📚 Essential Maths 7H Homework Answers: Key Knowledge Points Explained | Essential Maths 7H 作业答案知识点精讲
This article breaks down the core mathematical ideas behind typical Essential Maths 7H homework questions. Rather than just listing answers, we focus on the reasoning and methods that lead to those answers, covering number operations, algebra, geometry, probability and statistics. Each section explains a key topic with worked examples, common mistakes and tips to deepen understanding.
本文拆解 Essential Maths 7H 典型作业题背后的核心数学思想。我们不只罗列答案，而是聚焦于推导答案的推理过程与方法，涵盖数的运算、代数、几何、概率与统计。每一节通过例题、常见错误与技巧讲解一个关键知识点，帮助加深理解。
1. Working with Negative Numbers | 负数运算
When adding or subtracting negative numbers, imagine moving along a number line. Adding a negative number means moving left; subtracting a negative number means moving right. For multiplication and division, remember that two same signs give a positive result, while two different signs give a negative result.
加减负数时，可想象在数轴上移动。加上一个负数意味着向左移动；减去一个负数意味着向右移动。乘除运算时，记住同号得正，异号得负。
Example: Calculate −7 + (−3) and 5 − (−2). The first: start at −7, add −3 means move 3 left → −10. The second: start at 5, subtract −2 becomes move 2 right → 7. For −4 × 6, different signs give −24. For (−12) ÷ (−3), same signs give +4.
例子：计算 −7 + (−3) 与 5 − (−2)。第一个：从 −7 出发，加上 −3 向左移 3 格 → −10。第二个：从 5 出发，减去 −2 变成向右移 2 格 → 7。对于 −4 × 6，异号得 −24。对于 (−12) ÷ (−3)，同号得 +4。
A common mistake is confusing ‘double negatives’ in subtraction. Remind yourself that a number line always works, or use the rule: a − (−b) = a + b.
常见错误是混淆减法中的“双重负号”。提醒自己数轴永远有效，或用规则：a − (−b) = a + b。
2. Simplifying Algebraic Expressions | 化简代数式
In algebra, only like terms can be combined. Like terms have exactly the same variable part, such as 3x and 5x. Add or subtract the coefficients while keeping the variable unchanged. Terms with different variables or powers cannot be added together.
代数中，只有同类项才能合并。同类项具有完全相同的字母部分，例如 3x 和 5x。将系数相加或相减，字母部分保持不变。字母或指数不同的项不能相加。
Example: Simplify 4a + 7b − 2a + 3b. Group like terms: (4a − 2a) + (7b + 3b) = 2a + 10b.
例子：化简 4a + 7b − 2a + 3b。合并同类项：(4a − 2a) + (7b + 3b) = 2a + 10b。
When expanding brackets, multiply each term inside by the term outside. For 5(2x − 3), multiply 5 by 2x → 10x, and 5 by −3 → −15, giving 10x − 15.
去括号时，将括号外的项乘以括号内的每一项。对于 5(2x − 3)，5 乘 2x 得 10x，5 乘 −3 得 −15，结果为 10x − 15。
Always write simplified expressions in standard form, with terms in alphabetical order or descending powers. Avoid leaving ‘1x’ — write just x.
始终按标准形式书写化简后的表达式，按字母顺序或降幂排列。避免留下“1x”——直接写成 x。
3. Solving One-Step Equations | 解一步方程
To solve a one-step equation, you need to isolate the variable by performing the inverse operation on both sides. If the equation shows addition, subtract; if it shows multiplication, divide. Whatever you do to one side, you must do to the other to keep the balance.
解一步方程，需要通过执行逆运算来分离变量，并在等号两边同时操作。如果方程出现加法，就做减法；如果是乘法，就做除法。无论对一边做什么，另一边也必须做同样运算以保持平衡。
Example: x + 9 = 15. Inverse of +9 is −9, so x + 9 − 9 = 15 − 9 → x = 6. For 4y = 32, divide both sides by 4 → y = 8.
例子：x + 9 = 15。+9 的逆运算是 −9，所以 x + 9 − 9 = 15 − 9 → x = 6。对于 4y = 32，两边同除以 4 → y = 8。
Check your solution by substituting the value back into the original equation. If both sides match, the solution is correct.
通过把值代回原方程检验解。如果两边相等，解就是正确的。
4. Angles in Triangles and Quadrilaterals | 三角形与四边形的内角
The angles inside a triangle always add up to 180°. This fact can be used to find a missing angle when the other two are known. In any quadrilateral, the interior angle sum is 360°.
三角形内角和总是 180°。知道其中两个角时，可用这一事实求出第三个角。任意四边形的内角和为 360°。
Example (Triangle): Two angles are 67° and 54°. The third angle = 180 − 67 − 54 = 59°.
例子（三角形）：两个角分别为 67° 和 54°。第三个角 = 180 − 67 − 54 = 59°。
Example (Quadrilateral): Angles of 95°, 80° and 110° are given. The fourth angle = 360 − (95 + 80 + 110) = 75°.
例子（四边形）：已知三个角 95°、80° 和 110°。第四个角 = 360 − (95 + 80 + 110) = 75°。
Be careful when working with isosceles triangles: the base angles are equal. If you know the vertex angle, subtract from 180° and divide the result by 2 to find each base angle.
处理等腰三角形时要小心：底角相等。如果知道顶角，从 180° 中减去顶角，再将结果除以 2 即得每个底角的度数。
5. Operations with Fractions | 分数运算
Adding and subtracting fractions require a common denominator. Multiply the numerator and denominator of each fraction by whatever number turns the denominator into the least common multiple. For multiplication, simply multiply the numerators and multiply the denominators. For division, flip the second fraction (the divisor) and multiply.
分数加减需要公分母。将每个分数的分子和分母乘以能把分母变成最小公倍数的数。乘法时，分子乘分子，分母乘分母。除法时，将第二个分数（除数）翻转然后相乘。
Example (Addition): 2/3 + 4/5. LCM of 3 and 5 is 15. (2×5)/(3×5) + (4×3)/(5×3) = 10/15 + 12/15 = 22/15 = 1 7/15.
例子（加法）：2/3 + 4/5。3 和 5 的最小公倍数是 15。(2×5)/(3×5) + (4×3)/(5×3) = 10/15 + 12/15 = 22/15 = 1 7/15。
When multiplying a mixed number, first convert it to an improper fraction. 2 1/3 × 3/4 becomes 7/3 × 3/4 = 21/12 = 7/4 = 1 3/4.
乘带分数时，先化成假分数。2 1/3 × 3/4 变成 7/3 × 3/4 = 21/12 = 7/4 = 1 3/4。
Always simplify answers to their lowest terms by dividing numerator and denominator by their highest common factor.
始终将答案化简到最简形式，分子分母同除它们的最大公因数。
6. Fractions, Decimals and Percentages | 分数、小数与百分数转换
To convert a fraction to a decimal, divide the numerator by the denominator. To change a decimal to a percentage, multiply by 100. Converting a percentage to a fraction means writing it over 100 and simplifying.
分数化小数，用分子除以分母。小数化百分数，乘以 100。百分数化为分数，写成分母为 100 的分数再化简。
Example: 3/8 = 3 ÷ 8 = 0.375; 0.375 × 100 = 37.5%. Going the other way: 65% = 65/100 = 13/20.
例子：3/8 = 3 ÷ 8 = 0.375；0.375 × 100 = 37.5%。反方向：65% = 65/100 = 13/20。
Recognising common equivalents helps speed up homework: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%, 1/3 ≈ 0.333… = 33.3%.
熟记常见等价关系能加快作业速度：1/2 = 0.5 = 50%，1/4 = 0.25 = 25%，3/4 = 0.75 = 75%，1/3 ≈ 0.333… = 33.3%。
7. Introduction to Probability | 概率入门
Probability is a number between 0 and 1 that describes how likely an event is to happen. It can be written as a fraction, decimal or percentage. The probability of an impossible event is 0; the probability of a certain event is 1.
概率是介于 0 和 1 之间的一个数，描述事件发生的可能性大小。可用分数、小数或百分数表示。不可能事件的概率是 0；必然事件的概率是 1。
For equally likely outcomes, probability = (number of favourable outcomes) / (total number of possible outcomes). Example: rolling a fair 6-sided dice, P(even number) = 3/6 = 1/2.
对于等可能结果，概率 = （成功结果数）/（所有可能结果数）。例子：抛一个均匀六面骰子，掷出偶数的概率 = 3/6 = 1/2。
The sum of the probabilities of all possible outcomes is always 1. If the probability of winning a game is 0.3, the probability of not winning is 1 − 0.3 = 0.7.
所有可能结果的概率之和总是 1。若赢得比赛的概率是 0.3，则不胜的概率是 1 − 0.3 = 0.7。
Draw a probability scale from 0 to 1, marking events such as ‘sun will rise tomorrow’ near 1, ‘flipping a head’ at 0.5, and ‘rolling a 7 on a standard dice’ at 0.
绘制从 0 到 1 的概率标尺，将“明天太阳升起”标在接近 1 处，“抛硬币得正面”标在 0.5 处，“掷普通骰子得 7”标在 0 处。
8. Averages and Range | 平均数与范围
The three common averages are the mean, the median and the mode. The mean is found by adding all values and dividing by the number of values. The median is the middle value when data are ordered. The mode is the most frequent value.
三种常见的平均数是均值、中位数和众数。均值是将所有数值相加再除以数值的个数。中位数是将数据排序后位于中间的值。众数是出现次数最多的值。
The range measures spread: range = largest value − smallest value. A larger range indicates greater variability.
范围衡量离散程度：范围 = 最大值 − 最小值。范围越大表示变异性越大。
Example: Data set: 5, 8, 12, 8, 14, 7. Mean = (5+8+12+8+14+7) ÷ 6 = 54 ÷ 6 = 9. Order: 5, 7, 8, 8, 12, 14. Median: average of 3rd and 4th = (8+8)/2 = 8. Mode = 8 (occurs twice). Range = 14 − 5 = 9.
例子：数据集：5, 8, 12, 8, 14, 7。均值 = (5+8+12+8+14+7) ÷ 6 = 54 ÷ 6 = 9。排序：5, 7, 8, 8, 12, 14。中位数：第 3 和第 4 个数的平均 = (8+8)/2 = 8。众数 = 8（出现两次）。范围 = 14 − 5 = 9。
When there is an even number of data points, the median is the mean of the two middle numbers. Always list data in order first.
当数据个数为偶数时，中位数是中间两个数的平均数。务必先将数据按大小排列。
9. Sequences and Patterns | 数列与规律
A sequence is a list of numbers following a rule. The rule often involves adding or multiplying by a constant. To find the next term, apply the rule. To find the nth term of a linear sequence, use the formula: nth term = (difference × n) + adjustment.
数列是按照一定规律排列的一列数。规律通常涉及加上或乘以一个常数。要找下一项，应用规律即可。要找线性数列的第 n 项，使用公式：第 n 项 =（公差 × n）+ 调整数。
Example: Sequence 3, 7, 11, 15 … The difference is +4 each time. The 0th term (before the first) would be −1, so nth term = 4n − 1. Check: n=1 gives 3, n=2 gives 7, etc.
例子：数列 3, 7, 11, 15 … 每次差为 +4。第 0 项（第一项前）为 −1，所以第 n 项 = 4n − 1。检验：n=1 得 3，n=2 得 7，等等。
A common approach is to write the position numbers (1, 2, 3, …) above the terms. Multiply the position by the common difference, then ask what you must add or subtract to get the first term.
一个常见方法是将位置数（1, 2, 3, …）写在项的上方。用位置数乘以公差，再思考需要加上或减去多少才能得到第一项。
10. Coordinates and Straight-Line Graphs | 坐标与直线图
Coordinates are written in the form (x, y), where x is the horizontal position and y is the vertical position. The origin (0,0) is where the axes cross. Moving right increases x; moving up increases y.
坐标以 (x, y) 形式书写，x 表示水平位置，y 表示垂直位置。原点 (0,0) 是两轴相交的地方。向右 x 增加；向上 y 增加。
To plot a point, start at the origin, move along the x-axis first, then up or down for y. For a straight-line graph like y = 2x + 1, create a table of x-values, calculate y, plot the points and join them with a ruler.
描点绘图时，从原点出发，先沿 x 轴移动，再根据 y 值上下移动。对于 y = 2x + 1 这样的直线图，建立一个 x 值的表格，计算 y 值，描点后用直尺连线。
The equation of a straight line is often written as y = mx + c, where m is the gradient (steepness) and c is the y-intercept (where the line crosses the y-axis). In y = 3x − 2, the gradient is 3 and the intercept is −2.
直线方程常写作 y = mx + c，其中 m 是斜率（坡度），c 是 y 轴截距（直线与 y 轴的交点）。在 y = 3x − 2 中，斜率为 3，截距为 −2。
11. Rounding and Estimation | 四舍五入与估算
Rounding makes numbers easier to work with. To round to a given place value, look at the digit immediately to its right. If that digit is 5 or more, round up; if it is 4 or less, round down (leave the target digit unchanged).
四舍五入使数字更便于处理。四舍五入到某一位时，看它右边的那一位数字。如果该数字是 5 或更大，则进 1；如果是 4 或更小，则舍去（目标位数字不变）。
Example: Round 2847 to the nearest hundred. The tens digit is 4, which is less than 5, so the hundreds digit stays 8, and all digits to the right become zero → 2800.
例子：把 2847 四舍五入到百位。十位数字是 4，小于 5，所以百位数字 8 保持不变，右边数字全变零 → 2800。
Estimation involves rounding numbers before calculating to get an approximate answer quickly. For 624 × 30.5, round 624 to 600 and 30.5 to 30, estimate: 600 × 30 = 18 000. The exact answer is 19 032, so the estimate is close.
估算是先对数字四舍五入再计算，快速得到近似答案。对于 624 × 30.5，将 624 四舍五入为 600，30.5 为 30，估算：600 × 30 = 18 000。精确答案是 19 032，估算值很接近。
Estimation is a powerful checking tool. If your calculated answer is far from the estimate, you’ve likely made an error.
估算是很有用的检验工具。如果你的计算结果与估算值相差甚远，很可能出错了。
12. Units of Measurement and Conversions | 测量单位与换算
Understanding metric units is essential. Length: 10 mm = 1 cm, 100 cm = 1 m, 1000 m = 1 km. Mass: 1000 g = 1 kg, 1000 kg = 1 tonne. Capacity: 1000 ml = 1 litre, and for cubic measure, 1 cm³ = 1 ml.
理解公制单位很关键。长度：10 毫米 = 1 厘米，100 厘米 = 1 米，1000 米 = 1 千米。质量：1000 克 = 1 千克，1000 千克 = 1 吨。容量：1000 毫升 = 1 升，体积量度中，1 立方厘米 = 1 毫升。
To convert from a smaller unit to a larger unit, divide; from larger to smaller, multiply. For example, 4500 m into kilometres: divide by 1000 → 4.5 km. 0.8 kg into grams: multiply by 1000 → 800 g.
从小单位换算成大单位，除以进率；从大单位换算成小单位，乘以进率。例如，4500 米化为千米：除以 1000 → 4.5 千米。0.8 千克化为克：乘以 1000 → 800 克。
When solving word problems, make sure all units are the same before calculating. If a length is given in metres and centimetres, choose one unit and convert consistently.
解应用题时，确保所有单位一致再计算。如果一个长度既给出米又给出厘米，选择一种单位并全都换算一致。
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