12.10输入输出表

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  输入输出表

  

  [Figure 1]

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  Sonja’s 4-H club has spent the last two months collecting food donations for their community service project. The club plans to distribute food bags to needy families during the thanks giving holiday. Sonja is in charge of preparing the food bags. She has enlisted the help of several of her club members. The first day, Sonja and two other club members prepared 9 bags. The second day, Sonja prepared 3 bags by herself.  On the third day, Angela helped Sonja and the two of them prepared 6 bags. On the fourth day, five members showed up to help Sonja and the six of them prepared 18 bags. Sonja organized this data into and input-out table.
  ::Sonja的4-H俱乐部在过去两个月里为社区服务项目募集了食物捐赠。俱乐部计划在感谢节日期间向贫困家庭分发食品袋。Sonja负责准备食品袋。她已经请了几个俱乐部成员的帮助。第一天,Sonja和另外两名俱乐部成员准备了9袋。第二天,Sonja自己准备了3袋。第三天,Angela帮助Sonja,他们两人准备了6袋。第四天,5名成员前来帮助Sonja,他们中的6人准备了18袋。Sonja整理了这个数据并将其输入表格。

  Input	Output
  1	3
  2	6
  3	9
  6	18

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  Do you see a  pattern ? How can Sonja use the information in the table to find a pattern rule for preparing the bags?
  ::你看到一个模式吗?Sonja如何利用表格中的信息来找到一个模式规则来准备袋子?

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  In this concept, you will learn to write an  expression  for an  input-output table .
  ::在此概念中,您将学会写出输入产出表的表达式。

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  Writing Expressions for Input-Output Tables
  ::输入-产出表格的书写表达式

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  A  pattern  is something that repeats in a specific way. Patterns are everywhere in life. They exist in nature and in machinery and even in temperatures. Detecting patterns is one of the things that mathematicians and scientists do every day. They look for patterns in the way that things are created or counted and then they can draw conclusions based on those patterns.
  ::一种模式是一种以特定方式重复的模式。 模式存在于生活中的任何地方。 它们存在于自然和机械中,甚至温度中。 检测模式是数学家和科学家每天做的事情之一。 他们寻找创造或计算事物的方式的模式, 然后他们就可以根据这些模式得出结论。

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  A pattern functions according to a rule. The rule tells us how the pattern repeats.
  ::模式性函数根据规则运行。 规则告诉我们模式性重现的方式 。

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  This group of numbers is a pattern.
  ::此组数字是一个模式 。

  2, 4, 6, 8, 10. . . . .

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  Once you have a pattern, you can establish a rule about the pattern. This pattern counts by two’s. You could also say that two is added to each previous  term  to get the next term in the pattern.
  ::一旦您有一个模式, 您可以建立关于模式的规则。 这个模式由两个计算。 您也可以说, 上一个任期中每个任期都增加两个, 以获得模式中的下一个任期 。

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  In this example, you could use a  variable  to represent the  terms  in the list. Let’s use  $\begin{array}{r}x\end{array}$ .
  ::在此示例中, 您可以使用变量来代表列表中的术语。 让我们使用 x 。

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  $\begin{array}{r}x=\text{term in the pattern}\end{array}$
  ::x = 模式中的术语

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  The word “term” refers to the numbers 2, 4, 6 and so on.
  ::“术语”一词指数字2、4、6等。

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  Next, you can add more to the variable. Since you add two to each term to get the next term, then you can say that  $\begin{array}{r}x\end{array}$  plus two is the rule.
  ::下一步,您可以在变量中添加更多内容。由于您在每个术语中添加两个来获得下一学期,那么您可以说x+2是规则。

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  $\begin{array}{r}\text{Rule}:x+2\end{array}$
  ::规则:x+2

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  Now let’s check the rule to be sure that it works for each term in the list. 2, 4, 6, 8, 10 . . . 
  ::现在让我们检查一下这个规则, 以确保它适用于清单中的每个术语。 2, 4, 6, 8, 10.

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  If you take 2 and  substitute  it for  $\begin{array}{r}x,\end{array}$  then  $\begin{array}{r}2+2=4\end{array}$ , so the rule works here.
  ::如果你拿2来代替x, 那么2+2=4, 规则在这里有效 。

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  If you take 4 and substitute it for  $\begin{array}{r}x,\end{array}$  then  $\begin{array}{r}4+2=6\end{array}$ , so the rule works here.
  ::如果你拿4来代替x, 那么4+2=6, 规则在这里有效 。

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  If you take 6 and substitute it for  $\begin{array}{r}x,\end{array}$  then  $\begin{array}{r}6+2=8\end{array}$ , so the rule works here also.
  ::如果您拿 6 来代替 x, 那么 6 + 2 = 8, 规则在此也有效 。

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  An easier way to figure this out would be to use a table called an input/output table.
  ::比较容易找出答案的方法是使用一个称为输入/产出表的表格。

  Input	Output
  2	4
  4	6
  6	8
  8	10

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  Let’s see if the rule  $\begin{array}{r}x+2\end{array}$  works for this table.
  ::让我们看看规则 x + 2 是否为本表工作 。

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  A term has been put into the table; that is the input. Then a term comes out; that is the output. The rule tells us what happened to the input to equal the output.
  ::一个术语被放入表格;这就是输入。 然后一个术语出来; 这就是输出。 规则告诉我们输入发生什么, 以等值输出 。

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  Two can be added to each term in the input column to equal the output column so the rule  $\begin{array}{r}x+2\end{array}$  works.
  ::输入栏中每个术语中可以添加两个,以等同输出列,因此规则 x+ 2 工作。

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  You can write rules by examining the patterns in input/output tables.
  ::您可以通过检查输入/产出表格中的模式来撰写规则。

  Input	Output
  0	0
  1	3
  2	6
  3	9

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  Figuring out a rule is a little like deciphering a puzzle. You have to think of what happened to one term to equal another term.
  ::寻求一条规则有点像解开一个谜题。你必须想想一个术语发生什么,才能等得上另一个术语。

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  In this table, the term in the input column was multiplied by 3 to get the number in the output column.
  ::在本表中,输入栏中的术语乘以3,以获得产出栏中的数字。

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  You can write the rule as an expression.
  ::您可以将此规则写成表达式 。

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  If the input column is  $\begin{array}{r}x\end{array}$ , then  $\begin{array}{r}3x\end{array}$  is the rule for this table.
  ::如果输入列为 x,则此表格的规则为3 x。

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  $\begin{array}{r}\text{Rule}=3x\end{array}$
  ::规则=3x

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  Sometimes rules are a bit more complicated. Sometimes, there can be two  operations  in a rule.
  ::有时规则比较复杂,有时规则中可能有两种操作。

  Input	Output
  3	7
  4	9
  5	11
  7	15

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  Look for a pattern in this table.
  ::在此表格中查找一个模式 。

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  You should see that the input was multiplied by two and then one was added to get the output. You can write the rule as an expression, using a variable as the input.
  ::您应该看到输入乘以 2, 然后添加一个来获取输出。 您可以将规则写成表达式, 使用变量作为输入 。

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  $\begin{array}{r}\text{Rule}=2x+1\end{array}$
  ::规则=2x+1

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  The input-output relationship of terms is a  function , which is when one variable or terms depends on another according to a rule.
  ::术语的输入-产出关系是一个函数,即一个变量或术语根据规则依赖于另一个变量或术语。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were given a problem about Sonja and her 4-H club’s community service project.
  ::Sonja和她4H俱乐部的社区服务项目,

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  How can Sonja use the information presented in the table below to come up with a pattern rule that explains how the number of people preparing bags affects the output or number of bags prepared?
  ::Sonja如何利用下表所列信息提出一种模式规则,解释准备袋的人数如何影响准备袋的输出或数量?

  Input	Output
  1	3
  2	6
  3	9
  6	18

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  First, look at the table and ask yourself, “What happened to  $\begin{array}{r}x\end{array}$  (input) to get  $\begin{array}{r}y\end{array}$  (output)?”
  ::首先,看看桌子,问自己,“x(投入)为了获得(产出)发生了什么?”

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  What happened to 1 to get 3? What happened to 2 to get 6? If you look carefully, you will see that the input value is multiplied by 3 to get the output value.
  ::发生什么了 1 到 3 ? 发生什么了 2 到 6 ? 如果您仔细看, 你会看到输入值乘以 3 以获得输出值 。

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  You can write the pattern rule as,  $\begin{array}{r}3x\end{array}$ .
  ::您可以以 3 x 的形式写入模式规则。

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  Next, see if the rule  $\begin{array}{r}3x\end{array}$  works for each term in the table by plugging the input into the expression and seeing if it equals the listed output.
  ::下一步,请看看规则3 x 是否对表格中的每个术语起作用,方法是将输入插入表达式,看看它是否等于所列输出。

  $$\begin{array}{r}\begin{array}{rcl}& & 3x\\ & & 3(1)\\ & & 3(\text{output})\end{array}\end{array}$$

  $$\begin{array}{r}\begin{array}{rcl}& & 3x\\ & & 3(2)\\ & & 6(\text{output})\end{array}\end{array}$$

  $$\begin{array}{r}\begin{array}{rcl}& & 3x\\ & & 3(3)\\ & & 9(\text{output})\end{array}\end{array}$$

  $$\begin{array}{r}\begin{array}{rcl}& & 3x\\ & & 3(6)\\ & & 18(\text{output})\end{array}\end{array}$$

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  The answer is correct.
  ::答案是正确的。

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  Example 2
  ::例2

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  Write a rule for the table below.
  ::为下表写一条规则。

  Input	Output
  2	6
  3	8
  4	10
  5	12

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  First, look at the table and ask yourself, “What happened to  $\begin{array}{r}x\end{array}$  to get  $\begin{array}{r}y\end{array}$ ?”
  ::首先,看看桌子,问自己,“x怎么了?”

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  If you look carefully, you will see that the value in the input column was multiplied by 2 and then 2 was added to that value.
  ::如果您仔细看一看, 就会看到输入列中的值乘以 2, 然后将 2 乘以该值 。

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  Next, let  $\begin{array}{r}x\end{array}$  be the input value and write the expression.
  ::下一步,让 x 成为输入值,然后写入表达式。

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  $\begin{array}{r}2x+2\end{array}$
  ::2 x + 2

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  The answer is  $\begin{array}{r}2x+2\end{array}$ .
  ::答案是 2 x + 2 。

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  Then, see if the rule  $\begin{array}{r}2x+2\end{array}$  work for each term in the table by plugging the input into the expression and seeing if it equals the listed output.
  ::然后, 看看规则 2 x + 2 是否对表格中的每个术语有效, 将输入插入表达式, 并查看它是否等于列出的输出 。

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  Substitute the input values in for  $\begin{array}{r}x\end{array}$  in the expression  $\begin{array}{r}2x+2\end{array}$  to see if you get the results in the output column.
  ::替换表达式 2 x + 2 中的 x 输入值,以查看是否在输出列中取得结果。

  $$\begin{array}{r}\begin{array}{rcl}& & 2x+2\\ & & 2(2)+2\\ & & 4+2\\ & & 6\end{array}\end{array}$$

   

  $$\begin{array}{r}\begin{array}{rcl}& & 2x+2\\ & & 2(3)+2\\ & & 6+2\\ & & 8\end{array}\end{array}$$

   

  $$\begin{array}{r}\begin{array}{rcl}& & 2x+2\\ & & 2(4)+2\\ & & 8+2\\ & & 10\end{array}\end{array}$$

   

  $$\begin{array}{r}\begin{array}{rcl}& & 2x+4\\ & & 2(4)+4\\ & & 8+4\\ & & 12\end{array}\end{array}$$

  The results check out.

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  Example 3
  ::例3

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  Write a rule for the table below.
  ::为下表写一条规则。

  Input	Output
  10	6
  9	5
  8	4
  7	3

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  First, look at the table and ask yourself, “What happened to  $\begin{array}{r}x\end{array}$  (input) to get  $\begin{array}{r}y\end{array}$  (output)?”
  ::首先,看看桌子,问自己,“x(投入)为了获得(产出)发生了什么?”

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  If you look carefully, you will see that 4 is subtracted from the input value to get the output value.
  ::如果您仔细看一看, 就会看到从输入值中减去 4 , 以获得输出值 。

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  Next, let  $\begin{array}{r}x\end{array}$  be the input value and write the expression.
  ::下一步,让 x 成为输入值,然后写入表达式。

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  $\begin{array}{r}x-4\end{array}$
  ::x - 4 个

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  The answer is  $\begin{array}{r}x-4\end{array}$ .
  ::答案是x-4。

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  Then, see if the rule  $\begin{array}{r}x-4\end{array}$  work for each term in the table by plugging the input into the expression.
  ::然后,看看规则 x - 4 是否通过将输入插入表达式来对表格中每个术语起作用。

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  Substitute the input values in for  $\begin{array}{r}x\end{array}$  in the expression  $\begin{array}{r}x-4\end{array}$  to see if you get the results in the output column.
  ::替换表达式 x - 4 中的 x 的输入值,以查看是否在输出列中取得结果。

  $$\begin{array}{r}\begin{array}{rcl}& & x-4\\ & & 10-4\\ & & 6(\text{output})\end{array}\end{array}$$

  $$\begin{array}{r}\begin{array}{rcl}& & x-4\\ & & 9-4\\ & & 5(\text{output})\end{array}\end{array}$$

  $$\begin{array}{r}\begin{array}{rcl}& & x-4\\ & & 8-4\\ & & 4(\text{output})\end{array}\end{array}$$

  $$\begin{array}{r}\begin{array}{rcl}& & x-4\\ & & 7-4\\ & & 3(\text{output})\end{array}\end{array}$$

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  The answer is correct.
  ::答案是正确的。

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  Example 4
  ::例4

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  Write a rule for the table below.
  ::为下表写一条规则。

  Input	Output
  2	4
  4	8
  6	12
  7	14

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  First, look at the table and ask yourself, “What happened to  $\begin{array}{r}x\end{array}$  (input) to get  $\begin{array}{r}y\end{array}$  (output)?”
  ::首先,看看桌子,问自己,“x(投入)为了获得(产出)发生了什么?”

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  If you look carefully, you will see that the input value is multiplied by 2 to get the output value.
  ::如果您仔细看一看, 就会看到输入值乘以 2 以获得输出值 。

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  Next, let  $\begin{array}{r}x\end{array}$  be the input value and write the expression.
  ::下一步,让 x 成为输入值,然后写入表达式。

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  $\begin{array}{r}(x{)}^2\end{array}$  or  $\begin{array}{r}2x\end{array}$ .
  :x) 2或2x.

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  The answer is  $\begin{array}{r}2x\end{array}$ .
  ::答案是 2 x 。

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  Then, see if the rule  $\begin{array}{r}2x\end{array}$  works for each term in the table by plugging the input into the expression to see if you get the output value.
  ::然后,看看规则 2 x 是否对表格中的每个术语有效, 将输入插入表达式, 以查看您是否获得输出值 。

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  Substitute the input values in for  $\begin{array}{r}x\end{array}$  in the expression  $\begin{array}{r}2x\end{array}$  to see if you get the results in the output column.
  ::将表达式 2 x 中的 x 的输入值替换为输入值,以便查看您是否在输出列中取得了结果。

  $$\begin{array}{r}\begin{array}{rcl}& & 2x\\ & & 2(4)\\ & & 8(\text{output})\end{array}\end{array}$$

  $$\begin{array}{r}\begin{array}{rcl}& & 2x\\ & & 2(5)\\ & & 10(\text{output})\end{array}\end{array}$$

  $$\begin{array}{r}\begin{array}{rcl}& & 2x\\ & & 2(6)\\ & & 12(\text{output})\end{array}\end{array}$$

  $$\begin{array}{r}\begin{array}{rcl}& & 2x\\ & & 2(7)\\ & & 14(\text{output})\end{array}\end{array}$$

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  The answer is correct.
  ::答案是正确的。

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  Example 5
  ::例5

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  Write a rule for the table below.
  ::为下表写一条规则。

  Input	Output
  0	5
  1	6
  2	7
  4	9

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  First, look at the table and ask yourself, “What happened to  $\begin{array}{r}x\end{array}$  (input) to get  $\begin{array}{r}y\end{array}$  (output)?”
  ::首先,看看桌子,问自己,“x(投入)为了获得(产出)发生了什么?”

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  If you look carefully, you will see that 5 is added to the input value to get the output value.
  ::如果您仔细看一看, 将会看到输入值中添加 5 5 , 以获得输出值 。

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  Next, let  $\begin{array}{r}x\end{array}$  be the input value and write the expression.
  ::下一步,让 x 成为输入值,然后写入表达式。

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  $\begin{array}{r}x+5\end{array}$
  ::x+5x+5

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  The answer is  $\begin{array}{r}x+5\end{array}$ .
  ::答案是 x + 5 。

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  Now, see if the rule  $\begin{array}{r}x+5\end{array}$  work for each term in the table by plugging the input into the expression and seeing if it equals the listed output?
  ::现在, 看看规则 x + 5 是否对表格中的每个术语有效, 将输入插入表达式, 并查看它是否等于所列输出 ?

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  Substitute the input values in for  $\begin{array}{r}x\end{array}$  in the expression  $\begin{array}{r}x+5\end{array}$  to see if you get the results in the output column.
  ::替换表达式 x + 5 中的 x 的输入值,以查看是否在输出列中取得结果。

  $$\begin{array}{r}\begin{array}{rcl}& & x+5\\ & & 5+0\\ & & 5(\text{output})\end{array}\end{array}$$

  $$\begin{array}{r}\begin{array}{rcl}& & x+5\\ & & 1+5\\ & & 6(\text{output})\end{array}\end{array}$$

  $$\begin{array}{r}\begin{array}{rcl}& & x+5\\ & & 2+5\\ & & 7(\text{output})\end{array}\end{array}$$

  $$\begin{array}{r}\begin{array}{rcl}& & x+5\\ & & 3+5\\ & & 8(\text{output})\end{array}\end{array}$$

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  The answer is correct.
  ::答案是正确的。

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  Review
  ::回顾

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  There are three parts to each problem below. First, write an expression for each input-output table. Next, plug in the next two integer inputs into the expression to determine their outputs.
  ::下面的每个问题有三个部分。 首先, 为每个输入输出表写一个表达式。 下一步, 在表达式中插入下两个整数输入, 以确定其输出 。

  1.   

  Input	Output
  1	4
  2	5
  3	6
  4	7

  2.   

  Input	Output
  2	4
  3	6
  4	8
  5	10

  3.    

  Input	Output
  1	3
  2	6
  4	12
  5	15

  4.     

  Input	Output
  9	7
  7	5
  5	3
  3	1

  5.     

  Input	Output
  8	12
  9	13
  11	15
  20	24

  6.     

  Input	Output
  3	21
  4	28
  6	42
  8	56

  7.    

  Input	Output
  2	5
  3	7
  4	9
  5	11

  8.      

  Input	Output
  4	7
  5	9
  6	11
  8	15

  9.     

  Input	Output
  5	14
  6	17
  7	20
  8	23

  10.     

  Input	Output
  4	16
  5	20
  6	24
  8	32

  播放段落

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  播放段落

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