📚 Year 9 Maths Support Answers: Worked Examples for Core Skills | Year 9 数学支持答案：核心技能的分步示例
This support-answers guide for Year 9 maths walks through common question types and the working you need to show. Use it to check method, not just final answers.
本 Year 9 数学支持答案指南逐步讲解常见题型与需要展示的解题过程，用于检查方法而不仅仅是最终答案。
1. Negative Numbers and Order of Operations | 负数与运算顺序
Worked question: Evaluate (-3) × 4 + 12 ÷ (-2) - (-5).
例题：计算 (-3) × 4 + 12 ÷ (-2) - (-5)。
ision gives -6. Now work left to right: -12 + (-6) = -18, then -18 - (-5) = -18 + 5 = -13.
支持答案：使用 BIDMAS 规则。乘法得 -12；除法得 -6。然后从左到右计算：-12 + (-6) = -18，接着 -18 - (-5) = -18 + 5 = -13。
(-3) × 4 + 12 ÷ (-2) - (-5) = -13
Common mistake: writing -18 - (-5) as -23. Subtracting a negative is adding the positive value.
常见错误：把 -18 - (-5) 写成 -23。减去一个负数等于加上它的相反数。
2. Fractions, Decimals and Percentages | 分数、小数与百分比
a decimal and a percentage. Support answer: 7 ÷ 8 = 0.875; percentage = 0.875 × 100 = 87.5%.
例题：将 7/8 写成小数和百分比。支持答案：7 ÷ 8 = 0.875；百分比 = 0.875 × 100 = 87.5%。
Worked question: Order 3/5, 0.58 and 62% from smallest to largest. Convert all to percentages: 3/5 = 60%, 0.58 = 58%, 62% stays 62%. Order: 0.58, 3/5, 62%.
例题：将 3/5、0.58 和 62% 从小到大排列。全部转换为百分比：3/5 = 60%，0.58 = 58%，62% 不变。顺序为：0.58、3/5、62%。
Remember to use a common form when comparing; decimals are often fastest for calculator use.
比较时要使用统一形式；使用计算器时，小数通常最快。
3. Algebraic Simplification and Expansion | 代数化简与展开
Worked question: Expand and simplify 3(2x - 1) + 2(x + 4).
例题：展开并化简 3(2x - 1) + 2(x + 4)。
6x - 3 + 2x + 8. Collect like terms: 6x + 2x = 8x and -3 + 8 = 5, so the answer is 8x + 5.
支持答案：逐项展开括号：6x - 3 + 2x + 8。合并同类项：6x + 2x = 8x，-3 + 8 = 5，因此答案为 8x + 5。
3(2x - 1) + 2(x + 4) = 8x + 5
on: Factorise 4y + 10. The highest common factor of 4 and 10 is 2, so 4y + 10 = 2(2y + 5).
例题：因式分解 4y + 10。4 和 10 的最大公因数是 2，因此 4y + 10 = 2(2y + 5)。
4. Solving Linear Equations | 解一元一次方程
Worked question: Solve 5x - 7 = 3x + 9.
例题：解方程 5x - 7 = 3x + 9。
d 7 to both sides: 5x = 3x + 16. Subtract 3x from both sides: 2x = 16. Divide by 2: x = 8.
支持答案：两边加 7：5x = 3x + 16。两边减 3x：2x = 16。两边除以 2：x = 8。
5x - 7 = 3x + 9 → x = 8
he original equation: left side 5(8) - 7 = 33, right side 3(8) + 9 = 33, so it is correct.
务必把 x = 8 代入原方程检验：左边 5(8) - 7 = 33，右边 3(8) + 9 = 33，因此正确。
5. Ratio and Proportion | 比例与比例问题
Worked question: Divide £240 in the ratio 3 : 5. Total parts = 3 + 5 = 8. One part = £240 ÷ 8 = £30. First amount = 3 × £30 = £90; second amount = 5 × £30 = £150.
例题：按 3 : 5 的比例分配 £240。总份数 = 3 + 5 = 8。一份 = £240 ÷ 8 = £30。第一份 = 3 × £30 = £90；第二份 = 5 × £30 = £150。
ked question: A recipe for 6 people needs 300 g of flour. How much is needed for 10 people? Find flour per person: 300 g ÷ 6 = 50 g. For 10 people: 50 g × 10 = 500 g.
例题：一份 6 人食谱需要 300 克面粉。10 人需要多少？先求每人用量：300 克 ÷ 6 = 50 克。10 人需要：50 克 × 10 = 500 克。
For ratio problems, never add the parts incorrectly when one share is given; always identify the value of one part first.
解决比例问题时，如果已知某一份量，切勿错误相加；始终先求出一份的值。
6. Area and Perimeter of Compound Shapes | 复合图形的面积与周长
Worked question: Find the area of an L-shape made by joining a 5 cm by 3 cm rectangle and a 2 cm by 2 cm square on one corner.
例题：求一个由 5 厘米 × 3 厘米矩形和一个角上连接的 2 厘米 × 2 厘米正方形组成的 L 形的面积。
rectangle = 5 × 3 = 15 cm². Area of square = 2 × 2 = 4 cm². Total area = 15 + 4 = 19 cm².
支持答案：将图形分成两个矩形。第一个矩形面积 = 5 × 3 = 15 平方厘米。正方形面积 = 2 × 2 = 4 平方厘米。总面积 = 15 + 4 = 19 平方厘米。
Total area = 15 cm² + 4 cm² = 19 cm²
For perimeter, add only the outer edges of the combined shape; do not include the internal sides where the two rectangles touch.
计算周长时，只加组合图形的外边；不要包含两个矩形相接的内部边。
7. Volume and Surface Area of Prisms | 棱柱体积与表面积
Worked question: A triangular prism has a right-angled triangle cross-section with base 4 cm, height 3 cm and prism length 10 cm. Find the volume and total surface area.
例题：一个三棱柱的横截面是底为 4 厘米、高为 3 厘米的直角三角形，棱柱长度为 10 厘米。求体积和总表面积。
oss-section = 0.5 × 4 × 3 = 6 cm². Volume = cross-section area × length = 6 × 10 = 60 cm³.
支持答案：横截面积 = 0.5 × 4 × 3 = 6 平方厘米。体积 = 横截面积 × 长度 = 6 × 10 = 60 立方厘米。
The hypotenuse of the triangle is √4² + 3² e cross-section is 4 + 3 + 5 = 12 cm. Surface area = 2 × 6 + 12 × 10 = 12 + 120 = 132 cm².
三角形的斜边为 √4² + 3² = 5 厘米，所以横截面周长为 4 + 3 + 5 = 12 厘米。表面积 = 2 × 6 + 12 × 10 = 12 + 120 = 132 平方厘米。
Volume = 60 cm³, Surface area = 132 cm²
Write the correct units: area in cm², volume in cm³, and surface area also in cm².
写出正确单位：面积用 cm²，体积用 cm³，表面积也用 cm²。
8. Pythagoras' Theorem | 勾股定理
Worked question: Find the hypotenuse of a right-angled triangle with legs 6 cm and 8 cm.
例题：求直角边分别为 6 厘米和 8 厘米的直角三角形的斜边。
r: Use c² = a² + b². Substitute: c² = 6² + 8² = 36 + 64 = 100. Therefore c = √100 = 10 cm.
支持答案：使用 c² = a² + b²。代入：c² = 6² + 8² = 36 + 64 = 100。因此 c = √100 = 10 厘米。
c = √6² + 8² = √100 = 10 cm
Pythagoras' theorem only applies to right-angled triangles. Check the right angle is between the two shorter sides.
勾股定理只适用于直角三角形。请确认直角位于两条较短边之间。
Worked question: Is a triangle with sides 5 cm, 12 cm and 13 cm right-angled? Check 5² + 12² = 25 + 144 = 169 = 13², so yes.
例题：三边分别为 5 厘米、12 厘米和 13 厘米的三角形是直角三角形吗？检验 5² + 12² = 25 + 144 = 169 = 13²，所以是的。
9. Probability of Combined Events | 组合事件概率
Worked question: Two fair six-sided dice are rolled. Find the probability that the sum is 7.
例题：掷两枚公平的六面骰子。求点数和为 7 的概率。
Support answer: List the successful outcomes: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). There are 6 successful outcomes out of 36 total outcomes.
支持答案：列出成功结果：(1,6)、(2,5)、(3,4)、(4,3)、(5,2)、(6,1)。共有 6 个成功结果，总结果数为 36。
P(sum = 7) = 6/36 = 1/6
ilities: P(A and B) = P(A) × P(B). For mutually exclusive events, add their probabilities.
对于相互独立的事件 A 和 B，概率相乘：P(A 且 B) = P(A) × P(B)。对于互斥事件，则概率相加。
Always simplify fractions in probability answers unless the question asks for a decimal or percentage.
除非题目要求小数或百分比，概率答案中的分数要化简。
10. Averages and Range | 平均数与极差
Worked question: Find the mean, median, mode and range of the data set: 5, 9, 12, 12, 16, 20.
例题：求数据集 5、9、12、12、16、20 的平均数、中位数、众数和极差。
rdered data is already given; median = (12 + 12) ÷ 2 = 12. Mode = 12. Range = 20 - 5 = 15.
支持答案：平均数 = (5 + 9 + 12 + 12 + 16 + 20) ÷ 6 = 74 ÷ 6 ≈ 12.3。数据已按顺序排列；中位数 = (12 + 12) ÷ 2 = 12。众数 = 12。极差 = 20 - 5 = 15。
Mean ≈ 12.3, Median = 12, Mode = 12, Range = 15
If the data set has an even number of values, the median is the mean of the two middle values.
如果数据个数为偶数，中位数就是中间两个数的平均数。
11. Straight Line Graphs | 直线图像
Worked question: Draw the graph of y = 2x + 1 for x values -2, -1, 0, 1, 2.
例题：画出 y = 2x + 1 在 x = -2、-1、0、1、2 时的图像。
2(-2) + 1 = -3; for x = -1, y = -1; for x = 0, y = 1; for x = 1, y = 3; for x = 2, y = 5.
支持答案：逐个代入 x 值：当 x = -2，y = 2(-2) + 1 = -3；当 x = -1，y = -1；当 x = 0，y = 1；当 x = 1，y = 3；当 x = 2，y = 5。
The graph is a straight line with gradient 2 and y-intercept 1. The gradient is the coefficient of x.
图像是一条斜率为 2、y 截距为 1 的直线。斜率就是 x 的系数。
y = 2x + 1: gradient = 2, y-intercept = 1
12. Exam Technique and Checking Answers | 考试技巧与答案检查
Always show each step of working, even when the final answer is wrong; method marks can still be awarded.
始终展示每个解题步骤，即使最终答案错误；方法分仍然可以获得。
Check units, simplify fractions, and reread the question to make sure the answer is in the required form.
检查单位、化简分数，并重新读题，确保答案符合要求的形式。
Use estimation to see if a numerical answer is sensible, and substitute solutions back into equations where possible.
使用估算来判断数值答案是否合理，并在可能时将解代回方程检验。
If you finish early, revisit the questions you found hardest and verify your calculations line by line.
如果提前完成，重新检查你觉得最难的题目，并逐行验证计算。
Published by TutorHao | Maths Revision Series | aleveler.com
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