学习目标
- Recognise, explain, and apply “Quadratics”.
- Recognise, explain, and apply “Inequalities”.
- Recognise, explain, and apply “Functions”.
- Recognise, explain, and apply “Graph transformations”.
- Recognise, explain, and apply “Coordinate geometry”.
A Level Year 12 / Mathematics / Curriculum
Pure Mathematics 1
Pure Mathematics 1: structured theory, worked examples, answered practice, and a mastery checklist for A Level Year 12.
核心概念与检查
- I choose an appropriate statistic.
- I calculate and interpret probabilities.
- Every solution or explanation for “Pure Mathematics 1” 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。
Data and population
Before calculating, identify what each value measures, the population, and whether the sample is representative.
Statistic with interpretation
Mean, median, range, and probability answer different questions. Give the value together with a contextual interpretation.
Scale and probability check
A graph needs a valid scale and a probability must lie from 0 to 1. These checks immediately expose inconsistent answers.
Quadratics
State what each value measures, organise it in a table or graph, then choose the appropriate statistic. Finish with estimation, an inverse operation, or substitution as a check.
Inequalities
State what each value measures, organise it in a table or graph, then choose the appropriate statistic. Finish with estimation, an inverse operation, or substitution as a check.
Functions
State what each value measures, organise it in a table or graph, then choose the appropriate statistic. Finish with estimation, an inverse operation, or substitution as a check.
Graph transformations
State what each value measures, organise it in a table or graph, then choose the appropriate statistic. Finish with estimation, an inverse operation, or substitution as a check.
Coordinate geometry
State what each value measures, organise it in a table or graph, then choose the appropriate statistic. 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 data example
For the data 3, 6, 9, 8, 6, find the mean and range.
- 1Sum = 32 and count = 5.
- 2Mean = 32/5.
- 3Range = maximum - minimum.
Mean 6.4, range 6.
Reasoning example
Before calculating, explain the key idea from “Quadratics” 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 “Quadratics”, but also why it is valid here.
Pure Mathematics 1
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
Data: 8, 5, 8, 5. Find mean and range.
提示
Mean = sum/count.
答案指南
mean = 6.5, range = 3.
基础熟练5 分钟
Order 6, 4, 8, 2, 4 and find the median.
提示
The median is the middle ordered value.
答案指南
2, 4, 4, 6, 8; median = 4.
应用6 分钟
A bag contains 7 blue and 4 yellow counters. Find P(blue).
提示
Favourable outcomes / total outcomes.
答案指南
P = 7/11.
应用7 分钟
In 90 trials, an outcome occurred 18 times. Find relative frequency and percentage.
提示
Divide frequency by number of trials.
答案指南
18/90 = 0.2 = 20%.
推理8 分钟
Values 6, 3 have frequencies 3, 6. Find the weighted mean.
提示
Sum value×frequency and divide by total frequency.
答案指南
(6·3 + 3·6)/9 = 4.00.
推理9 分钟
Two samples have the same mean but ranges 9 and 45. Which is more consistent and why?
提示
Lower spread means greater consistency.
答案指南
The sample with range 9 is more consistent because its values are less spread out.
考试题型10 分钟
Choose a suitable graph for “Radians” and justify axes, scale, and title.
提示
Decide whether data are categorical, discrete, or continuous.
答案指南
A full response matches graph type to the data, labels axes and units, uses an even scale, and includes a clear title.
考试题型12 分钟
Explain why correlation between two variables does not prove causation.
提示
A third factor or reverse relationship may exist.
答案指南
Correlation shows co-variation, not a controlled mechanism. Establishing causation needs a designed experiment and control of confounders.
常见错误
- Using the mean when outliers distort the result.
- Using the wrong denominator in a probability.
- Drawing a graph without title, units, or consistent scale.
掌握度检查
- I choose an appropriate statistic.
- I calculate and interpret probabilities.
- I check sample and graph quality.
Pure Mathematics 1
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: Data: 4, 6, 4, 6. Find mean and range.
提示
Mean = sum/count.
答案指南
mean = 5.0, range = 2.
基础10 分钟
Homework 2: Order 10, 7, 21, 3, 7 and find the median.
提示
The median is the middle ordered value.
答案指南
3, 7, 7, 10, 21; median = 7.
进阶12 分钟
Homework 3: A bag contains 3 blue and 5 yellow counters. Find P(blue).
提示
Favourable outcomes / total outcomes.
答案指南
P = 3/8.
进阶15 分钟
Homework 4: In 60 trials, an outcome occurred 12 times. Find relative frequency and percentage.
提示
Divide frequency by number of trials.
答案指南
12/60 = 0.2 = 20%.
挑战18 分钟
Homework 5: Values 9, 4 have frequencies 4, 9. Find the weighted mean.
提示
Sum value×frequency and divide by total frequency.
答案指南
(9·4 + 4·9)/13 = 5.54.
挑战20 分钟
Homework 6: Two samples have the same mean but ranges 5 and 30. Which is more consistent and why?
提示
Lower spread means greater consistency.
答案指南
The sample with range 5 is more consistent because its values are less spread out.
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. Data: 9, 6, 9, 6. Find mean and range.
2 分答案指南
mean = 7.5, range = 3.
2. Order 8, 5, 15, 3, 5 and find the median.
3 分答案指南
3, 5, 5, 8, 15; median = 5.
3. A bag contains 8 blue and 5 yellow counters. Find P(blue).
3 分答案指南
P = 8/13.
4. In 40 trials, an outcome occurred 8 times. Find relative frequency and percentage.
4 分答案指南
8/40 = 0.2 = 20%.
5. Values 7, 4 have frequencies 4, 7. Find the weighted mean.
4 分答案指南
(7·4 + 4·7)/11 = 5.09.
6. Two samples have the same mean but ranges 3 and 18. Which is more consistent and why?
4 分答案指南
The sample with range 3 is more consistent because its values are less spread out.
7. Choose a suitable graph for “Radians” and justify axes, scale, and title.
5 分答案指南
A full response matches graph type to the data, labels axes and units, uses an even scale, and includes a clear title.
8. Explain why correlation between two variables does not prove causation.
5 分答案指南
Correlation shows co-variation, not a controlled mechanism. Establishing causation needs a designed experiment and control of confounders.
9. Design a study about “Sector area”: population, sample, variables, and possible bias.
5 分答案指南
A complete design defines population, random/stratified sampling, measurable variables, sample size, and a specific bias with mitigation.
10. Evaluate a conclusion about “Trigonometric identities” using sample size, spread, and limitations.
5 分答案指南
The evaluation links the claim to representativeness, uncertainty/spread, and at least one limitation of data collection.
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。
Mathematics
A Level Year 12: 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。
Quadratics
Inequalities
Functions
Graph transformations
Coordinate geometry
Circular measure
Radians
Arc length
Sector area
Trigonometric identities
Trigonometric equations
Binomial expansion in AS scope
Series where applicable
Differentiation
Tangents and normals
Stationary points
Integration
Area under a curve
重点关注
通常容易出错或需要更多复习的部分。
I choose an appropriate statistic.
I calculate and interpret probabilities.
I check sample and graph quality.
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。
Data: 8, 5, 8, 5. Find mean and range.
Order 6, 4, 8, 2, 4 and find the median.
A bag contains 7 blue and 4 yellow counters. Find P(blue).
In 90 trials, an outcome occurred 18 times. Find relative frequency and percentage.
Values 6, 3 have frequencies 3, 6. Find the weighted mean.
Two samples have the same mean but ranges 9 and 45. Which is more consistent and why?
Choose a suitable graph for “Radians” and justify axes, scale, and title.
Explain why correlation between two variables does not prove causation.
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。