Mathematics: Numeration and Number Theory

• Standard 1: Problem Solving
• Standard 2: Communication
• Standard 3: Reasoning
• Standard 4: Connections
• Standard 5: Estimation & Computation
• Standard 7: Geometry
• Standard 8: Measurement
• Standard 9: Statistics
• Standard 10: Probability
• Standard 11: Algebra
• Standard 12: Patterns, Relations, Functions
• Introduction to Mathematics
• Table of Contents - All Standards

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Content Standard: Numeration and Number Theory

Students will select and apply appropriate numerical representations and theory to solve problems in real world situations using a variety of mathematical and technological strategies.

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Fourth Grade Performance Standards: Numeration and Number Theory

By the end of grade Four, students will:

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)

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

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

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

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

Add and subtract fractions with like denominators

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

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Eighth Grade Performance Standards: Numeration and Number Theory

By the end of grade Eight, students will:

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)

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)

Generate and explain equivalencies among fractions, decimals, and percents

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

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)

Model and solve problems involving number-theory concepts such as

• prime and composite numbers
• divisibility and remainders
• greatest common factors
• least common multiples

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)

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Grade Level Performance Standards: Grade Six: Numeration and Number Theory

The student will:

Demonstrate relationships between fractions, decimals and percents.

Demonstrate the relevance of fractions and decimals to daily life.

Explain the attributes of prime and composite numbers.

Put numbers in categories such as prime, composite, square and triangular; justify choice of categories.

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Grade Level Performance Standards: Grade Seven: Numeration and Number Theory

The student will:

Compare and order positive and negative rational numbers.

Demonstrate and discuss the relationships among percents, fractions and decimals.

Represent fractions, decimals and percents in a variety of meaningful situations using concrete and abstract representations.

Represent and compare the size of large and small numbers using exponential and scientific notation.

Use relationships among ratio, percent and proportion to solve problems.

Use prime factorization in mathematical problem situations.

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Grade Level Performance Standards: Grade Eight: Numeration and Number Theory

The student will:

Demonstrate the hierarchy of the real number system using a Venn diagram or set notation.

Compare and order rational and irrational numbers on number lines.

Use exponential and scientific notation to solve problems.

Explain and use properties of numbers (e.g., divisibility, prime factorization, commutativity, associativity, distributivity).

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Twelfth Grade Performance Standards: Numeration and Number Theory

By the end of grade Twelve, students will:

Use complex counting procedures such as union and intersection of sets and arrangements (permutations and combinations) to solve problems

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

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)

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)

Create and critically evaluate numerical arguments presented in a variety of classroom and real-world situations (e.g., political, economic, scientific, social)

Routinely assess the acceptable limits of error when

• evaluating strategies
• testing the reasonableness of results
• using technology to carry out computations

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