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  • 界定斜坡

    Firefighters
    ::消防员

    Firefighters use proportional relationships in surprising ways. They use them to place a ladder against a wall, keep track of water usage and to operate the fire hose. When a firefighter is operating a hose, they set the pressure to different amounts based on what floor they are aiming the hose at. The higher the floor the more pressure they will need to reach the floor. You likely see this when you operate the hose at your house : the more you turn the handle, the farther the hose will reach.
    ::消防员使用比例关系的方式令人惊讶。 他们使用比例关系来将梯子放在墙上, 跟踪用水情况并操作消防管。 当消防员在操作水管时, 他们根据他们瞄准水管的地板将压力设定到不同程度。 地板越高, 他们越需要到达地板的压力。 当你在家里操作水管时, 你可能会看到这一点: 你越是转动水管, 水管就会到达越远 。

    Firefighters need to be very precise and use just the right amount of pressure to put out the fire but not cause excessive damage to the structural integrity of the house. To accomplish this, they use a pressure valve with a pressure gauge that measures the pressure in pressure per square inch (psi). Firefighters use a proportional relationship to figure out how much pressure to add for each floor. For each floor off the ground, they must add 5 psi. While this relationship is between pressure added and floor number are proportional, the relationship between pressure and floor level is not. This may seem confusing but to understand the difference between a proportional relationship and a linear relationship you will need to question the assumptions that  you have been making thus far.
    ::消防员需要非常精确, 并且仅仅使用适当的压力来扑灭火, 而不是对房屋的结构完整性造成过度损害。 为了做到这一点, 他们使用压力阀, 压力计能测量每平方英寸( psi) 的压力压力。 消防员使用比例关系来计算每层的加压。 对于地面的每层, 他们必须添加 5 psi 。 虽然这种关系是在加压和地板数之间, 但压力和地板层之间的关系并不相称 。 这似乎令人困惑, 要理解比例关系和线性关系之间的差别, 您需要质疑您迄今所作的假设 。

    Discussion Questions
    ::讨论问题 讨论问题

    1. What equation can you use to represent the relationship between pressure added and floor level?
      ::您可以用什么方程式来表示加压和地板层之间的关系?
    2. What would the psi be if you are fighting a fire on the first floor?
      ::如果你在一楼打火,会怎么样?

    Babysitting Revisited
    ::重新审查的婴儿看护

    T o begin to understand the firefighter question, revisit the babysitting example from :
    ::校对:Portnoy

    Emily has recently begun babysitting for family and friends to earn money. She charges $8 per hour.
    ::Emily最近开始照看家人和朋友赚钱,她每小时收费8美元。

    You  were able to write the equation y = 8 x  to model this situation. However, what if you changed it to the following:
    ::您能够写入 y=8x 等式来模拟此情形。 但是, 如果您将其修改为以下内容 :

    Emily has recently begun babysitting for family and friends to earn money. She charges $8 per hour plus a flat fee of $10. What would the new equation be?
    ::Emily最近开始照看家人和朋友赚钱。她每小时收费8美元加上10美元的固定费用。新的公式是什么?

    Use the interactive below to explore what this would look like in a table and on a graph.
    ::使用下面的交互方式来探索在表格和图表中会是什么样子。

    INTERACTIVE
    Babysitting Cost
    minimize icon
    • Move the red point to see how much it costs to hire a babysitter for different lengths of time.
      ::将红点移到不同的时间段雇用保姆要花多少钱。
    • Fill in the table by typing in five different hour to dollar ratios.
      ::通过打字填充表格,以五个不同的小时与美元的比率填充表格。
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    Proportional Relationships vs. Linear Relationships
    ::比例关系与线性关系

    The reason why you have been having so much trouble writing proportional equations so far is that a proportional relationship must always have a starting amount of 0. If the starting amount is a number other than 0, then the relationship is called a linear relationship. A linear relationship  is a relationship that traces a line when plotted. As you may be able to tell by the first four letters in LINEar, the word linear means arranged in a straight line. Proportional relationships are a specific type of linear  relationship where the starting amount is 0. All proportional relationships are linear because they form a straight line when graphed. However, not all linear relationships are proportional because they do not have to start at 0.
    ::您迄今为止在撰写比例方程式方面一直有这么多麻烦的原因是比例关系必须总是有0的起始数。 如果起始数不是0, 则关系被称为线性关系。 线性关系是一种在绘制时可以追溯线条的关系。 正如您在 Linear 前四封信中可以用直线排列的单词。 比例关系是一种特定类型的线性关系, 起始数为0。 所有比例关系都是线性关系, 因为它们在绘制图表时形成直线线线。 但是, 并非所有线性关系都是按比例的, 因为它们不需要在 0 开始 。

    In the example of the firefighter, you were dealing with a linear relationship between pressure and floor number. The pressure started at a given non-zero number and then increase by 5 psi for every floor. 
    ::在消防员的例子中, 您处理的是压力和地板号码之间的线性关系。 压力是从给定的非零数开始的, 然后每层加5 psi。

    Slope vs. The Constant of Proportionality
    ::曲线相对于比例的常数

    Like a proportional relationship, a linear relationship increases or decreases by a constant rate of change . In a proportional relationship, this is referred to as the constant of proportionality . In a linear relationship, this is referred to as the or the unit rate . The slope is the steepness of a line. The slope, like the constant of proportionality, represents how quickly an amount is increasing or decreasing. The slope has the same meaning as the constant of proportionality with the only exception being that  the relationship  does not need to start at 0. Mathematicians will often use the terms slope or unit rate in place of the constant of proportionality.
    ::类似比例关系,线性关系增加或减少以恒定变化率。在比例关系中,这被称为比例的常数。在线性关系中,这被称为单位率。在线性关系中,这被称为单位率或单位率。斜度是指线的陡峭度。斜度与比例的常数一样,代表一个数额的增减速度。斜度与比例的常数具有相同的含义,唯一的例外是这种关系不需要从0开始。数学家往往使用术语斜度或单位率来取代比例的常数。

    Use the interactive below to help understand the relationship between slope and the constant of proportionality. 
    ::利用下面的交互作用帮助理解斜坡与相称性不变之间的关系。

    INTERACTIVE
    Identifying Functions
    minimize icon
    • Select all of the boxes that describe the function on the screen. 
      ::选择屏幕上描述函数的所有框 。
    • Click "Check" to see if you got it right!
      ::点击"检查"看是否正确!
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    Discussion Questions
    ::讨论问题 讨论问题

    1. What is the difference between slope and constant of proportionality?
      ::斜坡与相称性不变之间的差别是什么?
    2. What is the difference between a proportional equation and a linear equation ?
      ::比例方程和线性方程之间的差别是什么?

        Summary
    ::摘要

    • A linear relationship is a relationship that traces a straight line when plotted.
      ::线性关系是一种在绘制时追踪直线的关系。
    • Proportional relationships are a specific type of linear relationship that always starts at zero.
      ::比例关系是一种特定类型的线性关系,总是从零开始。
    • The slope is the steepness of a line, similar to the constant of proportionality.
      ::斜坡是一条线的陡坡,与相称性不变相似。