📚 Year 7 CAIE Mathematics: Summer Prep and Bridging Course | Year 7 CAIE 数学：暑期预习与衔接课程
The summer break is the perfect time to solidify your primary maths skills and get a head start on the Year 7 CAIE curriculum. This bridging course will guide you through the essential topics you will encounter, from number and algebra to geometry and data, ensuring a smooth and confident transition into lower secondary mathematics.
暑假是巩固小学数学技能、提前预习 Year 7 CAIE 课程的绝佳时机。本衔接课程将带你梳理你将遇到的核心主题，从数与代数到几何与数据，确保平稳、自信地过渡到初中数学学习。
1. Mastering Place Value and Number Sense | 掌握位值与数感
Place value forms the bedrock of all number work. In Year 7, you must confidently read, write, and order whole numbers up to at least ten million, recognising the value of each digit from millions down to ones, and extending to decimals with tenths, hundredths, and thousandths.
位值是所有数字运算的基石。在 Year 7，你必须能自信地读写和排序至少到一千万的整数，能识别从百万位到个位每一位的数字价值，并延伸到十分位、百分位和千分位的小数。
Comparing numbers always starts with the highest place value. For example, 2,345,678 is less than 2,354,999 because the ten thousands digit 4 is smaller than 5, even though the hundreds digits differ.
比较数字时始终先从最高位值开始。例如，2,345,678 小于 2,354,999，因为万位上的 4 小于 5，即使后面的百位数字不同。
Rounding is a vital estimation skill. To round 47,352 to the nearest thousand, look at the hundreds digit: it is 3, so we round down to 47,000. If it were 5 or more, we would round up.
四舍五入是一项重要的估算技能。将 47,352 四舍五入到千位，看百位数字：百位是 3，所以舍去变为 47,000。若百位是 5 或更大，则向前进一。
Negative numbers appear on the number line to the left of zero. In Year 7, you will order, add, and subtract integers such as -3, 0, and 5, using contexts like temperature or elevation.
负数出现在数轴上零的左边。在 Year 7，你会利用温度或海拔等情境，对像 -3、0 和 5 这样的整数进行排序、加法和减法运算。
2. Efficient Addition and Subtraction | 高效的加法与减法
Mental strategies remain essential. You should be able to add and subtract two-digit and three-digit numbers mentally using partitioning, compensation, or counting on. For larger numbers or decimals, the standard column method ensures accuracy.
心算策略依然重要。你应能通过拆分、补偿或继续数数的方法，心算两位数和三位数的加减。对于更大的数或小数，标准竖式可以确保计算准确。
When using column addition with decimals, align the decimal points and fill empty spaces with zeros as placeholders. For example, 45.6 + 2.78 is rewritten as 45.60 + 2.78, giving 48.38.
使用竖式计算小数加法时，要对齐小数点，并用零占满空位。例如，45.6 + 2.78 写成 45.60 + 2.78，得到 48.38。
Subtraction often requires borrowing. If you have 5.04 – 2.37, you borrow across the decimal point: 5.04 becomes 4.14 (since 1 one = 10 tenths, but we already have zero tenths, so we borrow from the ones to make ten tenths, leaving 4 ones and 14 hundredths), then subtract to get 2.67.
减法常需要借位。计算 5.04 – 2.37 时，需跨小数点借位：5.04 变为 4.14（因为 1 个一等于 10 个十分之一，但十分位是 0，所以从个位借 1 变成 10 个十分之一，个位剩 4，百分位有 14 个百分之一），然后相减得 2.67。
Always estimate first to check the reasonableness of your answer. For 789 + 213, think 800 + 200 = 1000, so an answer around 1002 is sensible.
养成先估算再检查答案是否合理的习惯。计算 789 + 213 时，可以看作 800 + 200 = 1000，因此得到 1002 左右的答案是合理的。
3. Strengthening Multiplication and Division | 加强乘法与除法
By Year 7, you should know all multiplication facts up to 12 × 12 fluently. These facts unlock efficient long multiplication and division. For instance, 24 × 35 can be broken into 24 × 5 = 120 and 24 × 30 = 720, then summed to 840.
到了 Year 7，你应该已经熟练掌握了 12 × 12 以内的所有乘法口诀。这些口诀能让多位数乘除法更高效。例如，24 × 35 可拆分为 24 × 5 = 120 和 24 × 30 = 720，然后相加得到 840。
Long multiplication with decimals requires careful placement of the decimal point. Multiply 3.6 by 2.4 as 36 × 24 = 864. Since there is one decimal digit in each factor (total two), the product is 8.64.
小数乘法需要注意小数点的位置。计算 3.6 × 2.4，先看作 36 × 24 = 864。因为两个因数各有一位小数（共两位），所以积是 8.64。
Division can be thought of as repeated subtraction or sharing. The bus stop method (short division) is efficient for dividing by a single digit. For 789 ÷ 3, set up: 3 ⟌ 789. 7 hundreds ÷ 3 = 2 hundreds remainder 1 hundred, which becomes 18 tens, giving 6 tens remainder 0 tens, then 9 ones ÷ 3 = 3 ones, so 263.
除法可视为重复减法或均分。短除法（“站台法”）对除以一位数很高效。计算 789 ÷ 3，竖式为 3 ⟌ 789。7 个百除以 3 得 2 个百余 1 个百，1 个百化作 18 个十，得 6 个十余 0 个十，再将 9 个一除以 3 得 3 个一，因此商为 263。
Interpreting remainders is important: in real-life problems, a remainder of 2 might mean you need an extra box (round up) or simply state what is left.
对余数的解读很重要：在实际问题中，余数为 2 可能意味着需要多加一个盒子（进位），或只是指明剩下多少。
4. Understanding Fractions and Decimals | 理解分数与小数
Year 7 deepens your understanding of equivalent fractions. You will use multiplication and division to find common denominators and to simplify fractions. For example, 4/6 simplifies to 2/3 (dividing numerator and denominator by 2), and 2/3 is equivalent to 8/12.
Year 7 将加深你对等值分数的理解。你会用乘法和除法寻找公分母并化简分数。例如，4/6 可化简为 2/3（分子分母同除以 2），而 2/3 又等于 8/12。
Converting between fractions and decimals is routine. To turn 3/8 into a decimal, divide 3 by 8 using division: 3.000 ÷ 8 = 0.375. Common equivalents such as 1/4 = 0.25, 1/2 = 0.5, and 3/4 = 0.75 should be memorised.
分数与小数之间的转换是常规操作。把 3/8 化为小数，用 3 除以 8：3.000 ÷ 8 = 0.375。像 1/4 = 0.25、1/2 = 0.5、3/4 = 0.75 这样的常见等式应当熟记。
Adding and subtracting fractions with different denominators involves finding the least common multiple. For 1/4 + 2/3, the LCM of 4 and 3 is 12, so we convert to 3/12 + 8/12 = 11/12.
异分母分数加减时，需要找到最小公倍数。计算 1/4 + 2/3，4 和 3 的最小公倍数是 12，所以转化为 3/12 + 8/12 = 11/12。
Decimals and fractions also appear in measurement and money. A length of 0.75 metres is the same as 3/4 m, and £2.50 is 2 1/2 pounds.
小数和分数也出现在度量与货币中。0.75 米就等于 3/4 米，而 2.50 英镑就是 2 1/2 英镑。
5. Introduction to Percentages | 百分数入门
A percentage is a fraction with a denominator of 100. In Year 7, you will learn to interpret percentages as ‘out of 100’ and convert between fractions, decimals, and percentages. For instance, 40% = 40/100 = 2/5 = 0.4.
百分数是分母为 100 的分数。在 Year 7，你将学习将百分数理解为“每一百份中的多少”，并在分数、小数和百分数之间进行转换。例如，40% = 40/100 = 2/5 = 0.4。
You will find percentages of quantities without a calculator by using benchmark percentages. To find 15% of 200, first find 10% (20), then 5% is half of that (10), so 20 + 10 = 30.
你将学会在不使用计算器的情况下，通过基准百分数求一个数量的百分之几。求 200 的 15%，先找出 10% 是 20，那么 5% 是它的一半 10，所以 20 + 10 = 30。
Percentage change can be introduced through simple discounts or increases. A video game costing £50 with a 10% discount has a price reduction of £5, so the sale price is £45.
通过简单的折扣或上涨可以引入百分比变化。一款视频游戏售价 50 英镑，打 10% 折扣，即减价 5 英镑，因此售价为 45 英镑。
Understanding that percentage and proportion are linked will help later with ratio and proportion topics.
理解百分数与比例相互关联，对接下来的比和比例学习会有帮助。
6. Algebraic Thinking: Expressions and Equations | 代数思维：表达式与方程
Algebra uses letters to represent unknown numbers or variables. In Year 7, you will write simple expressions like ‘3 more than n’ as n + 3, or ‘twice a number add five’ as 2x + 5.
代数用字母表示未知数或变量。在 Year 7，你会写出像“比 n 多 3”写成 n + 3，或“一个数的两倍加五”写成 2x + 5 这样的简单表达式。
Substitution means replacing letters with given values. If a = 4 and b = 2, then the expression 3a – 2b becomes 3 × 4 – 2 × 2 = 12 – 4 = 8.
代入就是用给定数值替换字母。若 a = 4、b = 2，则表达式 3a – 2b 变为 3 × 4 – 2 × 2 = 12 – 4 = 8。
Solving equations involves finding the value of the unknown that makes the equation true. For a simple equation like
x + 7 = 15
, we subtract 7 from both sides to get x = 8.
解方程就是找出使等式成立的未知数的值。对于像
x + 7 = 15
这样的简单方程，两边同时减去 7 得到 x = 8。
You will also encounter two-step equations such as 2y – 3 = 9. Add 3 to both sides (2y = 12), then divide by 2: y = 6. Balancing both sides equally is the golden rule.
你还会遇到像 2y – 3 = 9 这样的两步方程。先两边加 3（2y = 12），再除以 2：y = 6。保持等式两边平衡是黄金法则。
7. Exploring Shapes and Angles | 探索图形与角度
Year 7 geometry classifies angles by size: acute (less than 90°), right angle (90°), obtuse (between 90° and 180°), and reflex (greater than 180°). Measuring and drawing angles with a protractor is a core skill.
Year 7 的几何会根据大小对角分类：锐角（小于 90°）、直角（90°）、钝角（90° 到 180° 之间）和优角（大于 180°）。用量角器测量和绘制角是一项核心技能。
Angles on a straight line add up to 180°, and angles around a point total 360°. If one angle on a straight line is 135°, the adjacent angle must be 45°.
直线上的角之和为 180°，一个点周围的角度之和为 360°。如果直线上一个角是 135°，相邻的角就一定是 45°。
Properties of triangles and quadrilaterals are introduced. A triangle’s interior angles always sum to 180°. An isosceles triangle has two equal sides and two equal angles.
我们将初步了解三角形和四边形的性质。三角形内角和总是 180°。等腰三角形有两条边相等，且两个底角相等。
Symmetry is explored through reflection and rotational symmetry. A regular pentagon has 5 lines of symmetry and rotational symmetry of order 5 (meaning it looks the same 5 times in a full turn).
我们会通过反射对称和旋转对称来探索对称性。正五边形有 5 条对称轴，且旋转对称的阶为 5（即旋转一整圈时可以重合 5 次）。
8. Measuring Length, Area, and Volume | 测量长度、面积与体积
Perimeter is the distance around a shape. For a rectangle with length 8 cm and width 5 cm, P = 2 × (8 + 5) = 26 cm. You will also find perimeters of compound shapes by adding all outer edges.
周长是形状一周的长度。一个长 8 厘米、宽 5 厘米的长方形，周长 P = 2 × (8 + 5) = 26 厘米。你还将通过相加所有外围边长求得组合图形的周长。
Area measures the surface inside a shape. The area of a rectangle is length × width. For the same rectangle, A = 8 × 5 = 40 cm². Triangles are half of a rectangle, so A = ½ × base × height.
面积测量的是形状内部的表面大小。长方形面积 = 长 × 宽，同一个长方形面积 A = 8 × 5 = 40 cm²。三角形可以看作长方形的一半，因此面积 A = ½ × 底 × 高。
Volume for cuboids is found by multiplying length × width × height. A box with dimensions 4 cm, 3 cm, and 2 cm has a volume of 24 cm³. Use counting cubes to visualise the space occupied.
长方体的体积通过长 × 宽 × 高计算。一个尺寸为 4 cm、3 cm、2 cm 的盒子，体积为 24 cm³。可以用数小立方体的方式来形象化所占空间。
Unit conversions are part of measurement: 1 m = 100 cm, 1 km = 1000 m, and 1 litre = 1000 ml. Apply these to real-life contexts like distances and capacity.
单位换算也是测量的一部分：1 m = 100 cm，1 km = 1000 m，1 升 = 1000 毫升。将这些应用到距离和容积等现实情境中。
9. Interpreting Data and Averages | 解读数据与平均数
Year 7 introduces various statistical graphs: bar charts for categorical data, line graphs for time series, and pictograms. Reading and interpreting scales accurately is essential when extracting information.
Year 7 会介绍多种统计图表：用于分类数据的条形图、用于时间序列的折线图，以及象形统计图。在提取信息时，准确读取并解读刻度至关重要。
The mode is the most frequent value. The median is the middle value when data is ordered. The mean is the total of all values divided by the number of values. For the data set 2, 3, 3, 7, 10, the mode is 3, median is 3, and mean is (2+3+3+7+10) ÷ 5 = 5.
众数是出现最频繁的值。中位数是将数据排序后居中的数值。平均数是所有数据的总和除以数据的个数。对于数据集 2, 3, 3, 7, 10，众数是 3，中位数是 3，平均数是 (2+3+3+7+10) ÷ 5 = 5。
A frequency table can help organise data before plotting. Consider the following example showing favourite colours:
频率表可以在绘图前帮我们整理数据。想想下面这个展示最喜爱颜色的例子：
| Colour | Frequency |
|---|---|
| Red | 6 |
| Blue | 10 |
| Green | 4 |
From this table, you can see that blue is the mode and can easily construct a bar chart.
通过这张表，你可以看到蓝色是众数，也能轻松地构建条形图。
Probability at a basic level uses words such as ‘likely’, ‘unlikely’, ‘certain’, and ‘impossible’, and may involve the probability scale from 0 to 1.
基础概率会用到“很可能”、“不太可能”、“一定”和“不可能”等词汇，并可能涉及 0 到 1 的概率标度。
10. Problem-Solving Toolkit | 问题解决工具箱
Throughout Year 7, you will be encouraged to apply maths to unfamiliar problems. The key is to read the problem carefully, identify what is given and what is asked, and choose a suitable strategy: working backwards, drawing a diagram, looking for patterns, or making a list.
整个 Year 7，你将被鼓励将数学应用于不熟悉的问题。关键是要仔细读题，找出已知条件和所求，然后选择合适的策略：逆向求解、画示意图、寻找规律或列出清单。
Estimation and mental checking are part of the toolkit. Before calculating the cost of 6 items at £2.95 each, round to £3: 6 × 3 = £18, so the exact answer of £17.70 is reasonable.
估算和心算验算是工具箱的一部分。计算单价 2.95 英镑的 6 件商品总价前，先四舍五入为 3 英镑：6 × 3 = 18 英镑，这样精确答案 17.70 英镑就合理了。
Using mathematical vocabulary precisely helps communicate reasoning. Words like ‘product’, ‘quotient’, ‘sum’, and ‘difference’ must be used correctly when explaining solutions.
准确使用数学词汇有助于表达推理过程。在解释解题步骤时，必须正确使用“积”、“商”、“和”、“差”等术语。
Practice with past paper-style questions or puzzles strengthens logical thinking. The more you connect different topics – like using fractions to solve a proportion problem – the more confident you become.
用真题风格的题目或趣味谜题进行练习，可以强化逻辑思维。你越能将不同主题联系起来（比如用分数解决比例问题），就会变得越自信。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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