Rational Numbers: Structure, Comparison, and Absolute Value

Identify prime, composite, and square numbers, apply comprehensive divisibility rules, and find prime factorizations using exponents.

Find the greatest common factor and least common multiple using prime powers and apply the distributive property to factor numerical expressions.

Execute the standard algorithm for multi-digit whole number division and interpret remainders in diverse real-world contexts.

Represent, compare, and order positive and negative integers on horizontal and vertical lines, and interpret absolute value as magnitude in context.

Add and subtract integers fluently using counter models, number lines, and algebraic rules, and solve real-world multi-step problems.

Multiply and divide integers fluently using patterns, models, and sign rules, and evaluate multi-step numerical expressions.

Define rational numbers as quotients of integers, plot mixed numbers and finite decimals on number lines, compare, and order them.

Add, subtract, multiply, and divide multi-digit decimals fluently, calculate with powers of 10, and estimate to verify reasonableness.

Add and subtract proper fractions, improper fractions, and mixed numbers fluently with like and unlike denominators, including signed fractions.

Multiply and divide fractions and mixed numbers, understand scaling effects, and apply reciprocal division algorithms.

Evaluate integer and whole-number powers, simplify expressions using exponent product and quotient laws, and evaluate zero/negative exponents.

Calculate square and cube roots of perfect powers, evaluate roots of negative integers and fractions, and solve geometric problems.

Read, write, and compare very large and very small quantities in standard and scientific notation (a · 10^k).

Synthesize multi-step arithmetic across fractions, decimals, integers, and powers to solve complex real-world modeling problems.