Choosing a strategy

Classroom resources

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Students will be able to

• I can choose a quick strategy for a calculation and explain why it is quick.
• I can use any order for multiplying, a friendly fraction, or splitting a number.
• I can estimate to check that my answer is about right.

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 multiplying fractions and percentages (previous lessons).

The one idea: there is more than one right way, and the best one depends on the numbers. Students own their working: the reasoning checks never mark the free working, only the answer and the reason at the key step.

This node integrates everything in the unit; there is no single new misconception. Listen for students who use the long way every time, and ask what made the quick way possible.

Running the lesson

1. Predict: ask a student who chose halving to explain why it works.
2. Run: ask which other percentages are friendly fractions (10%, 20%, 50%, 75%).
3. Investigate: share two different routes for 4 × 2.7 × 25 and compare.
4. Modify: for 37% of 50, show that percentages can swap: 37% of 50 = 50% of 37.
5. Make: collect pairs that give 18 and sort them by how quick they are.

The hard part (on purpose)

Pausing to look for a quick route instead of starting the long way. Give ten seconds of no-writing time to spot a friendly number before anyone calculates.

When students get stuck

Answers to the checks

1. 25% of 48: What is the quickest way to work out 25% of 48?
   All three give 12, but halving twice is quickest: 25% is a quarter, and 48 halves easily (24, then 12).
2. a quarter of the bar: The bar is 48. Set it to 25%. What fraction of the bar is shaded? Was your prediction right?
   
   A quarter: 48 ÷ 4 = 12. 50% would be a half (24), and 75% three quarters (36).
3. Three moves: Work out $4 \times 2.7 \times 25$ without a calculator. Write your steps.
   $4 \times 25 = 100$ (multiplying can be done in any order, so pair the 4 and the 25); $100 \times 2.7 = 270$ (multiplying by 100 makes each digit 100 times bigger)
   4 × 25 = 100, and 100 × 2.7 = 270.
4. Change the form: Work out $0.75 \times 36$ without a calculator. Write your steps.
   $0.75 = \tfrac{3}{4}$ (a decimal can be written as a fraction); $\tfrac{1}{4}$ of 36 is 9, so $\tfrac{3}{4}$ of 36 is 27 (36 divides by 4, so the fraction is quick)
   0.75 = 3/4. 36 ÷ 4 = 9, × 3 = 27.
5. Split: Which way works out $25 \times 3.2$ by splitting?
   $25 \times 3 + 25 \times 0.2 = 75 + 5 = 80$
   3.2 = 3 + 0.2, so 25 × 3.2 = 25 × 3 + 25 × 0.2 = 75 + 5 = 80 (the distributive law).
6. Three quarters: The bar is 36. Find 75% of it. Work it out in your head first.
   
   A quarter of 36 is 9, so three quarters is 27.
7. A longer product: Work out $0.25 \times 3.6 \times 40$ without a calculator.
   36
   0.25 × 40 = 10 (a quarter of 40), and 10 × 3.6 = 36.
8. A fraction of a number: Work out $\tfrac{3}{5}$ of 45 without a calculator.
   27
   45 ÷ 5 = 9, × 3 = 27.
9. An awkward percentage: Which is the quickest way to find 37% of 50?
   Halve 37 (50 is half of 100): 18.5
   37% of 100 is 37, so 37% of 50 is half: 18.5. (Percentages can swap: 37% of 50 = 50% of 37.)
10. Reorder: Work out $8 \times 1.25$.
    10
    1.25 = 1 1/4, so 8 + 2 = 10. Or 8 × 1.25 = 8 × 5/4 = 10.
11. Friendly percentage: Find 20% of 35.
    7
    20% is a fifth: 35 ÷ 5 = 7.
12. Near a round number: Work out $99 \times 0.4$. (Try $100 \times 0.4$ first.)
    39.6
    99 × 0.4 = 100 × 0.4 − 1 × 0.4 = 40 − 0.4 = 39.6.
13. Review: percentages: Review: a $40 bag is 15% off. What is the sale price, in dollars?
    34
    10% is $4 and 5% is $2, so $6 off: $34. Or 40 × 0.85 = 34.
14. 18 in your head: Choose an amount and a percentage you can do in your head that give exactly 18.
    
    For example 50% of 36, 25% of 72, 10% of 180, 20% of 90, 75% of 24, 30% of 60, 90% of 20 or 15% of 120.

Solutions

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

one change at a time › Three quarters

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

one change at a time › A longer product

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

one change at a time › A fraction of a number

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

one change at a time › An awkward percentage

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

18 in your head

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

Stretch and support

Stretch: Make a poster of your own efficiency rules: when to use a fraction, a decimal, or a percentage, with an example of each.

Support: Keep a table of friendly percentages and their fractions visible (10% = 1/10, 25% = 1/4, …).

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 "Choosing a strategy".

The idea: Efficient strategy selection: I can Students choose the most efficient calculation strategy for a given rational number problem, drawing on number properties and representation choices.

What understanding looks like:
- Selects and applies an efficient strategy (not just the standard algorithm) for a given computation.
- Justifies why the chosen strategy is efficient.
- Applies at least one number property (commutative, associative, distributive) explicitly.

Mistakes students often make here:
- A student always multiplies the long way.
- A student splits a product into a sum but forgets to multiply every part (25 × 3 + 0.2).
- A student's answer is far off and they do not notice.

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.
- If I seem to be missing the idea that comes before this one (Operations with percentages), check it quickly, then come back.
- 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 "Choosing a strategy".

Exit ticket

Work out 5 × 3.6 × 2 and 40% of 15, each in a quick way. Say which move you used.

5 × 2 = 10, then × 3.6 = 36 (any order). 40% = 2/5: 15 ÷ 5 × 2 = 6 (change the form).

What each step is for

• 25% of 48: Committing to the quickest method for 25% of 48 shows whether they see 25% as a quarter or only as a decimal to multiply.
• a quarter of the bar: Seeing 25% as a quarter of the bar connects the percentage to the friendly fraction that makes it quick.
• three moves: Each move has a reason: any order for multiplying, a fraction that suits the numbers, and splitting with the distributive law. Students write their own way and choose the reason for the key step.
• one change at a time: One change at a time: three quarters on the board, a longer product to reorder, a fraction of a whole number, then choosing a strategy for an awkward percentage.
• quick ways: A reorder, a friendly percentage, a split near a round number and a review, mixed; students choose the strategy each time.
• 18 in your head: Building several percentage-of-amount pairs that give 18, each chosen to be easy in your head, makes strategy choice the task.

Where it leads

Next: Calculating with negative fractions and decimals (AC9M8N04), where the same moves work with signs.

Where it leads: mental strategies in financial maths (AC9M7N09) and algebraic manipulation, where the distributive law becomes expanding brackets.