1.7 描述模式的单词

Section outline

• Select section General

  Collapse Expand

  General

  Collapse all Expand all

  • Select activity 新闻通告

    

    新闻通告 Forum
• Select section 描述模式的单词

  Collapse Expand

  描述模式的单词

  播放段落

  Describing Patterns 
  ::描述模式

  播放段落

  In algebra, an equation is a mathematical expression that contains an equals sign. An equation indicates that two expressions represent the same value, like  $\begin{array}{r}y=12x,\end{array}$ where $\begin{array}{r}y\end{array}$  and $\begin{array}{r}12x\end{array}$  are equal to each other. An inequality is a mathematical expression that contains inequality signs . Inequalities are used to tell us that an expression is either larger or smaller than another expression. For example, $\begin{array}{r}y<12x\end{array}$  indicates that the value of  $\begin{array}{r}y\end{array}$  is less than the value of $\begin{array}{r}12x.\end{array}$  Equations and inequalities can contain both variables and constants .
  ::在代数中, 方程式是一个数学表达式, 包含等号。 方程式表示两个表达式代表相同的值, 如 y=12x, y和 12x 等。 不平等是一个数学表达式, 包含不平等迹象。 不平等用来告诉我们一个表达式大于或小于另一个表达式。 例如, y < 12x 表示y 的值小于 12x 的值。 等式和不平等可以包含变量和常数 。

  播放段落

  Variables are usually identified by a letter and are used to represent unknown values. These quantities can change because they depend on other numbers in the problem.
  ::变量通常用字母识别,并用于代表未知值。这些数量可以变化,因为它们取决于问题中的其他数字。

  播放段落

  Constants are quantities that remain unchanged. Numbers like $\begin{array}{r}2,-3,\frac{3}{4},\end{array}$ and $\begin{array}{r}\pi \end{array}$ are examples of constants, because they always represent the same value.
  ::常数是数量不变的。数字如 2-3,34 和 是常数的例子,因为它们总是代表相同的值。

  播放段落

  Equations and inequalities are used as a shorthand notation for situations that involve numerical data. They are very useful because most problems require several steps to arrive at a solution, and it becomes tedious to repeatedly write out the situation in words.
  ::对于涉及数字数据的情况,均分和不平等被用作简称,非常有用,因为大多数问题需要若干步骤才能找到解决办法,反复用文字写出有关情况是令人厌烦的。

  播放段落

  Here are some examples of equations:
  ::以下是一些方程式的例子:

  播放段落

  $$\begin{array}{r}3x-2=5x+9=2x+5\frac{x}{3}=15x^2+1=10\end{array}$$

  
  ::3x-2=5x+9=2x+5x3=15x2+1=10

  播放段落

  To write an inequality, we use the following symbols:
  ::为了写出不平等,我们使用以下符号:

  播放段落

  > greater than
  ::> 大于

  播放段落

  $\begin{array}{r}\ge \end{array}$ greater than or equal to
  ::大于或等于

  播放段落

  < less than
  ::<小于

  播放段落

  $\begin{array}{r}\le \end{array}$ less than or equal to
  ::大于或等于

  播放段落

  $\begin{array}{r}\ne \end{array}$ not equal to
  ::不等于

  播放段落

  Here are some examples of inequalities:
  ::以下是一些不平等的例子:

  播放段落

  $$\begin{array}{r}3x<54-x\le 2x{x}^2+2x-1>0\frac{3x}{4}\ge \frac{x}{2}-3\end{array}$$

  
  ::3x 3 < 54- x=2x2+2x-1>03x4xx2-3

  播放段落

  Arguably the most important skill in algebra is the ability to translate a word problem into an appropriate equation or inequality so you can find the solution easily. The first two steps are defining the variables and translating the word problem into a mathematical equation.
  ::代数中最重要的技能可以说是将单词问题转化为合适的方程或不平等,这样你就可以很容易地找到解决办法。 前两个步骤是定义变量,并将单词问题转化为数学方程。

  播放段落

  Defining the variables means that we assign letters to any unknown quantities in the problem.
  ::定义变量意味着我们给问题的任何未知数量分配字母。

  播放段落

  Translating means that we change the word problem into a mathematical statement containing variables and mathematical operations , generally with an equal sign or an inequality sign.
  ::翻译意味着我们把字的问题 变成一个数学说明, 包含变量和数学操作, 通常有一个平等的符号或不平等的符号。

  播放段落

  Defining Variables and Translating into Equations 
  ::定义变量和转换成方形

  播放段落

  Define the variables and translate the following statements into equations. Notice that "is" and "was" indicate an equals or inequality sign.
  ::定义变量, 并将以下的语句转换为方程 。 注意“ 是” 和“ 是” 表示一个相等或不平等的符号 。

  播放段落

  a) A number plus 12 is 20.
  :a) 数字加12为20。

  播放段落

  Define
  ::定义定义

  播放段落

  Let $\begin{array}{r}n=\end{array}$ the number we are seeking.
  ::让我们寻找的号码。

  播放段落

  Translate
  ::翻译

  播放段落

  A number plus 12 is 20.
  ::数字加12是20

  播放段落

  $$\begin{array}{r}n+12=20\end{array}$$

  
  ::n+12=20

  播放段落

  b) 9 less than twice a number is 33.
  :b) 9个不到数字的两倍是33。

  播放段落

  Define
  ::定义定义

  播放段落

  Let $\begin{array}{r}n=\end{array}$ the number we are seeking.
  ::让我们寻找的号码。

  播放段落

  Translate
  ::翻译

  播放段落

  9 less than twice a number is 33.
  ::9个不到两倍 数字是33

  播放段落

  This means that twice the number, minus 9, is 33.
  ::这意味着数字的两倍,负9是33。

  播放段落

  $$\begin{array}{r}2n-9=33\end{array}$$

  
  ::2n-9=33

  播放段落

  c) $20 was one quarter of the money spent on the pizza.
  :c) 20美元是披萨花在比萨上的钱的四分之一。

  播放段落

  Define
  ::定义定义

  播放段落

  Let $\begin{array}{r}m=\end{array}$ the money spent on the pizza.
  ::让我们花在披萨上的钱。

  播放段落

  Translate
  ::翻译

  播放段落

  $20 was one quarter of the money spent on the pizza.
  ::20美元是披萨花在比萨上的钱的四分之一

  播放段落

  $$\begin{array}{r}20=\frac{1}{4}m\end{array}$$

  
  ::20=14米

  播放段落

  Often word problems need to be reworded before you can write an equation.
  ::语言问题通常需要重新措辞,才能写出方程式。

  播放段落

  Solving Word Problems 
  ::解决单词问题

  播放段落

  Find the solution to the following problems.
  ::找到解决下列问题的办法。

  播放段落

  a) Shyam worked for two hours and packed 24 boxes. How much time did he spend on packing one box?
  :a) Shyam工作了两个小时,包装了24个箱子,他花了多少时间包装一个箱子?

  播放段落

  Define
  ::定义定义

  播放段落

  Let $\begin{array}{r}t=\end{array}$ time it takes to pack one box.
  ::让我们花时间打包一个盒子

  播放段落

  Translate
  ::翻译

  播放段落

  Shyam worked for two hours and packed 24 boxes. This means that two hours is 24 times the time it takes to pack one box.
  ::Shyam工作了两个小时,包装了24个箱子,这意味着两个小时是包装一个箱子的时间的24倍。

  播放段落

  $$\begin{array}{r}2=24t\end{array}$$

  
  ::2=24吨

  播放段落

  Solve
  ::解决

  播放段落

  $$\begin{array}{rl}t=\frac{2}{24}& =\frac{1}{12}\text{hours}\\ \frac{1}{12}\times 60\text{minutes}& =5\text{minutes}\end{array}$$

  
  ::t=224=112 112x60 分钟=5分钟

  播放段落

  Answer
  ::答复答复答复

  播放段落

  Shyam takes 5 minutes to pack a box.
  ::希亚姆要花5分钟才能装箱

  播放段落

  b) After a 20% discount, a book costs $12. How much was the book before the discount?
  :b) 折价20%后,一本书花费12美元,在折价之前那本书多少钱?

  播放段落

  b) Define
  ::b) 界定

  播放段落

  Let $\begin{array}{r}p=\end{array}$ the price of the book before the discount.
  ::让我们在折扣之前先付书价吧

  播放段落

  Translate
  ::翻译

  播放段落

  After a 20% discount, the book costs $12. This means that the price minus 20% of the price is $12.
  ::折价20 % 后, 书价12美元。 这意味着价格减去价格的20%是 12 美元。

  播放段落

  $$\begin{array}{r}p-0.20p=12\end{array}$$

  
  ::p - 0. 20p=12

  播放段落

  Solve
  ::解决

  播放段落

  $$\begin{array}{rl}p-0.20p& =12\\ 0.8p& =12\\ p=\frac{12}{0.8}& =15\end{array}$$

  
  ::p-0.20p=120.8p=12p=120.8=15

  播放段落

  Answer
  ::答复答复答复

  播放段落

  The price of the book before the discount was $15.
  ::折扣前的书价是15美元

  播放段落

  Check
  ::支票支票支票支票

  播放段落

  If the original price was $15, then the book was discounted by 20% of $15, or $3.
  ::如果最初价格是15美元,那么这本书被折价20%,即15美元或3美元。

  播放段落

  $\begin{array}{r}\mathrm{\$}15-3=\mathrm{\$}12.\end{array}$ The answer checks out .
  ::15 -3美元=12美元。 答案检查出来。

  播放段落

  Defining Variables and Translating into Inequalities 
  ::界定变量并将变量转化为不平等

  播放段落

  Define the variables and translate the following statements into inequalities.
  ::界定变量,并将以下陈述转化为不平等。

  播放段落

  a) The sum of 5 and a number is less than or equal to 2.
  :a) 5和数之和小于或等于2。

  播放段落

  Define
  ::定义定义

  播放段落

  Let $\begin{array}{r}n=\end{array}$ the unknown number.
  ::让 n= 未知的数字 。

  播放段落

  Translate
  ::翻译

  播放段落

  $$\begin{array}{r}5+n\le 2\end{array}$$

  
  ::5+N%2

  播放段落

    Define
  ::定义定义

  播放段落

  Let $\begin{array}{r}n=\end{array}$ the unknown number.
  ::让 n= 未知的数字 。

  播放段落

  Translate
  ::翻译

  播放段落

  $$\begin{array}{r}5+n\le 2\end{array}$$

  
  ::5+N%2

  播放段落

  b) The distance from San Diego to Los Angeles is less than 150 miles.
  :b) 从圣地亚哥到洛杉矶的距离不到150英里。

  播放段落

  Define
  ::定义定义

  播放段落

  Let $\begin{array}{r}d=\end{array}$ the distance from San Diego to Los Angeles in miles.
  ::让我们从圣地亚哥到洛杉矶的距离 以英里计

  播放段落

  Translate
  ::翻译

  播放段落

  $$\begin{array}{r}d<150\end{array}$$

  
  ::d <150

  播放段落

  c) Diego needs to earn more than 82 points on his test to receive a $\begin{array}{r}B\end{array}$ in his algebra class.
  :c) 迭戈需要在其代数类接受B的测试中赚取82分以上。

  播放段落

  Define
  ::定义定义

  播放段落

  Let $\begin{array}{r}x=\end{array}$ Diego’s test grade.
  ::让我们来完成Diego的测试等级。

  播放段落

  Translate
  ::翻译

  播放段落

  $$\begin{array}{r}x>82\end{array}$$

  
  ::x>82

  播放段落

  d) A child needs to be 42 inches or taller to go on the roller coaster.
  :d) 儿童上过山车需要42英寸或更高。

  播放段落

  Define
  ::定义定义

  播放段落

  Let $\begin{array}{r}h=\end{array}$ the height of child in inches.
  ::让我们把孩子的身高以英寸计。

  播放段落

  Translate:
  ::翻译 :

  播放段落

  $$\begin{array}{r}h\ge 42\end{array}$$

  
  ::h42

  播放段落

  Examples
  ::实例

  播放段落

  Define the variables and translate the following statements into inequalities.
  ::界定变量,并将以下陈述转化为不平等。

  播放段落

  Example 1
  ::例1

  播放段落

  Jose took 5 train trips in a day, some of which cost $2.75 and some of which cost $3.95. His total cost was $9.45.
  ::Jose一天有5次列车旅行,其中一些费用为2.75美元,有些费用为3.95美元,总费用为9.45美元。

  播放段落

  Let $\begin{array}{r}t\end{array}$ be the number of train rides that cost $2.75. Then $\begin{array}{r}5-t\end{array}$ is the number of train rides that cost $3.95. Then we get:
  ::费用为2.75美元的火车乘车数, 费用为3.95美元的火车乘车数,然后是:

  播放段落

  $\begin{array}{r}2.75t+3.95(5-t)=\mathrm{9.45.}\end{array}$
  ::2.75t+3.95(5-t)=945。

  播放段落

  Example 2
  ::例2

  播放段落

  The product of 3 and some number is more than the sum of 24 and that number.
  ::3和某些数字的产物大于24和24之和。

  播放段落

  Let $\begin{array}{r}n\end{array}$ be "some number." Then the product of 3 and $\begin{array}{r}n\end{array}$ may be written as $\begin{array}{r}3n\end{array}$ . The sum of 24 and $\begin{array}{r}n\end{array}$ may be written as  $\begin{array}{r}24+n.\end{array}$ Since the statement specifies that the product is greater than the sum, we get:
  ::那么3和n的产物可以写成3n。24和n的总和可以写成24+n。

  播放段落

  $\begin{array}{r}3n>24+n\end{array}$
  ::3n>24+n

  播放段落

  Review
  ::回顾

  播放段落

  For questions 1-10, define the variables and translate the following statements into equations or inequalities. Then solve for the value of the variable.
  ::对于问题1-10, 定义变量, 并将以下的语句转换成方程式或不平等。 然后解决变量的价值 。

  播放段落
  1. Peter’s Lawn Mowing Service charges $10 per job plus $0.20 per square yard. Peter earns $25 for a job.
     ::彼得的草坪Mowing Service为每份工作收费10美元,每平方码场加0.20美元。 彼得为一份工作挣25美元。
  播放段落
  2. Renting the ice-skating rink for a birthday party costs $200 plus $4 per person. The rental costs $324 in total.
     ::租一个生日派对的溜冰场 每人200美元加4美元 租金共324美元
  播放段落
  3. Renting a car costs $55 per day plus $0.45 per mile. The cost of the rental is $100.
     ::租车费用为每天55美元,每英里0.45美元,租金为100美元。
  播放段落
  4. Nadia gave Peter 4 more blocks than he already had. He already had 7 blocks.
     ::Nadia给了Peter4个街区 比他已经多4个街区了 他已经有7个街区了
  播放段落
  5. A bus can seat 65 passengers or fewer.
     ::公共汽车可以坐65名乘客或少于65名乘客。
  播放段落
  6. The sum of two consecutive integers is less than 54.
     ::两个连续整数的总和小于54。
  播放段落
  7. The product of a number and 3 is greater than 30.
     ::数字和3的产值大于30。
  播放段落
  8. An amount of money is invested at 5% annual interest. The interest earned at the end of the year is greater than or equal to $250.
     ::投资金额按年利息5%计算,年底所得利息超过或等于250美元。
  播放段落
  9. You buy hamburgers at a fast food restaurant. A hamburger costs $0.49. You have at most $3 to spend. Write an inequality for the number of hamburgers you can buy.
     ::你在一个快餐餐厅买汉堡包。汉堡包要花0.49美元,你最多要花3美元。写一个可以买汉堡包数量不平等的字。
  播放段落
  10. Mariel needs at least 7 extra credit points to improve her grade in English class. Additional book reports are worth 2 extra credit points each. Write an inequality representing the number of book reports Mariel needs to do.
      ::Mariel至少需要7个额外信贷点来提高英语班的分数。 额外的书籍报告每份都值2个额外信贷点。 写一份不平等报告代表Mariel需要做的书籍报告数量。

  播放段落

  Review (Answers) 
  ::回顾(答复)

  播放段落

  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 。