1.16 利用战略和计划解决现实世界问题

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  利用战略和计划解决现实世界问题

  

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  Cameron is going to visit his grandparents. His flight leaves at 9:00 pm. He must be at the airport two hours before departure and he lives one hour from the airport. Cameron knows that he will need one and a half hours to pack and get ready for his trip. Cameron arrives home from school at 4:00 pm. How can Cameron figure out if he will make his flight in time?
  ::卡梅伦要去拜访他的祖父母。他的航班在晚上9点起飞。他必须在出发前2小时在机场,他住在机场1小时。卡梅伦知道他需要1个半小时打包和准备旅行。卡梅伦下午4点从学校回家。卡梅伦怎么知道他能否及时赶到飞机?

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  In this concept, you will learn to solve real-world problems by using strategies and a plan.
  ::在这个概念中,你将学会通过战略和计划解决现实世界的问题。

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  Using Strategies to Solve Problems
  ::使用解决问题战略

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  When problem-solving, you will use strategies as part of a plan. For each situation, you will be asked to read and understand a given problem. You will make a plan to solve by choosing appropriate strategies. There are multiple ways to solve a word problem . Therefore , you will need to consider and compare different approaches for each problem given.
  ::当解决问题时, 您将使用策略作为计划的一部分。 对于每种情况, 都会要求您阅读和理解给定的问题 。 您将会通过选择合适的策略来制定解决计划 。 解决单词问题的多种方法 。 因此, 您需要考虑和比较给定的每个问题的不同方法 。

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  First, always read the problem completely. Do not skip words or sections since they may be key to your understanding of what the problem is asking you to do. Make note of the question being asked by highlighting it. Another tip is to underline or circle all clues that may help you with planning and strategy.
  ::首先, 总是完整地阅读问题 。 不要跳过文字或章节, 因为它们可能是您了解问题所在的关键 。 请注意问题所在, 突出它 。 另一个提示是强调或环绕所有线索, 帮助您规划和策略 。

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  Let’s apply this strategy to an example.
  ::让我们把这个策略应用到一个例子上。

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  A lizard ate a total of five hundred flies on five consecutive nights. Each night he ate twenty-five more than the night before. How many flies did the lizard eat each night?
  ::蜥蜴连续五个晚上共吃了五百只苍蝇 每天晚上吃二十五只 比前天晚上多吃二十五只 蜥蜴每晚吃几只苍蝇?

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  For this problem, you are to determine the number of flies the lizard ate each night. Here is what you are told:
  ::对于这个问题,你要确定蜥蜴每天晚上吃苍蝇的数量。以下是你们被告知的:

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  • You are told that the lizard ate a total of five hundred flies over the course of five nights.
    ::你被告知蜥蜴在五个晚上里 共吃了五百只苍蝇
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  • You are told that the lizard eats twenty-five more flies each night than the night before.
    ::你被告知蜥蜴每晚比前天晚上多吃25只苍蝇
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  • You should know the word consecutive means a logical sequence or succession. In this case it means one night after the other.
    ::您应该知道连续的单词意指逻辑序列或继承。 在这种情况下,它意指一个又一个晚上的接一个晚上。

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  Next, make a plan to solve the problem. Use the information listed above to create a verbal model and use this model to write and solve an equation to answer the highlighted question.
  ::下一步, 制定解决问题的计划。 使用上面列出的信息创建语言模型, 并使用此模型写入和解析方程式, 以解答突出的问题 。

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  Next, write the verbal model.
  ::下一个,写出口头模型。

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  Number of flies eaten on night one + (number of flies eaten on night one + twenty-five) + (number of flies eaten on night one + twenty-five + twenty-five) + (number of flies eaten on night one + twenty-five + twenty-five + twenty-five) + (number of flies eaten on night one + twenty-five + twenty-five + twenty-five + twenty-five) = total number of flies eaten over the five nights (500).
  ::1+夜间1+(1+25+25+25+25)+(1+25+25+25+25)夜间食用苍蝇的数量)+(1+25+25+25+25+25+25)夜间食用苍蝇的数量)+(1+25+25+25+25+25)夜间食用苍蝇的数量)+(1+25+25+25+25+25+25)+=5夜晚食用苍蝇的总数(500)。

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  Next, name the variable . Let  $\begin{array}{r}x\end{array}$  represent the number of flies eaten on night one.
  ::下一个点名变量。 让 x 代表晚上1点吃过的苍蝇数量 。

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  Next, write a single- variable equation to represent the verbal model.
  ::接下来,写一个单变量方程式来代表语言模型。

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  $$\begin{array}{r}x+(x+25)+(x+25+25)+(x+25+25+25)+(x+25+25+25+25)=500\end{array}$$

  
  ::x+(x+25)+(x+25)+(x+25)+(x+25+25)+(x+25+25+25)++(x+25+25+25)=500

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  Next, simplify the equation on the left side of the equal sign by performing the addition in the parenthesis.
  ::接下来,通过执行括号中的添加,简化等式左侧的等式。

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  $$\begin{array}{r}x+(x+25)+(x+50)+(x+75)+(x+100)=500\end{array}$$

  
  ::x+(x+25)+(x+50)+(x+75)+(x+100)=500

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  Remember that t here is an "understood 1" in front of each set of parenthesis.
  ::记住每组括号前 都有一个"明白1"

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  Next, apply the to clear the parenthesis.
  ::下一步,应用括号来清除括号。

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  $$\begin{array}{r}x+x+25+x+50+x+75+x+100=500\end{array}$$

  
  ::x+x+25+x+50+x+75+x+100=500

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  Then  simplify the left side of the equation.
  ::然后简化方程的左侧。

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  $$\begin{array}{r}5x+250=500\end{array}$$

  
  ::5x+250=500

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  Next, isolate the variable by subtracting 250 from both sides of the equation.
  ::接下来,从方程的两侧减去250,将变量分离出来。

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  $$\begin{array}{r}5x+250-250=500-250\end{array}$$

  
  ::5x+250-250=500-250

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  Next, simplify both sides of the equation.
  ::接下来,简化方程式的两边

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  $$\begin{array}{r}5x=250\end{array}$$

  
  ::5x=250

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  Then  divide both sides of the equation by 5.
  ::然后将等式两侧除以5。

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  $$\begin{array}{r}\begin{array}{rcl}\frac{{}^1\cancel{5}x}{\cancel{5}}& =& \frac{250}{5}\\ x& =& 50\end{array}\end{array}$$

  
  ::15x5=2505x=50

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  Remember,  $\begin{array}{r}x\end{array}$  represents the number of flies the lizard ate on night one.
  ::记住, x 表示蜥蜴在一夜之间吃掉的苍蝇数量 。

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  Next, substitute $\begin{array}{r}x=50\end{array}$  into the expression representing each night to determine the number of flies the lizard ate on each of the five consecutive nights.
  ::接下来,将x=50改为代表每晚的表达式,以确定蜥蜴在连续五个夜晚中每一晚吞食的苍蝇数量。

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  Night one =  $\begin{array}{r}x=50\end{array}$
  ::一晚 = x= 50

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  Night two =  $\begin{array}{r}x+25=50+25=75\end{array}$
  ::二号夜=x+25=50+25=75

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  Night three =  $\begin{array}{r}x+25+25=50+25+25=100\end{array}$
  ::三夜=x+25+25=50+25+25=100

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  Night four =  $\begin{array}{r}x+25+25+25=50+25+25+25=125\end{array}$
  ::4夜=x+25+25+25=50+25+25+25+25=125

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  Night five =  $\begin{array}{r}x+25+25+25+25=50+25+25+25+25=150\end{array}$
  ::夜间5 = x+25+25+25+25=50+25+25+25+25+25+25+25+25=150

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  Then, check to determine if the solutions make a true statement. Substitute $\begin{array}{r}x=50\end{array}$  into the original equation.
  ::然后检查以确定解决方案是否真实声明。 替换 x=50 到原始方程中 。

  $$\begin{array}{r}\begin{array}{rcl}& & 50+(50+25)+(50+25+25)+(50+25+25+25)+(50+25+25+25+25)=500\\ & & 50+75+100+125+150=500\\ & & 500=500\end{array}\end{array}$$

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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 Cameron trying  to figure out if he will get to the airport on time. This is his timeline:
  ::早些时候,你遇到一个问题 卡梅伦想弄清楚 他是否能准时到达机场

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  • Cameron arrived home at 4:00 pm.
    ::卡梅伦下午4点回家
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  • He needs one and a half hours to pack.
    ::他需要一个半小时的时间收拾行李
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  • He has a one hour drive to get to the airport.
    ::他有一个一小时的车程 去机场。
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  • Cameron must be at the airport two hours prior to his departure time.
    ::Cameron必须在出发前两小时到达机场
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  • His flight leaves at 9:00 pm.
    ::他的航班晚上9点起飞

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  Next, Cameron has to make a plan of what he can do with all of this information.
  ::接下来,卡梅伦必须制定计划 说明他能用所有这些信息做什么

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  Working backwards from the departure time will tell him what time he should leave the house.
  ::离开时倒行逆施,会告诉他何时离开家。

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  $$\begin{array}{r}\begin{array}{rcl}& & \text{Departure time}-2\text{ hours}=\text{Check-in time}\\ & & 9:00\text{ pm}-2:00\text{ hours}=7:00\text{ pm}\\ \\ & & \text{Check-in time}-1\text{ hour drive}=\text{Leave home time}\\ & & 7:00\text{ pm}-1:00\text{ pm}=6:00\text{ pm}\\ \\ & & \text{Leave home time}-\text{Pack time}=\text{Time to arrive home}\\ & & 6:00\text{ pm}-1\frac{1}{2}\text{ hours}=4:30\text{ pm}\\ & & 6:00\text{ pm}-1:30=4:30\text{ pm}\end{array}\end{array}$$

   
  ::离家时间-2小时=在岗时间9:00:下午-2:00=7:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00=7:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:下午6:00:1112:00:4:30:00:00:00:00:下午1:30:30:下午4:30:00:00:00:00

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  Cameron arrived home at 4:00 pm so he should be able to make his flight.
  ::卡梅伦下午4点回家 这样他就能起飞了

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  Example 2
  ::例2

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  A train’s caboose is 12 feet long. Each of the train’s eight cars are twice the length of the caboose. What is the length of the entire train?
  ::一列火车的卡布罗斯长12英尺,火车的8辆汽车每辆都比卡布罗斯长两倍。 整列火车的长度是多少?

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  First, read the problem. Highlight the question and underline key words that will help you to answer the question being asked.
  ::首先,读读问题。突出问题,突出关键词,帮助回答提出的问题。

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  Next, confirm what the problem is asking you to do and list the information given in the problem.
  ::接下来,确认问题是什么 要求你做什么 并列出问题中的信息。

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  You are asked to determine the length of the entire train. You were told some information.
  ::要求您确定整个列车的长度。 有人告诉您一些信息 。

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  • The caboose is 12 feet long.
    ::卡布欧斯有12英尺长
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  • There are an additional eight cars.
    ::另外还有八辆车
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  • Each car is twice the length of the caboose.
    ::每辆车的长度是卡布罗斯车长度的两倍

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  Next, create a plan for solving the problem.
  ::下一个,制定解决问题的计划。

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  You can draw a diagram and use a verbal model to visualize the information given in the problem. Then, write an equation to determine the length of the entire train.
  ::您可以绘制图表, 并使用语言模型来直观问题中提供的信息。 然后, 写一个方程来决定整个列车的长度 。

  

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  Verbal Model: Eight trains twice the length of the caboose + the length of the caboose = length of the entire train
  ::方形:八列列车,长度是卡布罗斯的两倍,+卡布罗斯的长度=整个列车的长度

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  Next, name the variable. Let “ $\begin{array}{r}x\end{array}$ ”  represent the length of the entire train.
  ::接下来,给变量命名。让“x”代表整个列车的长度。

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  Next, write and solve a single-variable equation to represent the verbal model.
  ::接下来,写和解决一个单变量方程式,以代表语言模型。

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  $$\begin{array}{r}8(2\cdot 12)+12=x\end{array}$$

  
  ::8(212)+12=x

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  First, perform the multiplication is the parenthesis $\begin{array}{r}(2\cdot 12)=24\end{array}$  and write the new equation.
  ::首先,执行乘法是括号 (212) =24, 并写新方程 。

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  $$\begin{array}{r}8(24)+12=x\end{array}$$

  
  ::8(24)+12=x

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  Next, multiply: $\begin{array}{r}8(24)=192\end{array}$  to clear the parenthesis. Write the new equation.
  ::下一个乘以: 8(24) = 192 清除括号。 写入新公式 。

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  $$\begin{array}{r}192+12=x\end{array}$$

  
  ::192+12=x

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  Then, add the numbers on the left side of the equation.  $\begin{array}{r}192+12=204\end{array}$
  ::然后添加方程左侧的数字。 192+12=204

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  $$\begin{array}{r}204=x\end{array}$$

  
  ::204=x

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  The answer is 204.
  ::答案是204

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  The length of the entire train is 204 feet.
  ::整列火车的长度是204英尺

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  Example 3
  ::例3

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  A ten year old's heart beats approximately 85 times per minute. How many times does the heart beat in 24 seconds?
  ::十岁小孩的心跳大约每分钟85次,24秒内心跳多少次?

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  First, read the problem. Highlight the question and underline key words that will help you to answer the question being asked.
  ::首先,读读问题。突出问题,突出关键词,帮助回答提出的问题。

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  Next, confirm what the problem is asking you to do and list the information given in the problem.
  ::接下来,确认问题是什么 要求你做什么 并列出问题中的信息。

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  You are asked to determine the number of heart beats in 24 seconds. You were told some information.
  ::请您在24秒内确定心脏跳动的数量。 有人告诉您一些信息 。

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  • The heart beats 85 times per minute.
    ::心跳每分钟85次

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  Next, create a plan for solving the problem.
  ::下一个,制定解决问题的计划。

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  You are being asked to find an unknown rate (number of heart beats in 24 seconds) which tells you to write a proportion to solve the problem.
  ::您被要求找到一个未知的速率( 心跳数为24秒) , 要求您撰写一个比例来解决问题 。

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  Next, name the variable in the proportion.
  ::下一步,按比例命名变量。

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  Let  $\begin{array}{r}x\end{array}$  represent the number of heartbeats in 24 seconds.
  ::x 表示24秒的心跳数 。

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  Next, write and solve the proportion. Remember, there are 60 seconds in one minute.
  ::接下来,写和解答比例。记住,一分钟里有60秒。

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  First, write down what you know $\begin{array}{r}(85\text{ beats}=60\text{ seconds})\end{array}$  and express it over what you want to know $\begin{array}{r}(x=24\text{ seconds})\end{array}$ .
  ::首先,写下你所知道的(85拍=60秒),用您想知道的(x=24秒)表达出来。

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  $$\begin{array}{r}\frac{85\text{ beats}}{x}=\frac{60\text{ seconds}}{24\text{ seconds}}\end{array}$$

  
  ::85拍x=60秒24秒

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  Next, cross out the seconds on the right side of the proportion. Write the new proportion.
  ::下一步, 划出比例右侧的秒数。 写入新比例 。

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  $$\begin{array}{r}\begin{array}{rcl}\frac{85\text{ beats}}{x}& =& \frac{60\cancel{\text{ seconds}}}{24\cancel{\text{ seconds}}}\\ \frac{85\text{ beats}}{x}& =& \frac{60}{24}\end{array}\end{array}$$

  
  ::85拍sx=60秒24秒85拍sx=6024

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  Next, multiply both sides of the proportion by $\begin{array}{r}24x\end{array}$  and write the new equation.
  ::接下来,将比例的两边乘以 24x 并写下新的方程。

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  $$\begin{array}{r}\begin{array}{rcl}24\cancel{x}\left(\frac{85\text{ beats}}{\cancel{x}}\right)& =& \cancel{24}x\left(\frac{60}{\cancel{24}}\right)\\ 2040& =& 60x\end{array}\end{array}$$

  
  ::24x( 85 beatsx) = 24x( 6024) 2040= 60x

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  Then, divide both sides of the equation by 60.
  ::然后,将等式的两边除以60。

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  $$\begin{array}{r}\begin{array}{rcl}\frac{2040\text{ beats}}{60}& =& \frac{{}^1\cancel{60}x}{\cancel{60}}\\ 34\text{ beats}& =& x\end{array}\end{array}$$

  
  ::2040拍60=160x6034拍=x

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  The answer is 34.
  ::答案是34岁

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  The answer is $\begin{array}{r}x=34\end{array}$  beats in 24 seconds.
  ::答案是x=34 24秒后按下 。

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  Review
  ::回顾

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  Read each problem and then answer the questions below the problem.
  ::阅读每个问题,然后回答问题下面的问题。

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  Ted is starting a collection of rare coins. The first week he purchases 1 coin. During the second week he purchases 4 new coins. During the third he purchases 9 new coins. In the fourth week he adds 16 new coins to his collection. If this continues, how many weeks will it take until Ted owns 82 coins?
  ::Ted正在开始收集稀有硬币。 第一个星期他购买了1枚硬币。第二周他购买了4枚新硬币。第三周他购买了9枚新硬币。第四周他又在收藏中添加了16枚新硬币。如果这种情况继续下去,要花多少周时间直到Ted拥有82枚硬币?

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  1. Which strategy should Ted use to solve this problem?
  ::1. Ted应采用何种战略解决这一问题?

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  2. What could Ted draw to help him with his solution?
  ::2. Ted能画出什么来帮助他解决问题?

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  3. How long will it take Ted to collect 125 coins?
  ::3. Ted要多久才能收集125枚硬币?

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  Savannah wants to buy a pair of jeans that cost $59.00. They are on sale for 25% off.
  ::萨凡纳想买一条牛仔裤,价格59美元,卖价25%。

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  4. Which strategy could Savannah use to calculate the price?
  ::4. 萨凡纳可采用何种战略来计算价格?

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  5. What is the amount of the discount?
  ::5. 折扣数额是多少?

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  6. If the original price was $59.00, how much are the jeans after the discount is applied?
  ::6. 如果原价为59.00美元,在适用折扣后牛仔裤多少钱?

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  Carlos was in charge of organizing cookies for a bake sale. He organized them into bundles of six cookies. When he was done, he had 15 bundles of cookies. How many cookies did he start with?
  ::Carlos负责组织一个面包销售的饼干,他把它们分成6个饼干包。当他完成后,他有15个饼干包。他开始吃多少饼干?

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  7. Which strategy could you use to solve this problem?
  ::7. 贵国可采用何种战略来解决这一问题?

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  8. Write an equation to describe the problem.
  ::8. 写一个方程来描述问题。

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  9. Solve the equation.
  ::9. 解决等式。

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  10. How many cookies did Carlos begin with?
  ::10. Carlos开始吃多少饼干?

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  Veronica made brownies. She made twice as many brownies as Carlos had cookies. How many brownies did she make?
  ::维罗妮卡做了巧克力蛋糕 她做的蛋糕比卡洛斯做的饼干多两倍

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  11. Which strategy could you use to solve this problem?
  ::11. 你能采用什么战略来解决这个问题?

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  12. Write an equation to describe the problem.
  ::12. 写一个方程来描述问题。

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  13. Solve the equation.
  ::13. 解决等式。

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  14. How many brownies did Veronica make?
  ::14. 维罗妮卡做了多少巧克力蛋糕?

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  If Veronica sold half of the cookies that she made, how many would be sold? If she charged $1.50 per brownie, how much money would she make?
  ::如果维罗妮卡把一半饼干卖了 卖了多少?

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  15. Which strategy could you use to solve this problem?
  ::15. 你能采用什么战略来解决这个问题?

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  16. How much money would she make?
  ::16 她能挣多少钱?

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  Review (Answers)
  ::回顾(答复)

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  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 。