📚 Year 8 AQA Further Maths: Case Study Practical Exercises | Year 8 AQA 进阶数学：案例分析实战演练
In Year 8 Further Maths, you will often encounter real‑world problems that require you to apply multiple mathematical concepts. These case studies test your ability to model situations, reason logically and communicate solutions clearly. This article presents a series of hands‑on case studies covering algebra, geometry, number theory and probability, designed to mirror the AQA approach. Work through each example to build your problem‑solving toolkit.
在八年级进阶数学中，你经常会遇到需要运用多种数学概念的现实世界问题。这些案例分析考验你建立模型、逻辑推理和清晰表达解决方案的能力。本文提供一系列动手案例分析，涵盖代数、几何、数论和概率，旨在反映 AQA 的风格。通过逐一练习，构建你的解题工具箱。
1. What is a Case Study in Further Maths? | 什么是进阶数学中的案例分析？
A case study is an extended, context‑driven problem where mathematics is used to answer a practical question. Unlike routine exercises, case studies require you to identify the relevant data, choose appropriate strategies and present your reasoning step by step. They often combine topics — for example, algebra and geometry — to solve a single scenario.
案例分析是一个扩展的、情境驱动的问题，要用数学回答一个实际问题。与常规练习不同，案例分析要求你识别相关数据，选择恰当的策略，并逐步展示推理过程。它们往往结合多个主题——例如代数和几何——来解决一个单一情境。
2. Case Study 1: Maximising the Volume of an Open Box | 案例一：最大化无盖盒子的体积
A manufacturer wants to create an open box from a rectangular sheet of card measuring 30 cm by 20 cm. Squares of side length x cm are cut from each corner, and the sides are folded up. What value of x gives the largest possible volume for the box?
某制造商想用一张 30 cm × 20 cm 的矩形卡纸制作一个无盖盒子。从每个角剪去边长为 x cm 的正方形，然后将侧面折起。x 取何值能使盒子的体积最大？
2x)(20 − 2x). We need to maximise V for 0 < x < 10 (since the smaller dimension is 20 cm).
底面的长和宽分别变为 (30 − 2x) 和 (20 − 2x)，高为 x。因此体积 V 为 V = x(30 − 2x)(20 − 2x)。我们需要在 0 < x < 10 范围内最大化 V（因为较短的边是 20 cm）。
m (rejecting the larger root outside the domain). The maximum volume is approximately 1056 cm³.
通过展开并求导可找到最优的 x。展开得 V = 4x³ − 100x² + 600x。令导数 dV/dx = 12x² − 200x + 600 = 0，解得 x ≈ 3.92 cm（舍去超出定义域的另一根）。最大体积约 1056 cm³。
This optimisation problem blends area, volume, algebraic expansion and simple differentiation — a typical Further Maths challenge.
这个优化问题融合了面积、体积、代数展开和简单求导——一个典型的进阶数学挑战。
3. Case Study 2: Cracking a Caesar Cipher with Modular Arithmetic | 案例二：用模运算破解凯撒密码
In cryptography, a Caesar cipher shifts each letter by a fixed number of positions. Suppose we intercept the message 'WKH GRJ LV KLGGHQ' and suspect a shift of 3 backwards (or 23 forwards). Use modular arithmetic to decode it and explain why mod 26 is essential.
Published by TutorHao | Year 8 进阶数学 Revision Series | aleveler.com
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