Grade 9: Number

Warm up with a few quick number warm-ups on the board while the class settles, so every child starts thinking about the skill.

Exponent (index) laws are shortcuts for multiplying, dividing and raising powers with the same base. Irrational numbers are numbers, like most square roots and pi, that cannot be written as an exact fraction and whose decimals never terminate or repeat. Model the method clearly, thinking aloud:

• Start with the multiplication law: a^m x a^n = a^(m+n), showing why by writing out the repeated multiplication in full for small numbers.
• Introduce the division law (subtract exponents) and the power-of-a-power law (multiply exponents) the same way, always checking with a full expansion first.
• Teach square numbers and roots before irrational numbers, so 'rational' (a perfect square's root) has a clear contrast with 'irrational' (any other root).
• Classify a mix of numbers (fractions, decimals, square roots, pi) as rational or irrational, justifying each with the fraction/decimal test.
• Keep base numbers small and positive until the three exponent laws are automatic, then introduce zero and negative exponents.

Work this through step by step on the board, then have the class talk you through a second one.

• Simplify 3^4 x 3^2
• Add the exponents (same base): 3^(4+2) = 3^6
• Is sqrt(9) rational or irrational? sqrt(9) = 3, a whole number, so rational
• Is sqrt(7) rational or irrational? 7 is not a perfect square, so irrational

Do the first few questions of the practice worksheet together, one child explaining each step. Check for understanding before releasing the class to work alone.

Students complete the practice worksheet independently while you circulate and support.

Circulate and look for these, they are the usual sticking points:

• Multiplying the exponents instead of adding them for a^m x a^n (that rule is for (a^m)^n).
• Assuming every square root is irrational, forgetting perfect squares like sqrt(16) = 4 are whole numbers.
• Believing a long decimal is automatically irrational, when a long REPEATING decimal is still rational.
• Adding the bases instead of keeping the base the same and combining only the exponents.

Pull the class back together. Ask one child to explain number in their own words, pose a single check question everyone answers on a mini whiteboard, and name what you will build on next lesson.

Use the independent worksheet as the evidence. A child who can complete it accurately and explain one answer has met the objective; anyone who cannot needs the easier level and a short reteach next session.