📚 Essential Maths Book 8 Support (Feb 21): Building Confidence in Year 8 Maths | 八年级数学基础支持（2021年2月版）：建立数学信心
This revision guide walks through the core skills in the Essential Maths Book 8 Support edition (February 2021). It is written for Year 8 learners who need clear steps, simple examples and regular practice to strengthen their mathematical understanding.
本复习指南带你梳理《Essential Maths Book 8 Support》（2021年2月版）中的核心技能。它专为需要清晰步骤、简单示例和定期练习来夯实数学基础的八年级学生编写。
1. Understanding Place Value and Integers | 理解位值与整数
Whole numbers are made from digits placed in columns: units, tens, hundreds, thousands and beyond. In Year 8 support work, you must be able to read, write, order and compare large numbers fluently, such as 340 715 or 1 208 046.
整数由位于不同数位的数字组成：个位、十位、百位、千位等。在八年级基础练习中，你需要能够熟练地读、写、排列和比较大数，例如 340 715 或 1 208 046。
For example, in 340 715 the digit 4 has a value of 40 000 because it is in the ten-thousands column. The digit 7 has a value of 700, because it is in the hundreds column. Writing the number in a place-value table helps prevent mistakes with zeros.
例如，在 340 715 中，数字 4 的值是 40 000，因为它位于万位。数字 7 的值是 700，因为它位于百位。把数字写进数位表有助于避免零带来的错误。
Integers include positive and negative whole numbers. We often use a number line to compare them. Remember that on a horizontal number line, numbers increase to the right, so -3 < 2 and 5 > -1.
整数包括正整数和负整数。我们常用数轴来比较它们。记住，在水平数轴上，数字向右逐渐增大，所以 -3 < 2，而 5 > -1。
2. Order of Operations and Directed Numbers | 运算顺序与有向数
The order of operations helps us calculate correctly when a problem has more than one operation. In the UK, many students remember this as BODMAS or BIDMAS: Brackets, Orders (or Indices), Division and Multiplication, Addition and Subtraction.
当一个算式包含多个运算时，运算顺序能帮助我们正确地计算。在英国，许多学生用 BODMAS 或 BIDMAS 来记忆：括号、幂（或指数）、除法和乘法、加法和减法。
Division and multiplication have equal priority, so we work from left to right. The same is true for addition and subtraction. This rule prevents different people getting different answers to the same question.
除法和乘法具有相同的优先级，因此我们按从左到右的顺序计算。加法和减法也是如此。这条规则可以避免不同的人对同一个算式得出不同的答案。
If we incorrectly added first, we would get 3 + 4 = 7 and then 7 × 5 = 35, which is wrong.
例如，计算 3 + 4 × 5 时，我们先算乘法：4 × 5 = 20，再加 3 得到 23。如果错误地先算加法，就会得到 3 + 4 = 7，再算 7 × 5 = 35，这是错误的。
2) = 6 - 2 = 4. Subtracting a negative number is the same as adding: 5 - (-3) = 5 + 3 = 8.
有向数包括正数和负数。加上一个负数等于减去它的正数：6 + (-2) = 6 - 2 = 4。减去一个负数等于加上它的正数：5 - (-3) = 5 + 3 = 8。
3. Fractions: Simplifying and Calculating | 分数：化简与运算
Equivalent fractions represent the same part of a whole even though they look different. To simplify a fraction, divide the numerator and denominator by their highest common factor. For example, 12/18 simplifies to 2/3 because both 12 and 18 can be divided by 6.
等值分数表示相同的部分，但它们看起来不同。化简分数时，把分子和分母同时除以它们的最大公因数。例如，12/18 化简为 2/3，因为 12 和 18 都可以除以 6。
mple, 1/4 + 2/4 = 3/4, but 1/3 + 1/4 needs a common denominator of 12: 4/12 + 3/12 = 7/12.
分数相加或相减时，它们必须有相同的分母。如果分母不同，先为每个分数找出等值分数。例如，1/4 + 2/4 = 3/4，但 1/3 + 1/4 需要公分母 12：4/12 + 3/12 = 7/12。
its reciprocal. For example, 2/3 × 3/5 = 6/15 = 2/5, and 1/2 ÷ 1/4 = 1/2 × 4/1 = 4/2 = 2.
分数相乘时，分子乘分子，分母乘分母。除以一个分数时，乘以它的倒数。例如，2/3 × 3/5 = 6/15 = 2/5，而 1/2 ÷ 1/4 = 1/2 × 4/1 = 4/2 = 2。
Fraction multiplication: a/b × c/d = (a × c) / (b × d)
分数乘法：a/b × c/d = (a × c) / (b × d)
4. Decimals and Percentages | 小数与百分数
Decimal numbers extend place value into tenths, hundredths and thousandths. It is useful to know the value of each decimal digit. For example, in 3.476, the digit 4 is in the tenths column, the 7 is in the hundredths column and the 6 is in the thousandths column.
小数把位值延伸到十分位、百分位和千分位。了解每个小数位的值很有用。例如，在 3.476 中，数字 4 在十分位，数字 7 在百分位，数字 6 在千分位。
Rounding decimals is a key skill. To round 3.476 to two decimal places, look at the third decimal digit. Since it is 6, we round up, so 3.476 rounds to 3.48.
小数取整是一项关键技能。把 3.476 保留两位小数时，看第三位小数。由于它是 6，我们要进位，所以 3.476 约等于 3.48。
of expressing the same amount. You should know common conversions by heart: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%, 1/10 = 0.1 = 10% and 1/5 = 0.2 = 20%.
分数、小数和百分数是表示相同数量的三种方式。你应该熟记常见换算：1/2 = 0.5 = 50%，1/4 = 0.25 = 25%，3/4 = 0.75 = 75%，1/10 = 0.1 = 10%，1/5 = 0.2 = 20%。
f an amount, convert the percentage to a fraction or decimal and multiply. For example, 20% e an amount by a percentage, work out the increase and then add it to the original amount.
求一个数的百分之几时，把百分数转换为分数或小数再相乘。例如，60 的 20% = 0.2 × 60 = 12。要把一个数增加某个百分比，先算出增加量，再加到原来的数上。
5. Ratio and Proportion | 比与比例
A ratio compares two or more quantities. Ratios can be simplified by dividing each part by the same number. For example, 8:12 simplifies to 2:3 because both parts can be divided by 4.
比用于比较两个或多个数量。比可以通过把每一部分都除以同一个数来化简。例如，8:12 可以化简为 2:3，因为两部分都可以除以 4。
When sharing an amount in a ratio, first find the total number of parts. Then divide the amount by the total number of parts to find the value of one part. Finally multiply by the number of parts for each share.
按比例分配一个数量时，先求总份数。然后用总量除以总份数，得到一份的值。最后乘以每部分对应的份数。
For example, to share £60 in the ratio 2:3, the total number of parts is 2 + 3 = 5. One part is £60 ÷ 5 = £12. The two shares are 2 × £12 = £24 and 3 × £12 = £36.
例如，按 2:3 的比例分配 60 英镑，总份数是 2 + 3 = 5。每份是 £60 ÷ 5 = £12。两部分分别是 2 × £12 = £24 和 3 × £12 = £36。
Proportion is used when two quantities change in the same ratio. If 3 pens cost £1.50, then 6 pens cost double, which is £3.00. Finding the unit rate first is often the easiest strategy.
当两个量以相同比例变化时，就会用到比例。如果 3 支笔售价 £1.50，那么 6 支笔的价格是双倍，即 £3.00。先求出单位价格通常是最简单的策略。
6. Algebraic Expressions and Substitution | 代数表达式与代入
Algebra uses letters to represent unknown numbers or changing quantities. An expression such as 3a + 2b combines numbers and letters using operations. You cannot solve an expression, but you can simplify or evaluate it.
代数用字母表示未知数或变化的量。像 3a + 2b 这样的表达式用运算把数字和字母组合在一起。表达式无法求解，但可以化简或求值。
example, 3a + 2a = 5a, but 3a + 2b cannot be simplified because the letters are different.
同类项具有完全相同的字母部分。只有同类项可以合并。例如，3a + 2a = 5a，但 3a + 2b 不能化简，因为字母不同。
hen calculating the result. If x = 4, then the value of 3x + 2 is 3 × 4 + 2 = 12 + 2 = 14.
代入是指用数字替换字母，然后计算结果。如果 x = 4，那么 3x + 2 的值就是 3 × 4 + 2 = 12 + 2 = 14。
, then 2(y + 5) = 2 × (3 + 5) = 2 × 8 = 16. Always show each step clearly in your working.
处理表达式时要小心使用括号。如果 y = 3，那么 2(y + 5) = 2 × (3 + 5) = 2 × 8 = 16。解题时每一步都要写清楚。
7. Solving Linear Equations | 解线性方程
An equation shows that two expressions are equal. To solve an equation, we find the value of the unknown letter that makes the equation true. Keep the equation balanced by doing the same operation to both sides.
方程表示两个表达式相等。解方程就是求出使方程成立的未知字母的值。通过对方程两边进行相同的运算来保持等式平衡。
5 - 5 = 12 - 5, so x = 7. If the equation is 3x = 18, divide both sides by 3 to get x = 6.
使用逆运算把字母单独留在一边。如果方程是 x + 5 = 12，两边都减去 5：x + 5 - 5 = 12 - 5，所以 x = 7。如果方程是 3x = 18，两边都除以 3，得到 x = 6。
Subtract x from both sides to get x + 1 = 6, then subtract 1 from both sides to get x = 5.
对于字母在等号两边的方程，把含字母的项移到一边，数字项移到另一边。例如，2x + 1 = x + 6。两边都减去 x，得到 x + 1 = 6，再两边都减去 1，得到 x = 5。
Balanced equation: if
Published by TutorHao | Year 8 Mathematics Revision Series | aleveler.com 更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图