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    Josephine, a local artist, has been commissioned to enlarge a quilt hanging. The new quilt cannot exceed 7.5 feet in length or 6.5 feet in width due to the space available for hanging. After much thought, Josephine has decided to increase the size of the 4 foot by 3 foot quilt by adding a uniform, quilted strip around the existing one. Josephine thinks that a 1.25 feet wide strip will work, but how can Josephine use a variable expression to figure this out?
    ::当地艺术家约瑟芬(Josephine)被委托扩大一件缝衣吊挂。 新的缝衣因可悬吊的空间而不能超过7.5英尺长或6.5英尺宽。 经过深思熟虑后,约瑟芬决定增加4英尺长3英尺长的缝衣,在原有的缝衣上加一条制服和被缝的条纹。 约瑟芬认为宽1.25英尺宽的条纹会有效,但约瑟芬如何用变体表达来解决这个问题呢?

    In this concept, you will learn to write and evaluate variable expressions for given situations.
    ::在此概念中,您将学习写作和评估特定情况下的变量表达式。

    Variable Expressions for Given Situations
    ::给定情况变量表达式

    Words can be translated into mathematical expressions to represent real-world problems. Then, mathematical operations can be used to solve the problem. You must be able to recognize the operations being presented by the words and be able to use symbols to display them. All unknown quantities can be represented by variables.
    ::单词可以转换为数学表达式, 以代表现实世界的问题。 然后, 数学操作可以用来解决问题 。 您必须能够识别单词所展示的操作, 并且能够使用符号来显示它们。 所有未知的数量都可以用变量来表示 。

    Let’s look at an example.
    ::让我们举个例子。

    Complete the following table with variable expressions having either one or two variables as needed.
    ::填写下表,可变表达式按需要有一个或两个变量。

    Words to be Translated Translations Variable Expression
    “Six times a number decreased by 7”    
    “The sum of Sam’s age and Joe’s age”    
    “The value, in cents, of y  dimes”    
    “Three times the length less twice the width”    
    Words to be Translated Translations Variable Expression
    “Six times a number decreased by 7” Six times a number 6 n     decreased by     seven 7 6 n 7
    “The sum of Sam’s age and Joe’s age” The sum of +     Sam's age s     and     Joe's age j   s + j
    “The value, in cents, of  y  dimes” The value in cents 0.10     of     y  dimes y 0.10 y
    “Three times the length less twice the width” Three times the length 3 l     less     twice the width 2 w 3 l 2 w

    Once you have written a variable expression to represent the words, you can evaluate the expression for a given value of the variable or values of the variables.
    ::一旦您写了代表单词的变量表达式,您就可以对变量或变量的值的给定值或变量的值来评估表达式。

    Let’s look at this problem.
    ::让我们来看看这个问题。

    Evaluate “Four times a number reduced by “five” when the number is 4.
    ::在数字为4时,评价“四倍于数字减少“五”的“四倍”。

    First, write a variable expression.
    ::首先,写一个变量表达式。

    There are two operations presented by the words. The word “times” tells you there is multiplication and the words “reduced by” tells you there is subtraction .
    ::“时间”一词表示有乘法,“减少”一词表示有减法。

    Four times a number 4 n reduced by five 5

    ::4乘以4 乘以4 乘以5 5 乘以4 乘以4 乘以4 乘以4 乘以5 乘以5

    The variable expression is 4 n 5 .
    ::变量表达式为 4n-5。

    First, evaluate the expression when n = 4 .
    ::首先, 当 n=4 时对表达式进行评价 。

    Next, substitute the value n = 4  into the expression.
    ::接下来,将数值 n=4 替换为表达式。

    4 ( 4 ) 5

    Next, multiply 4 ( 4 ) = 16  to clear the parenthesis. Write the new expression.
    ::下一步,乘以 4(4)=16来清除括号。写新表达式。

    16 5

    Then, subtract:
    ::然后,减去:

    16 5 = 11

    The answer is 11.
    ::答案是11个

    Examples
    ::实例

    Example 1
    ::例1

    Earlier, you were given a problem about  Josephine and her quilt enlargement.
    ::早些时候,你得到一个问题 关于约瑟芬 和她的裁缝扩大。

    In order to enlarge the quilt she has to add two strips of the same width to the width of the quilt and two more strips of the same width to the length of the quilt. To figure out if a strip 1.25 feet wide will work, Josephine has to write two variable expressions – one for the length and one for the width.
    ::为了扩大裁缝,她必须在裁缝的宽度上加两个相同宽度的条纹,而在裁缝的长度上加两个相同宽度的条纹。 要弄清楚宽1.25英尺的条纹是否有效,Josephine必须写两个变量表达式 — — 一个是长度的条纹,另一个是宽度的条纹。

    For the length of 4 feet:
    ::四英尺长:

    First, name the variable. Let “ x ” represent the width of the strip.
    ::首先,给变量命名。让“x”代表条宽度。

    Next, include any operation on the variable.  2 x
    ::下一步, 包括变量 2x 上的任何操作

    Next, include any constant associated with the variable: 4
    ::下一步,包括变量 4 中与变量有关的任何常数

    Next, represent the operation between the constant and the variable by its symbol. +
    ::下一个,按其符号代表常数和变量之间的操作。+

    Then, write the variable expression for the length.
    ::然后,写出长度的变量表达式。

    The answer is 4 + 2 x .
    ::答案是4+2x。

    For the width of 3 feet:
    ::三英尺宽度:

    First, name the variable. Let “ x ” represent the width of the strip.
    ::首先,给变量命名。让“x”代表条宽度。

    Next, include any operation on the variable.  2 x
    ::下一步, 包括变量 2x 上的任何操作

    Next, include any constant associated with the variable: 3
    ::下一步,包括变量3中与变量3有关的任何常数

    Next, represent the operation between the constant and the variable by its symbol. +
    ::下一个,按其符号代表常数和变量之间的操作。+

    Then, write the variable expression for the length.
    ::然后,写出长度的变量表达式。

    The answer is 3 + 2 x .
    ::答案是3+2x。

    The space for hanging the quilt is 7.5 feet in length and 6.5 feet in width.
    ::上吊盖的面积为7.5英尺长,宽6.5英尺宽。

    First, substitute x = 1.25  into the variable expression for the length.
    ::首先,将长度的变量表达式中的 x=1.25 替换为变量表达式。

    4 + 2 ( 1.25 )

    Next, multiply 2 ( 1.25 ) = 2.50  to clear parenthesis.
    ::下一个乘以2(1.25)=2.50来清除括号。

    4 + 2.50

    Then, add:
    ::然后,加上:

    4 + 2.50 = 6.50

    The answer is 6.5.
    ::答案是6.5。

    First, substitute x = 1.25  into the variable expression for the width.
    ::首先,将宽度的变量表达式中的 x=1.25 替换为变量表达式。

    3 + 2 ( 1.25 )

    Next, multiply 2 ( 1.25 ) = 2.50  to clear parenthesis.
    ::下一个乘以2(1.25)=2.50来清除括号。

    3 + 2.50

    Then, add:
    ::然后,加上:

    3 + 2.50 = 5.50

    The answer is 5.5.
    ::答案是5.5。

    Josephine can hang her enlarged quilt in the available space.
    ::约瑟芬可以把宽宽的毛衣挂在空隙里

    Example 2
    ::例2

    Grace is saving money to purchase a new bike that costs one hundred eighty-five dollars. She has twenty-five dollars saved already and plans to save twenty dollars of her babysitting money each week. Write a variable expression to model the problem.
    ::格蕾丝在节省钱购买一辆新自行车,费用为一百八十五美元。 她已经节约了二十五美元,并计划每周节省20美元看孩子的钱。 写一个可变的表达方式来模拟这个问题。

    First, name the variable. Let ‘ w ’ represent the number of weeks she has to save twenty dollars.
    ::首先, 点出变量名称。 让“ w” 代表她要节省20美元的周数 。

    Next, include the dollar value with the variable.
    ::下一步,在变量中包括美元值。

    20 w

    ::20w 20w

    Next, add the money she has already saved. +25
    ::下一步,加上她已经储蓄的钱。+25

    20 w + 25

    ::20w+25 20w+25

    Then, evaluate the variable expression
    ::然后,对变量表达式进行评价

    20 w + 25  for w = 7  and w = 8 .
    ::W=7和W=8,20w+25。

    If one of these given values result in the variable expression being $185.00, then you know that Grace has to save twenty dollars for that number of weeks to have enough money to buy her new bike.
    ::如果其中一个给定值导致变数表达式为185.0美元,那么你知道Grace要为这个星期节省20美元 才能有足够的钱买下她的新自行车。

    First, substitute w = 7  into the expression.
    ::首先,在表达式中替换 w=7。

    20 ( 7 ) + 25

    Next, multiply: 20 ( 7 ) = 140  to clear the parenthesis.
    ::下一个乘以:20(7)=140来清除括号。

    140 + 25

    Then, add:
    ::然后,加上:

    140 + 25 = 165

    The answer is $165.00
    ::答案是$165.00

    First, substitute w = 8  into the expression.
    ::首先,将 w=8 替换为表达式。

    20 ( 8 ) = 25

    Next, multiply: 20 ( 8 ) = 160  to clear the parenthesis.
    ::下一个乘以: 20(8)=160来清除括号。

    160 + 25

    Then, add:
    ::然后,加上:

    160 + 25 = 185

    The answer is $185.00.
    ::答案是$185.00

    Grace must save $20.00 for eight weeks to have enough money to buy the bike.
    ::格蕾丝得在八周内节省20美元 才能有足够的钱买自行车

    Example 3
    ::例3

    Write a variable expression to model “Twice the length increased by three times the width” and evaluate it when the length is 12 and the width is 9.
    ::给模型写一个变量表达式,“将长度增加三倍,宽度增加三倍”,并在长度为12和宽度为9时进行评估。

    First, name the variables. Let “ l ” represent the length. Let “ w ” represent the width.
    ::首先,给变量命名。让“l”代表长度。让“w”代表宽度。

    Next, include any operations on the variables.
    ::下一步,包括变量上的任何操作。

    2 l  and  3 w
    ::2-1和3w

    Next, represent the operation between the variables by its symbol. +
    ::下一步,按变量的符号表示变量之间的操作。+

    Then, write the variable expression.
    ::然后,写下变量表达式。

    Twice the length 2 l increased by + three times the width 3 w
     
    ::宽度乘以 3 倍 宽度乘以 3w

    The answer is 2 l + 3 w .
    ::答案是 2l+3w。

    First, substitute the values l = 12  and w = 9  into the expression.
    ::首先,将值 I=12 和 w=9 替换为表达式。

    2 ( 12 ) + 3 ( 9 )

    Next, multiply 2 ( 12 ) = 24  to clear the parenthesis.
    ::下一个乘以2(12)=24来清除括号。

    24 + 3 ( 9 )

    Next, multiply 3 ( 9 ) = 27  to clear the parenthesis.
    ::下一个乘以3(9)=27来清除括号。

    24 + 27

    Then, add:
    ::然后,加上:

    24 + 27 = 51

    The answer is 51.
    ::答案是51岁

    Example 4
    ::例4

    Jake and his Uncle Mark were sitting on the front step discussing what Jake was learning in Math this year. Jake looked up at his uncle and said, “Uncle Mark, do you know what one-fourth my age will be eight years from now?”
    ::Jake和他的叔叔Mark坐在前台讨论Jake今年在数学中学了什么。Jake抬头看着他叔叔说,“叔叔马克,你知道从现在起我的四分之一年龄是8岁吗? ”

    First, name the variable. Let “ a ” represent Jake’s age now.
    ::首先,给变量起个名字。让“a”现在代表Jake的年龄。

    Next, write the expression to represent “ a ” eight years from now.
    ::接下来写“a”表示八年后的“a”表示。

    a + 8

    ::a+8 +8

    Then, include any operation performed on the variable.  1 4 (   )
    ::然后,包括变量14egg 上进行的任何操作

    1 4 ( a + 8 )

    ::14(a+8)

    Evaluate the expression when a = 16 .
    ::当 a=16 时评价表达式 。

    First, substitute the value a = 16  into the expression.
    ::首先,将a=16的值替换为表达式。

    1 4 ( 16 + 8 )

    Next, perform the operation in the parenthesis.  16 + 8 = 24
    ::下一步,在括号内执行操作。 16+8=24

    1 4 ( 24 )

    Then, multiply 1 4 ( 24 ) = 6  to clear the parenthesis.
    ::然后乘以14(24)=6来清除括号。

    The answer is 6.
    ::答案是6个

    Review
    ::回顾

    Evaluate each verbal phrase.
    ::评估每个口头短语。

    1. The sum of a number 12 and twelve.
    ::1. 数字12和12之和。

    2. The difference between a number 12 and eight.
    ::2. 12和8之间的差额。

    3. Three times a number 12.
    ::3. 3乘以12乘以3乘以12 乘以3乘以12

    4. A number 12 squared plus five.
    ::4. 12号方形加5号。

    5. A number 12 divided by two plus seven.
    ::5. 12号除以2+7。

    6. Four times the quantity of a number 12 plus six.
    ::6. 数量为12+6的四倍。

    7. A number 12 times two divided by four.
    ::7. 12乘以2除以4。

    8. A number 12 times six plus the same number times two.
    ::8. 乘以12乘以6乘以12乘以6乘以相同乘以2。

    9. A number 12 squared plus seven take away four.
    ::9. 12平方加7除以4。

    10. A number 12 divided by three plus twelve.
    ::10. 12号除以3+12。

    Evaluate each expression when x = 2  and y = 4 .
    ::x=2 和 y4 时评价每个表达式 。

    11.  4 x + 6 y 4
    ::11. 4x+6y-4

    12.  6 y + x 9
    ::12.-6y+x-9

    13.  4 x y 6 x y
    ::13.-4xy-6xy

    14.  9 x y + 8
    ::14. 9xy+8

    15.  8 x 12 y + 15
    ::15.-8x-12y+15

    Review (Answers)
    ::回顾(答复)

    Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
    ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。

     

    Resources
    ::资源