7.4 解决乘数和分裂等号-interactive

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  解决乘数和分裂等号

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  The Hyperloop
  ::超音环

  

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  Planes have made it easy to travel great distances in a relatively short amount of time. While planes may have solved the problem of long distance travel, they have not solved the problem of traveling intermediate distances. Traveling from Los Angeles to San  Francisco by car, two major cities in the same state, would take approximately six hours. Traveling from Los Angeles to San Francisco by plane would only take 1 hour and 30 minutes. However, when you factor in the time it takes to get to the airport, get through security, board the plane, get off the plane, and travel to your desired destination in the city, it will nearly take the same amount of time to travel the same distance by plane.
  ::飞机虽然解决了长途旅行的问题,但并没有解决中途旅行的问题。乘车从洛杉矶到旧金山,同一州的两个大城市,大约需要6个小时。乘飞机从洛杉矶到旧金山只需要1小时30分钟。然而,考虑到到达机场、通过安全系统、登机、下机和前往你所期望的城市目的地的时间,从洛杉矶到旧金山几乎需要同样时间。

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  This challenge is where hyperloops come in. A hyperloop is a magnetic train that can travel at high speeds through an underground tunnel. Several companies are currently developing hyperloops with the goal of introducing them to the public in the next decade. Hyperloops are projected to move at speeds of up to 700 mph, about 150 mph faster than an average passenger plane. If the distance from LA to San Francisco is about 378 miles, how long will it take to travel from LA to San Francisco by hyperloop?
  ::这个挑战就在于超大螺旋桨的出现。超大螺旋桨是一个磁性列车,可以高速通过地下隧道。一些公司目前正在开发超大螺旋桨,目的是在未来十年内向公众介绍。超大螺旋桨的移动速度预计高达700米,比平均客机速度快约150米。如果从洛杉矶到旧金山的距离大约378英里,那么用超大螺旋桨从洛杉矶到旧金山需要多长时间?

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  Discussion Question
  ::讨论问题

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  If the distance from LA to San Francisco is about 378 miles, how long will it take to travel from LA to San Francisco by hyperloop?
  ::如果从洛杉矶到旧金山的距离 大约378英里, 从洛杉矶到旧金山 需要多久才能通过超频流?

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  Understanding Multiplication Equations
  ::乘数等数

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  Answering the question  above will require knowledge of multiplication equations. Multiplication equations have many similarities with addition equations.
  ::回答上述问题需要了解乘法方程,乘法方程与增加方程有许多相似之处。

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  Use multiplication and division in the interactive below to solve multiplication equations.
  ::在下文互动中使用乘法和除法来解析乘法方程式。

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  Just as  you used the additive inverse to solve an addition equation ,  you   can use the multiplicative inverse to solve a multiplication equation. The multiplicative   inverse  of a number is its  reciprocal . When a number is multiplied by its reciprocal, the result will be 1. For example,  $\begin{align*}5 \cdot \tfrac{1}{5} = 1\end{align*}$ .
  ::正如您使用添加反向来解答添加方程式, 您可以使用倍数反向来解答倍数方程式。 数字的倍数反向是对应的。 当数字乘以对等数时, 结果将是 1 。 例如, 515= 1 。

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  A n equation is just a statement that two things (one on each side of the equal sign) are equal! In order for that to remain true, w hen you perform an operation on one side of an equation  to  cancel out a value,  you must perform the same operation to the other side .
  ::等式只是一个声明, 表示两样东西( 相等符号的两边各一个) 相等 。 为了保持真实性, 当您在等式的一边执行一个操作来取消一个值时, 您必须执行对另一方的相同操作 。

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  The next interactive shows the relationship between positive integers and their multiplicative inverses.
  ::下一个互动显示正整数与其倍数反差之间的关系。

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  Another way to look at multiplying by the multiplicative inverse is as dividing by the original number. Multiplication and division are inverse operations .
  ::另一种看待乘以倍数反向的乘法是除以原数。乘法和除法是反向操作。

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  Example
  ::示例示例示例示例

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  How many 44-cent stamps can you buy with $11?
  ::11美元能买多少44%的邮票?

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  Since we are repeatedly buying 44-cent stamps and adding the costs, this equation can be represented with a multiplication equation (remember that multiplication is repeated addition).
  ::由于我们一再购买44%的邮票并增加成本,这个等式可以用乘法表示(记住乘法是重复加法)。

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  $$\begin{align*}0.44n = 11\end{align*}$$
  ::0.44n=11

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  To get the variable by itself,  divide both sides by 0.44:
  ::将两边除以0.44:

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  $$\begin{align*}\cancel{0.44} \! \cdot n ~ & {= 11} \\ {\div \cancel{0.44}} ~~~~~\; & ~~~{\div~ 0.44} \\ \hline {n} ~ & {= 25}\end{align*}$$
  ::0.44n = 110.44 0.44n = 25

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  Answer: You can buy 25 44-cent stamps with $11.
  ::答:你可以买25,44美分的邮票,

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  The Hyperloop Continued
  ::超螺旋继续

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  The opening activity asked you to find the time it would take to travel 378 miles at 700 mph. You  will need to use the formula $\begin{align*}\text{Distance}=\text{Rate}\cdot \text{Time}\end{align*}$  to determine the time.
  ::开始活动要求您找到在700米时行驶378英里所需的时间。 您需要使用“ 距离=RateTime” 公式来确定时间 。

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  • Distance: 378
    ::距离:378
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  • Rate : 700
    ::费率:700
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  • Time:  $\begin{align*}x\end{align*}$  
    ::时间: x

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  $$\begin{align*}{\text{Distance }} ~ & {= \text{Rate} \cdot \text{Time}} \\ {378} ~ & { = 700 \cdot t} \\ {\tfrac{378}{700}} ~ & {= t}\\ {0.54} ~ & {= t}\end{align*}$$
  ::距离=Rate-Time 378=700_t-378700=t0.54=t

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  It will take 0.54 hours, or about a half hour, to travel from San Francisco to Los Angeles at 700 mph.
  ::从旧金山到洛杉矶需要0.54小时,大约半小时, 每天700英里。

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  In the following interactive use multiplicative inverses to solve how long it would take to travel distances at different speeds.
  ::在接下来的交互式使用多倍反射中,解决以不同速度行走距离需要多长时间的问题。

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  Division Equations
  ::平方

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  Some multiplication equations can be written using division. For example, the equation  $\begin{align*}\tfrac{1}{2}\cdot x=8\end{align*}$  can also be written as  $\begin{align*}\tfrac{x}{2} = 8.\end{align*}$
  ::一些乘法方程式可以用除法写入。例如,公式 12x=8 也可以以 x2=8 写成。

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  Use multiplication and division in the interactive below to solve different division equations.
  ::在下文互动中使用乘法和除法解决不同的分裂方程式。

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  Discussion Questions
  ::讨论问题 讨论问题

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  1. How did you determine the value of x in the interactive?
     ::您如何确定互动中的 x 值 ?
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  2. If you have ¼ of a variable on one side and you add three more fourths to that side of the balance beam, what operation can be used to represent this?
     ::如果您在一边有一个变量, 并在平衡光束的那一面再增加四分之三, 那么什么操作可以用来代表这个操作 ?
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  3. How can you check that a value is an answer to an equation?
     ::你怎么能确认一个值是方程式的答案?
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  4. Becky claims that both multiplication and division problems can be solved using multiplication. Is she correct? Support your answer with evidence.
     ::Becky声称乘法和分法问题都可以通过乘法解决。 她正确吗? 用证据支持你的回答。

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  Example
  ::示例示例示例示例

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  Ohm’s Law states the relationship between voltage, current, and resistance is:
  ::电压、电流和阻力之间的关系是:

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  $$\begin{align*}\text{Resistance} = \frac{\text{Voltage}}{\text{Current}}\end{align*}$$
  ::抵抗力量=志愿力量

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  Find the voltage of a toy robot with a circuit that draws 4 amps of current through a resistance of 3 Ohms.
  ::找到一个玩具机器人的电压 电路通过3兆米的阻力 吸引4兆米的电流

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  B egin by identifying each value in the equation :  $\begin{align*}\text{Resistance} = \frac{\text{Voltage}}{\text{Current}}\end{align*}$
  ::开始首先确定方程中的每个值:

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    •  Resistance: 3
  ::• 抵抗:3

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    •  Voltage:  $\begin{align*}x\end{align*}$
  ::• 电压:x

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    •  Current: 4 
  ::• 目前:4

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  Substitute  the values  you know into the equation:  $\begin{align*}3 = \frac{x}{4}\end{align*}$
  ::将您所知道的值替换为方程: 3=x4

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  You  can look at this equation as  $\begin{align*}3 = x \div 4, ~\text{or as} ~ 3 = \tfrac{1}{4} \cdot x.\end{align*}$
  ::您可以以 3=x4 或 3=14x 来查看此方程式 。

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  S olve this equation by multiplying both sides by 4:
  ::通过将双方乘以4 来解决这个方程式:

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  $$\begin{align*}{4 \cdot 3} & ~ {= \frac{x}{\cancel{4}} \cdot \cancel{4}}\\ {12} ~ & {=x}\end{align*}$$
  ::43 =x4412 =x

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  Answer: 12 Volts
  ::答复:12伏

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   Summary
  ::摘要

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  • Multiplication  and  division equations can be used to solve real-world problems
    ::乘数和分裂方程式可用于解决现实世界的问题
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  • The  multiplicative inverse of a number is the reciprocal, because a number is  multiplied by its reciprocal  will produce 1 .
    ::乘以乘以乘以乘以乘以乘以乘以乘以1,乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以1。
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  • When you use the  multiplicative inverse to cancel a number, make sure to do it to both sides of the equation.
    ::当您使用倍数反转来取消数字时, 一定要对等方程的两侧进行此操作 。