课程： 4.1 比例关系图-interactive

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比例关系图

Doctors, Nurses, and First Responders
::医生、护士和第一反应者

Linear relationships are used in nearly every professional field, and there is a very high probability that you will use them regularly over the course of your life. You may have to interpret them from a graph or table, make a graph to visualize a relationship or write an equation to represent a situation. You will be looking at how a variety of professions use linear relationship, but the professional area which you will be focusing on is that of doctors, nurses, and first responders. First responders, who are the first to arrive at emergency situations, include EMS, police officers, and firefighters. Doctors, nurses, and first responders need to use their knowledge of linear relationships on a daily basis. Speed and accuracy are extremely important. Over the course of the chapter, you will look at these situations in greater detail, and you will have the opportunity to see how math is used to save lives.
::几乎在每一个专业领域都使用线性关系,而且你一生中经常使用线性关系的可能性非常大。你可能需要用图表或表格来解释这些关系,可能需要用图表来解释这些关系,可能需要用图表来描述某种关系,或者写一个方程式来代表一种情况。你将研究各种职业如何使用线性关系,但你将关注的专业领域是医生、护士和第一反应者。第一反应者是首先到达紧急情况的人,包括应急系统、警官和消防员。医生、护士和第一反应者需要每天利用他们对线性关系的知识。速度和准确性是极为重要的。在本章过程中,你将更详细地审视这些情况,你将有机会看到数学是如何用来拯救生命的。

Proportional Relationships
::比例关系

A proportional relationship is when two quantities vary directly. When one quantity changes, the other will change by a proportional amount. For example, if one number doubles, the other number will double. You have experienced these relationships in the past when using ratio tables.
::比例关系是指当两个数量直接变化时。当一个数量变化时,另一个将改变一个比例数量。例如,如果一个数字翻一番,另一个数字将翻一番。在过去使用比率表时,你经历过这些关系。

Example
::示例示例示例示例

Emily has recently begun babysitting for family friends to earn money. She charges $8 per hour. Use the interactive below to show Emily’s earnings as the number of hours that she babysits increases. Graph at least 5 points.
::Emily最近开始为家人朋友做保姆赚钱。她每小时收费8美元。使用下面的交互方式来显示Emily的收入是她照看孩子的小时数的增加。图至少5点。

Discussion Questions
::讨论问题讨论问题

What challenges did your choice of scale present?
::你对规模的选择带来了哪些挑战?

What strategies did you use for determining where the points went?
::你用什么战略来确定要点的去向?

Did you notice a pattern while you were plotting points? What was it and how might you be able to use this pattern when plotting in the future?
::在你绘制点数时,你注意到一个图案吗?是什么图案,以及你如何在未来绘制图案时使用这个图案?

Dilation
::关系

The concept of proportional relationships was first discussed in the chapter Similarity, in which a scale factor was used to dilate shapes. When a shape is dilated, every side of the shape scales by a proportional amount. This proportional scaling also happens when something is made from a blueprint. By knowing the blueprint dimensions and the actual dimensions, you can graph the relationship. In the table below you have measurements from the blueprint of a house and the actual house.
::比例关系的概念首先在“相似性”章中讨论,该章使用一个比例因子来放大形状。当形状膨胀时,形状的每个侧面都会以比例数量成比例比例。这种比例比例比例的缩放也会在蓝图制作时发生。通过了解蓝图的维度和实际维度,您可以对关系进行图解。在下表中,您可以从房屋和实际房屋的蓝图中进行测量。

Measurement	Blueprint Size ( feet )	Actual Size (feet)
Height	1	35
The Height of a Door Frame	0.2	7
Length	1.2	42
Width	0.8	28
The H eight of the 1st Floor	0.4	14

Use the interactive below to plot these points, and then l abel the $\begin{aligned} x - \end{aligned}$ axis and $\begin{aligned} y - \end{aligned}$ axis.
::使用下面的交互键绘制这些点, 然后标记 X 轴和 Y 轴。

Discussion Questions
::讨论问题讨论问题

If a construction worker needed to quickly estimate the length of a room which is 0.6 feet on the blueprint, how could this graph be helpful?
::如果建筑工人需要迅速估计蓝图上0.6英尺的房间长度,这个图有什么用?

How was the scale of this graph different from the graph in the previous activity?
::此图的尺度与上一个活动中的图表有何不同?

What did you notice about how the points were arranged on the coordinate plane ?
::你注意到坐标飞机上如何安排点数了吗?

What was the starting point of this graph? What was the starting point of the previous graph?
::此图的起点是什么? 前一图的起点是什么 ?

Heart Rate Continued
::心脏病发病率持续

The equation for a proportional relationship is $\begin{aligned} y = k x \end{aligned}$ where $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ are the related quantities and $\begin{aligned} k \end{aligned}$ is the constant of proportionality . This type of relationship forms a straight line when graphed. The constant of proportionality is the constant ratio between $\begin{aligned} y \end{aligned}$ and $\begin{aligned} x . \end{aligned}$ This ratio represents the rate at which the proportion is increasing or decreasing. In the first example, where Emily babysits for $8 an hour, the constant of proportionality is 8 because the amount of money she makes from babysitting will always be 8 times more than the number of hours she works. The equation which models this situation is $\begin{aligned} y = 8 x \end{aligned}$ with $\begin{aligned} x \end{aligned}$ representing the number of hours worked and $\begin{aligned} y \end{aligned}$ representing the amount of money earned. Each $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ value form a point $\begin{aligned} (x, y) \end{aligned}$ which you can graph as you saw in the second activity.
::比例关系方程式是 y=kx, 其中 x 和 y 是相关数量, k 是相称性的常数。这种类型的关系在图形绘制时形成一条直线。比例的常数是 y 和 x 之间的常数比。这个比例表示比例在上升或下降的速率。在第一个例子中, Emily 每小时8美元做婴儿, 比例的常数是 8, 因为她从照看孩子中挣得的钱总比她工作时数多8倍。以 x 表示工作小时数和 y 表示挣取的钱数的公式是 y= 8x 。每个 x 和 y 值组成一个点( x y) , 您可以在第二个活动中看到这个点( x y) 。

To determine the constant of proportionality for any proportional relationship, you can divide the $\begin{aligned} y \end{aligned}$ value by its corresponding $\begin{aligned} x \end{aligned}$ value. This formula represents a rearrangement of the relationship from above solved for $\begin{aligned} k : k = \frac{y}{x} . \end{aligned}$
::要确定任何比例关系是否相称的常数,您可以将 Y 值除以相应的 x 值。此公式代表从上方重新排列 k:k=yx 的解析关系。

Example
::示例示例示例示例

Identify the constant of proportionality and proportional equation that will produce the point (5, 9)?
::确定得出点(5,9)的相称性和比例等式的常数(5,9)?

To determine the constant of proportionality put the $\begin{aligned} x - \end{aligned}$ value and $\begin{aligned} y - \end{aligned}$ value into the general formula $\begin{aligned} k = \frac{y}{x} . \end{aligned}$
::为确定相称性的常数,将x值和y值放入一般公式=yx。

\begin{aligned} k = \frac{9}{5} \\ k = 1.8 \end{aligned}

::k=95k=1.8 (k=95k=1.8)

You can then use this to write the equation which represents the relationship between $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ .
::然后您可以使用此选项来写入代表 x 和 y 之间关系的方程式。

\begin{aligned} y = 1.8 x \end{aligned}

::y= 1. 8x y= 1. 8x

When you go to a doctor’s office for a check-up, it will often begin with a nurse measuring your heart rate. Your heart rate is determined by the relatively proportional relationship between your heartbeat and time. Doctors measure your heart rate based on how many beats occur in a minute. A healthy heart rate ranges between 60 and 100 beats per minute.
::当你去医生办公室接受检查时,通常从护士测量你的心跳开始。你的心跳取决于心跳和时间之间的相对比例关系。医生根据一分钟中跳动的次数来测量你的心跳。健康的心跳每分钟60至100下。

Example
::示例示例示例示例

Matt is a nurse and wants to find a patient's heart rate in beats per minute. He measures 43 beats in 30 seconds. What is the constant of proportionality, and an equation that he c ould use to represent the patient's heart rate?
::马特是一名护士,想在每分钟的节拍中找到病人的心跳。他用30秒的节拍量了43次。什么是相称性的常数,什么是他可以用来代表病人心跳的方程式?

To find the constant of proportionality you need to plug the information into the equation $\begin{aligned} y = k x . \end{aligned}$ First, solve for $\begin{aligned} k : \end{aligned}$ $\begin{aligned} k = \frac{y}{x} . \end{aligned}$ Let $\begin{aligned} x \end{aligned}$ represent the number of minutes and $\begin{aligned} y \end{aligned}$ represent the number of beats. However, if you plug in 30 for $\begin{aligned} x, \end{aligned}$ your answer will be incorrect , since 30 is the number of seconds , not minutes! That means your answer will be in beats per second. Fix this by first converting 30 seconds to 0.5 minutes.
::要在等式=kx中找到匹配性常数, 您需要将信息插入信息。首先, 解答 k: k=yx 。让 x 代表分钟数, y 代表节拍数。但是, 如果您在 30 中插入 x , 您的回答将是错误的, 因为 30 是秒数, 不是分钟。这意味着您的回答将以每秒的节拍方式进行。请先将 30 秒转换为 0. 5 分钟。

\begin{aligned} k & = \frac{43}{0.5} \\ k & = 86 \end{aligned}

::k=430.5 k=86

The constant of proportionality is 86, and you can plug this into the general equation to obtain the following:

\begin{aligned} y = 86 x \end{aligned}

::相称性不变值为 86, 您可以在一般方程中插入此值, 以便获得以下条件: y=86x

Use the interactive below to graph the relationship between minutes and the number of beats based on the equation.
::使用下面的交互效果来绘制分钟和基于方程式的节拍次数之间的关系图。

Discussion Questions
::讨论问题讨论问题

What does the constant of proportionality mean in the context of the problem?
::相称性的一贯性在问题的背景中意味着什么?

Why did you write x and y instead of putting numbers writing numbers?
::你为什么写 x 和 y 而不是写数字写号?

In 100 minutes, how many times would you expect the patient's heart to beat?
::在100分钟内, 你会指望病人的心脏跳动多少次?

Does the patient have a healthy heart rate?
::病人的心跳是否健康?

All of the graphs have been straight lines, do you think it is a coincidence? Why?
::所有的图表都是直线,你认为这是巧合吗?

Summary
::摘要

The equation for a proportional relationship is $\begin{aligned} y = k x \end{aligned}$ . Where $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ are related quantities and $\begin{aligned} k \end{aligned}$ is the constant of proportionality.
::比例关系方程式是y=kx。 x 和 y 是相关数量, k 是比例性常数。

The graphs of proportional equations are straight lines that go through the points $\begin{aligned} (0, 0) \end{aligned}$ and $\begin{aligned} (1, k) \end{aligned}$ .
::比例方程式图是穿过点数(0,0)和点数(1,k)的直线。

To determine the constant of proportionality use the equation $\begin{aligned} k = \frac{y}{x} \end{aligned}$ .
::为确定相称性的常数,使用方程式 k=yx。

章节大纲

常规

比例关系图

Doctors, Nurses, and First Responders ::医生、护士和第一反应者

Proportional Relationships ::比例关系

Dilation ::关系

Heart Rate Continued ::心脏病发病率持续

Summary ::摘要

Doctors, Nurses, and First Responders
::医生、护士和第一反应者

Proportional Relationships
::比例关系

Dilation
::关系

Heart Rate Continued
::心脏病发病率持续

Summary
::摘要