7.2 溶解等量-interactive

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  溶解等量

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  Kaleidoscopes Revisited
  ::重新检查的心血管镜

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  Formulas are at the heart of equations. A formula is an expression or equation that uses mathematical symbols to express a relationship. The formula s that we use come from business, physics, chemistry, statistics, and many more professional fields.  You  may have used equations without realizing it while exploring various formulas. 
  ::公式是方程式的核心。 公式是一种表达式或方程式, 使用数学符号来表达关系。 我们使用的公式来自商业、物理、化学、统计和更多的专业领域。 您可能在探索各种公式时使用方程式而没有意识到它。

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  T he lesson  explored the math behind kaleidoscopes. Use the interactive to make a kaleidoscope and  see how the number of reflections is related to the angle measure between mirrors.  Afterwards, you will substitute the number of reflections into an equation to find the angles between them.
  ::课程在 kaleidoscope 后面探索了数学。 使用互动来制作一个 kaleidoscope , 并查看反射数与镜面之间的角度量有何关联 。 然后, 您将用方程式替换反射数, 以找到它们之间的角 。

   

   

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  To find the angle measure between mirrors, divide 360° by the number of reflections. You  can represent this relationship using the equation below where  $\begin{align*}a\end{align*}$  is the angle measure between mirrors and  $\begin{align*}r\end{align*}$  is the number of reflections.
  ::要在镜中找到角量度, 请用反射数除以 360 ° 。 您可以使用下面的方程式来表示此关系, 在反射数中, 镜中和 r 之间的角度度度度是反射数 。

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  $$\begin{align*}{a = \frac{360}{r}}\end{align*}$$
  ::a=360r

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  Suppose   you were designing a kaleidoscope and wanted  the angle measure between mirrors to be  18° . How could  you determine the number of reflections ?
  ::假设你正在设计一个千兆字眼, 并且希望镜面之间的角度量为 18 °。 您如何确定反射数 ?

   

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  What is an Equation?
  ::什么是方程?

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  Start by looking at what an equation is . The equation below represents a formula used to determine the perimeter of a square where  $\begin{align*}P\end{align*}$  is perimeter and  $\begin{align*}s\end{align*}$  is the length of a side:
  ::从观察方程开始。下面的方程代表一个公式,用来确定P是周边和s是侧面长度的方形的周边:

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  $$\begin{align*}{P = 4s}\end{align*}$$
  ::P=4s

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  Example
  ::示例示例示例示例

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  If the perimeter of a square is 20 inches, what is the length of each side of the square?
  ::如果一个广场的周界是20英寸,那么广场两侧的长度是多少?

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  B egin by substituting 20 for  $\begin{align*}P,\end{align*}$  since  you know that the perimeter of the square is 20 inches.
  ::首先用P20代替P 因为你知道广场的周界是20英寸

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  $$\begin{align*}20 = 4s\end{align*}$$
  ::20=4s 20=4s

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  You  will explore methods for solving equations in  other sections, but  you do not need them to solve this equation. An equation is a statement that two expressions are equal.  You know that the left side of the equation is twenty, so  you must identify what value of  $\begin{align*}s\end{align*}$  will make the right side of the equation equal to 20. To determine this,  you can use your knowledge of multiplication facts:
  ::您将探索解析其他部分中方程式的方法, 但您不需要这些方程式来解析此方程式。 方程式是一个两个表达式相等的语句。 您知道方程式的左侧是 20 , 所以您必须确定 s 的值会使方程式的右侧等于 20 。 要确定这一点, 您可以使用您对乘法事实的知识 :

  $$\begin{align*}4 \cdot 1 =& \:4\\ 4 \cdot 2 =& \:8\\ 4 \cdot 3 =& \:12\\ 4 \cdot 4 =& \:16\\ 4 \cdot 5 =& \:20\end{align*}$$

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  The side length of the square must be  five inches since  $\begin{align*}4 \cdot 5 = 20.\end{align*}$
  ::方形的侧长必须自4°5=20起为5英寸。

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  The solution to an equation is the value that makes both sides equal.
  ::等式的解决方案是让双方平等的价值。

   

   

   

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  Kaleidoscopes Continued
  ::继续

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  In the opening activity,  you considered the equation  $\begin{align*}a = \frac{360}{r}\end{align*}$  where  $\begin{align*}a\end{align*}$  is the angle measure between mirrors and  $\begin{align*}r\end{align*}$  is the number of reflections.  You were asked to determine how many reflections would be needed for an angle of 1 8 °.
  ::在开始活动时,您会考虑方程式 a=360r, 其中镜像与 r 之间的角度为反射数。 您被要求确定18 °角需要多少反射 。

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  You know the angle between mirrors is this scenario is 18° so  you can substitute 18 for  $\begin{align*}a:\end{align*}$
  ::你知道镜子之间的角是18° 这样你就可以用18°代替:

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  $$\begin{align*}18 = \frac{360}{r}\end{align*}$$
  ::18=360r

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  Then  explore possible values for  $\begin{align*}r\end{align*}$  to determine  what value, when divided into 360, will equal 18.
  ::然后为r探索可能的数值,以确定当分为360时,什么值等于18。

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  Use the interactive below to determine which value of $\begin{align*}r\end{align*}$  can be substituted to solve the equation. Drag the red slider to change the value of  $\begin{align*}r\end{align*}$  to make the left side of the equation equal 18.
  ::使用下面的交互作用来确定 r 的哪个值可以替代解方程式。拖曳红色滑块以修改 r 的值,使方程式的左侧等于 18 。

   

   

   

   

   

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  Checking Solutions
  ::检查解决方案

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  When solving an equation, you are finding the values that make the equation true.  To check,  substitute  the solution back into the equation  and verify that  both sides result in the same number.
  ::当解析方程式时, 您正在找到使方程式成为真实的值。 要检查, 将解决方案重新转换到方程式中, 并验证双方的结果是否相同 。

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  Some equations may have multiple instances of a variable on either side of the equal sign. What would a solution look like in th at situation ?
  ::有些方程式在等号的两侧可能有多个变量实例。 在这种情况下,解决方案会是什么样子?

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  Use the interactive below to substitute values for $\begin{align*}x\end{align*}$  to see if they make the equation true.
  ::使用下面的交互效果来替换 x 的值, 看看它们是否将方程式变为真实 。

   

   

   

   

   

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  Astronomical Order Continued
  ::天文秩序继续

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  T he relationship between the  period  it takes an object in our solar system to orbit the sun in years, $\begin{align*}T,\end{align*}$  and the distance from that object to the S un in astronomical units,  $\begin{align*}a\end{align*}$   (1 AU ≈ the average distance from the Sun to the Earth), can be expressed using the equation $\begin{align*}{T}^{2} = {a}^{3}.\end{align*}$   You can use this equation to find period,  $\begin{align*}T,\end{align*}$  if you know the distance ,  $\begin{align*}a,\end{align*}$  or to find  $\begin{align*}a\end{align*}$   if you know   $\begin{align*}T.\end{align*}$
  ::太阳系中一个物体绕太阳运行所需时间与天文单位中该物体与太阳之间的距离(一个从太阳到地球的平均距离为1个AU = = = = = a3)之间的关系,可以用方程式T2=a3表示。 您可以使用此方程式找到时间段, T, 如果你知道距离, a 或知道T, 或找到一个。

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  In the interactive below, you will determine the orbital period of an asteroid by finding what value can be substituted to make the equation $\begin{align*}{T}^{2} = {a}^{3}\end{align*}$  true.
  ::在下文互动中,您将找到可以替代什么价值来使方程式T2=a3成为真实,从而确定小行星的轨道周期。

   

   

   

   

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  Discussion Questions
  ::讨论问题 讨论问题

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  1. Given the equation $\begin{align*}p^2=a^3,\end{align*}$  if the value of a is -4, what is the value of p?
     ::考虑到p2=a3等式,如果a值为4,那么p值是多少?
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  2. Will this equation work with any negative number? Why?
     ::这个等式能用负数吗?

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  When Johannes Kepler made this discovery, he believed that he had discovered the order to the universe! It was not until about 60 years later, when Isaac Newton discovered gravity, that we had an explanation for Kepler’s discovery.
  ::当约翰尼斯·开普勒发现了这个发现时,他相信他发现了宇宙的秩序!直到大约60年后,艾萨克·牛顿发现了引力,我们才对开普勒的发现做出解释。

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  The time it takes an object in our solar system to orbit the S un in years,  $\begin{align*}p,\end{align*}$  and the distance from that object to the sun in astronomical units,  $\begin{align*}a,\end{align*}$  are proportional as a result of gravity. 
  ::太阳系中一个天体运行太阳轨道的时间, p, 以及天文单位从该天体到太阳的距离, a 由重力成比例。

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   Summary
  ::摘要

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  • An equation is a statement that two expressions are equal.
    ::等式是两个表达式相等的语句 。
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  • The solution to an equation is a value that makes both sides equal.
    ::等式的解决方案是一种使双方平等的价值。
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  • Check for solutions by substituting values for the variable into the equation. If both sides are equal, then the value is a solution. 
    ::通过将变量的值替换为等式来检查解决方案。如果两边相等,则该值是一个解决方案。