📚 7C Homework Prelims: Core Skills Revision | 7C 作业预备考试：核心技能复习
This revision guide covers the essential skills you need for the 7C homework prelims. Work through each section carefully, and make sure you can explain every method in your own words before the test.
本复习指南涵盖 7C 作业预备考试所需的核心技能。请仔细学习每一节，并确保在考试前能够用自己的话解释每一种方法。
1. Integers and Order of Operations | 整数与运算顺序
Always follow the order of operations: brackets first, then powers, then multiplication or division from left to right, and finally addition or subtraction from left to right.
始终遵循运算顺序：先算括号，再算乘方，然后从左到右计算乘除，最后从左到右计算加减。
Worked example: Calculate −6 + 4 × (8 − 3)² ÷ 5.
例题：计算 −6 + 4 × (8 − 3)² ÷ 5。
−6 + 4 × (8 − 3)² ÷ 5 = −6 + 4 × 5² ÷ 5 = −6 + 4 × 25 ÷ 5 = −6 + 100 ÷ 5 = −6 + 20 = 14
With negative numbers, remember that adding a negative is the same as subtracting, and subtracting a negative is the same as adding.
对于负数，请记住：加上一个负数等于减去这个数，减去一个负数等于加上这个数。
- −3 + (−5) = −8
- 4 − (−6) = 4 + 6 = 10
The same sign rules apply when multiplying or dividing: two same signs give a positive answer, while two different signs give a negative answer.
乘除时符号规则相同：同号得正，异号得负。
2. Fractions: Simplify and Compare | 分数：化简与比较
To simplify a fraction, divide the numerator and the denominator by their highest common factor. For example, 18/24 simplifies to 3/4 because both 18 and 24 divide by 6.
化简分数时，将分子和分母同时除以它们的最大公因数。例如，18/24 化简为 3/4，因为 18 和 24 都能被 6 整除。
To compare fractions, rewrite them with a common denominator. Find the lowest common multiple of the denominators, convert each fraction, then compare the numerators.
比较分数时，先将它们改写为同分母。找出分母的最小公倍数，转换每个分数，然后比较分子。
| Fraction | Common denominator 6 | Comparison |
|---|---|---|
| 1/2 | 3/6 | 3/6 < 4/6 |
| 2/3 | 4/6 | So 1/2 < 2/3 |
When adding or subtracting fractions, always change them to a common denominator first. Only add or subtract the numerators; keep the denominator the same.
加减分数时，一定要先化为同分母。只加减分子，分母保持不变。
3. Decimals and Place Value | 小数与位值
Place value tells you the value of each digit in a decimal number. Moving one place left multiplies by 10, and moving one place right divides by 10.
位值告诉你小数中每个数字的值。向左移动一位乘以 10，向右移动一位除以 10。
To multiply a decimal by 10, 100 or 1000, move the digits to the left by 1, 2 or 3 places. To divide by 10, 100 or 1000, move the digits to the right.
将小数乘以 10、100 或 1000 时，把数字向左移动 1、2 或 3 位。除以 10、100 或 1000 时，把数字向右移动。
Example: 3.46 × 100 = 346, while 78.5 ÷ 1000 = 0.0785.
例如：3.46 × 100 = 346，而 78.5 ÷ 1000 = 0.0785。
When ordering decimals, compare the digits column by column from left to right. Adding zeros at the end of a decimal does not change its value, which can help with comparison.
排列小数顺序时，从左到右逐位比较。小数末尾加零不改变其值，这有助于比较。
4. Percentages, Fractions and Decimals | 百分比、分数与小数的互化
A percentage is a fraction out of 100. To convert a percentage to a fraction, write it over 100 and simplify if possible.
百分比是分母为 100 的分数。将百分数转换为分数时，写成以 100 为分母的形式，并在可能时化简。
To convert a percentage to a decimal, divide by 100. To convert a decimal to a percentage, multiply by 100.
将百分数转换为小数时除以 100。将小数转换为百分数时乘以 100。
| Percentage | Fraction | Decimal |
|---|---|---|
| 25% | 25/100 = 1/4 | 0.25 |
| 60% | 60/100 = 3/5 | 0.6 |
| 12.5% | 12.5/100 = 1/8 | 0.125 |
To find a percentage of an amount, convert the percentage to a decimal and multiply. For example, 15% of 80 is 0.15 × 80 = 12.
求一个数的百分比时，将百分数转换为小数后相乘。例如，80 的 15% 是 0.15 × 80 = 12。
To increase or decrease by a percentage, calculate the percentage change and add it to or subtract it from the original amount. Alternatively, use a single multiplier such as 1.15 for a 15% increase or 0.85 for a 15% decrease.
按百分比增加或减少时，先计算变化量，再加到原数上或从原数中减去。也可以用单一的乘数，例如增加 15% 用 1.15，减少 15% 用 0.85。
5. Algebraic Expressions | 代数表达式
An expression contains numbers, letters and operations but no equals sign. You can simplify it by collecting like terms: terms that have exactly the same letter part.
表达式包含数字、字母和运算，但没有等号。可以通过合并同类项来化简：同类项是指字母部分完全相同的项。
Example: Simplify 3a + 5b − 2a + 7b.
例题：化简 3a + 5b − 2a + 7b。
3a + 5b − 2a + 7b = (3a − 2a) + (5b + 7b) = a + 12b
When multiplying terms, multiply the numbers first and then write the letters in alphabetical order. For example, 4x × 3y = 12xy.
相乘时，先乘数字，再按字母顺序写出字母。例如，4x × 3y = 12xy。
To expand a bracket, multiply every term inside the bracket by the term outside. For example, 5(2x − 3) = 10x − 15.
去括号时，将括号外的项乘以括号内的每一项。例如，5(2x − 3) = 10x − 15。
6. Solving Linear Equations | 解一元一次方程
An equation has an equals sign and can be solved by using inverse operations to isolate the variable. Whatever you do to one side, you must do to the other side.
方程含有等号，可以通过逆运算来求解，使未知数单独在一边。对等式一边做什么，另一边也必须做什么。
Worked example: Solve 2x + 5 = 17.
例题：解方程 2x + 5 = 17。
2x + 5 = 17
2x = 17 − 5
2x = 12
x = 12 ÷ 2
x = 6
Always check your solution by substituting it back into the original equation. For the example above, 2(6) + 5 = 12 + 5 = 17, so the answer is correct.
始终将解代回原方程进行检查。对于上面的例子，2(6) + 5 = 12 + 5 = 17，因此答案正确。
If the variable appears on both sides, collect the variable terms on one side and the number terms on the other before solving.
如果未知数出现在两边，先把含未知数的项移到一边，把数字项移到另一边，然后再求解。
7. Angles and Lines | 角与直线
Angles on a straight line add up to 180°, and angles around a point add up to 360°. Vertically opposite angles are always equal.
直线上的角之和为 180°，一个点周围的角之和为 360°。对顶角始终相等。
Angles at a point on one side of a straight line are supplementary, so if you know one angle, subtract it from 180° to find the other.
直线同侧的角互为补角，因此如果已知其中一个角，用 180° 减去它即可求出另一个角。
In a triangle, the interior angles always sum to 180°. In a quadrilateral, the interior angles sum to 360°.
三角形的内角和始终为 180°。四边形的内角和为 360°。
When two lines cross, the opposite angles are called vertically opposite angles and are equal. This fact is often used to find missing angles without any calculation other than recognition.
当两条直线相交时，相对的角称为对顶角并且相等。这一事实常用于直接辨认出缺失的角，无需额外计算。
8. Coordinates and Graphs | 坐标与图像
A point is written as an ordered pair (x, y). The x-coordinate tells you how far to move left or right from the origin, and the y-coordinate tells you how far to move up or down.
点写成有序数对 (x, y)。x 坐标表示从原点向左或向右移动的距离，y 坐标表示向上或向下移动的距离。
The origin is (0, 0), where the horizontal x-axis and vertical y-axis intersect.
原点是 (0, 0)，即水平 x 轴与竖直 y 轴的交点。
When plotting coordinates, move along the x-axis first, then move parallel to the y-axis. Plot points carefully with a sharp pencil to avoid losing marks.
描点时，先沿 x 轴移动，再平行于 y 轴移动。用削尖的铅笔仔细描点，以免失分。
You should also be able to read coordinates from a grid and identify the midpoint of two points by finding the averages of the x-coordinates and y-coordinates separately.
你还需要能从网格中读取坐标，并通过分别求 x 坐标和 y 坐标的平均值来找出两点之间的中点。
9. Averages and Range | 平均数与极差
The mean is found by adding all values and dividing by the number of values. The range is the difference between the largest and smallest values.
平均数的计算方法是把所有数值相加，再除以数值的个数。极差是最大值与最小值之间的差。
Worked example: Find the mean and range of 4, 7, 8, 9, 12.
例题：求 4、7、8、9、12 的平均数和极差。
Mean = (4 + 7 + 8 + 9 + 12) ÷ 5 = 40 ÷ 5 = 8
Range = 12 − 4 = 8
The median is the middle value when data are ordered, and the mode is the most frequent value. These are all types of average, but they measure the centre in different ways.
中位数是数据排序后中间的值，众数是出现次数最多的值。这些都是不同类型的平均数，但它们以不同的方式衡量数据的中心。
When a data set has an even number of values, find the median by taking the mean of the two middle values.
当数据集中有偶数个值时，取中间两个值的平均数作为中位数。
10. Problem-Solving Strategy | 解题策略
Read the problem carefully and underline key numbers and words. Identify what is unknown, choose a method, estimate the answer first, then calculate accurately.
仔细阅读题目，在关键数字和词语下划线。确定未知量，选择方法，先估算答案，再进行准确计算。
Write down each step clearly. If you make an error, you will be able to trace your working and correct it. This also helps you gain method marks in the prelims.
清楚地写出每一步。如果出错，你可以追查计算过程并加以改正。这也有助于你在预备考试中获得步骤分。
For multi-step problems, break them into smaller parts. Solve each part one at a time, and keep track of units such as cm, kg, £ or degrees.
对于多步问题，将其分解为较小的部分。一次解决一个部分，并注意单位，如厘米、千克、英镑或度数。
If you have time, use a different method to check your answer, such as substituting back into an equation or rounding to estimate the result.
如果有时间，用另一种方法检查答案，例如代入方程进行回代，或通过四舍五入进行估算。
11. Checking Your Work | 检查你的作业
Before you finish, check every calculation. For addition, use subtraction as a reverse check; for multiplication, use division.
在完成之前，检查每一道计算。加法的逆运算是减法，可以用减法检查；乘法的逆运算是除法，可以用除法检查。
For fraction and decimal problems, ask whether your answer is reasonable. For example, 0.75 of a number should be less than the whole number.
对于分数和小数问题，问自己答案是否合理。例如，一个数的 0.75 应该小于原数。
Finally, make sure you have answered the exact question. If the question asks for a percentage, do not leave your answer as a decimal unless it has been converted correctly.
最后，确保你回答的是题目要求的内容。如果题目要求百分数，不要把答案写成小数，除非已经正确转换。
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