Year 7 SQA Advanced Mathematics: Summer Preparation and Bridging Course | SQA 数学进阶：暑期预习与衔接课程（7年级）

📚 Year 7 SQA Advanced Mathematics: Summer Preparation and Bridging Course | SQA 数学进阶：暑期预习与衔接课程（7年级）

Welcome to your summer bridging programme for Year 7 Advanced Mathematics under the Scottish SQA framework. The move from primary to secondary mathematics can feel like a giant leap, but with the right preparation you will build confidence, deepen your understanding and be ready to tackle the challenges of the Broad General Education (BGE) and beyond. In this article we break down the key concepts you will encounter in Year 7, show how they connect to what you already know, and provide practical study strategies for the summer break.

欢迎来到为苏格兰 SQA 体系下的 7 年级进阶数学设计的暑期衔接课程。从小学到中学数学的过渡可能会让人感觉像是一次巨大的飞跃，但通过正确的准备，你将建立信心、加深理解，并能从容应对广泛通用教育阶段乃至未来的挑战。本文将分解你在 7 年级将会遇到的关键概念，展示它们与你已有知识的联系，并为你提供暑期实用的学习策略。

1. Bridging from Arithmetic to Algebra | 从算术到代数的衔接

In primary school you used empty boxes or blanks for missing numbers, such as 3 + __ = 10. In Year 7 those blanks are replaced by letters like x, y or n. This is the beginning of algebra, a way of writing general rules and solving problems when some quantities are unknown. The key is to treat a letter as a number you have not yet found.

在小学你使用方框或空格来表示缺失的数字，例如 3 + __ = 10。到了 7 年级，这些空格被字母如 x、y 或 n 取代。这就是代数的开端——一种书写一般性规则并在某些量未知时解决问题的方法。关键是把这个字母当作一个尚未发现的数字来对待。

Start by translating simple sentences into algebraic expressions: ‘a number increased by 5’ becomes x + 5, ‘twice a number’ becomes 2y. Practice substituting values into expressions like 3a + 2 when a = 4. Remember that in algebra we usually write the number before the letter (4x not x4) and we omit the multiplication sign (so 5 × m is written as 5m).

从将简单的语句翻译成代数表达式开始：‘一个数增加 5’ 写成 x + 5，‘一个数的两倍’ 写成 2y。练习当 a = 4 时计算表达式 3a + 2 的值。记住在代数中我们通常把数字写在字母前面（4x 而不是 x4），并省略乘号（因此 5 × m 写作 5m）。

Words	Expression
A number plus 7	n + 7
5 less than k	k – 5
The product of t and 9	9t
p divided by 4	p/4

2. Integers and Order of Operations | 整数与运算顺序

Year 7 extends your number system to include negative integers confidently. You need to be able to add, subtract, multiply and divide with positive and negative numbers, using a number line as a visual aid. Remember that subtracting a negative is the same as adding the positive: 5 − (−3) = 5 + 3 = 8.

7 年级将你的数系扩展到能自信地处理负整数。你需要能够正负数进行加、减、乘、除运算，并利用数轴作为可视化辅助。记住，减去一个负数等同于加上它的正数：5 − (−3) = 5 + 3 = 8。

The correct order of operations is vital when calculations mix different operations. In Scotland we often use BODMAS: Brackets, Orders (powers and roots), Division and Multiplication (left to right), Addition and Subtraction (left to right). For example, 2 + 3 × 4 is 14, not 20, because multiplication is performed before addition.

当计算涉及多种运算时，正确的运算顺序至关重要。在苏格兰我们常用 BODMAS 规则：括号、阶（幂和方根）、除法和乘法（从左到右）、加法和减法（从左到右）。例如 2 + 3 × 4 等于 14 而不是 20，因为乘法先于加法进行。

English: Calculate (−6) × 3 ÷ (−2). First −6 × 3 = −18, then −18 ÷ (−2) = 9.
中文：计算 (−6) × 3 ÷ (−2)。先 −6 × 3 = −18，再 −18 ÷ (−2) = 9。
English: Evaluate 4 + 5² − 6 ÷ 3. Orders first: 5² = 25, then division: 6 ÷ 3 = 2, so 4 + 25 − 2 = 27.
中文：求 4 + 5² − 6 ÷ 3 的值。先算幂：5² = 25，然后除法：6 ÷ 3 = 2，因此 4 + 25 − 2 = 27。

3. Fractions, Decimals and Percentages | 分数、小数与百分数

Fluency in converting between fractions, decimals and percentages is one of the most important mathematical skills in Year 7. You should know common equivalents such as 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, and 1/3 ≈ 0.333… = 33½%. Being able to move between these forms allows you to choose the most efficient method for each problem.

在分数、小数和百分数之间进行熟练转换是 7 年级最重要的数学技能之一。你需要知道常见的等值关系，如 1/2 = 0.5 = 50%，1/4 = 0.25 = 25%，以及 1/3 ≈ 0.333… = 33½%。能够在这几种形式之间转换能让你为每个问题选择最高效的方法。

To add or subtract fractions, first find a common denominator. For multiplication, simply multiply numerators together and denominators together. When dividing by a fraction, flip the second fraction and multiply (keep-change-flip). With decimals, pay close attention to place value when adding and subtracting; with multiplication, count the decimal places in the factors.

要对分数进行加减，首先要找到公分母。做乘法时，直接将分子相乘、分母相乘。除以一个分数时，将第二个分数颠倒再相乘（保持、改变、翻转）。处理小数时，加减运算要格外注意位值；乘法要数清因数中的小数位数。

Example: A shirt is priced at £40. During a sale, it is reduced by 15%. What is the sale price? 15% of £40 = 0.15 × 40 = £6, so sale price is £40 − £6 = £34. This combines percentages with decimal multiplication.

示例：一件衬衫标价 £40。促销期间降价 15%。售价是多少？£40 的 15% = 0.15 × 40 = £6，因此售价为 £40 − £6 = £34。这道题将百分数与小数乘法结合在了一起。

4. Equations and Simple Inequalities | 简易方程与不等式入门

An equation is like a balance scale: whatever you do to one side, you must do to the other to keep it balanced. The goal is to isolate the unknown on one side. Start with one-step equations, such as x + 7 = 15, then move to two-step equations like 3y − 4 = 11. Always check your solution by substituting it back into the original equation.

方程就像一架天平：你对一边做什么，就必须对另一边做同样的事以保持平衡。目标是将未知数单独移到一边。先从一步方程开始，如 x + 7 = 15，然后过渡到两步方程，如 3y − 4 = 11。始终将解代入原方程进行检验。

Inequalities use symbols like < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to). Solving an inequality is similar to solving an equation, with one crucial difference: when you multiply or divide by a negative number, you must reverse the inequality sign. For instance, −2x < 8 becomes x > −4 after dividing by −2 and flipping the sign.

不等式使用诸如 <（小于）、>（大于）、≤（小于等于）、≥（大于等于）等符号。解不等式与解方程类似，但有一个关键区别：当你乘以或除以一个负数时，必须反转不等号。例如，−2x < 8 在除以 −2 并反转符号后变为 x > −4。

English: Solve 2a + 3 = 13. Subtract 3: 2a = 10, then divide by 2: a = 5. Check: 2(5) + 3 = 13.
中文：解 2a + 3 = 13。减 3：2a = 10，再除以 2：a = 5。检验：2(5) + 3 = 13。
English: Solve 5 − m ≤ 2. Subtract 5: −m ≤ −3, multiply by −1 and reverse sign: m ≥ 3.
中文：解 5 − m ≤ 2。减 5：−m ≤ −3，乘以 −1 并反转符号：m ≥ 3。

5. Sequences and Patterns | 序列与模式

In Year 7 you will explore number sequences and learn to find the rule that generates them. A sequence is an ordered list of numbers, often following a term-to-term rule (how you get from one term to the next) or a position-to-term rule (the nth term). Being able to describe a pattern in words and then with algebra is a key algebraic thinking skill.

在 7 年级你会探究数列，并学会找出生成它们的规则。数列是一组有序的数字，通常遵循一项到下一项的递推规则或位置到项的规则（第 n 项）。能够先用文字描述规律，再用代数表达是一项关键的代数思维能力。

Start with linear sequences where the difference between terms is constant. For the sequence 4, 7, 10, 13, … the term-to-term rule is ‘add 3’. The position-to-term rule (nth term) can be written as 3n + 1, because when n = 1, 3(1) + 1 = 4; when n = 2, 3(2) + 1 = 7, and so on. Understanding the connection between the common difference and the coefficient of n is essential.

从项差恒定的线性序列开始。对于序列 4, 7, 10, 13, …，递推规则是‘加 3’。位置到项的规则（第 n 项）可以写作 3n + 1，因为当 n = 1 时，3(1) + 1 = 4；当 n = 2 时，3(2) + 1 = 7，以此类推。理解公差与 n 的系数之间的联系至关重要。

Also practise with patterns of shapes, matches or dots. Being able to draw the next diagram and then explain how many tiles are needed for the 10th or 100th pattern using the nth term reinforces the bridge between concrete and abstract thinking.

同时也要练习用形状、火柴棍或圆点构成的模式。能够画出下一个图形，然后利用第 n 项解释第 10 个或第 100 个图形需要多少块材料，这会巩固具象与抽象思维之间的桥梁。

6. Angles and Basic Geometry | 角度与基础几何

Geometry in Year 7 builds on your knowledge of shape and space. You will classify angles as acute, obtuse, right, reflex or straight, and you will learn to estimate and measure angles accurately with a protractor. The key angle facts to memorise include: angles on a straight line sum to 180°, angles around a point sum to 360°, and vertically opposite angles are equal.

7 年级的几何学建立在你已有的形状与空间知识之上。你将把角分为锐角、钝角、直角、优角和周角，并学习使用量角器准确地估测和测量角度。需要记住的关键角度事实包括：直线上的角之和为 180°，一点周围的角之和为 360°，对顶角相等。

In triangles, the sum of the interior angles is always 180°. Know the properties of different triangles: equilateral (all sides and angles equal, each 60°), isosceles (two equal sides and two equal base angles), and scalene (all sides and angles different). You will also begin to work with parallel lines and recognise alternate and corresponding angles.

在三角形中，内角之和恒为 180°。要了解不同三角形的性质：等边三角形（所有边和内角相等，每个角 60°）、等腰三角形（两条边相等，两个底角相等）以及不等边三角形（所有边和角都不相等）。你还将开始接触平行线并识别交错角与同位角。

When giving reasons for angle calculations, use precise language: ‘angles on a straight line sum to 180°’, ‘angles in a triangle sum to 180°’, or ‘alternate angles on parallel lines are equal’. This attention to justification is a valuable skill for SQA-style exams later on.

在为角度计算提供理由时，要使用精确的语言：‘直线上的角之和为 180°’，‘三角形内角之和为 180°’，或‘平行线上的交错角相等’。这种对证明过程的注重是未来 SQA 考试中一项宝贵的技能。

7. Perimeter, Area and Volume | 周长、面积与体积

Measurement moves from counting squares to using formulae in Year 7. Perimeter is the distance around a shape; for a rectangle, P = 2(l + w). Area is the space inside a 2D shape; for a rectangle, A = l × w; for a triangle, A = ½ × base × height; for a parallelogram, A = base × vertical height. Always include the correct units (mm, cm, m, km for length; mm², cm², m² for area).

在 7 年级，测量从数方格发展为使用公式。周长是围绕一个形状的距离；对于矩形，P = 2(l + w)。面积是二维图形内部的空间；对于矩形，A = l × w；对于三角形，A = ½ × 底 × 高；对于平行四边形，A = 底 × 垂直高度。永远要标注正确的单位（长度用 mm、cm、m、km；面积用 mm²、cm²、m²）。

For compound shapes, split the shape into rectangles or triangles, calculate each area separately and then add or subtract as needed. Volume of a cuboid is found by multiplying length × width × height, V = l × w × h, with units such as cm³. Recognise that capacity in litres is linked to volume: 1 litre = 1000 cm³.

对于组合图形，将图形分割为矩形或三角形，分别计算各部分的面积，然后根据需要相加或相减。长方体的体积通过长 × 宽 × 高得出，V = l × w × h，单位例如 cm³。要认识到以升为单位的容量与体积相关：1 升 = 1000 cm³。

A typical problem: A rectangular garden is 8 m long and 5 m wide. A path 1 m wide runs around the inside edge. Find the area of the path. The outer rectangle area is 8 × 5 = 40 m². The inner lawn is 6 m by 3 m (subtracting 1 m from each side twice), area = 18 m². Area of path = 40 − 18 = 22 m².

一道典型的题目：一个长方形花园长 8 m、宽 5 m。一条 1 m 宽的小径沿内侧边缘环绕。求小径的面积。外边缘矩形面积为 8 × 5 = 40 m²。内边缘草坪将每边减去两个 1 m 后为 6 m × 3 m，面积 = 18 m²。小径面积 = 40 − 18 = 22 m²。

8. Statistics and Probability | 统计与概率

In Year 7 you will collect, organise and interpret data using a variety of charts and graphs, including bar charts, line graphs, pictograms and pie charts. You should be able to calculate the mean, median, mode and range of a data set and understand what each average tells you about the data.

在 7 年级你将使用多种统计图表收集、整理和解读数据，包括条形图、折线图、象形图和饼图。你应该能够计算一组数据的平均数、中位数、众数和极差，并理解每种平均数所传达的不同信息。

Probability is introduced as a measure of how likely an event is, on a scale from 0 (impossible) to 1 (certain). The probability of an event is calculated as: number of favourable outcomes ÷ total number of possible outcomes. Practise expressing probabilities as fractions, decimals and percentages, and understand that the probabilities of all possible outcomes must sum to 1.

概率被引入作为衡量事件发生可能性的工具，其尺度从 0（不可能）到 1（确定）。事件的概率计算公式为：有利结果的数量 ÷ 可能结果的总数。练习将概率表示为分数、小数和百分数，并理解所有可能结果的概率之和必须为 1。

For example, if you roll a fair six-sided die, the probability of rolling a number greater than 4 is: favourable outcomes {5, 6} = 2, total outcomes = 6, so P(>4) = 2/6 = 1/3. In experimental probability, you compare theoretical probability with results from an actual experiment and discuss why they might differ.

例如，抛一个均匀的六面骰子，掷出大于 4 的点的概率为：有利结果 {5, 6} = 2，总结果数 = 6，因此 P(>4) = 2/6 = 1/3。在实验概率中，你要比较理论概率与实际实验得到的结果，并讨论它们可能不同的原因。

9. Ratio and Proportion | 比率与比例

Ratio compares the sizes of two or more quantities, while proportion describes how a quantity changes in relation to another. You will learn to simplify ratios by dividing by the highest common factor, just like simplifying fractions. A ratio of 12:8 simplifies to 3:2 (both divided by 4).

比率用来比较两个或多个量的大小，而比例则描述一个量相对于另一个量如何变化。你将学习通过除以最大公因数来化简比，就像约简分数一样。比 12:8 可化简为 3:2（两者同除以 4）。

Proportional reasoning allows you to solve problems such as ‘If 5 pens cost £3.50, how much do 8 pens cost?’ Find the unit cost first: £3.50 ÷ 5 = £0.70 per pen, then multiply by 8: £5.60. Or use a scaling method: the multiplier from 5 to 8 is 8/5, so cost = 8/5 × £3.50 = £5.60.

比例推理让你能够解决诸如‘如果 5 支笔花费 £3.50，那么 8 支笔需要多少钱？’的问题。先求出单位价格：£3.50 ÷ 5 = 每支 £0.70，然后乘以 8 得到 £5.60。也可以使用缩放法：从 5 到 8 的乘数是 8/5，因此费用 = 8/5 × £3.50 = £5.60。

Recipe problems are common: A recipe for 6 people uses 300 g of flour. How much flour is needed for 10 people? Ratio 6:10 simplifies to 3:5. Flour for 6 people is 300 g, so 1 part = 300 ÷ 6 = 50 g; for 10 people: 10 × 50 = 500 g.

菜谱问题很常见：一份供 6 人食用的菜谱使用 300 g 面粉。10 人份需要多少面粉？6:10 可化简为 3:5。6 人份用面粉 300 g，因此每一份为 300 ÷ 6 = 50 g；10 人份：10 × 50 = 500 g。

10. Problem-Solving and Mathematical Reasoning | 问题解决与逻辑推理

Year 7 advanced mathematics places a strong emphasis on applying your skills to unfamiliar problems. This might involve multi-step word problems, puzzles or investigations. A good strategy is to read the problem carefully, identify what is known and what is unknown, and decide which mathematical tools to use.

7 年级进阶数学非常强调将你的技能应用于不熟悉的问题。这可能涉及多步文字题、智力游戏或探究活动。一个好的策略是仔细审题，分辨已知和未知信息，并决定使用哪些数学工具。

Practise breaking down problems into smaller steps. If the problem says, ‘Tom thinks of a number, multiplies it by 3, subtracts 7 and gets 14’, you can translate this into an equation: 3x − 7 = 14, then solve to find x = 7. Using worked examples and trying similar problems is an effective way to build logical reasoning.

练习将问题分解为较小的步骤。如果题目说：‘汤姆想一个数，将它乘以 3，减去 7 后得到 14’，你可以将其转化为方程：3x − 7 = 14，然后解出 x = 7。使用例题并尝试类似问题是培养逻辑推理能力的有效方法。

Explain your thinking out loud or in writing. Being able to justify each step—’I added 5 to both sides because I need to isolate x’—deepens understanding and prepares you for the reasoning required in later SQA examinations.

出声或在纸上解释你的思考过程。能够为每一步提供理由——‘我在等式两边加上 5，因为我需要把 x 单独分离出来’——这将加深理解并为你日后 SQA 考试所需的推理做好准备。

11. Using a Calculator Wisely | 合理使用计算器

While mental arithmetic and written methods are essential, a calculator is a powerful tool when used correctly. In Year 7 you should learn to use a scientific calculator for BODMAS calculations, fractions, percentages, and powers. Always check that your answer is sensible by estimating first.

虽然心算和笔算方法必不可少，但当正确使用时计算器是一个强大的工具。在 7 年级你应该学会使用科学计算器进行 BODMAS 计算、分数、百分数和幂运算。永远通过先做估算来检查你的答案是否合理。

For instance, to calculate √(5.76 + 8.2)², you might estimate: 5.76 + 8.2 ≈ 14, and the square root is about 3.7. Then use the calculator carefully with brackets: √( (5.76 + 8.2)² ) actually gives you 14. You can also explore memory functions and the fraction key (a b/c) to save time.

例如，要计算 √(5.76 + 8.2)²，你可以先估算：5.76 + 8.2 ≈ 14，平方根大约为 3.7。然后仔细地使用带括号的计算器：√( (5.76 + 8.2)² ) 实际上得到 14。你还可以探索记忆功能和分数键（a b/c）以节省时间。

Be careful with negative numbers: use the (−) key rather than the minus key for entering a negative number. This prevents errors when squaring negatives: (−4)² = 16, whereas −4² is interpreted as −(4²) = −16.

处理负数时要小心：使用 (−) 键而不是减号键来输入一个负数。这样可以避免在平方负数时出错：(−4)² = 16，而 −4² 会被理解为 −(4²) = −16。

12. Summer Study Plan and Mindset | 暑期学习计划与心态

To make the most of your summer, aim for little and often rather than last-minute cramming. Spend 15–30 minutes a day, three to five days a week, reviewing one topic at a time. Use a mix of online practice platforms, workbooks, and real-life maths like cooking, budgeting and measuring around the home.

为了充分利用暑假，要追求少量多次，而不是临时抱佛脚。每天花 15-30 分钟，每周三到五天，每次复习一个主题。结合使用在线练习平台、练习册以及生活中的数学活动，如烹饪、做预算和在家中进行测量。

Try keeping a ‘maths diary’ where you record something you learned or a problem you solved each day. This builds a growth mindset—the belief that you can improve with effort. Mistakes are a natural part of learning; when you get a problem wrong, take time to understand why and correct it.

试着写一本‘数学日记’，每天记录你学到的一点内容或解决的一个问题。这会培养一种成长型思维模式——即相信通过努力可以进步的信念。错误是学习过程中自然的一部分；当你做错一道题时，花时间弄清楚为什么错了并改正它。

Before the new term, review your times tables up to 12 × 12 and your number bonds. These facts should be automatic so that your working memory is free for higher-level reasoning. A confident start in Year 7 Advanced Mathematics is built on both solid foundational skills and a positive, resilient attitude.

在新学期开始前，复习巩固 12 × 12 以内的乘法表和数字组合。这些知识应该达到自动化的程度，这样你的工作记忆就可以腾出来进行更高层次的推理。7 年级进阶数学的一个自信开头是建立在扎实的基本功和积极、有韧性的态度之上的。

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Year 7 SQA Advanced Mathematics: Summer Preparation and Bridging Course | SQA 数学进阶：暑期预习与衔接课程（7年级）