---
marp: true
theme: default
html: true
---

# Choosing a strategy

Number Foundations · Fractions and decimals

<!-- Speaker note: Students compare strategies for calculations with fractions, decimals and percentages: changing the order (any order for multiplying), changing the form (0.75 = 3/4), and splitting (the distributive law), with an estimate to check (AC9M7N06, node 4). About 40 minutes (Predict and Run 8 min · Investigate 12 min · Modify 10 min · Practise and Make 10 min). -->

---

## Today

**We are learning to** the same calculation can be quick or slow. Change the order (4 × 2.7 × 25 = 100 × 2.7), change the form (0.75 × 36 = 3/4 of 36), or split a number (12 × 2.5 = 24 + 6). Then check the answer is about right.

- 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.

<!-- Speaker note: 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. -->

---

## Why it matters

> Two students work out 25% of 48. One writes a long multiplication; the other halves twice in their head. Both get 12. Which would you rather do in a shop?

<!-- Speaker note: Every calculation with fractions, decimals and percentages can be done the long way. A good strategy makes it quick, and it usually comes from one of three moves: - **Change the order.** Multiplying can be done in any order: $4 \times 2.7 \times 25 = (4 \times 25) \times 2.7 = 100 \times 2.7$. - **Change the form.** $0.75 = \tfrac{3}{4} = 75\%$: use whichever suits the numbers. - **Split a number.** $12 \times 2.5 = 12 \times 2 + 12 \times 0.5 = 24 + 6$. Then **check**: is the answer about the right size? -->

---

<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**What is the quickest way to work out 25% of 48?**

- Work out 0.25 × 48 as a long multiplication
- Work out 48 × 25, then divide by 100
- They are all as quick
- Halve 48, then halve again

<!-- Speaker note: What happens: All three give 12, but halving twice is quickest: 25% is a quarter, and 48 halves easily (24, then 12). Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask a student who chose halving to explain why it works. -->

---

<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fnumber%2Fchoosing-a-strategy%2Fstep-2&code=widget%3A+percent%0Aq%3A+48%0Aslider+p%3A+0%2C+100%2C+0%2C+5%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/number/choosing-a-strategy/?full=step-2#step-2)

<!-- Speaker note: Look at where 50% and 25% sit on the bar. -->

<!-- Speaker note: Teacher: ask which other percentages are friendly fractions (10%, 20%, 50%, 75%). -->

<style scoped>pre, marp-pre { font-size: 0.5em; line-height: 1.25; } section { font-size: 25px; }</style>

---

<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* All three give 12, but halving twice is quickest: 25% is a quarter, and 48 halves easily (24, then 12).

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

---

<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Three moves

Here are the three moves, each with a check.

Your working can be short notes. Then choose the reason for the key step.

$$4 \times 2.7 \times 25 = (4 \times 25) \times 2.7 = 270$$

1) $0.75 \times 36 = \tfrac{3}{4} \times 36 = 27$ — Change the form: 36 ÷ 4 = 9, × 3 = 27. Check: a bit less than 36.
2) $12 \times 2.5 = 12 \times 2 + 12 \times 0.5 = 24 + 6 = 30$ — Split a number: 2.5 = 2 + 0.5. Check: between 12 × 2 and 12 × 3.

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write 4 × 2.7 × 25 = (4 × 25) × 2.7 = 270 and say "Change the order: 4 × 25 = 100. Check: 2.7 is about 3, and 100 × 3 = 300." Line 1: write 0.75 × 36 = 3 ÷ 4 × 36 = 27 and say "Change the form: 36 ÷ 4 = 9, × 3 = 27. Check: a bit less than 36." Line 2: write 12 × 2.5 = 12 × 2 + 12 × 0.5 = 24 + 6 = 30 and say "Split a number: 2.5 = 2 + 0.5. Check: between 12 × 2 and 12 × 3." -->

<!-- Speaker note: Teacher: share two different routes for 4 × 2.7 × 25 and compare. -->

<style scoped>pre, marp-pre { font-size: 0.5em; line-height: 1.25; } section { font-size: 27px; }</style>

---

<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Three moves

**Work out $4 \times 2.7 \times 25$ without a calculator. Write your steps.**

* Answer: $4 \times 25 = 100$ (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)

<!-- Speaker note: 4 × 25 = 100, and 100 × 2.7 = 270. -->

---

<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Change the form

**Work out $0.75 \times 36$ without a calculator. Write your steps.**

* Answer: $0.75 = \tfrac{3}{4}$ (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)

<!-- Speaker note: 0.75 = 3/4. 36 ÷ 4 = 9, × 3 = 27. -->

---

<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Split

**Which way works out $25 \times 3.2$ by splitting?**

- $25 \times 3 + 25 \times 0.2 = 75 + 5 = 80$
- $25 \times 3 + 0.2 = 75.2$
- $25 \times 3 \times 0.2 = 15$
- $20 \times 3.2 + 5 \times 3 = 79$

* Answer: $25 \times 3 + 25 \times 0.2 = 75 + 5 = 80$

<!-- Speaker note: 3.2 = 3 + 0.2, so 25 × 3.2 = 25 × 3 + 25 × 0.2 = 75 + 5 = 80 (the distributive law). -->

---

<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## One change at a time



- **Three quarters:** Each task changes one thing.
- **A longer product:** 
- **A fraction of a number:** 
- **An awkward percentage:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/number/choosing-a-strategy/?full=step-4#step-4)

<!-- Speaker note: What this step is for: 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. Hints, in order: 75% is three quarters. For students who need more support: Keep a table of friendly percentages and their fractions visible (10% = 1/10, 25% = 1/4, …). For early finishers: Make a poster of your own efficiency rules: when to use a fraction, a decimal, or a percentage, with an example of each. 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: A student always multiplies the long way. Ask: Is any number here a friendly fraction, or a pair that makes 10 or 100? Fix: Look for 25 and 4, 0.5, 0.25, 0.75, 10% and 50% first. / A student splits a product into a sum but forgets to multiply every part (25 × 3 + 0.2). Ask: What is 3.2 made of? Does each part get multiplied by 25? Fix: Every part of the split is multiplied. / A student's answer is far off and they do not notice. Ask: Roughly how big should it be? Fix: Estimate first: round the numbers and compare. -->

<!-- Speaker note: Teacher: for 37% of 50, show that percentages can swap: 37% of 50 = 50% of 37. -->

---

<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Reorder

**Work out $8 \times 1.25$.**

* Answer: 10

<!-- Speaker note: 1.25 = 1 1/4, so 8 + 2 = 10. Or 8 × 1.25 = 8 × 5/4 = 10. -->

---

<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Friendly percentage

**Find 20% of 35.**

* Answer: 7

<!-- Speaker note: 20% is a fifth: 35 ÷ 5 = 7. -->

---

<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Near a round number

**Work out $99 \times 0.4$. (Try $100 \times 0.4$ first.)**

* Answer: 39.6

<!-- Speaker note: 99 × 0.4 = 100 × 0.4 − 1 × 0.4 = 40 − 0.4 = 39.6. -->

---

<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review: percentages

**Review: a $40 bag is 15% off. What is the sale price, in dollars?**

* Answer: 34

<!-- Speaker note: 10% is $4 and 5% is $2, so $6 off: $34. Or 40 × 0.85 = 34. -->

---

<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: 18 in your head

Then find two more, and say which was quickest and why.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/number/choosing-a-strategy/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Building several percentage-of-amount pairs that give 18, each chosen to be easy in your head, makes strategy choice the task. Hints, in order: 50% of what is 18? / Or 10% of something, or 25% of something. -->

<!-- Speaker note: Teacher: collect pairs that give 18 and sort them by how quick they are. -->

---

<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## 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).

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this 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. -->