both repeating and growing patterns are generated.

Level II Third - Fifth Grade

1. Describes, extends, and makes generalizations about geometric and numeric patterns.

2. Represents and analyzes patterns and functions; using words, tables, and graphs.

Level III Sixth - Eighth Grade

1. Represents, analyzes, and generalizes a variety of patterns; using tables, graphs, words, and, when possible, symbolic rules.

2. Relates and compares different forms of representation for a relationship.

3. Identifies functions as linear/nonlinear and contrasts their properties from tables, graphs, or equations.

Level IV Ninth - Twelfth Grade

1. Generalizes patterns using explicitly and recursively defined functions.

2. Understands, selects, converts flexibly among, and uses various representations of relations and functions.

3. Analyzes functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local/global behavior.

4. Understands and performs transformations by arithmetically combining, composing, and inverting commonly used functions; or using technology to perform more complicated symbolic expressions.

5. Understands and compares the properties of classes of functions (e.g. exponential, polynomial, rational, logarithmic, periodic functions).

6. Interprets the representations of functions of two variables.

Standard #5

The student will be able to represent and analyze mathematical situations and structures using algebraic symbols.

Benchmarks

Level I Pre - Kindergarten - Second Grade

1. Illustrates general principles and properties of operations (e.g. commutativity) using specific numbers.

2. Uses concrete, pictorial, and verbal representations to develop an understanding of invented and conventional symbolic notations.

Level II Third - Fifth Grade

1. Identifies such properties as commutativity, associativity, and distributivity and uses them to compute with whole numbers.

2. Represents the idea of a variable as an unknown quantity using a letter or symbol.

3. Expresses mathematical relationships using equations.

Level III Sixth - Eighth Grade

1. Develops an initial conceptual understanding of different uses of variables.

2. Explores relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope.

3. Uses symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships.

4. Recognizes and generates equivalent forms for simple algebraic expressions and solves linear equations.

Level IV Ninth - Twelfth Grade

1. Understands the meaning of equivalent forms of expressions, equations, inequalities, and relations. The student also writes equivalent forms of equations, inequalities, and systems of equations and solves them with fluency (e.g. mentally or with paper-and-pencil in simple cases, using technology in other cases).

2. Uses symbolic algebra to represent and explain mathematical relationships.

3. Uses a variety of symbolic representations (e.g. equations) for functions and relations.

4. Judges the meaning, utility, and reasonableness of the results of symbol manipulations; including those carried out by technology.

Standard #6

The student will be able to use mathematical models to understand and represent quantitative relationships.

Benchmarks

Level I Pre - Kindergarten - Second Grade

1. Models situations that involve the addition and subtraction of whole numbers with objects, pictures, and symbols.

Level II Third - Fifth Grade

1. Models problem situations with objects and uses representations (e.g. graphs, tables, and equations) to draw conclusions.

Level III Sixth - Eighth Grade

1. Models and solves contextualized problems using various representations (e.g. graphs, tables, equations).

Level IV Ninth - Twelfth Grade

1. Identifies essential quantitative relationships in a situation and determines the class/classes of functions that might model the relationships.

2. Uses symbolic expressions (e.g. iterative and recursive forms) to represent relationships arising from various contexts.

3. Draws reasonable conclusions about a situation being modeled.

Standard #7

The student will be able to