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KS3 Year 7 / Mathematics / Curriculum

Ratio, proportion and rates

Ratio, proportion and rates: structured theory, worked examples, answered practice, and a mastery checklist for KS3 Year 7.

学习目标

Recognise, explain, and apply “Simplifying ratios”.
Recognise, explain, and apply “Sharing in a ratio”.
Recognise, explain, and apply “Direct proportion in tables”.
Recognise, explain, and apply “Unitary method”.
Recognise, explain, and apply “Simple scale problems”.

核心概念与检查

I can convert safely between number forms.
I can explain why the chosen operation is appropriate.
Every solution or explanation for “Ratio, proportion and rates” should include a method, justification, and final check.

单元

核心理论

练习前按清晰顺序掌握本章核心概念。

01

Represent before calculating

Write the number in a useful form, check sign and place value, and only then choose the operation. A clear representation reduces mechanical errors.

02

Equivalent forms

Fractions, decimals, percentages, ratios, and powers can describe the same quantity. Changing form is a solving tool, not a separate trick.

03

Reasonableness check

Estimate before and after the exact calculation. The sign, order of magnitude, and unit must agree with the problem context.

04

Simplifying ratios

Use a suitable representation, state the rule, and keep every numerical or algebraic step equivalent. Finish with estimation, an inverse operation, or substitution as a check.

05

Sharing in a ratio

Use a suitable representation, state the rule, and keep every numerical or algebraic step equivalent. Finish with estimation, an inverse operation, or substitution as a check.

06

Direct proportion in tables

Use a suitable representation, state the rule, and keep every numerical or algebraic step equivalent. Finish with estimation, an inverse operation, or substitution as a check.

07

Unitary method

Use a suitable representation, state the rule, and keep every numerical or algebraic step equivalent. Finish with estimation, an inverse operation, or substitution as a check.

08

Simple scale problems

Use a suitable representation, state the rule, and keep every numerical or algebraic step equivalent. Finish with estimation, an inverse operation, or substitution as a check.

Mathematics

例题讲解

逐步查看方法，并理解每一步为何成立。

Worked number example

Calculate and simplify 5/35 + 7/35.

1The denominators are equal, so add the numerators.
2Write (5 + 7)/35.
3Simplify by a common factor and check that the value is reasonable.

12/35

Reasoning example

Before calculating, explain the key idea from “Simplifying ratios” and which conditions must be checked.

1Define the idea in one clear sentence.
2Connect it to a representation, law, or formula.
3State a restriction, unit, or final check that makes the solution valid.

The answer should show not only which rule is used for “Simplifying ratios”, but also why it is valid here.

Ratio, proportion and rates

含答案练习

八道分层练习，从基础熟练到考试应用。请先独立完成，再查看提示或答案。

基础熟练4 分钟

Calculate 6 + 6 × 12 and justify the order of operations.

提示

Multiplication precedes addition.

答案指南

6 + 6 · 12 = 78.

基础熟练5 分钟

Find 9/27 of 81.

提示

Multiply by the fraction and simplify first.

答案指南

9/27 · 81 = 27.

应用6 分钟

Increase 250 by 25%.

提示

Use multiplier 1 + 25/100.

答案指南

250 · 1.25 = 312.5.

应用7 分钟

Divide 120 in the ratio 8:2.

提示

There are 10 ratio parts.

答案指南

120 : 10 = 12; 96 and 24.

推理8 分钟

Compare 4/8 and 4/16 using a common denominator or decimals.

提示

Use cross-products.

答案指南

4 · 16 = 64, 4 · 8 = 32; 4/8 > 4/16.

推理9 分钟

Write 4200000 in standard form.

提示

The first factor must be at least 1 and less than 10.

答案指南

4200000 = 4.2 × 10^6.

考试题型10 分钟

Estimate 294 ÷ 29.700000000000003, then calculate to one decimal place.

提示

Round to compatible numbers before calculating exactly.

答案指南

Estimate ≈ 300 ÷ 30 = 10.0. Exact ≈ 9.9.

考试题型12 分钟

Test the claim 6² + 5² = 11² and explain.

提示

Calculate both sides separately.

答案指南

36 + 25 = 61, while 11² = 121; false.

常见错误

Dropping brackets or using the wrong order of operations.
Converting a fraction or percentage without checking equivalence.
Answering without an estimate, unit, or sign check.

掌握度检查

I can convert safely between number forms.
I can explain why the chosen operation is appropriate.
I check the result with estimation or an inverse operation.

Ratio, proportion and rates

章节作业

六道不同作业，从基础到挑战，包含预计用时、提示和答案指南。

基础8 分钟

Homework 1: Calculate 9 + 2 × 11 and justify the order of operations.

提示

Multiplication precedes addition.

答案指南

9 + 2 · 11 = 31.

基础10 分钟

Homework 2: Find 5/20 of 80.

提示

Multiply by the fraction and simplify first.

答案指南

5/20 · 80 = 20.

进阶12 分钟

Homework 3: Increase 480 by 40%.

提示

Use multiplier 1 + 40/100.

答案指南

480 · 1.40 = 672.0.

进阶15 分钟

Homework 4: Divide 84 in the ratio 4:3.

提示

There are 7 ratio parts.

答案指南

84 : 7 = 12; 48 and 36.

挑战18 分钟

Homework 5: Compare 7/12 and 5/35 using a common denominator or decimals.

提示

Use cross-products.

答案指南

7 · 35 = 245, 5 · 12 = 60; 7/12 > 5/35.

挑战20 分钟

Homework 6: Write 600000 in standard form.

提示

The first factor must be at least 1 and less than 10.

答案指南

600000 = 0.6 × 10^6.

50 分钟 / 40 分

完整章节测试

带计时器、完整分值和参考答案的章节自测。

Start the timer when ready, work without notes, show every step, and open model answers only after finishing.

1. Calculate 7 + 2 × 9 and justify the order of operations.

2 分

答案指南

7 + 2 · 9 = 25.

2. Find 3/12 of 48.

3 分

答案指南

3/12 · 48 = 12.

3. Increase 360 by 30%.

3 分

答案指南

360 · 1.30 = 468.0.

4. Divide 144 in the ratio 9:3.

4 分

答案指南

144 : 12 = 12; 108 and 36.

5. Compare 5/10 and 5/25 using a common denominator or decimals.

4 分

答案指南

5 · 25 = 125, 5 · 10 = 50; 5/10 > 5/25.

6. Write 1600000 in standard form.

4 分

答案指南

1600000 = 1.6 × 10^6.

7. Estimate 392 ÷ 39.6, then calculate to one decimal place.

5 分

答案指南

Estimate ≈ 400 ÷ 40 = 10.0. Exact ≈ 9.9.

8. Test the claim 7² + 6² = 13² and explain.

5 分

答案指南

49 + 36 = 85, while 13² = 169; false.

9. Create and solve a two-step exam-style problem about “Unitary method” using 3, 3, and 9.

5 分

答案指南

A valid response uses the data in two justified steps and includes an independent check; for example 3 × 3 = 9, then 9 + 6 = 15.

10. Write a revision rule for “Simple scale problems”, one example, and one counterexample.

5 分

答案指南

Full response: an accurate rule for “Simple scale problems”, a numerical example that satisfies it, and an explained counterexample.

单元

官方来源与核验

课程参考来源；教学顺序请始终与学校和老师确认。

UK Department for Education - MathematicsOfficial national curriculum progression and attainment guidance.

Mathematics

KS3 Year 7: 数学与物理的结构化路线，包含章节、练习和下一步。

本章覆盖内容

结构跟随教材，用来帮助组织学习。

Simplifying ratios

Sharing in a ratio

Direct proportion in tables

Unitary method

Simple scale problems

重点关注

通常容易出错或需要更多复习的部分。

I can convert safely between number forms.

I can explain why the chosen operation is appropriate.

I check the result with estimation or an inverse operation.

来源、日常资料与学习资源

从教材、daily material、PDFs、videos 和 worked examples 开始。

Start from the official textbook or specification referenced on the subject page.

Use the notes and examples as support, not as a replacement for the official syllabus.

Check the current syllabus version before exam preparation.

按小主题练习

进入 full tests 前先做 targeted practice，让覆盖情况更清楚。

Calculate 6 + 6 × 12 and justify the order of operations.

Find 9/27 of 81.

Increase 250 by 25%.

Divide 120 in the ratio 8:2.

Compare 4/8 and 4/16 using a common denominator or decimals.

Write 4200000 in standard form.

Estimate 294 ÷ 29.700000000000003, then calculate to one decimal place.

Test the claim 6² + 5² = 11² and explain.

Mocks 与进度检查

如何衡量本章进度，以及何时加入综合 mock。

Start with untimed practice by subtopic.

Move to a short timed checkpoint only after completing the mastery checklist.

Record each error with the correct method and revisit it after 48 hours.

下一步

完成本章后该做什么，以及它如何连接到下一单元。

Complete the practice without support.

Explain the core method aloud in under two minutes.

Continue to the next chapter or request targeted tutor support.

说明：每学年的正式考试范围请始终以学校、老师和当前官方来源为准。