Graphing Number Patterns on the Coordinate Plane | 在坐标平面上绘制数字模式

📚 Graphing Number Patterns on the Coordinate Plane | 在坐标平面上绘制数字模式

In this lesson, we explore how to turn a number pattern into a visual graph using the coordinate plane. You will learn to write ordered pairs, plot points, and use patterns to predict future points.

在本课中，我们将学习如何利用坐标平面把数字模式转换成直观的图形。你将学会写出有序对、描点，并利用模式预测后续的点。

1. Ordered Pairs and Coordinate Grid | 有序对与坐标格

A coordinate grid is a flat surface formed by a horizontal number line and a vertical number line. Each point on the grid is identified by an ordered pair, written as (x, y).

坐标格是由一条水平数轴和一条竖直数轴构成的平面。平面上的每一个点都可以用一个有序对 (x, y) 来表示。

The first number, x, tells you how far to move right or left from the origin. The second number, y, tells you how far to move up or down.

第一个数 x 表示从原点向右或向左移动的距离；第二个数 y 表示向上或向下移动的距离。

The x-coordinate is always written first: (x, y).
x 坐标永远写在前面： (x, y)。
The point (0, 0) is called the origin.
点 (0, 0) 被称为原点。

2. The x-Axis and y-Axis | x 轴与 y 轴

The horizontal number line is called the x-axis. The vertical number line is called the y-axis. Both axes meet at the origin.

水平的数轴叫做 x 轴，竖直的数轴叫做 y 轴。两条轴在原点相交。

Positive numbers are to the right of the origin on the x-axis, and above the origin on the y-axis. Negative numbers are to the left or below the origin.

在 x 轴上，正数在原点的右边；在 y 轴上，正数在原点的上方。负数则在原点的左边或下方。

x-axis: y = 0 | y-axis: x = 0

x 轴： y = 0 | y 轴： x = 0

3. Plotting Points | 描点

To plot the point (3, 2), start at the origin. Move 3 units to the right along the x-axis, then move 2 units up. Place a dot there.

要描出点 (3, 2)，先从原点出发，沿 x 轴向右移动 3 个单位，再向上移动 2 个单位，并在该处画一个点。

Step 1: Start at (0, 0).
第 1 步：从 (0, 0) 出发。
Step 2: Move horizontally by the x-coordinate.
第 2 步：按 x 坐标水平移动。
Step 3: Move vertically by the y-coordinate.
第 3 步：按 y 坐标竖直移动。

Remember that the order of the numbers matters. The point (3, 2) is not the same as (2, 3).

请记住，数字的顺序非常重要。点 (3, 2) 与 (2, 3) 是不同的点。

4. Reading Points from a Graph | 从图上读取点

When you see a point on a coordinate grid, you can read its coordinates by moving straight down to the x-axis and straight across to the y-axis.

当你看到坐标格上的一个点时，可以通过竖直向下到 x 轴、水平向左到 y 轴来读出它的坐标。

For example, a point located directly above 4 on the x-axis and to the right of 5 on the y-axis has coordinates (4, 5).

例如，一个点位于 x 轴 4 的正上方、且与 y 轴 5 在同一水平线上，那么它的坐标是 (4, 5)。

Point	Coordinates
A	(4, 5)
B	(1, 6)
C	(0, 0)

Here, point A is at (4, 5), point B is at (1, 6), and point C is the origin.

这里，点 A 在 (4, 5)，点 B 在 (1, 6)，而点 C 是原点。

5. Number Patterns as Ordered Pairs | 数字模式转化为有序对

A number pattern can be written as a set of ordered pairs. Suppose you have two lists of numbers that are connected by a rule.

数字模式可以写成一组有序对。假设你有两组数字，它们之间通过一个规则相联系。

For example, the rule y = 2x + 1 gives these pairs:

例如，规则 y = 2x + 1 可以得到下面这些有序对：

x	0	1	2	3
y	1	3	5	7
Pair	(0,1)	(1,3)	(2,5)	(3,7)

Each x-value is paired with its corresponding y-value to form an ordered pair.

每一个 x 值与对应的 y 值组合在一起，就构成了一个有序对。

6. Graphing a Pattern | 绘制模式

To graph a number pattern, plot each ordered pair on the coordinate grid. Then observe how the points are arranged.

要绘制数字模式，把每个有序对描在坐标格上，然后观察这些点的排列方式。

If the points lie on a straight line, the pattern is called a linear pattern.

如果这些点连成一条直线，这个模式就叫做线性模式。

y = 2x + 1 produces points along a straight line.

y = 2x + 1 所产生的点位于同一条直线上。

Plot all given pairs carefully.
仔细描出所有给出的有序对。
Connect the points in order if the data are continuous.
如果数据是连续的，按顺序连接各点。
Do not connect points if the pattern describes separate individual values.
如果模式表示的是彼此独立的数值，则不要连接各点。

7. Predicting the Next Point | 预测下一个点

Once a pattern is graphed, you can use it to predict the next point. Look at the change between consecutive x-values and the change between consecutive y-values.

一旦把模式绘制成图，就可以利用它来预测下一个点。观察相邻 x 值的变化以及相邻 y 值的变化。

In the table for y = 2x + 1, as x increases by 1, y increases by 2. So the next point after (3, 7) would be (4, 9).

在 y = 2x + 1 的表格中，x 每次增加 1，y 就增加 2。因此 (3, 7) 之后的下一个点是 (4, 9)。

Check: y = 2(4) + 1 = 9 → (4, 9)

验证： y = 2×4 + 1 = 9 → (4, 9)

8. Real-World Applications | 现实应用

Coordinate graphs are used to show real-world relationships, such as the cost of items, distance over time, or temperature changes.

坐标图可以用来表示现实中的关系，比如商品价格、距离随时间的变化，或者温度变化。

Suppose a taxi charges a base fee of $3 and then $2 per mile. The pattern is represented by y = 2x + 3, where x is miles and y is cost in dollars.

假设出租车起步价为 3 美元，之后每英里收费 2 美元。这个关系可以用 y = 2x + 3 表示，其中 x 是英里数，y 是费用（美元）。

Miles (x)	0	1	2	3
Cost (y)	3	5	7	9

Graphing this pattern helps us quickly see how the cost grows.

绘制这个模式可以帮助我们快速看出费用是如何增长的。

9. Common Mistakes to Avoid | 常见错误

Swapping the order: writing (y, x) instead of (x, y). Always read the x-coordinate first.
颠倒顺序：写成 (y, x) 而不是 (x, y)。一定要先看 x 坐标。
Forgetting the origin: every point is measured from (0, 0).
忘记原点：所有点都是从 (0, 0) 开始计量的。
Using different scale lengths on the two axes.
两条轴上的单位长度不一致。
Plotting fractional coordinates incorrectly.
分数坐标描点不正确。

For example, the point (½, 2) should be halfway between 0 and 1 on the x-axis, then go exactly 2 units up.

例如，点 (½, 2) 应该在 x 轴上 0 和 1 的正中间，然后竖直向上移动 2 个单位。

10. Practice Problems | 练习

Try these problems to check your understanding.

尝试下面的练习来检查你的理解。

Write the ordered pair for a point 4 units right and 3 units up from the origin.
写出从原点向右 4 个单位、再向上 3 个单位所对应点的有序对。
Plot the points (2, 1), (4, 2), (6, 3). Describe the pattern.
描出 (2, 1)、(4, 2)、(6, 3)，并描述这个模式。
Use the rule y = x − 1 to make a table with x = 1, 2, 3, 4 and list the ordered pairs.
利用规则 y = x − 1，以 x = 1、2、3、4 制作表格并列出有序对。

Answers: 1. (4, 3). 2. The points lie on a straight line; y is half of x. 3. (1,0), (2,1), (3,2), (4,3).

答案：1. (4, 3)。2. 这些点在同一条直线上；y 是 x 的一半。3. (1,0)、(2,1)、(3,2)、(4,3)。

11. Summary | 总结

In this lesson, we learned how to use ordered pairs on a coordinate plane to represent and graph number patterns. We explored plotting points, reading coordinates, predicting the next point, and applying these ideas to real situations.

在本课中，我们学习了如何在坐标平面上使用有序对来表示并绘制数字模式。我们探索了描点、读取坐标、预测下一个点，以及把所学知识应用到现实情境中。

Remember the key steps: start at the origin, move along the x-axis first, then move along the y-axis. Always keep the order (x, y).

请记住关键步骤：从原点出发，先沿 x 轴移动，再沿 y 轴移动。始终都要保持 (x, y) 的顺序。

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Graphing Number Patterns on the Coordinate Plane | 在坐标平面上绘制数字模式