- 1. Squares by Sight
Define a square number geometrically as the area of a square with an integer side length, generate square numbers from 1x1 to 10x10, and recognize perfect squares.
- 2. Extending Squares to 15² and Digit Patterns
Recall square numbers from $11^2$ to $15^2$, identify the last-digit pattern of square numbers, and use this pattern to quickly check if a large number could be a perfect square.
- 3. The Language of Powers
Correctly name the base, exponent, and power in an expression like $2^3 = 8$; read and write expressions using "to the power of", "squared", and "cubed"; and distinguish between the exponent and the power.
- 4. Expanded Form and Index Form
Convert fluently between repeated multiplication (expanded form) and index notation in both directions, including mixed-base expressions.
- 5. Cube Numbers — Seeing in 3D
Define a cube number geometrically as the volume of a cube with an integer edge length, generate and recall all 10 cube numbers ($1^3$ to $10^3$), and distinguish cube numbers from square numbers.
- 6. Evaluating Any Power
Evaluate any power with a positive integer base by systematically expanding to repeated multiplication and computing step by step; fill in blanks when the exponent or the base is unknown.
- 7. Comparing Sizes of Powers
Determine which of two power expressions is larger without fully evaluating both, by reasoning about the effect of increasing the base or the exponent.
- 8. The Multiplication Rule for Indices — Discovery
State and apply the rule $a^m \times a^n = a^{m+n}$ for numerical bases; explain why the rule works by expanding powers; and identify expressions where the rule applies versus where it does not.
- 9. Applying the Multiplication Rule — Same Base Only
Simplify products of powers with the same base, handle implicit exponents of 1, and identify when the multiplication rule does not apply.
- 10. The Division Rule for Indices — Discovery
State and apply the rule $a^m \div a^n = a^{m-n}$ for numerical bases; explain it through cancellation in expanded form; and begin to notice what happens when the denominator's exponent is larger.
- 11. Mixed Index Laws and Chained Operations
Apply both the multiplication and division rules for indices in the same expression; simplify chained products and quotients; and discover the zero exponent rule.
- 12. Square Roots — Undoing the Square
Evaluate the square root of any perfect square up to 225 by recognizing it as the inverse of squaring; write paired facts; and use the operator diagram to show the relationship.
- 13. Cube Roots — Undoing the Cube
Evaluate the cube root of any perfect cube from $1^3$ to $10^3$; write paired cube and cube-root facts; and correctly use the notation $\sqrt[3]{x}$.
- 14. Mixed Roots and Estimation
Evaluate square and cube roots of perfect powers fluently; determine the index of a root from context; and bracket non-perfect roots between consecutive integers.
- 15. Higher-Order Roots — The nth Root as the Inverse of the nth Power
Evaluate 4th, 5th, and 6th roots of simple perfect powers by reversing the power; correctly use the notation $\sqrt[4]{x}$, $\sqrt[5]{x}$, $\sqrt[6]{x}$; and state the inverse relationship $\sqrt[n]{a^n} = a$.
- 16. Areas, Volumes and the Square-Cube Link
Apply squaring to find the area of a square, and square roots to find its side; apply cubing to find the volume of a cube, and cube roots to find its edge; and connect geometric units to their exponents.
- 17. Powers and Roots of Multiples of 10
Evaluate powers like $40^2$, $200^3$, $0.2^2$ and roots like $\sqrt{900}$, $\sqrt[3]{512,000}$, $\sqrt{0.0016}$ by decomposing the number into a product involving a power of 10.
- 18. Prime Numbers — Definition and the Hundred Square
Define a prime number precisely; identify all primes up to 100 using the hundred square; explain why 1 is not prime, why 2 is the only even prime, and why a composite number is not prime.
- 19. Prime Numbers — Recall, Divisibility Tests and Reasoning
Recall primes less than 100; apply divisibility tests (2, 3, 5, 7) to determine if a number is composite; and solve reasoning puzzles involving primes, squares, and cubes.
- 20. Prime Factorisation — Composing and Decomposing
Express any positive integer as a product of prime factors using a factor tree; write the result using index notation; and state that every integer has a unique prime factorisation.
- 21. Using Prime Factorisations — Factors, Factor Pairs and HCF
Use a prime factorisation to identify factors of a number, and find the Highest Common Factor (HCF) of two numbers by comparing their prime factorisations.
- 22. Unit Synthesis — Powers, Roots and Primes Connected
Apply all concepts from this unit in connected, multi-step problems; recognize how index laws, roots, and prime factorisations interlock to solve complex reasoning puzzles.
