课程： 4.6 计算曲线-interactive

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计算曲线
Measuring Slope
::测量斜体

I n , you learned that is the steepness of a line. In this section, you will be looking at how you can measure the steepness of a line. This concept pops up in a wide range of professions.
::在其中,您知道这是一条线的陡峭之处。在本节中,您将查看如何测量一条线的陡峭之处。这个概念出现在广泛的职业中。

Discussion Questions
::讨论问题讨论问题

In the picture above, the red line appears to be steeper than the blue line. How can you measure this? What tools can you use?
::在以上图片中, 红线似乎比蓝线要陡峭。您如何测量这个 ? 您可以使用什么工具 ?

Find something in your classroom that you can measure the slope of. What was the slope? What tools did you use to find it? What methods did you use? Did you use any units?
::在教室里找到一些可以测量斜坡的东西。斜坡是什么? 您使用什么工具来找到它? 您使用什么方法? 您是否使用过任何单位 ?

Ladders
::梯子

When you think of how to measure steepness, your mind might immediately jump to measuring the angle of the line. Using angles is common; however, roofs, ramps, and many other real-world items have their steepness measured using slope. The reason for this is that when measuring steepness with an angle, you have to consider what the initial angle is which can lead to confusion. Additionally, what if the line represented how much money you make an hour instead of a physical object? Would it make sense to say that you make 45 degrees of dollars per hour? This could be used to mean that you make $1 per hour.
::当您想到如何测量陡度时, 您的头脑可能会立即跳跃来测量线的角。使用角度是常见的; 但是, 屋顶、坡道和许多其他真实世界的物品会用斜度测量其陡度。原因是当用角度测量陡度时, 您必须考虑最初的角度是什么会导致混乱。此外, 如果线代表了您每小时挣多少钱而不是物理物体? 说您每小时挣45 度的美元是否合理? 这可以用来表示您每小时赚1美元。

The slope of a line is measured as a ratio of how much it rises to how much it runs. The rise is how much a line goes up or down in a given segment. The run is how much a line goes left or right in a given segment.
::线的斜坡测量为线的升幅与运行量之比。升幅是某一段线向上或向下多少。升幅是某一段线向上或向下多少。升幅是某一段线向左或向右多少。

$\begin{aligned} Slope = \frac{Rise}{Run} \end{aligned}$
$\begin{align*}\text{Slope} = \frac{\text{Rise}}{\text{Run}}\end{align*}$
::斜斜线= RiseRun

When a firefighter needs to place a ladder against a wall, it is important to get the angle just right. They climb to dangerous heights and need to make sure that the ladder is positioned perfectly to keep them safe. When a firefighter places a ladder against a wall, it must have a slope of 4. To understand what this means let’s write 4 as a fraction : $\begin{align*}\tfrac{4}{1}.\end{align*}$ Any whole number can be written as a fraction by putting a 1 in the denominator because any number divided by 1 is itself. A slope of $\begin{align*}\tfrac{4}{1}\end{align*}$ means that for every 4 feet up the ladder goes it must come 1 foot out from the building. This slope ensures that the ladder will not tip backward or slip out from underneath the firefighter. Use the interactive below to explore this situation.
::当消防员需要将梯子放在墙上时,必须正确地抓住角度。他们爬到危险的高度,需要确保梯子处于完全安全的位置。当消防员将梯子放在墙上时,必须有一个4的斜坡。要理解这意味着将4分写成一个分数:41。任何整数都可以通过在分数中加上一个分数来作为一个分数来写成一个分数,因为数字除以1是其本身。一个41分数意味着每4英尺的梯子必须从建筑物中爬出1英尺。这个斜坡可以确保梯子不会从消防员的下面向后爬或滑出。使用下面的交互方式来探索这种情况。

Discussion Questions
::讨论问题讨论问题

When you moved the ladder did the steepness of the ladder change?
::当你移动梯子的时候梯子的陡峭程度有没有改变?

How could the rise and run change without changing the slope?
::在不改变坡度的情况下,如何能不改变坡度而改变上升和变化呢?

Roofs
::屋顶

Roofs on homes are required to have a minimum slope (also referred to as pitch) to prevent collapse under heavy snowfall and extreme weather. The typical minimum slope for a roof with shingles is $\begin{align*}\tfrac{1}{6}\end{align*}$ . Use the interactive to determine which roofs meet the minimum requirement. You will need to find the slope of the roof and determine if it is greater than or equal to $\begin{align*}\tfrac{1}{6}\end{align*}$ . Remember that the slope is a ratio of the rise to the run.
::住家屋顶必须有一个最低限度的斜坡(也称为斜坡),以防止在大雪降雪和极端天气下坍塌。典型的顶顶顶带闪烁的最小斜坡是16。使用互动的斜坡来确定哪些屋顶符合最低要求。您需要找到屋顶的斜坡,并确定它是否大于或等于16。请记住斜坡是上升与运行的比率。

Discussion Question
::讨论问题

Jeff is an engineer and needs to measure the slope of a roof. Which tool would be more effective for doing so, a protractor or a tape measure?
::Jeff是工程师,需要测量屋顶的坡度。哪种工具会更有效,比如减速器或磁带测量器?

Measuring Slope on a Coordinate Plane
::坐标平面上的测量曲线

When the slope of an object is being measured by a computer program or the slope does not represent a physical object, it is measured on a coordinate plane . To help visualize this, imagine that you are walking from one point to another but you can only move up, down, left, or right. The direction that you go in will be expressed as a ratio of steps up or down to steps left or right. To identify the direction , use positives, and negatives. When you take steps up or to the right, say that these are positive because you are going in the positive direction of the axes. When you take steps down or to the left, say that these are negative because you are going in the negative direction of the axes. Use the interactive below to guide the character from point A to point B.
::当一个对象的斜坡由计算机程序测量,或者斜坡不代表物理对象时,它会被测量在坐标平面上。为了帮助视觉化,想象一下您正在从一个点向另一个点向另一个点行走,但只能向上、向下、向左或向右移动。你进入的方向将表现为向上或向下到左或向右的阶梯之比。要辨明方向,使用正数和负数。在向上或向右走的阶梯时,表示这些是正数,因为您正朝着轴向正方向走。在向下或向左走的阶梯时,表示这些是负数,因为您正在向轴向负方向行走。使用下面的交互作用来引导从A点到B点的字符。

Discussion Questions
::讨论问题讨论问题

How many different ways could you get from point A to point B in the interactive?
::从互动的A点到B点你能得到多少种不同的方法?

Will going up and down first versus right and left first result in the same slope? How are the 2 different paths related?
::首先向上向下向上对右和向左向上向后向同一斜坡?两种不同路径之间有何关联?

If you went from point B to point A, you would travel downward and to the left. Will this be the same slope?
::如果你从B点到A点,你会向下和向左走。这是同一个斜坡吗?

Why will moving downward and to the left will result in a positive slope ?
::为什么向下和向左移动将导致一个积极的斜坡?

Consider two unknown points (a,b) and (c,d). How would you find the rise? How would you find the run?
::想想两个未知点(a、b和c、d)。你怎么找到上升点呢?你怎么找到运行点呢?

The Slope Formula
::斜坡公式

If a person or computer program needs to calculate the slope without having to graph them, you can use the slope formula :
::如果一个人或计算机程序需要计算斜坡而无需图解,您可以使用斜坡公式:

$\begin{align*}m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\end{align*}$
::m=y2 - y1x2 - x1

The 1 and 2 are subscript and refer to the first and second point.
::第1和第2项是下标,指第1和第2点。

$\begin{align*}m\end{align*}$ is the slope
::m 是斜坡

$\begin{align*}{x}_{1}\end{align*}$ is the $\begin{align*}x\text{-}\end{align*}$ value from the first point.
::x1 是从第一点开始的 x 值。

$\begin{align*}{y}_{1}\end{align*}$ is the $\begin{align*}y\text{-}\end{align*}$ value from the first point.
::y1 是从第一点开始的 Y 值。

$\begin{align*}{x}_{2}\end{align*}$ is the $\begin{align*}x\text{-}\end{align*}$ value from the second point.
::x2 是从第二点起的 x 值。

$\begin{align*}{y}_{2}\end{align*}$ is the $\begin{align*}y\text{-}\end{align*}$ value from the second point.
::y2 是从第二点开始的 Y 值。

Example
::示例示例示例示例

Use the slope formula to find the slope of the line formed by the points (1, 3) and (7, 10)
::使用斜度公式查找由点(1、3和7、10)组成的线条的斜度

First, you must identify the values:
::首先,你必须确定值:

$\begin{aligned} x_{1} y_{1} x_{2} y_{2} \\ (1, 3) (7, 10) \end{aligned}$
$\begin{align*}\space{x}_{1}\space\space\space {y}_{1}\space\space \space{x}_{2} \space\space\space{y}_{2}\\ (1, \space3)\space \space(7, \space10)\end{align*}$
::x1 y1 x2 y2( 1, 1, 3) (7, 10)

Next, you can substitute the values into the formula and solve. Note that if you subtract a negative value, it will cause a double negative.
::接下来, 您可以将值替换为公式并解析。请注意, 如果您减去负值, 就会导致双负值。

$\begin{aligned} m & = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \\ m & = \frac{10 - 3}{7 - 1} \\ m & = \frac{7}{6} \end{aligned}$
$\begin{align*}m&=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\\ \\ m&=\frac{10-3}{7-1}\\ \\ m &= \frac{7}{6}\end{align*}$
::my2-y1x2-x1m=10-37-1m=76

If a slope has a negative in the numerator and denominator, they will cancel, and the answer will be positive. Use the interactive below to find the slope of the line formed by the chosen points.
::如果一个斜坡在分子和分母中出现负值, 它们将会取消, 答案将是正值。使用下面的交互作用来找到所选点组成的线条的斜度。

Similar Slope Triangles
::相似的斜坡三角形

The slope is the same between any two distinct points on a non- vertical line in the coordinate plane. Use the similar triangles in the interactive below to explain why the slope is the same between them.
::斜度与坐标平面上非垂直线上任何两个不同点的斜度相同。使用下方互动的类似三角形来解释为什么斜度是它们之间的相同点。

Summary
::摘要

The slope of a line is the vertical change over the horizontal change: $\begin{align*}\frac{rise}{run}\end{align*}$ .
::线的斜坡是水平变化的垂直变化:上升。

Use the formula: $\begin{align*}\frac{y_2-y_1}{x_2-x_1}\end{align*}$ to calculate the slope, using two points that fall on the same line.
::使用公式:y2-y1x2-x1计算斜度,使用同一线上的两个点计算斜度。

The equation for any line that goes through the origin is $\begin{align*}y=mx, \end{align*}$ where $\begin{align*}m \end{align*}$ is the slope.
::任何直线的方程式是y=mx, m 是斜坡。

章节大纲

常规

计算曲线

Measuring Slope ::测量斜体

Ladders ::梯子

Roofs ::屋顶

Measuring Slope on a Coordinate Plane ::坐标平面上的测量曲线

The Slope Formula ::斜坡公式

Similar Slope Triangles ::相似的斜坡三角形

Summary ::摘要

Measuring Slope
::测量斜体

Ladders
::梯子

Roofs
::屋顶

Measuring Slope on a Coordinate Plane
::坐标平面上的测量曲线

The Slope Formula
::斜坡公式

Similar Slope Triangles
::相似的斜坡三角形

Summary
::摘要