A Level Year 13 / Mathematics / Curriculum
Pure Mathematics 3: structured theory, worked examples, answered practice, and a mastery checklist for A Level Year 13.
单元
练习前按清晰顺序掌握本章核心概念。
Mark every given on the diagram and state the target clearly. Start from a relationship that connects exactly those quantities.
Every geometric step needs a property, theorem, or definition. A result without justification is not a complete proof.
Length, area, volume, and angle use different units. Round only at the end and keep sufficient intermediate precision.
Draw and label every given before choosing a property or formula. Finish with estimation, an inverse operation, or substitution as a check.
Draw and label every given before choosing a property or formula. Finish with estimation, an inverse operation, or substitution as a check.
Draw and label every given before choosing a property or formula. Finish with estimation, an inverse operation, or substitution as a check.
Draw and label every given before choosing a property or formula. Finish with estimation, an inverse operation, or substitution as a check.
Draw and label every given before choosing a property or formula. Finish with estimation, an inverse operation, or substitution as a check.
Mathematics
逐步查看方法，并理解每一步为何成立。
A right triangle has perpendicular sides 12 cm and 16 cm. Find its hypotenuse and area.
Hypotenuse 20 cm, area 96 cm².
Before calculating, explain the key idea from “Advanced functions” and which conditions must be checked.
The answer should show not only which rule is used for “Advanced functions”, but also why it is valid here.
Pure Mathematics 3
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。
A 5 cm by 2 cm rectangle: find perimeter and area.
P = 2(l+w), A = lw.
P = 14 cm, A = 10 cm².
A triangle has base 32 cm and height 4 cm. Find its area.
A = 1/2 bh.
A = 1/2 · 32 · 4 = 64 cm².
Two angles in a triangle are 40° and 60°. Find the third.
Triangle angles sum to 180°.
80°.
A right triangle has legs 9 cm and 12 cm. Find the hypotenuse.
Use a²+b²=c².
15 cm.
A circle has radius 5 cm. Find circumference and area in terms of π.
C = 2πr, A = πr².
C = 10π cm, A = 25π cm².
Scale 1:600. What real length in metres does 2 cm represent?
Multiply in cm, then convert to metres.
2 · 600 = 1200 cm = 12 m.
A cuboid is 9 cm × 4 cm × 13 cm. Find its volume.
V = lwh.
V = 468 cm³.
Points A(0,0) and B(5,6). Find AB.
AB = √(Δx²+Δy²).
AB = √61.
Pure Mathematics 3
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。
Homework 1: A 8 cm by 3 cm rectangle: find perimeter and area.
P = 2(l+w), A = lw.
P = 22 cm, A = 24 cm².
Homework 2: A triangle has base 20 cm and height 5 cm. Find its area.
A = 1/2 bh.
A = 1/2 · 20 · 5 = 50 cm².
Homework 3: Two angles in a triangle are 70° and 20°. Find the third.
Triangle angles sum to 180°.
90°.
Homework 4: A right triangle has legs 12 cm and 16 cm. Find the hypotenuse.
Use a²+b²=c².
20 cm.
Homework 5: A circle has radius 6 cm. Find circumference and area in terms of π.
C = 2πr, A = πr².
C = 12π cm, A = 36π cm².
Homework 6: Scale 1:900. What real length in metres does 3 cm represent?
Multiply in cm, then convert to metres.
3 · 900 = 2700 cm = 27 m.
50 分钟 / 40 分
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. A 6 cm by 3 cm rectangle: find perimeter and area.
2 分P = 18 cm, A = 18 cm².
2. A triangle has base 45 cm and height 5 cm. Find its area.
3 分A = 1/2 · 45 · 5 = 112.5 cm².
3. Two angles in a triangle are 50° and 20°. Find the third.
3 分110°.
4. A right triangle has legs 12 cm and 16 cm. Find the hypotenuse.
4 分20 cm.
5. A circle has radius 6 cm. Find circumference and area in terms of π.
4 分C = 12π cm, A = 36π cm².
6. Scale 1:700. What real length in metres does 3 cm represent?
4 分3 · 700 = 2100 cm = 21 m.
7. A cuboid is 3 cm × 5 cm × 8 cm. Find its volume.
5 分V = 120 cm³.
8. Points A(0,0) and B(6,2). Find AB.
5 分AB = √40.
9. Similar shapes have length scale factor 1:4. If the small area is 9 cm², find the large area.
5 分9 · 4² = 144 cm².
10. Explain which givens and theorem are needed for a complete argument about “Parametric equations”.
5 分Full response: a labelled diagram, givens, an applicable theorem with its conditions, and a logically derived conclusion.
单元
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 13: 数学与物理的结构化路线，包含章节、练习和下一步。
结构跟随教材，用来帮助组织学习。
通常容易出错或需要更多复习的部分。
从教材、daily material、PDFs、videos 和 worked examples 开始。
进入 full tests 前先做 targeted practice，让覆盖情况更清楚。
如何衡量本章进度，以及何时加入综合 mock。
完成本章后该做什么，以及它如何连接到下一单元。
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。
