📚 Ordered Pairs and Coordinate Plane | 有序数对与坐标平面
An ordered pair tells you exactly where a point sits on a coordinate grid. In IGCSE Mathematics and in enVision G5 Topic 17-02, you learn to plot points such as (3, 4) and to read coordinates back from a graph. This revision note builds that skill from the first quadrant to all four quadrants, adds distances, midpoints, reflections and translations, and links everything to exam-style questions.
有序数对能精确地告诉你一个点在坐标网格上的位置。在 IGCSE 数学以及 enVision G5 第 17-02 课中，你将学习如何描出 (3, 4) 这样的点，并从图中读回坐标。本文从第一象限逐步扩展到四个象限，并加入距离、中点、对称和平移，把这一技能与考试题型完整联系起来。
1. What Is an Ordered Pair? | 什么是有序数对？
An ordered pair is a pair of numbers written as (x, y). The first number is the x-coordinate, also called the abscissa: it tells you how far to move horizontally from the origin. The second number is the y-coordinate, also called the ordinate: it tells you how far to move vertically. Positive x means move right, negative x means move left, positive y means move up, and negative y means move down.
有序数对是写成 (x, y) 的一对数。第一个数是 x 坐标（也叫横坐标）：它表示从原点出发沿水平方向移动多远。第二个数是 y 坐标（也叫纵坐标）：它表示沿竖直方向移动多远。x 为正表示向右移动，x 为负表示向左移动；y 为正表示向上移动，y 为负表示向下移动。
The order matters: (3, 4) and (4, 3) are different points. Never swap the two numbers, and never drop the brackets when writing an answer.
顺序非常重要：(3, 4) 和 (4, 3) 是两个不同的点。写答案时不要交换这两个数，也不要漏掉括号。
(x, y) = (horizontal movement, vertical movement)
2. The Coordinate Plane: Axes, Origin and Quadrants | 坐标平面：坐标轴、原点与象限
A coordinate plane (or Cartesian plane) is formed by two perpendicular number lines. The horizontal line is the x-axis, the vertical line is the y-axis, and the point where they cross is the origin, labelled (0, 0). Both axes are drawn to a stated scale, for example 1 unit per grid square or 2 units per grid square.
坐标平面（也叫直角坐标平面）由两条互相垂直的数轴组成。水平线为 x 轴，竖直线为 y 轴，它们相交的点叫做原点，记作 (0, 0)。两条轴都按题目给定的刻度来画，例如每格 1 个单位或每格 2 个单位。
The axes divide the plane into four regions called quadrants. Quadrants are numbered with Roman numerals, starting from the top right and going anticlockwise.
两条轴把平面分成四个区域，称为象限。象限用罗马数字编号，从右上角开始按逆时针方向排列。
| Quadrant | Position | Signs of (x, y) | 象限 | 位置 | (x, y) 的符号 |
|---|---|---|---|---|---|
| I | top right | x > 0, y > 0 | 第一象限 | 右上 | x > 0，y > 0 |
| II | top left | x < 0, y > 0 | 第二象限 | 左上 | x < 0，y > 0 |
| III | bottom left | x < 0, y < 0 | 第三象限 | 左下 | x < 0，y < 0 |
| IV | bottom right | x > 0, y < 0 | 第四象限 | 右下 | x > 0，y < 0 |
In many early exercises you only use the first quadrant, but IGCSE questions often test all four, so knowing the signs is essential.
在许多基础练习中，你只使用第一象限；但 IGCSE 题目经常考查全部四个象限，所以掌握符号规律非常关键。
3. Plotting Points Step by Step | 分步描点
Plotting a point is a two-step movement that always starts at the origin. First move along the x-axis, then move parallel to the y-axis. Working in that fixed order prevents almost every plotting error.
描点是一个两步移动的过程，永远从原点出发。先沿 x 轴移动，再平行于 y 轴移动。固定使用这个顺序，几乎可以避免所有描点错误。
- Step 1: start at (0, 0).
- Step 2: move right if x is positive, left if x is negative.
- Step 3: move up if y is positive, down if y is negative.
- Step 4: mark the point with a small cross and label it with its letter and coordinates.
- 第一步：从 (0, 0) 出发。
- 第二步：x 为正向右移动，x 为负向左移动。
- 第三步：y 为正向上移动，y 为负向下移动。
- 第四步：用小叉标出该点，并标注字母和坐标。
Worked example: plot A(3, 4), B(−2, 5), C(−4, −3) and D(5, −1). A lies in quadrant I, B in quadrant II, C in quadrant III and D in quadrant IV. If you plot them correctly and join A to B to C to D to A, you get a quadrilateral whose vertices sit in four different quadrants.
例题：描出 A(3, 4)、B(−2, 5)、C(−4, −3) 和 D(5, −1)。A 在第一象限，B 在第二象限，C 在第三象限，D 在第四象限。如果描点正确，并按 A→B→C→D→A 连接，你会得到一个四个顶点分别位于四个象限的四边形。
4. Reading Coordinates Back from a Graph | 从图中读取坐标
Reading coordinates is the reverse process. Start at the given point, drop a perpendicular to the x-axis to read the x-coordinate, then drop a perpendicular to the y-axis to read the y-coordinate. Always read the value of the grid line the point sits on, not the gap between two lines.
读坐标是描点的逆过程。从给定点出发，先向 x 轴作垂线读出 x 坐标，再向 y 轴作垂线读出 y 坐标。要读点所在的那条网格线的数值，而不是两条线之间的空隙。
grid square is worth 2 units, a point three squares to the right of the y-axis has x = 6, not 3.
有两个检查能抓住大多数错误。第一，检查符号：第二象限的点必须是负 x、正 y。第二，检查刻度：如果每格代表 2 个单位，那么位于 y 轴右侧三格的点，其 x = 6，而不是 3。
0, and a point on the x-axis has y = 0. The origin is the only point where both are zero.
恰好落在 y 轴上的点，x = 0；落在 x 轴上的点，y = 0。原点是唯一两个坐标都为零的点。
5. Scales, Signs and Quadrant Checks | 刻度、符号与象限检查
Exam graphs frequently use a scale other than 1 unit per square, and examiners deliberately use it to test whether you read the axis labels. Before plotting, always find the value of one grid square by looking at the numbers on the axes.
考试中的图象经常采用每格不等于 1 个单位的刻度，命题人正是借此考查你是否认真读了坐标轴上的数字。描点之前，一定要先看清一格代表多少。
- If the axis reads 0, 5, 10, 15 and there are five squares between each label, one square is 1 unit.
- If the axis reads 0, 2, 4, 6 and there are two squares between each label, one square is 1 unit.
- If the axis reads 0, 10, 20 and there are two squares between each label, one square is 5 units.
- 若坐标轴标出 0、5、10、15，且相邻数字之间有五格，则一格代表 1 个单位。
- 若坐标轴标出 0、2、4、6，且相邻数字之间有两格，则一格代表 1 个单位。
- 若坐标轴标出 0、10、20，且相邻数字之间有两格，则一格代表 5 个单位。
Sign checking is equally important. If a question asks in which quadrant the point (−7, 12) lies, you only need the signs: x < 0 and y > 0, so the point lies in quadrant II. No drawing required.
符号判断同样重要。如果题目问点 (−7, 12) 位于哪个象限，你只需要看符号：x < 0 且 y > 0，因此它在第二象限，不需要画图。
6. Points on the Axes and Special Cases | 坐标轴上的点与特殊情况
Points that lie on an axis belong to no quadrant, because a quadrant requires both coordinates to be non-zero. This is a favourite exam trap, especially in multiple-choice questions.
位于坐标轴上的点不属于任何象限，因为一个象限要求两个坐标都不为零。这是考试中常见的陷阱，尤其在选择题里。
- (a, 0) lies on the x-axis; for example (4, 0) and (−3, 0).
- (0, b) lies on the y-axis; for example (0, 7) and (0, −2).
- (0, 0) is the origin and lies on both axes.
- (a, 0) 位于 x 轴上，例如 (4, 0) 和 (−3, 0)。
- (0, b) 位于 y 轴上，例如 (0, 7) 和 (0, −2)。
- (0, 0) 是原点，同时位于两条轴上。
= −2. Recognising this helps you plot coordinates that satisfy a simple rule, such as plotting every point (n, 2n) for n = 1, 2, 3, 4.
由此可得到两类很有用的直线。所有 x = 3 的点组成竖直线 x = 3；所有 y = −2 的点组成水平线 y = −2。识别这一点有助于描出满足简单规律的点，例如描出 n = 1, 2, 3, 4 时所有形如 (n, 2n) 的点。
7. Distances and Midpoints on the Grid | 网格上的距离与中点
Once points are plotted, you can measure lengths directly from the grid. For two points with the same y-coordinate, the horizontal distance is the difference of the x-coordinates; for two points with the same x-coordinate, the vertical distance is the difference of the y-coordinates. Take the absolute value so the answer is positive.
描出点之后，可以直接在网格上量出长度。若两点的 y 坐标相同，水平距离等于 x 坐标之差；若两点的 x 坐标相同，竖直距离等于 y 坐标之差。要取绝对值，使答案为正。
Horizontal distance = |x₂ − x₁| Vertical distance = |y₂ − y₁|
For a diagonal distance, use Pythagoras' theorem on the right-angled triangle formed by the two points. For example, between (1, 2) and (4, 6) the horizontal gap is 3 and the vertical gap is 4, so the distance is √3² + 4² = √25 = 5.
若要求斜向距离，就用两点构成的直角三角形配合勾股定理。例如 (1, 2) 与 (4, 6) 之间水平差为 3、竖直差为 4，所以距离为 √3² + 4² = √25 = 5。
d = √((x₂ − x₁)² + (y₂ − y₁)²)
The midpoint of a segment is the average of the two endpoints:
线段的中点是两个端点坐标的平均值：
M = ( (x₁ + x₂) ÷ 2 , (y₁ + y₂) ÷ 2 )
So the midpoint of (1, 2) and (4, 6) is (2.5, 4). If a question gives the midpoint and one endpoint, work backwards by doubling the midpoint and subtracting the known endpoint.
因此 (1, 2) 与 (4, 6) 的中点是 (2.5, 4)。如果题目给出中点和其中一个端点，就用中点乘 2 再减去已知端点来反推另一个端点。
8. Reflections and Symmetry | 对称与反射
Reflections on the coordinate plane follow simple sign rules, and remembering them saves a lot of time in the exam. Reflect the point across the stated line and the other coordinate stays the same.
坐标平面上的对称（反射）遵循简单的符号规律，记住这些规律可以在考试中节省大量时间。按题目给出的对称轴反射后，另一个坐标保持不变。
- Reflection in the x-axis: (x, y) → (x, −y).
- Reflection in the y-axis: (x, y) → (−x, y).
- Reflection in the origin: (x, y) → (−x, −y).
- Reflection in the line y = x: (x, y) → (y, x).
- 关于 x 轴对称：(x, y) → (x, −y)。
- 关于 y 轴对称：(x, y) → (−x, y)。
- 关于原点对称：(x, y) → (−x, −y)。
- 关于直线 y = x 对称：(x, y) → (y, x)。
For example, reflecting (3, 4) in the x-axis gives (3, −4), which lies in quadrant IV; reflecting it in the y-axis gives (−3, 4), which lies in quadrant II. If you reflect a shape, the image is congruent to the object: lengths and angles do not change, only the position and orientation.
例如，把 (3, 4) 关于 x 轴反射得到 (3, −4)，位于第四象限；关于 y 轴反射得到 (−3, 4)，位于第二象限。图形反射后与原图形全等：边长和角度都不变，改变的只是位置和方向。
9. Translations and Simple Coordinate Mapping | 平移与简单的坐标变换
A translation slides a shape without turning or flipping it. It is described either in words, such as 3 right and 2 down, or as a column vector. In coordinate terms, a translation adds a fixed amount to each x-coordinate and a fixed amount to each y-coordinate.
平移是把图形整体滑动，不旋转也不翻转。它可以用文字描述，例如“向右 3、向下 2”，也可以写成列向量。用坐标表示时，平移就是给每个 x 坐标加上同一个数，给每个 y 坐标加上同一个数。
(x, y) → (x + a, y + b)
aps to (7, 3). The image of a segment under a translation is parallel to the original segment and exactly the same length, which is why translations preserve area and perimeter.
若 a = 5、b = −4，则点 (2, 7) 变为 (7, 3)。线段平移后与原线段平行且长度完全相同，所以平移不改变图形的面积和周长。
A common exam question gives the image and asks for the translation. Work out the horizontal change by subtracting the original x-coordinate from the image x-coordinate, then do the same for the y-coordinates. Always state the direction clearly, including negative signs.
常见的考题会给出象并要求求平移向量。用象的 x 坐标减去原图的 x 坐标得到水平变化量，对 y 坐标做同样处理。作答时要把方向说清楚，包括负号。
10. Patterns, Tables and Linear Relationships | 表格、规律与线性关系
Coordinate work connects naturally with tables of values. Write the values in a table with two rows, treat the top row as x and the bottom row as y, then plot each column as an ordered pair.
坐标知识与数值表紧密相连。把数值写进两行的表格，上面一行看作 x，下面一行看作 y，然后把每一列当作一个有序数对描点。
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y = 3x + 1 | 4 | 7 | 10 | 13 |
ase of 1 in x describes a linear relationship. A constant difference in y shows a linear pattern; anything involving squaring, such as y = x², gives a curve.
对应的有序数对是 (1, 4)、(2, 7)、(3, 10) 和 (4, 13)。描点后它们落在同一条直线上，因为 x 每增加 1、y 就增加 3 的固定变化描述的是线性关系。y 的差恒定为线性规律；而像 y = x² 这样涉及平方的关系画出的是曲线。
ion: for y = 3x + 1, the point (5, 16) works because 3 × 5 + 1 = 16, but (5, 17) does not.
这还能让你在不用画图的情况下判断点是否在直线上。把坐标代入方程即可：对于 y = 3x + 1，点 (5, 16) 成立，因为 3 × 5 + 1 = 16；而 (5, 17) 不成立。
11. Common Mistakes and Exam Technique | 常见错误与考试技巧
Most lost marks in coordinate questions come from a small set of repeated errors. Reading through this list before an exam is a fast way to protect your marks.
坐标题丢分大多来自几个反复出现的错误。考前过一遍下面的清单，是保住分数最快的办法。
- Plotting the pair in the wrong order, so (2, 6) becomes (6, 2).
- Turning left when x is negative is correct, but students often move left for a negative y as well.
- Assuming every grid square is 1 unit without reading the axis labels.
- Saying a point on an axis lies in a quadrant.
- Giving a negative distance; distances are always positive.
- Reflecting in the wrong axis and changing the wrong sign.
- Writing coordinates without brackets or without a comma.
- 把数对的顺序写反，把 (2, 6) 描成 (6, 2)。
- x 为负时向左移动是对的，但学生常把 y 为负也误当成向左移动。
- 不读坐标轴标签，就默认每格为 1 个单位。
- 把坐标轴上的点说成位于某个象限。
- 距离写成负数；距离永远为正。
- 关于错误的轴作对称，改错了符号。
- 写坐标时不加括号或漏掉逗号。
Good exam technique is simple: always draw the axes with a ruler, label both axes and the origin, mark points with a small cross rather than a dot, label each point with its letter, and finish by re-reading the coordinates from the graph to check your plot. Show the substitution when you test whether a point lies on a line, because method marks are usually available.
良好的应试技巧很简单：用直尺画坐标轴，标出两条轴和原点，用小小的叉而不是圆点来标点，给每个点标注字母，最后再从图上把坐标读一遍以检查描点是否正确。判断点是否在直线上时要写出代入过程，因为方法分通常可以拿到。
12. Practice Questions with Worked Answers | 练习题与解答
Try each question before reading the answer. These cover the whole topic and match the style of IGCSE and enVision G5 17-02 practice questions.
请先自己完成每一题，再看解答。这些题目覆盖整个主题，题型与 IGCSE 及 enVision G5 17-02 的练习题一致。
Question 1: In which quadrant does the point (−6, 9) lie?
第 1 题：点 (−6, 9) 位于第几象限？
Answer: x < 0 and y > 0, so it lies in quadrant II.
解答：x < 0 且 y > 0，所以位于第二象限。
Question 2: Find the distance between (2, 1) and (2, 7), and the midpoint of the segment joining them.
第 2 题：求 (2, 1) 与 (2, 7) 之间的距离，以及连接这两点的线段的中点。
Answer: the x-coordinates are equal, so the distance is |7 − 1| = 6. The midpoint is ((2 + 2) ÷ 2, (1 + 7) ÷ 2) = (2, 4).
解答：两个 x 坐标相同，所以距离为 |7 − 1| = 6。中点为 ((2 + 2) ÷ 2, (1 + 7) ÷ 2) = (2, 4)。
Question 3: Reflect the point (5, −3) in the y-axis and then in the x-axis. State the final coordinates.
第 3 题：先把点 (5, −3) 关于 y 轴反射，再关于 x 轴反射，写出最终的坐标。
Answer: reflecting in the y-axis gives (−5, −3); reflecting that in the x-axis gives (−5, 3). The combined effect is the same as a reflection in the origin.
解答：关于 y 轴反射得到 (−5, −3)；再关于 x 轴反射得到 (−5, 3)。两次反射合起来相当于关于原点对称。
Question 4: A point is translated by 4 right and 3 up. If the image is (9, 5), find the original point.
第 4 题：一个点经过“向右 4、向上 3”的平移后，象为 (9, 5)，求原图的坐标。
Answer: work backwards: 9 − 4 = 5 and 5 − 3 = 2, so the original point is (5, 2).
解答：反向推算：9 − 4 = 5，5 − 3 = 2，所以原图上的点是 (5, 2)。
Question 5: Does the point (7, 22) lie on the line y = 3x + 1?
第 5 题：点 (
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