Year 9 Mathematics Lesson Plans

45-minute lesson

Year 9 Mathematics: Index laws

Lesson objective

- Write repeated multiplication using index notation. - Apply the multiplication, division and power-of-a-power index laws. - Understand why any non-zero number to the power of zero equals 1. Success criteria: - I can identify the base and the index in an expression like 5^3. - I can simplify expressions such as 3x^4 × 5x^2 and 20a^7 ÷ 4a^3. - I can simplify a power of a power, such as (2m^3)^4. - I can evaluate expressions that include a zero index.

Materials

- Paper and pencil - A calculator (optional, for checking) - A sheet of paper to fold

Introduction

5 min

Introduce today's words: - Base: The number or pronumeral being multiplied repeatedly. In 2^5, the base is 2. - Index (power, exponent): The small raised number that shows how many times the base is used as a factor. In 2^5, the index is 5. - Index notation: A short way of writing repeated multiplication, such as writing 3 × 3 × 3 × 3 as 3^4. - Coefficient: The number multiplying a pronumeral. In 7x^2, the coefficient is 7. - Zero index: Any non-zero base raised to the power of 0 equals 1. Ask your child what they already know about index laws.

Explanation

9 min

Index notation is a shorthand for repeated multiplication. Instead of writing 2 × 2 × 2 × 2 × 2, we write 2^5 (read as "2 to the power of 5"). The 2 is the base and the 5 is the index. On this site we write powers with the ^ symbol, so x^3 means x × x × x. Multiplying with the same base: add the indices. For example, 2^3 × 2^4 means (2 × 2 × 2) × (2 × 2 × 2 × 2), which is seven 2s multiplied together, so 2^3 × 2^4 = 2^7. In general, a^m × a^n = a^(m + n). Dividing with the same base: subtract the indices. For example, 5^6 ÷ 5^2: six 5s on top and two 5s underneath; two pairs cancel, leaving four 5s, so the answer is 5^4. In general, a^m ÷ a^n = a^(m − n). Power of a power: multiply the indices. (x^2)^3 means x^2 × x^2 × x^2, which is x^6. In general, (a^m)^n = a^(m × n). When a bracket contains a product, every factor inside is raised to the power, so (3a)^2 = 3^2 × a^2 = 9a^2. The zero index: look at the pattern 2^3 = 8, 2^2 = 4, 2^1 = 2. Each time the index drops by 1 we divide by 2, so the next value is 2^0 = 1. The division law gives the same result: a^4 ÷ a^4 = a^0, and anything divided by itself is 1. So a^0 = 1 for any non-zero a. When terms have coefficients, deal with the numbers and the pronumerals separately. Multiply (or divide) the coefficients as normal numbers, then use the index laws for the pronumerals. - The index laws only work when the bases are the same. 2^3 × 3^2 cannot be combined into a single power.

Worked examples

7 min

Multiplying terms with coefficients Simplify 3x^4 × 5x^2. Step 1: Multiply the coefficients: 3 × 5 = 15. Step 2: The pronumeral parts have the same base, so add the indices: x^4 × x^2 = x^(4 + 2) = x^6. Step 3: Combine the two parts. Answer: 15x^6 Dividing terms with coefficients Simplify 20a^7 ÷ 4a^3. Step 1: Divide the coefficients: 20 ÷ 4 = 5. Step 2: Subtract the indices for the same base: a^7 ÷ a^3 = a^(7 − 3) = a^4. Step 3: Combine the two parts. Answer: 5a^4 Raising a product to a power Simplify (2m^3)^4. Step 1: Every factor inside the bracket is raised to the power of 4. Step 2: The number part: 2^4 = 2 × 2 × 2 × 2 = 16. Step 3: The pronumeral part: (m^3)^4 = m^(3 × 4) = m^12. Step 4: Combine the two parts. Answer: 16m^12

Guided practice (do together)

9 min

1. Simplify 2^3 × 2^4. (a) 2^7 (b) 2^12 (c) 4^7 (d) 2^1 2. Simplify y^9 ÷ y^3. (a) y^3 (b) y^6 (c) y^27 (d) y^12 3. Simplify (x^2)^5. (a) x^7 (b) x^25 (c) x^10 (d) 2x^5 4. What is the value of 7^0? (a) 0 (b) 7 (c) 70 (d) 1

Independent practice

11 min

5. Simplify (3a)^2. (a) 3a^2 (b) 6a (c) 9a^2 (d) 9a 6. Evaluate 5^0 + 2^3. 7. Simplify 6x^5 × 2x^3. Write powers using ^, for example x^2. 8. Write 3^4 as an ordinary number.

Questions to check understanding

- Can you identify the base and the index in an expression like 5^3? - Can you simplify expressions such as 3x^4 × 5x^2 and 20a^7 ÷ 4a^3? - Can you simplify a power of a power, such as (2m^3)^4? - Can you evaluate expressions that include a zero index? - What was the trickiest part today?

Answer guide

1. 2^7 — Same base, so add the indices: 3 + 4 = 7. The answer is 2^7. 2. y^6 — Same base, so subtract the indices: 9 − 3 = 6. The answer is y^6. 3. x^10 — For a power of a power, multiply the indices: 2 × 5 = 10. The answer is x^10. 4. 1 — Any non-zero number raised to the power of 0 equals 1. 5. 9a^2 — Both factors inside the bracket are squared: 3^2 × a^2 = 9a^2. 6. 9 — 5^0 = 1 and 2^3 = 8, so the total is 1 + 8 = 9. 7. 12x^8 — Multiply the coefficients (6 × 2 = 12) and add the indices (5 + 3 = 8) to get 12x^8. 8. 81 — 3^4 = 3 × 3 × 3 × 3 = 9 × 9 = 81.

Review

4 min

Recap the success criteria together. Watch for these common misconceptions: - Multiplying the base by the index, such as thinking 2^3 = 6. Write the repeated multiplication out in full (2 × 2 × 2 = 8) until the meaning is secure. - Multiplying the indices when multiplying terms, such as writing x^4 × x^2 = x^8. Expand both terms to count the factors and see that there are six xs, not eight. - Thinking that a^0 = 0. Use the halving pattern (8, 4, 2, 1) to show why the answer is 1. - Forgetting to apply the power to the coefficient, such as writing (3a)^2 = 3a^2. Expand it as 3a × 3a to see that the 3 is squared too.

Extension activities

- Fold a sheet of paper in half repeatedly. Record the number of layers after each fold as a power of 2 and predict how many layers there would be after 10 folds (2^10 = 1024). - Investigate what a negative index might mean by continuing the halving pattern below 2^0: 2^−1, 2^−2 and so on. - Find out how computer storage sizes relate to powers of 2, and write a short explanation for a younger sibling.

Suggested follow-up

- Revisit index laws tomorrow with two or three quick questions from memory. - Try the online practice check for this topic and look at any questions that need another go.

Tips for parents

- When your learner is unsure, ask them to write the expression out as repeated multiplication. This turns the rule back into something they can count. - Encourage them to check numerical answers with a calculator using the power key, but to work the index laws out by hand first. - Ask your learner to explain each law in their own words; explaining is a strong sign of real understanding.

Sample plan: Index laws