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KS3 Year 9 / Mathematics / Curriculum

Graphs and functions

Graphs and functions: structured theory, worked examples, answered practice, and a mastery checklist for KS3 Year 9.

学习目标

Recognise, explain, and apply “Linear graphs”.
Recognise, explain, and apply “Gradient and intercept”.
Recognise, explain, and apply “Quadratic graphs introduction”.
Recognise, explain, and apply “Reciprocal graphs introduction”.
Recognise, explain, and apply “Real-life graphs”.

核心概念与检查

I move between formula, table, and graph.
I interpret slope, intercepts, and key features.
Every solution or explanation for “Graphs and functions” 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

Input-output relationship

A function maps each allowed input to one output. Formula, table, and graph are different representations of the same relationship.

02

Parameters and graph

Parameters change slope, position, amplitude, or rate of change. Read the key features before plotting individual points.

03

Domain and interpretation

The domain states which inputs are allowed. In a real model, also check whether the output is physically meaningful.

04

Linear graphs

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

Gradient and intercept

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

Quadratic graphs introduction

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

Reciprocal graphs introduction

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

Real-life graphs

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 function example

For f(x) = 3x - 6, find f(5) and the x-intercept.

1Substitute x = 5: f(5) = 3·5 - 6.
2For the intercept set f(x) = 0: 3x - 6 = 0.
3Solve for x and check the point on the graph.

f(5) = 9, x-intercept = 6/3

Reasoning example

Before calculating, explain the key idea from “Linear graphs” 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 “Linear graphs”, but also why it is valid here.

Graphs and functions

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

Solve 9x + 5 = 50.

提示

Subtract 5 from both sides, then divide by 9.

答案指南

x = 5.

基础熟练5 分钟

Expand and simplify 5(x + 2) - 2x.

提示

Multiply every term inside the bracket.

答案指南

5x + 10 - 2x = 3x + 10.

应用6 分钟

Factorise 8x + 32.

提示

Take out the common factor 8.

答案指南

8(x + 4).

应用7 分钟

For f(x) = 4x + 6, find f(10) and solve f(x) = 30.

提示

Substitute, then solve the equation.

答案指南

f(10) = 46; x = 6.

推理8 分钟

A sequence starts 3, 10, 17. Find its nth term and 10th term.

提示

The common difference is 7.

答案指南

7n + -4; term 10 = 66.

推理9 分钟

Solve 3x + 5 > 20.

提示

Use equation steps; the coefficient of x is positive.

答案指南

x > 5.

考试题型10 分钟

Solve x + y = 8, x - y = 4.

提示

Add the two equations.

答案指南

x = 6, y = 2.

考试题型12 分钟

A learner writes 9(x + 4) = 9x + 4. Identify and correct the error.

提示

The outside factor multiplies every term.

答案指南

9(x + 4) = 9x + 36.

常见错误

Confusing f(x) with the product f times x.
Plotting without a scale or key points.
Ignoring domain and model restrictions.

掌握度检查

I move between formula, table, and graph.
I interpret slope, intercepts, and key features.
I state domain and relevant restrictions.

Graphs and functions

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: Solve 5x + 6 = 36.

提示

Subtract 6 from both sides, then divide by 5.

答案指南

x = 6.

基础10 分钟

Homework 2: Expand and simplify 8(x + 3) - 3x.

提示

Multiply every term inside the bracket.

答案指南

8x + 24 - 3x = 5x + 24.

进阶12 分钟

Homework 3: Factorise 4x + 20.

提示

Take out the common factor 4.

答案指南

4(x + 5).

进阶15 分钟

Homework 4: For f(x) = 7x + 2, find f(9) and solve f(x) = 16.

提示

Substitute, then solve the equation.

答案指南

f(9) = 65; x = 2.

挑战18 分钟

Homework 5: A sequence starts 4, 7, 10. Find its nth term and 10th term.

提示

The common difference is 3.

答案指南

3n + 1; term 10 = 31.

挑战20 分钟

Homework 6: Solve 6x + 6 > 42.

提示

Use equation steps; the coefficient of x is positive.

答案指南

x > 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. Solve 3x + 6 = 24.

2 分

答案指南

x = 6.

2. Expand and simplify 6(x + 3) - 3x.

3 分

答案指南

6x + 18 - 3x = 3x + 18.

3. Factorise 9x + 45.

3 分

答案指南

9(x + 5).

4. For f(x) = 5x + 2, find f(7) and solve f(x) = 12.

4 分

答案指南

f(7) = 37; x = 2.

5. A sequence starts 4, 12, 20. Find its nth term and 10th term.

4 分

答案指南

8n + -4; term 10 = 76.

6. Solve 4x + 6 > 30.

4 分

答案指南

x > 6.

7. Solve x + y = 10, x - y = 4.

5 分

答案指南

x = 7, y = 3.

8. A learner writes 3(x + 5) = 3x + 5. Identify and correct the error.

5 分

答案指南

3(x + 5) = 3x + 15.

9. Model this: 2 tickets cost x euros each plus a fixed 6 euros. Write the cost and find it for x = 8.

5 分

答案指南

C = 2x + 6; C(8) = 22.

10. Represent “Quadratic graphs introduction” as an equation, a graph, and a verbal description of the same model.

5 分

答案指南

Example y = 9x + 4: a line with gradient 9, intercept 4, meaning “starts at 4 and rises by 9 per unit”.

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 9: 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。

Linear graphs

Gradient and intercept

Quadratic graphs introduction

Reciprocal graphs introduction

Real-life graphs

Graphical solutions

Function notation introduction

重点关注

通常容易出错或需要更多复习的部分。

I move between formula, table, and graph.

I interpret slope, intercepts, and key features.

I state domain and relevant restrictions.

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。

Solve 9x + 5 = 50.

Expand and simplify 5(x + 2) - 2x.

Factorise 8x + 32.

For f(x) = 4x + 6, find f(10) and solve f(x) = 30.

A sequence starts 3, 10, 17. Find its nth term and 10th term.

Solve 3x + 5 > 20.

Solve x + y = 8, x - y = 4.

A learner writes 9(x + 4) = 9x + 4. Identify and correct the error.

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。