Year 7 CCEA Further Maths: Core Topics Summary | Year 7 CCEA 进阶数学：核心知识点梳理

📚 Year 7 CCEA Further Maths: Core Topics Summary | Year 7 CCEA 进阶数学：核心知识点梳理

This article provides a clear and concise overview of the core topics covered in Year 7 CCEA Further Mathematics. Mastering these building blocks will give you the confidence to tackle more challenging problems and develop strong mathematical reasoning skills.

本文清晰梳理了 Year 7 CCEA 进阶数学的核心知识点。掌握这些基础模块将助你从容应对更具挑战性的问题，并培养扎实的数学推理能力。

1. Number Systems and Place Value | 数制与位值

Whole numbers are written using the digits 0–9, with each position having a place value: units, tens, hundreds, thousands, and so on.

整数用数字0–9书写，每一个位置都有位值：个位、十位、百位、千位等。

Decimal numbers extend place value to the right of the decimal point as tenths, hundredths and thousandths. For example, in 23.456, the digit 4 is in the tenths place and 5 in the hundredths place.

小数将位值扩展到小数点右侧，分为十分位、百分位和千分位。例如，在23.456中，数字4在十分位，5在百分位。

3 271 = 3000 + 200 + 70 + 1

3 271 = 3000 + 200 + 70 + 1

To compare or order whole numbers, start by comparing the digits from the highest place value (leftmost), then move right.

比较或排序整数时，从最高位值（最左侧）开始比较数字，然后依次向右。

Rounding a number to a given place value makes it easier to work with. When rounding to the nearest ten, look at the units digit: 5 or more rounds up.

将数字四舍五入到指定位值可使其更易处理。舍入到最接近的十位时，看个位数字：5或以上则进位。

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2. Negative Numbers and the Number Line | 负数与数轴

Negative numbers are numbers less than zero, written with a minus sign, e.g., –3, –15. They are used to represent temperatures below zero, debts or depths below sea level.

负数是小于零的数，用减号表示，如–3、–15。它们用于表示零下温度、负债或海平面以下深度。

On a horizontal number line, numbers increase from left to right. Because –7 is to the left of –2, we say –7 < –2.

在水平数轴上，数字从左向右增大。因为–7在–2的左边，所以说–7 < –2。

When adding a negative number, move left on the number line. For instance, 5 + (–3) = 2.

加上一个负数时，在数轴上向左移动。例如，5 + (–3) = 2。

Subtracting a negative number is the same as adding its positive opposite: 6 – (–4) = 6 + 4 = 10.

减去一个负数等于加上它的正相反数：6 – (–4) = 6 + 4 = 10。

Multiplying or dividing two numbers with the same sign gives a positive result; with different signs gives a negative result. (–2) × (–5) = 10, but (–2) × 5 = –10.

同号两数相乘或相除得正；异号得负。(–2) × (–5) = 10，而 (–2) × 5 = –10。

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3. Factors, Multiples and Primes | 因数、倍数与质数

A factor of a number divides exactly into that number with no remainder. The factors of 18 are 1, 2, 3, 6, 9, 18.

一个数的因数是可以整除该数而没有余数的数。18的因数有1、2、3、6、9、18。

A multiple of a number is the result of multiplying that number by an integer. The first five multiples of 7 are 7, 14, 21, 28, 35.

一个数的倍数是该数乘以整数所得的结果。7的前五个倍数是7、14、21、28、35。

A prime number has exactly two distinct factors: 1 and the number itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19. The number 1 is not prime.

质数恰好只有两个不同的因数：1和它本身。例如：2、3、5、7、11、13、17、19。数字1不是质数。

Any composite number can be written as a product of prime factors. Use a factor tree to find them: 24 = 2³ × 3.

任何合数都可以写成质因数的乘积。利用因子树可以找出质因数：24 = 2³ × 3。

The highest common factor (HCF) of two numbers is the largest factor they share. The lowest common multiple (LCM) is the smallest multiple they share. For 8 and 12, HCF = 4 and LCM = 24.

两个数的最大公因数(HCF)是它们共有的最大因数。最小公倍数(LCM)是它们共有的最小倍数。对8和12，HCF = 4，LCM = 24。

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4. Powers, Roots and Order of Operations | 幂、方根与运算顺序

A power tells you how many times to multiply a number by itself. The square of 5 is 5² = 5 × 5 = 25, and the cube of 2 is 2³ = 2 × 2 × 2 = 8.

幂表示一个数自乘的次数。5的平方是5² = 5 × 5 = 25，2的立方是2³ = 2 × 2 × 2 = 8。

The square root of a number is a value that, when multiplied by itself, gives the original number. √49 = 7 because 7 × 7 = 49.

一个数的平方根是自乘后等于原数的值。√49 = 7，因为7 × 7 = 49。

a² = a × a    a³ = a × a × a    √a is the inverse of squaring.

a² = a × a    a³ = a × a × a    √a 是平方的逆运算。

When several operations are combined, follow BIDMAS (or BODMAS): Brackets, Indices, Division/Multiplication (left to right), Addition/Subtraction (left to right).

当一个算式包含多种运算时，遵循BIDMAS（或BODMAS）规则：先括号，再指数，接着乘除（从左到右），最后加减（从左到右）。

For example, 3 + 2 × (5 – 1)² = 3 + 2 × 4² = 3 + 2 × 16 = 3 + 32 = 35.

例如，3 + 2 × (5 – 1)² = 3 + 2 × 4² = 3 + 2 × 16 = 3 + 32 = 35。

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

Fractions represent parts of a whole. Equivalent fractions have the same value, e.g., 1/2 = 2/4 = 5/10.

分数表示整体的部分。等值分数具有相同的值，例如1/2 = 2/4 = 5/10。

To simplify a fraction, divide numerator and denominator by their highest common factor. 8/12 simplifies to 2/3.

化简分数时，将分子和分母同时除以它们的最大公因数。8/12 化简为 2/3。

To convert a fraction to a decimal, divide the numerator by the denominator. 3/4 = 0.75. Percents are fractions out of 100: 40% = 40/100 = 2/5.

将分数转换为小数，用分子除以分母。3/4 = 0.75。百分数是分母为100的分数：40% = 40/100 = 2/5。

To find a percentage of a quantity, multiply by the decimal equivalent. 20% of 60 = 0.2 × 60 = 12.

求一个量的百分之几，乘以对应的小数。60的20% = 0.2 × 60 = 12。

Ordering a mix of fractions, decimals and percentages is best done by converting them all to the same form, usually decimals.

要将分数、小数和百分数混合排序，最好将它们全部转化为同一种形式，通常转化为小数。

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6. Introduction to Algebra and Formulae | 代数与公式入门

In algebra, letters (variables) stand for unknown numbers. An expression is a combination of numbers, variables and operation signs, such as 4a + 3b – 2.

在代数中，字母（变量）代表未知数。表达式是数字、变量和运算符号的组合，如4a + 3b – 2。

The number in front of a variable is called the coefficient. In 7x, 7 is the coefficient.

变量前面的数字称为系数。在7x中，7是系数。

Like terms contain exactly the same variable part and can be collected (simplified): 5c + 2c = 7c, and 9y – 3y + 2 = 6y + 2.

同类项含有完全相同的变量部分，可以合并（化简）：5c + 2c = 7c，9y – 3y + 2 = 6y + 2。

Substitution means replacing the variable with a given number to find the value of an expression. If p = 5, then 3p – 4 = 3 × 5 – 4 = 11.

代入法是指用给定的数替换变量，从而求出表达式的值。若p = 5，则3p – 4 = 3 × 5 – 4 = 11。

A formula is a mathematical rule written with symbols. For example, the perimeter of a rectangle is P = 2l + 2w, where l is length and w is width.

公式是用符号表示的数学规则。例如，矩形的周长公式为 P = 2l + 2w，其中l是长，w是宽。

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7. Solving Linear Equations | 一元一次方程求解

An equation states that two expressions are equal. The goal is to find the value of the unknown that makes the equation true.

方程表示两个表达式相等。目标是求出使方程成立的未知数的值。

Use inverse operations to isolate the variable. For x + 8 = 15, subtract 8 from both sides: x = 7.

运用逆运算分离变量。对于 x + 8 = 15，两边同时减去8，得x = 7。

If the equation involves multiplication, divide both sides: 4x = 28 → x = 7. If there is a combination, undo the addition/subtraction first.

如果方程涉及乘法，两边同时相除：4x = 28 → x = 7。如果有加减乘除组合，先移去加减。

When the variable appears on both sides, collect the variable terms on one side. Solve: 3x + 2 = x + 10 → 2x = 8 → x = 4.

当变量出现在方程两边时，将含变量的项移到一边。求解：3x + 2 = x + 10 → 2x = 8 → x = 4。

Always check your solution by substituting it back into the original equation.

务必将求得的解代回原方程进行验证。

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8. Sequences and Patterns | 数列与规律

A sequence is an ordered list of numbers following a rule. The term-to-term rule tells you how to move from one term to the next. For 5, 9, 13, 17, …, the rule is ‘add 4’.

数列是按照一定规则排列的有序数字列表。项间法则告诉你如何从一项得到下一项。对于5, 9, 13, 17, …，法则是“加4”。

The position-to-term rule (nth term) lets you find any term directly. For a linear sequence with constant difference d, the nth term is often written as an = d × n + c.

位置法（第n项）让你能直接求出任意项。对于公差为d的线性数列，第n项常写作 an = d × n + c。

If the sequence 3, 7, 11, 15, … increases by 4 each time and the zero term would be –1, the nth term is 4n – 1.

若数列 3, 7, 11, 15, … 每次增加4，且第零项为–1，则第n项为 4n – 1。

To find the 10th term, substitute n = 10 into the expression: 4 × 10 – 1 =

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