Powers

Classroom resources

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Students will be able to

• I can write a repeated multiplication as a power, and expand a power.
• I can say what the base and the exponent mean.
• I can work out powers, including powers of 10.

Timing

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

Before class

Do the Run board yourself first. Students need squares (previous lessons) and primes and factor pairs (Year 6).

The one idea: the exponent counts how many times the base is multiplied. The common mistake is multiplying base and exponent (2⁵ = 10, Mia's mistake). Fluency with the notation is the hinge for prime factorisation in the next lesson.

The factor tree board is used again for prime factors and HCF and LCM.

Running the lesson

1. Predict: students who chose 6 multiplied 2 × 3. Ask them to write 2³ the long way.
2. Run: ask how many 2s the tree ended with, and where the 4 in 2⁴ comes from.
3. Investigate: link 10³ to place value: thousands.
4. Modify: for 2⁵ against 5², ask for a prediction before working them out.
5. Make: collect the three names for 64 and ask for a fourth (64¹).

The hard part (on purpose)

Reading the small raised number as a count, not a multiplier. Ask students to write the power the long way first, every time, before working it out.

When students get stuck

Answers to the checks

1. 2 cubed: What is $2^3$?
   8. 2³ = 2 × 2 × 2 = 8: three 2s multiplied. 6 is 2 × 3.
2. grow a tree for 16: Split 16 into factor pairs until every branch ends in a prime. How many 2s? Was your prediction for $2^3$ right?
   
3. Read the power: Work out $3^4$.
   81
   3 × 3 = 9, × 3 = 27, × 3 = 81.
4. Find the mistake: Mia works out $2^5$. Click the line where her working goes wrong.
   line 3
   Line 3: 2⁵ is five 2s multiplied, 2 × 2 × 2 × 2 × 2 = 32, not 2 × 5.
5. Match them: Match each power to what it means.
   $10^3$ → B. 10 × 10 × 10 = 1000; $3^2$ → D. 3 × 3 = 9; $2^3$ → A. 2 × 2 × 2 = 8; $5^1$ → C. just 5
   The exponent counts the factors. 10³ is 1 followed by three zeros; 5¹ is one 5.
6. Thirty-two: Grow a tree for 32. Write 32 as a power of 2.
   
   32 = 2 × 2 × 2 × 2 × 2 = 2⁵.
7. Write the power: Write $7 \times 7 \times 7$ as a power.
   $7^3$
   Three 7s multiplied: 7³ = 343.
8. Two bases: Work out $2^3 \times 3^2$.
   72
   2³ = 8 and 3² = 9, so 8 × 9 = 72.
9. Swap them: Edge case: which is bigger, $2^5$ or $5^2$?
   $2^5 = 32$ is bigger than $5^2 = 25$
   2⁵ = 32 and 5² = 25: the order matters.
10. Evaluate: Work out $4^3$.
    64
    4 × 4 × 4 = 64.
11. Powers of 10: Work out $10^5$.
    100000
    1 followed by five zeros: 100 000.
12. Write it: Write $2 \times 2 \times 5 \times 5 \times 5$ with exponents.
    $2^2 \times 5^3$
    Two 2s and three 5s: 2² × 5³ = 4 × 125 = 500.
13. Review: roots: Review: work out $\sqrt{144}$.
    12
    12 × 12 = 144.
14. 64 three ways: Grow a tree for 64.
    
    64 = 2⁶ = 4³ = 8²: six 2s, grouped in pairs (three 4s) or in threes (two 8s).

Solutions

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

one change at a time › Thirty-two

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

one change at a time › Write the power

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

one change at a time › Two bases

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

one change at a time › Swap them

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

64 three ways

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

Stretch and support

Stretch: The fold-the-paper hook: after how many folds is the stack more than 1000 layers? More than 1 000 000?

Support: Expand every power before working it out; use the tree board for powers of 2 and 3.

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 7 student in Australia. Help me understand ONE idea only, from the lesson "Powers".

The idea: Exponent notation for repeated multiplication: I can Students understand exponent notation as shorthand for repeated multiplication and can convert between expanded and exponent forms.

What understanding looks like:
- Correctly converts between repeated multiplication and exponent notation.
- States the meaning of the base and exponent in a given expression.
- Does not confuse aⁿ with a×n (e.g. does not evaluate 2³ as 6).

Mistakes students often make here:
- A student multiplies base and exponent (Mia's mistake).
- A student swaps base and exponent (7³ written 3⁷).
- A student adds different bases (2² × 5³ as 7⁵).

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 "Powers".

Exit ticket

Work out 3³ and write 5 × 5 × 5 × 5 as a power.

3³ = 27; 5⁴.

What each step is for

• 2 cubed: Committing to a value for 2³ shows whether they read the exponent as a multiplier (6) or as a count of factors (8).
• grow a tree for 16: Splitting 16 until every branch is a prime shows four equal factors, and the board writes them as 2⁴.
• base and exponent: The exponent counts factors of the base; Mia's mistake multiplies the base by the exponent. Powers of 10 count zeros, the same idea in place value.
• one change at a time: One change at a time: a bigger tree, writing a power from a product, two bases, then the exponent 1 and the order of base and exponent.
• powers: Evaluating, writing and comparing powers, mixed; the exponent as a multiplier is the planted slip; the review keeps roots fresh.
• 64 three ways: Growing a tree for 64 and writing it as a power of 2, of 4 and of 8 shows one number has several power names, all from the same six 2s.

Where it leads

Next: Prime factors (node 2), writing any number as a product of prime powers.

Where it leads: index laws and scientific notation (Year 8 and 9), and square numbers as powers with exponent 2.