一页教材，AI 改成三种程度的讲义和小测

The Foundation version uses short sentences, smaller steps, and an extra everyday example. The Standard version presents the same content as concise key points. The Extension version adds a “why” question and a variation that leads towards fractions and finding an unknown total.

Foundation Handout: Ratios

1. What is a ratio?

A ratio compares two quantities of the same kind. We write it with a colon, for example 12 : 8.

Order matters. Boys : girls means the number of boys comes first.

2. Simplifying a ratio

To simplify a ratio, divide both parts by their highest common factor (HCF).

The HCF is the greatest whole number that divides exactly into both numbers.

Simplify 12 : 8:

1. The factors of 12 are 1, 2, 3, 4, 6 and 12.
2. The factors of 8 are 1, 2, 4 and 8.
3. The HCF is 4.
4. Divide both parts by 4.

12 : 8 = (12 ÷ 4) : (8 ÷ 4) = 3 : 2

The numbers are smaller. The comparison stays the same.

Example 1: Boys and girls

A class has 12 boys and 18 girls.

The ratio of boys to girls is 12 : 18.

The HCF of 12 and 18 is 6.

12 : 18 = (12 ÷ 6) : (18 ÷ 6) = 2 : 3

For every 2 boys, there are 3 girls.

Everyday example: Fruit in a lunchbox

A lunchbox has 4 strawberries and 6 grapes.

Strawberries : grapes = 4 : 6.

Divide both parts by 2:

4 : 6 = 2 : 3

For every 2 strawberries, there are 3 grapes.

3. Dividing a quantity in a given ratio

Example 2: Share $60 in the ratio 2 : 3

The ratio tells us how many equal parts each share gets.

1. Total parts: 2 + 3 = 5.
2. One part: $60 ÷ 5 = $12.
3. First share: 2 × $12 = $24.
4. Second share: 3 × $12 = $36.

Check: $24 + $36 = $60.  
Also, 24 : 36 = 2 : 3.

Remember: total parts → one part → shares → check.

Quick Check

1. Simplify 6 : 9.
2. There are 8 red counters and 12 blue counters. Write red : blue in simplest form.
3. Share $20 in the ratio 1 : 3.

Answers and one-sentence explanations

1. 2 : 3. Divide both parts by their HCF, 3.
2. 2 : 3. Write 8 : 12, then divide both parts by 4.
3. $5 and $15. There are 4 total parts, so one part is $5 and the shares are 1 × $5 and 3 × $5.

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Standard Handout: Ratios

Key Points

- A ratio compares two quantities of the same kind. We write it with a colon, for example 12 : 8.
- Keep the quantities in the stated order.
- To simplify a ratio, divide both parts by their highest common factor (HCF).
- Equivalent ratios express the same comparison.

Simplifying a ratio

12 : 8 → HCF is 4 → 3 : 2

Example 1: Boys and girls

A class has 12 boys and 18 girls.

Boys : girls = 12 : 18 = 2 : 3

Divide both parts by the HCF, 6.

This means there are 2 boys for every 3 girls.

Dividing a Quantity in a Given Ratio

Use: total parts → one part → shares → check.

Example 2: Share $60 in the ratio 2 : 3

- Total parts: 2 + 3 = 5.
- One part: $60 ÷ 5 = $12.
- Shares: 2 × $12 = $24 and 3 × $12 = $36.
- Check the total: $24 + $36 = $60.
- Check the ratio: 24 : 36 = 2 : 3.

The numbers 2 and 3 count equal parts; they are not the dollar amounts.

Quick Check

1. Simplify 18 : 24.
2. A club has 15 boys and 20 girls. Write boys : girls in simplest form.
3. Share $84 in the ratio 3 : 4.

Answers and one-sentence explanations

1. 3 : 4. Divide both parts by their HCF, 6.
2. 3 : 4. The ratio 15 : 20 simplifies by dividing both parts by 5.
3. $36 and $48. There are 7 total parts, so one part is $12 and the shares are 3 × $12 and 4 × $12.

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Extension Handout: Ratios

Key Points

A ratio compares two quantities of the same kind. We write it with a colon, for example 12 : 8.

To simplify a ratio, divide both parts by their highest common factor (HCF).

12 : 8 → HCF is 4 → 3 : 2

Keep the quantities in the stated order.

Example 1: Boys and girls

A class has 12 boys and 18 girls.

Boys : girls = 12 : 18 = 2 : 3

The HCF is 6, so divide both parts by 6.

Why does simplifying work?

Both quantities are scaled by the same factor.

For example, dividing 12 and 18 by 6 gives 2 and 3. Their relative sizes stay the same, so the ratios are equivalent.

Dividing a Quantity in a Given Ratio

Use: total parts → one part → shares → check.

Example 2: Share $60 in the ratio 2 : 3

- Total parts: 2 + 3 = 5.
- One part: $60 ÷ 5 = $12.
- Shares: 2 × $12 = $24 and 3 × $12 = $36.
- Check: $24 + $36 = $60 and 24 : 36 = 2 : 3.

Next step: Ratios and fractions

In the ratio 2 : 3, there are 5 total parts.

- The first share is 2/5 of the total.
- The second share is 3/5 of the total.
- The first share is 2/3 of the second share.

The denominator depends on what you are comparing.

Variation: Work backwards

Two shares are in the ratio 2 : 3. The smaller share is $24. Find the total.

The smaller share represents 2 parts.

One part = $24 ÷ 2 = $12

Total = 5 × $12 = $60

You can find one part from a known share as well as from a known total.

Quick Check

1. Simplify 42 : 63 and explain why the comparison stays the same.
2. Two shares are in the ratio 3 : 5. What fraction of the total is the larger share?
3. Two shares are in the ratio 3 : 5. The larger share is $45. Find the smaller share and the total.

Answers and one-sentence explanations

1. 2 : 3. Dividing both parts by their HCF, 21, scales both quantities by the same factor.
2. 5/8. The larger share has 5 of the 8 total equal parts.
3. Smaller share: $27; total: $72. Five parts equal $45, so one part is $9, giving 3 × $9 and 8 × $9.

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Classroom Slide Outline

Slide 1 — A Ratio Compares Quantities

- A ratio compares two quantities of the same kind.
- Write a ratio with a colon: 12 : 8.
- Order matters: boys : girls.
- 12 boys and 18 girls → 12 : 18.

Slide 2 — Simplify Using the HCF

- Divide both parts by their highest common factor (HCF).
- 12 : 8 → HCF 4 → 3 : 2.
- 12 : 18 → HCF 6 → 2 : 3.
- Both quantities scale by the same factor; the comparison stays the same.

Slide 3 — Share $60 in the Ratio 2 : 3

- Total parts: 2 + 3 = 5.
- One part: $60 ÷ 5 = $12.
- Shares: 2 × $12 = $24; 3 × $12 = $36.
- Check: $24 + $36 = $60; 24 : 36 = 2 : 3.

Slide 4 — Connect Ratios to Fractions

- 2 : 3 has 5 total equal parts.
- First share: 2/5 of the total.
- Second share: 3/5 of the total.
- Why is the first share 2/3 of the second share, but 2/5 of the total?

Slide 5 — Practise and Work Backwards

- Simplify 6 : 9.
- Share $84 in the ratio 3 : 4.
- Challenge: shares are in the ratio 3 : 5, and the larger share is $45; find the total.
- Use: total parts → one part → shares → check.

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Teacher Notes

- All three versions retain the original definition, example numbers and core method.
- Foundation: use five equal boxes to model the total parts before calculating shares.
- Standard: ask students to check both the total and the ratio.
- Extension: emphasise the distinction between a fraction of the total and a comparison with the other share.
- Slide 5 answers: 2 : 3; $36 and $48; $72.
- Original-material corrections: None required. All calculations in the supplied page are correct.

The Foundation handout uses short sentences, smaller steps and an extra everyday example.
The Standard handout keeps the original level and organises the method into key points.
The Extension handout uses the same methods and examples, then explains why percentage changes cannot always be reversed using the same percentage.

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1. Foundation Handout

Percentages

A percentage（百分比） means “out of 100”.

25% = 25/100 = 0.25

A decimal（小数） is another way to write a percentage.

To change a percentage to a decimal, divide by 100.

Find a percentage of an amount

Change the percentage to a decimal. Then multiply（乘）.

Percentage amount = decimal × original amount（原来的量）

Example 1: Find 20% of 45 kg.

1. Change 20% to a decimal: 20% = 0.2.
2. Multiply: 0.2 × 45 = 9.
3. Write the unit: 9 kg.

Check: 10% of 45 kg is 4.5 kg.  
20% is twice 10%. So 20% is 9 kg.

Everyday example: You have $10. You save 50% of it.

50% = 0.5  
Amount saved = 0.5 × $10 = $5.

50% means half. Half of $10 is $5.

Percentage increase and decrease（百分比增加和减少）

First, find the percentage amount.

For an increase（增加）, add it.  
For a decrease（减少）, subtract it.

Example 2: A school bag costs $80. The price goes up by 15%. What is the new price?

1. Change: 15% = 0.15.
2. Find the increase: 0.15 × $80 = $12.
3. Add: $80 + $12 = $92.

Example 3: A $120 jacket is 25% off.

“25% off” means the price decreases by 25%.

1. Change: 25% = 0.25.
2. Find the discount（折扣金额）: 0.25 × $120 = $30.
3. Find the sale price（折后价）: $120 − $30 = $90.

Remember: Find the percentage amount first. Increase → add. Decrease → subtract.

Quick Check

1. Write 30% as a decimal.
2. Find 20% of $30. You may use 0.2 × 30.
3. A $40 game is 25% off. Find the discount and sale price. You may use 0.25 × 40.

Answers

1. 0.3. Divide 30 by 100 to change 30% to a decimal.
2. $6. Multiply 0.2 × $30 to find the percentage amount.
3. Discount: $10; sale price: $30. Find 0.25 × $40 = $10, then subtract $10 from $40.

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2. Standard Handout

Percentages

- A percentage（百分比） means “out of 100”.
- 25% = 25/100 = 0.25.
- To change a percentage to a decimal（小数）, divide by 100.

Find a percentage of an amount

Change the percentage to a decimal and multiply（乘）.

Percentage amount = decimal × original amount（原来的量）

Example 1: Find 20% of 45 kg.

20% = 0.2  
0.2 × 45 kg = 9 kg

Percentage increase and decrease（百分比增加和减少）

1. Find the percentage amount using the original amount.
2. For an increase（增加）, add the percentage amount.
3. For a decrease（减少）, subtract the percentage amount.

Example 2: A school bag costs $80. The price goes up by 15%. What is the new price?

Increase = 0.15 × $80 = $12  
New price = $80 + $12 = $92

Example 3: A $120 jacket is 25% off.

Discount（折扣金额） = 0.25 × $120 = $30  
Sale price（折后价） = $120 − $30 = $90

Remember: Increase → add. Decrease → subtract.

Quick Check

1. Find 35% of 60 kg.
2. A $60 ticket increases in price by 15%. Find the new price.
3. A $140 jacket is 25% off. Find the discount and sale price.

Answers

1. 21 kg. Calculate 0.35 × 60 kg.
2. $69. The increase is 0.15 × $60 = $9, so add $9 to $60.
3. Discount: $35; sale price: $105. Calculate 0.25 × $140 = $35, then subtract it from $140.

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3. Extension Handout

Percentages

A percentage（百分比） means “out of 100”.

25% = 25/100 = 0.25

To find a percentage of an amount, change the percentage to a decimal（小数） and multiply（乘）.

Percentage amount = decimal × original amount（原来的量）

Example 1: Find 20% of 45 kg.

20% = 0.2  
0.2 × 45 kg = 9 kg

Percentage increase and decrease（百分比增加和减少）

Find the percentage amount using the original amount.

- Increase（增加） → add.
- Decrease（减少） → subtract.

Example 2: A school bag costs $80. The price goes up by 15%. What is the new price?

Increase = 0.15 × $80 = $12  
New price = $80 + $12 = $92

Example 3: A $120 jacket is 25% off.

Discount（折扣金额） = 0.25 × $120 = $30  
Sale price（折后价） = $120 − $30 = $90

Why can we calculate the new price in one step?

The original amount is 100%.

After a 15% increase, the new amount is 115% of the original.

New price = 1.15 × $80 = $92

After a 25% decrease, 75% of the original remains.

Sale price = 0.75 × $120 = $90

The decimals 1.15 and 0.75 are multipliers（乘数）.

Next step: reverse percentages（逆向百分比）

Suppose the bag now costs $92 after a 15% increase.

Why is subtracting 15% of $92 the wrong way to find its original price?

The increase was 15% of $80, not 15% of $92. The amount used to calculate the percentage has changed.

Instead:

1.15 × original price = $92  
Original price = $92 ÷ 1.15 = $80

To reverse a percentage change, divide by its multiplier.

Quick Check

1. Find 35% of 60 kg.
2. A $200 item increases in price by 10%, then decreases by 10%. Find the final price. Does it return to $200?
3. A jacket costs $90 after a 25% discount. Find its original price.

Answers

1. 21 kg. Calculate 0.35 × 60 kg.
2. $198; no. Calculate $200 × 1.10 × 0.90, because the decrease is calculated from $220.
3. $120. The sale price is 75% of the original, so calculate $90 ÷ 0.75.

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Classroom Slides Outline

Slide 1 — Percentages Mean “Out of 100”

- A percentage（百分比） means “out of 100”.
- 25% = 25/100 = 0.25.
- Change a percentage to a decimal（小数）: divide by 100.
- Check: Write 50% and 20% as decimals.

Slide 2 — Find a Percentage of an Amount

- Percentage amount = decimal × original amount（原来的量）
- Multiply（乘） after changing to a decimal.
- Example 1: 20% of 45 kg = 0.2 × 45 kg = 9 kg.
- Everyday example: Save 50% of $10 → save $5.

Slide 3 — Percentage Increase

- Percentage increase（百分比增加）: find the increase, then add.
- Example 2: A school bag costs $80; its price rises by 15%.
- Increase = 0.15 × $80 = $12.
- New price = $80 + $12 = $92.

Slide 4 — Percentage Decrease

- Percentage decrease（百分比减少）: find the decrease, then subtract.
- Example 3: A $120 jacket is 25% off.
- Discount（折扣金额） = 0.25 × $120 = $30.
- Sale price（折后价） = $120 − $30 = $90.
- Remember: increase → add; decrease → subtract.

Slide 5 — Explain Why and Look Ahead

- 100% + 15% = 115% → multiplier（乘数） 1.15.
- 100% − 25% = 75% → multiplier 0.75.
- Why does a 10% increase followed by a 10% decrease change the starting amount?
- Reverse percentages（逆向百分比）: $92 ÷ 1.15 = $80.
- Complete your handout’s Quick Check.

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教师提醒

- 三版统一使用“percentage amount = decimal × original amount”，并保留原教材三个例子的全部数字。新增生活例子和小测另设数字。
- 基础组可先口头练习“百分数转小数”，再分开计算乘法与加减；英语较弱的学生可圈出 *of、goes up、off*，对应“乘、增加、减少”。
- 提高组完成第 5 页的推理与逆向百分比；其余学生以熟练掌握两步法为主。
- 原稿更正：Example 2 原写“$80 + $12 = $82”，已改为“$80 + $12 = $92”。