📚 Year 10 Edexcel Mathematics: Cross-Curricular Integrated Problem-Solving Training | Year 10 爱德思数学：跨学科综合题型训练
In Year 10 Edexcel Mathematics, you will increasingly encounter problems that blend mathematical concepts with real-world contexts from science, economics, geography, and technology. These cross-curricular questions test your ability to translate a situation into mathematical language and select the right techniques. This training article guides you through the most common cross-over areas, with worked examples and paired explanations to build your confidence.
在Year 10爱德思数学中，你会越来越多地遇到将数学概念与科学、经济、地理和技术等实际情境相结合的问题。这些跨学科题目考查你将现实情境转化为数学语言并选择正确方法的能力。本文通过典型领域、例题示范和中英对照讲解，帮助你进行针对性的综合题型训练，提升解题信心。
1. Interpreting Motion Graphs in Physics | 解读物理中的运动图像
Distance-time and velocity-time graphs are a classic cross-over between mathematics and physics. The gradient of a distance-time graph gives speed, while the gradient of a velocity-time graph gives acceleration. The area under a velocity-time graph represents distance travelled. You need to be able to read values, calculate gradients, and find areas of trapeziums or triangles to answer questions about motion.
距离-时间图和速度-时间图是数学与物理的经典跨学科结合点。距离-时间图的梯度表示速度，速度-时间图的梯度表示加速度。速度-时间图下方的面积则代表移动的距离。你需要能够读取数值、计算梯度，并求出梯形或三角形的面积来回答有关运动的问题。
Example: A car moves from rest. Its velocity-time graph shows a straight line from (0,0) to (8, 12). What is the acceleration and the distance covered in these 8 seconds?
例题：一辆车从静止开始运动，其速度-时间图显示了一条从(0,0)到(8,12)的直线。求这8秒内的加速度和行驶的距离。
= gradient of the line = change in velocity ÷ time taken = (12 - 0) ÷ (8 - 0) = 1.5 m/s².
步骤1：加速度 = 直线的梯度 = 速度变化量 ÷ 所用时间 = (12 - 0) ÷ (8 - 0) = 1.5 m/s²。
under the graph, which is a triangle. Area = ½ × base × height = ½ × 8 s × 12 m/s = 48 m.
步骤2：距离是图像下的面积，为一个三角形。面积 = ½ × 底 × 高 = ½ × 8 s × 12 m/s = 48 m。
2. Direct and Inverse Proportion in Economics | 经济学中的正比与反比
Many economic relationships are modelled using direct and inverse proportion. For example, the total cost of goods is directly proportional to the quantity when the unit price is fixed. Conversely, the time taken to complete a job is inversely proportional to the number of workers, assuming they work at the same rate. You must be able to set up proportionality equations and use the constant of proportionality to solve problems.
许多经济关系可以用正比例和反比例来建模。例如，当单价固定时，总成本与数量成正比。反之，完成一项工作所需的时间与工人数量成反比（假设工作效率相同）。你必须能够建立比例关系式，并利用比例常数来解决问题。
Example: If 5 kg of apples cost £4.00, how much will 9 kg cost? The cost is directly proportional to the mass.
例题：如果5公斤苹果的价格为4.00英镑，那么9公斤苹果需要多少钱？总价与质量成正比。
1: Write the proportion: Cost = k × mass. Using the given pair, 4.00 = k × 5, so k = 0.80 £/kg.
步骤1：列出比例式：总价 = k × 质量。代入已知数据：4.00 = k × 5，得 k = 0.80 英镑/千克。
Step 2: For 9 kg, Cost = 0.80 × 9 = £7.20.
步骤2：对于9千克，总价 = 0.80 × 9 = 7.20英镑。
3. Surface Area and Volume in Design Technology | 设计技术中的表面积与体积
In design and technology, you often need to calculate how much material is required for a product or how much it can hold. This involves the surface area and volume of 3D shapes such as cylinders, cones, and composite solids. Edexcel problems may ask you to work with curved surfaces or to optimise dimensions for cost efficiency.
在设计技术领域，你经常需要计算一个产品需要多少材料，或者它能容纳多少东西。这涉及圆柱、圆锥和组合体等三维形状的表面积和体积。爱德思题目可能会要求你使用曲面进行计算，或者优化尺寸以提高成本效益。
Example: A cylindrical can has a radius of 3 cm and a height of 10 cm. Find its volume and the area of sheet metal needed to make the can (including the top and bottom). Leave your answer in terms of π.
例题：一个圆柱形罐头的底面半径为3厘米，高为10厘米。求它的体积，以及制作这个罐子（包括顶面和底面）所需的金属板面积。答案用π表示。
Volume V = πr²h = π × 3² × 10 = 90π cm³.
体积 V = πr²h = π × 3² × 10 = 90π cm³。
Surface Area = 2πr² + 2πrh = 2π × 9 + 2π × 30 = 18π + 60π = 78π cm².
表面积 = 2πr² + 2πrh = 2π × 9 + 2π × 30 = 18π + 60π = 78π cm²。
4. Statistical Diagrams and Measures in Geography | 地理中的统计图表与统计量
Geography frequently uses bar charts, pie charts, scatter graphs, and cumulative frequency diagrams to present demographic or environmental data. You might be asked to estimate a median from a cumulative frequency curve, calculate the mean from a grouped frequency table, or describe correlation from a scatter graph. Accuracy in reading graphs and interpreting class intervals is key.
地理学经常使用条形图、饼图、散点图和累积频数图来呈现人口或环境数据。你可能会被要求从累积频数曲线上估算中位数，从分组频数表计算均值，或者根据散点图描述相关性。准确读取图表和解释组距是解题关键。
Example: The grouped frequency table shows the ages of residents in a village. Ages 0-20: 35 residents; 20-40: 60; 40-60: 80; 60-80: 45. Calculate an estimate of the mean age.
例题：分组频数表显示了一个村庄居民的年龄分布。0-20岁：35人；20-40岁：60人；40-60岁：80人；60-80岁：45人。计算平均年龄的估计值。
35 = 350, 30×60 = 1800, 50×80 = 4000, 70×45 = 3150. Total sum = 350+1800+4000+3150 = 9300.
步骤1：找出组中值：10, 30, 50, 70。各乘以对应频数：10×35 = 350，30×60 = 1800，50×80 = 4000，70×45 = 3150。总和 = 350+1800+4000+3150 = 9300。
Step 2: Total frequency = 35+60+80+45 = 220. Estimated mean = 9300 ÷ 220 ≈ 42.3 years.
步骤2：总频数 = 35+60+80+45 = 220。估计均值 = 9300 ÷ 220 ≈ 42.3 岁。
5. Algebraic Formulae in Physics | 物理中的代数公式
c relationships. In Year 10, you will rearrange such formulae to make one variable the subject, substitute values, and solve linear equations derived from them. Pay careful attention to units and significant figures in your final answers.
物理方程如 v = u + at、F = ma 和 E = mc² 本质上都是代数关系。在Year 10，你将重新整理这些公式，将某个变量作为主项，代入数值，并求解由公式导出的线性方程。解题时要特别注意最终答案的单位和有效数字。
arrange the formula to find the mass when a force of 15 N produces an acceleration of 3.0 m/s².
例题：作用在物体上的力 F 由公式 F = m × a 给出，其中 m = 4.5 kg，a = 2.0 m/s²。计算 F，然后将公式变形，求出当力为 15 N，加速度为 3.0 m/s² 时的质量。
Step 1: F = 4.5 × 2.0 = 9.0 N.
步骤1：F = 4.5 × 2.0 = 9.0 N。
Rearranging for m gives m = F / a. With F = 15 N and a = 3.0 m/s², m = 15 / 3.0 = 5.0 kg.
步骤2：将公式变形为 m = F / a。代入 F = 15 N，a = 3.0 m/s²，得 m = 15 / 3.0 = 5.0 kg。
6. Probability in Biology and Genetics | 生物学和遗传学中的概率
Probability appears in biology when predicting genetic traits. Using Punnett squares, you can determine the likelihood of offspring inheriting certain alleles. The mathematics involves sample space diagrams, probability fractions, and sometimes tree diagrams for multiple traits. You should be comfortable expressing probabilities as fractions, decimals, or percentages.
概率在生物学中用于预测遗传特征。使用庞纳特方格，你可以确定后代继承特定等位基因的可能性。数学部分涉及样本空间图、概率分数，有时会用到树状图来处理多种性状。你应该熟练地用分数、小数或百分比表示概率。
Example: In pea plants, tall (T) is dominant over short (t). Two heterozygous tall plants (Tt) are crossed. What is the probability that an offspring will be short?
例题：在豌豆植株中，高茎 (T) 对矮茎 (t) 为显性。两株杂合高茎植株 (Tt) 杂交。后代出现矮茎的概率是多少？
Step 1: Draw a Punnett square. Possible gametes are T and t from each parent. The combinations are TT, Tt, Tt, tt. All four outcomes are equally likely.
步骤1：绘制庞纳特方格。每个亲本产生的配子均为 T 和 t。组合为 TT、Tt、Tt、tt。四种结果出现的可能性相同。
ults in a short plant. Probability = number of favourable outcomes / total outcomes = 1/4.
步骤2：只有 tt 表现为矮茎。概率 = 有利结果数 / 总结果数 = 1/4。
7. Trigonometry in Navigation and Surveying | 航海与测量中的三角学
Trigonometric ratios come alive in navigation, where you calculate bearings, distances, and heights using sine, cosine, or tangent. Problems often involve right-angled triangles formed by paths or lines of sight. You may need to apply angle of elevation or depression, and always round answers as specified.
三角比在航海测量中变得生动，你可以利用正弦、余弦或正切计算方位角、距离和高度。这类问题通常涉及路径或视线构成的直角三角形。你可能需要运用仰角或俯角，并按规定进行取整。
Example: From a point 80 m from the base of a lighthouse, the angle of elevation to the top is 35°. Calculate the height of the lighthouse.
例题：在距离灯塔底部80米处，测得灯塔顶部的仰角为35°。计算灯塔的高度。
p 1: In the right-angled triangle, opposite = height h, adjacent = 80 m. tan 35° = h / 80.
步骤1：在此直角三角形中，对边 = 高 h，邻边 = 80 m。tan 35° = h / 80。
h = 80 × tan 35° ≈ 80 × 0.7002 = 56.02 m. Rounding to a suitable degree gives about 56 m.
步骤2：h = 80 × tan 35° ≈ 80 × 0.7002 = 56.02 m。合理四舍五入后约为56米。
8. Compound Interest in Financial Mathematics | 金融数学中的复利
Financial contexts require the compound interest formula A = P(1 + r/n)nt. In Year 10, questions are often simplified to annual compounding, so A = P(1 + r)t. You will need to substitute correctly, calculate powers, and sometimes rearrange to find the rate or time. Understanding percentage multipliers is crucial here.
金融情境中需要使用复利公式 A = P(1 + r/n)nt。在Year 10，题目通常简化为年复利，即 A = P(1 + r)t。你需要正确代入数值，计算幂，有时还需要重新整理公式以求出利率或时间。理解百分数乘数在此处至关重要。
Example: £500 is invested for 3 years at 4% compound interest per annum. Calculate the total amount after 3 years.
例题：将500英镑以每年4%的复利投资3年。计算3年后的总金额。
Step 1: Identify P = 500, r = 0.04, t = 3. Multiplier = 1 + r = 1.04.
步骤1：确定 P = 500，r = 0.04，t = 3。乘数 = 1 + r = 1.04。
Step 2: A = 500 × (1.04)³ = 500 × 1.124864 = £562.43 (rounded to the nearest penny).
步骤2：A = 500 × (1.04)³ = 500 × 1.124864 = 562.43英镑（四舍五入到分）。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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