- 1. The Three Families & Commutativity
Students will categorize mathematical operations into three distinct families, determine which operations are commutative (where order does not matter), and identify equivalent expressions based on their structural properties.
- 2. Subtraction Structure (Minuends & Subtrahends)
Students will identify minuends and subtrahends in subtraction expressions and manipulate their order to create equivalent calculations without violating structural rules.
- 3. Division Structure & Fraction Notation
Students will identify dividends and divisors, apply the swapping rule for multiple divisors, and use fraction notation to prove that dividing by two consecutive numbers is equivalent to dividing by their product.
- 4. Mixing Multiplication and Addition
Students will use area models to visually prove that multiplication must be evaluated before addition, and accurately calculate mixed arithmetic expressions containing both operation families.
- 5. Mixing the Four Basic Operations
Students will extend the hierarchy of operations to include division and subtraction, accurately evaluating mixed expressions by calculating the Multiplication Family before the Addition Family.
- 6. Integrating Exponents (The Top of the Hierarchy)
Students will understand that exponents represent repeated multiplication and must therefore be evaluated before standard multiplication, division, addition, or subtraction. Students will evaluate expressions using the complete three-tier hierarchy of operations.
- 7. Brackets – Strategically Grouping to Change the Order
Students will use brackets to group operations so that addition or subtraction happens before multiplication or division when needed, understanding when grouping changes the answer.
- 8. Implied Multiplication and Decimals
Students will recognize and evaluate implied multiplication using bracket notation, and confidently apply the order of operations to expressions containing decimal numbers.
- 9. Hidden Brackets – Invisible Grouping in Roots, Fractions and Exponents
Students will recognize when fraction bars, radical symbols, and exponent notation act as hidden brackets. Students will correctly evaluate these expressions by performing the grouped operations first.
- 10. Multiple Bracket Placements
Students will strategically place brackets in a static arithmetic expression to alter the order of operations and achieve different target numbers.
- 11. Missing Operations, Missing Numbers & the Zero Principle
Students will find the missing operations or missing numbers that make an equation true. They will work backwards using the order of operations and apply the zero product principle to solve quickly when the answer is zero.
- 12. Games, Puzzles and Real Applications
Students will use the full order of operations — including visible brackets, hidden brackets and the default hierarchy — to play games, solve rich puzzles and evaluate real mathematical formulas.
