Index laws

Classroom resources

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Students will be able to

• I can explain each index law by writing the powers out.
• I can use the laws to simplify powers of the same base.
• I can explain why a⁰ = 1.

Timing

Predict and Run 8 min · Investigate 10 min · Modify 10 min · Practise and Make 12 min

Before class

Do the Run board yourself first. Students need exponent notation (Number Foundations: Powers).

The one idea: the laws come from counting factors; the base never changes. The common mistake is multiplying the bases (Kai's mistake).

The board writes each power as a row of its factors: joined for a product, cancelled in pairs for a quotient, in groups for a power of a power. The quotient board keeps m at least n (negative indices are Year 9).

Wider: the pattern 2³, 2², 2¹, 2⁰ (halving each time) also gives 2⁰ = 1, and leads to 2⁻¹ = ½.

Running the lesson

1. Predict: ask a '12' student to write 2³ × 2⁴ out.
2. Run: ask what happens to the count when n goes up by 1.
3. Investigate: ask Kai how many 3s there are.
4. Modify: ask how (2²)³ differs from 2² × 2³.
5. Make: ask for all the ways they found.

The hard part (on purpose)

Keeping the base. Ask students to write the powers out as factors before they use any law.

When students get stuck

Answers to the checks

1. 2³ × 2⁴: 2³ × 2⁴ = 2 to what power?
   2⁷: three 2s times four 2s is seven 2s multiplied together. The base stays 2.
2. join the factors: Make 2³ × 2⁴. How many 2s are there? Was your prediction right?
   
   3 + 4 = 7 factors of 2: 2⁷ = 128.
3. Divide: 5⁶ ÷ 5² = 5 to what power?
   4
   Two pairs of 5s cancel, leaving four: 5⁴.
4. Find the mistake: Kai simplifies 3² × 3⁴. Click the line where his working goes wrong.
   line 3
   Line 3: the base stays 3. There are six 3s multiplied together: 3⁶ (= 729), not 9⁶.
5. Zero: Why is 5⁰ equal to 1?
   5³ ÷ 5³ = 1, and the quotient law gives 5³⁻³ = 5⁰
   Dividing anything by itself gives 1, and the law says it is a⁰.
6. Quotient: Make 2⁶ ÷ 2². What is left?
   
   Two pairs cancel: 2⁴ = 16.
7. Power of a power: Make (2²)³. How many 2s?
   
   3 groups of 2 factors: 2⁶ = 64.
8. Zero index: Show 5³ ÷ 5³. What is 5⁰?
   
   Everything cancels: 5⁰ = 1.
9. Power or product: Which is bigger: (2³)² or 2³ × 2²?
   (2³)², which is 2⁶ = 64
   (2³)² = 2⁶ = 64 multiplies the indices; 2³ × 2² = 2⁵ = 32 adds them.
10. Product: 7⁵ × 7³ = 7 to what power?
    8
    5 + 3 = 8: 7⁸.
11. Power of a power: (4²)⁵ = 4 to what power?
    10
    5 groups of 2: 2 × 5 = 10, so 4¹⁰.
12. Zero: What is 10⁰?
    1
    10⁰ = 1.
13. Review: powers: Review: what is 2⁵?
    32
    2 × 2 × 2 × 2 × 2 = 32.
14. 2⁹ two ways: Make a product of two powers of 2 equal to 2⁹, with the first index bigger than the second.
    
    2⁵ × 2⁴, or 2⁶ × 2³: the indices add to 9.

Solutions

One possible solution for each task. Other correct programs are fine.

each law on the board › Quotient

Open task: any program that does what the step asks.

each law on the board › Power of a power

Open task: any program that does what the step asks.

each law on the board › Zero index

Open task: any program that does what the step asks.

each law on the board › Power or product

Open task: any program that does what the step asks.

2⁹ two ways

Open task: any program that does what the step asks.

Stretch and support

Stretch: Continue the pattern 2⁴, 2³, 2², 2¹, 2⁰ by halving. What should 2⁻¹ and 2⁻² be?

Support: A card for each law with the expanded-form picture beside it.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 8 student in Australia. Help me understand ONE idea only, from the lesson "Index laws".

The idea: Establishing the exponent laws from expanded form: I can Students derive the product of powers, quotient of powers, and power of a power rules from expanded form, and state each rule.

What understanding looks like:
- Derives at least one exponent law from expanded form.
- States all three laws and the zero exponent correctly.
- Explains why a⁰ = 1.

Mistakes students often make here:
- A student multiplies the bases.
- A student multiplies indices for a product.
- A student says a⁰ = 0.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Index laws".

Exit ticket

Simplify 4⁶ ÷ 4² and (3²)⁴, leaving each as a power.

4⁴ and 3⁸.

What each step is for

• 2³ × 2⁴: Committing to the index of 2³ × 2⁴ shows whether they add, multiply, or multiply the bases too.
• join the factors: Building products of powers and counting the factors shows the indices add while the base stays.
• three laws and zero: Dividing cancels pairs, so indices subtract; the base never changes (Kai multiplies it); a⁰ = 1 follows from dividing a power by itself.
• each law on the board: Building a quotient, a power of a power and a zero index on the board, then telling the product and power laws apart.
• index laws: Each law once, with a zero index, mixed; multiplying the bases is the planted slip; the powers review keeps evaluating fresh.
• 2⁹ two ways: Finding a product that makes a target power uses the product law backwards.

Where it leads

This starts AC9M8N02.

Where it leads: combining the laws (next lesson), scientific notation, and negative indices (Year 9).