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  • 问题解决模型的比较

    lesson content

    Lisa's grandmother gave her some money for her birthday. Lisa decided to save it to help pay for a trip with her friends this summer. Since her birthday, Lisa has saved an additional $5 every week! It's been 8 weeks and Lisa has $95. How much money did Lisa's grandmother give her for her birthday?
    ::Lisa的祖母给了她一些生日钱。Lisa决定保留这笔钱来帮助支付她和朋友今年夏天的旅行费用。Lisa生日后,每周多省5美元!已经8周了,Lisa的祖母给她95美元。Lisa的祖母给了她多少钱给她过生日?

    In this concept, you will learn how to choose an appropriate strategy for solving a real world problem.
    ::在这个概念中,你们将学会如何选择一个适当的战略来解决真正的世界问题。

    Comparing Problem-Solving Models
    ::比较问题解决模型

    To start solving a real world problem it can help to ask yourself the following key questions.
    ::为了开始解决一个真正的世界问题,它可以帮助自己提出以下关键问题。

    Key Questions:
    ::关键问题:

    1. What am I trying to find out?
      ::我到底想查出什么来?
    2. What do I know?
      ::我知道什么? 我怎么会知道? What do I know?
    3. How can I solve the problem?
      ::我怎样才能解决问题?

    By reading the problem a few times you should be able to answer the first two key questions. However, sometimes answering the third key question is not that easy. It is often not obvious how to solve a problem! Here are some commonly used strategies for solving problems.
    ::通过阅读这个问题,您应该能够回答前两个关键的问题。 但是,有时回答第三个关键的问题并不那么容易。 解决一个问题的方法通常并不明显。 这是一些常用的解决问题的战略。

    1. Find a  Pattern  - Best used when there is a series of numbers and/or when you are being asked for a later quantity. For example, find the number in the tenth step.
      ::查找模式 - 在一系列数字和(或) 稍后要求数量时最佳使用。 例如, 在第十步中找到数字 。
    2. Guess and Check  - Best used when you are looking for one or two numbers and you think one of them might work. You can take a guess, try out a number and then adjust your answer from there.
      ::猜测和勾选 - 在您寻找一两个数字时使用得最好, 您认为其中之一可能有效 。 您可以进行猜测, 尝试一个数字, 然后从中调整答案 。
    3. Work Backwards  - Best used when you are given a total or final amount and you are looking for some partial amount or original amount.
      ::向后工作 - 当给您给定总金额或最终金额, 并且您正在寻找部分金额或原始金额时, 最佳使用 。
    4. Draw a Picture  - Best used when the problem describes some sort of visual.
      ::绘制图片 - 问题描述某种视觉效果时使用得最好 。
    5. Write an  Equation  - Best used when there is a missing quantity that needs to be figured out. Then you can write an equation and solve for the answer.
      ::写一个方程 - 当缺少数量需要解答时, 最常用的 。 然后您可以写一个方程, 解答答案 。
    6. Use a  Formula  - At this point, best used for area and perimeter problems.
      ::使用公式 - 此时此刻,最适合处理面积和周界问题。

    The more you practice solving problems, the quicker you will become at identifying the most appropriate strategy to use.
    ::越是练习解决问题,你就越快地找到最合适的战略。

    Here is an example.
    ::举一个例子。

    Ron arranged his herb garden rows in the following  order : 2 plants, 5 plants, 11 plants, 23 plants. How many plants will be in the fifth row?
    ::罗恩按以下顺序安排了他的草药园排:两株植物,五株植物,十一株植物,二十三株植物。第五排有多少株植物?

    First, ask yourself "what am I trying to find out?"
    ::首先,问自己 "我到底想了解什么?"

    You are trying to find out how many plants will be in the fifth row of Ron's herb garden.
    ::你想弄清楚罗恩草药园第五排有多少植物

    Next, ask yourself "what do I know?"
    ::接下来,问自己 "我知道什么?"

    You know the following pieces of information:
    ::您知道以下信息:

    • The first row has 2 plants.
      ::第一排有两株植物
    • The second row has 5 plants.
      ::第二排有5个工厂。
    • The third row has 11 plants.
      ::第三排有11个工厂。
    • The fourth row has 23 plants.
      ::第四排有23个工厂

    Then, ask yourself "how can I solve the problem?"
    ::然后问自己 "我怎样才能解决问题?"

    Notice that in this problem you were given a list of numbers and asked about a later quantity. This is a perfect time to use the  find a pattern  strategy. First, find the pattern and describe the pattern rule. Then, extend the pattern to figure out how many plants are in the fifth row.
    ::请注意, 在此问题上, 您得到了一个数字列表, 并询问了稍后的数量 。 这是使用模式策略的完美时间 。 首先, 找到模式, 描述模式规则 。 然后, 扩展模式以了解第五行有多少工厂 。

    Now, implement your plan. So far the pattern is 2, 5, 11, 23, . . .
    ::现在,执行你的计划,到目前为止的模式是 2,5,11,23,..

    The pattern rule is multiply by 2 and add 1.
    ::模式规则乘以 2 并增加 1。

    Next, extend the pattern.
    ::下一步,扩展图案。

    The answer is that there will be 47 plants in the fifth row.
    ::答案是第五排将有47个植物

    Remember that once you have an answer, you need to make sure you have actually answered the original question that was asked and that your answer seems realistic. You were trying to find out how many plants are in the fifth row and that's what you did. 47 plants is a lot, but it fits the pattern so your answer makes sense.
    ::记住一旦你得到答案, 你需要确定你实际上已经回答了最初提出的问题, 您的答案似乎很现实。 您试图找出第五排有多少植物, 这就是你所做的。 47个植物很多, 但是它符合模式, 所以答案是有道理的 。

    Examples
    ::实例

    Example 1
    ::例1

    Earlier, you were given a problem about Lisa and her saved money.
    ::早些时候,你得到一个问题 关于丽莎和她储蓄的钱。

    After her grandmother gave her some money for her birthday, Lisa has been saving $5 a week for 8 weeks. She now has $95! You wonder how much money Lisa got from her grandmother for her birthday.
    ::她奶奶给她一些生日钱之后,丽莎每周节省5美元8周。她现在有95美元!你想知道丽莎从她祖母那里为她的生日赚了多少钱。

    First, ask yourself "what am I trying to find out?"
    ::首先,问自己 "我到底想了解什么?"

    You are trying to find out how much money Lisa got from her grandmother for her birthday.
    ::你想知道Lisa从她祖母那里拿到多少钱 来庆祝她的生日

    Next, ask yourself "what do I know?"
    ::接下来,问自己 "我知道什么?"

    You know the following pieces of information:
    ::您知道以下信息:

    • Currently Lisa has $95 dollars.
      ::目前丽莎有95美元
    • Lisa is saving $5 each week.
      ::Lisa每周节省5美元
    • It has been 8 weeks.
      ::已经8个星期了
    • Lisa started out with just the money from her grandmother.
      ::Lisa一开始只是从她祖母那里拿钱

    Then, ask yourself "how can I solve the problem?"
    ::然后问自己 "我怎样才能解决问题?"

    Notice that in this problem you were given a final amount of $95 and asked about an initial amount. This is a good time to use the  work backwards  strategy. First, figure out how much money Lisa has saved over the 8 weeks since she got the money from her grandmother. Then, work backwards from the $95 to figure out how much money she must have started with from her grandmother.
    ::请注意, 在这个问题上, 您得到了最终的95美元金额, 并询问了初始金额。 这是使用倒置策略的好时机。 首先, 找出丽莎从祖母那里拿到钱后8周里储蓄了多少钱。 然后, 从95美元开始, 找出她必须从祖母那里拿到多少钱。

    Now, implement your plan. Lisa has saved $5 a week for 8 weeks.
    ::现在,执行你的计划 丽莎每周节省5美元8周

    $ 5 × 8 = $ 40

    So over the 8 weeks Lisa has saved an additional $40.
    ::所以在这8周里 丽莎又节省了40美元

    Next, work backwards. Lisa has $95 now. She added $40 to the money from her grandmother to get to the $95. That means you can subtract $40 form $95 to figure out how much money Lisa started with from her grandmother.
    ::接下来是倒着工作。丽莎现在有95美元了。她从祖母那里加了40美元 来拿到95美元,这意味着你可以减去40美元, 也就是95美元, 来弄清楚丽莎从祖母那里开始挣多少钱。

    $ 95 − $ 40 = $ 55

    The answer is Lisa got $55 from her grandmother.
    ::答案是丽莎从她祖母那里拿到了55美元

    Make sure you have actually answered the original question that was asked and that your answer seems realistic. You were trying to find out how much money Lisa's grandmother gave her for her birthday and you found out that Lisa got $55 from her grandmother. $55 is a realistic amount of money for a birthday present so your answer makes sense.
    ::确保你实际上已经回答了最初提出的问题,而且你的答复似乎很现实。你试图弄清楚丽莎的祖母给她的生日多少钱。你发现丽莎从她祖母那里拿到了55美元。55美元是一份生日礼物的实际金额,所以你的答案是有道理的。

    Example 2
    ::例2

    Ms. Powell wants to hang a large rectangular tapestry lengthwise on her living room wall. The tapestry has a perimeter of 42 feet and a width of 9 feet. Ms. Powell’s wall is 10 feet high. Will the length of the tapestry fit against the height of Ms. Powell’s ceiling?
    ::鲍威尔女士想在客厅墙上挂长长长的长长长长长的长长长长长的长长长长长的长长长长长长的长长长长长的长长长的长长的长长42英尺宽9英尺的长长长长的长长长长长长的长长长长长长的长长长长长的长长长长长长的长长长长的长长长与鲍威尔女士天花板的高度是否合适?

    First, ask yourself "what am I trying to find out?"
    ::首先,问自己 "我到底想了解什么?"

    You are trying to find out if the length of the tapestry is  less than  the height of Ms. Powell's ceiling which is 10 feet.
    ::你试图弄清楚 挂毯的长度 是否低于鲍威尔女士上限的高度 也就是10英尺

    Next, ask yourself "what do I know?"
    ::接下来,问自己 "我知道什么?"

    You know the following pieces of information:
    ::您知道以下信息:

    • The tapestry is a rectangle.
      ::挂毯是一个矩形。
    • The tapestry has a perimeter of 42 feet.
      ::挂毯周边42英尺
    • The tapestry has a width of 9 feet.
      ::挂毯宽度为9英尺
    • Ms. Powell's ceilings are 10 feet high.
      ::鲍威尔女士的天花板有10英尺高
    • The tapestry will be hung lengthwise.
      ::挂毯将被挂长 。

    Then, ask yourself "how can I solve the problem?"
    ::然后问自己 "我怎样才能解决问题?"

    Notice that this problem referenced perimeter and a rectangle, so this is a great time to use the  use a formula  strategy. First, use the rectangle perimeter formula to find the length of the tapestry. Then, see if the length of the tapestry is less than 10 feet.
    ::请注意这个问题指向了周边和矩形, 因此这是一个使用公式策略的好时机。 首先, 使用矩形周边公式来查找挂毯的长度 。 然后, 看看挂毯的长度是否小于 10 英尺 。

    Now, implement your plan. The rectangle perimeter formula is  P = 2 l + 2 w . You know  P = 42  and  w = 9 . You want to figure out the length,  l .
    ::现在,执行你的计划。矩形周边公式是 P = 2 l + 2 w. 你知道 P = 42 和 w = 9. 你想弄清楚长度, l 。

    Substitute  the values for  P  and  w  into the rectangle perimeter formula.
    ::将 P 和 w 的值替换为矩形周边公式。

    Now, solve for  2 l . "What number plus 18 is equal to 42?" You know that 24 plus 18 is equal to 42, so  2 l  must be equal to 24.
    ::现在,解决2 l。"什么数字加18等于42?" 你知道24加18等于42, 所以2 l必须等于24。

    2 l = 24
    ::2升=24

    Next you can solve for the length  l . "2 times what number is equal to 24?" You know that 2 times 12 is equal to 24 so  l  must be equal to 12.
    ::接下来您可以解答 l 的长度 。 “ 数字乘以2 等于 24 。 ” 你知道 2 乘以 12 等于 24 , 因此我必须等于 12 。

    l = 12
    ::l=12

    Ms. Powell's tapestry is 12 feet long. Since her ceilings are 10 feet high, the tapestry is too long to fit on the wall.
    ::鲍威尔女士的挂毯有12英尺长,因为她的天花板有10英尺高,挂毯太长,不能放在墙上。

    The answer is that Ms. Powell's tapestry is too long to fit on the wall.
    ::答案是,鲍威尔女士的挂毯太长,不能放在墙上。

    Now make sure you answered the question. The question asked if the length of the tapestry was less than the height of Ms. Powell's ceiling. You determined that no, the length of the tapestry is not less than the height of the ceiling. 12 feet for the length of a tapestry that you would put on the wall is realistic, so your answer makes sense.
    ::现在,请确保您回答这个问题。问的是,挂毯的长度是否低于鲍威尔女士天花板的高度。您确定,不,挂毯的长度不低于天花板的高度。您将挂在墙上的挂毯的长度为12英尺,这是现实的,所以您的答复是有道理的。

    Example 3
    ::例3

    Melissa has 144 cookies she wants to put evenly into 8 gift bags. How many cookies will go into each bag?
    ::Melissa有144个饼干,她想平均地把8个礼物袋装进8个礼物袋。

    First, ask yourself "what am I trying to find out?".
    ::首先,问自己"我到底想了解什么?"

    You are trying to find out how many cookies will go into each gift bag.
    ::你想弄清楚每个礼物袋里会有多少饼干

    Next, ask yourself "what do I know?".
    ::接下来,问自己 "我知道什么?"

    You know the following pieces of information:
    ::您知道以下信息:

    • There are 144 cookies total.
      ::共有144个饼干。
    • There are 8 gift bags.
      ::有8个礼品袋。
    • Each gift bag needs to have the same number of cookies.
      ::每个礼物袋都需要相同数量的饼干

    Then, ask yourself "how can I solve the problem?"
    ::然后问自己 "我怎样才能解决问题?"

    Notice that we have a missing quantity to figure out, the number of cookies in each bag. This is a good time to use the  write an equation  strategy. First, write an equation to show the relationship between the different numbers in the problem. Then, solve the equation.
    ::注意我们有一个缺失的数量, 每个包中缺少的饼干数量。 这是使用公式策略写入的好时机 。 首先, 写入一个公式来显示问题中不同数字之间的关系 。 然后, 解开公式 。

    Now, implement your plan. You know the number of cookies in each bag times the number of bags will equal the total number of cookies. The unknown quantity is the number of cookies in each bag so that will be your  variable .
    ::现在,执行你的计划。你知道每个袋子中的饼干数量是袋数的倍数,袋数将等于饼干的总数。未知的数量是每个袋子中的饼干数量,所以这将是你的变量。

    Let  x =   the number of cookies in each bag.
    ::让 x = 每个袋子中的饼干数量 。

    x ⋅ 8 = 144
    ::x 8 = 144

    Now, solve the equation. "What number times 8 is equal to 144?" If you need to, you can use your calculator to divide 144 by 8 to get the answer. 18 times 8 is equal to 144 so  x  is equal to 18.
    ::现在, 解答方程 。 “ 什么数字乘以8等于144 144 ” 。 如果您需要, 您可以使用计算器将144除以8来获得答案。 18乘以8等于144, 所以x等于18 。

    x = 18
    ::x=18x=18

    The answer is there will be 18 cookies in each bag.
    ::答案是每个袋子里会有18个饼干

    Next, make sure you have actually answered the original question that was asked and that your answer seems realistic. You were trying to find out how many cookies were in each bag and you did. 18 cookies in a bag is realistic so your answer makes sense.
    ::下一步, 请确认您实际上已经回答了最初提出的问题, 您的答案似乎很现实。 您试图找出每个包里有多少饼干, 您确实做了。 包里有18个饼干是现实的, 您的答案是有道理的 。

    Example 4
    ::例4

    A list of numbers is found in the problem. Which strategy might you use?
    ::在问题中发现了一个数字列表。 您可以使用哪种策略 ?

    When there is a list of numbers, the  find a pattern  strategy is a useful one to try.
    ::当有数字清单时,发现模式战略是有用的,可以尝试。

    The answer is the find a pattern strategy.
    ::答案是找到一个模式战略。

    Example 5
    ::例5

    A final quantity is given, but the initial amount is missing. Which strategy might you use?
    ::给定了最终数量, 但初始数量缺失 。 您可能使用哪种策略 ?

    When you know the final amount but not the starting amount, the  work backwards  strategy is a good one to try.
    ::当您知道最终金额而不是起始金额时, 倒向工作策略是值得尝试的好策略 。

    The answer is the work backwards strategy.
    ::答案是工作倒退战略。

    Review
    ::回顾

    Solve the following problems.
    ::解决以下问题。

    1. Mary went to the music store with her babysitting money. She bought two CDs for $12.50 each and two magazines for $4.25 each. She left the store with $10.25. How much money did she start with?
      ::她买了两张CD,每张12.50美元,两本杂志,每张4.25美元。她离开商店时有10.25美元。她先买了多少钱?
    2. Since he began his fitness routine, Mr. Trigg has measured his weight every week. His weights for the first six weeks are as follows: 236, 230, 232, 226, 228, 222. If the pattern continues, how much will he weigh in the tenth week?
      ::特里格先生自开始健身活动以来,每周都测量他的体重,头六周的体重如下:236、230、232、226、228、222。
    3. The area of City Park is  75   k m 2 . The length of the park is 15 feet. What is the perimeter of the park?
      ::城市公园的面积是75公里2米。 公园的长度是15英尺。 公园的周界是什么?
    4. A farmer planted corn, wheat, and cotton in a total of 88 fields. He planted 10 fields of corn. If he planted an equal number of fields for the other crops, how many fields of wheat did he plant?
      ::种植玉米、小麦和棉花的农民在总共88个田地种植玉米、小麦和棉花,他种植了10个田地的玉米,如果他为其他作物种植同等数量的田地,他种植了多少小麦?
    5. Mrs. Whitaker is mailing a pair of shoes to her daughter. She wants to fit the rectangular shoebox  inside  a larger square box. The area of the shoebox is  84   i n 2 ; the length of one side is 12 inches. One side of the larger square box measures 14 inches. Will the shoebox fit in the larger box? How do you know?
      ::Whitaker夫人正在给女儿寄一双鞋,她想把长方形鞋盒装在一个大方格箱内。鞋盒的面积是84 i n 2;一面的长度是12英寸。大方格箱的一面是14英寸。鞋盒在大方格内是否合适?你怎么知道?
    6. After a pin-ball game, the score board showed that the combined points of Peter, Ella, and Ned is 728. Ella scored half the points of Ned and Peter scored one-fourth the points of Ned. How many points did each player score?
      ::球球比赛结束后,计分板显示,彼得、埃拉和奈德的合并点是728。埃拉得一半奈德的分,彼得得四分之一奈德的分。每个球员得多少分?
    7. Tami made a total of $47 babysitting on New Year’s Eve. She made her hourly rate plus a $7 tip. If she worked 5 hours, what is her hourly rate?
      ::塔米在新年前夕共做了47个保姆。 她的小时工资加7美元小费。 如果她工作了5小时,她的小时工资是多少?
    8. A weightlifter lifts weights in the following order: 0.5 lb, 1.5 lb, 4.5 lb, 13.5 lb. How many pounds will he lift next?
      ::重量提升器按以下顺序提升重量:0.5磅、1.5磅、4.5磅、13.5磅。 下一个他将提多少磅?
    9. Figure A is a square, with a side that measures 9 cm. Figure B is a square with a side that measures 6 cm. Which figure has the greater area, Figure A or Figure B?
      ::图A是一个方形,一面测量9厘米。 图B是一个方形,一面测量6厘米。哪个数字的面积较大,图A或图B?
    10. Mr. Rowe and Mrs. Rowe are driving 959 miles to a beach vacation. They want to split the distance over 4 days, driving the exact same amount on the first three days and the remainder on the fourth day. If they drive 119 miles on the fourth day, how many miles will they drive on the first day?
      ::Rowe先生和Rowe夫人开着959英里的车去海滩度假,他们想将距离分成四天,头三天和第四天的车数相同,第四天开119英里,第一天开多少英里?
    11. Cedric spent $27.75 on pizza for his friends. Each cheese pizza cost $8 and each extra topping cost $0.75. If Cedric bought 3 cheese pizzas, how many extra toppings did he get?
      ::Cedric花27.75美元买披萨给朋友吃,每块芝士披萨要8美元,每块额外加点费用为7.75美元,如果Cedric买了3块芝士披萨,他又买了多少?
    12. What was the total cost without the toppings?
      ::全部成本是多少?
    13. If 29 people went to the zoo the first day and double went the second, how many went in both days combined?
      ::如果第一天有29人去动物园, 第二天翻了一倍, 那么这两天一共有多少人去了?
    14. If on the third day, double the people from the second day went, how many people went to the zoo on that day?
      ::如果在第三天,从第二天开始,人口翻了一番, 那天有多少人去了动物园?
    15. If ten less went to the zoo on the fourth day verses the third day, how many people went to the zoo on the fourth day?
      ::如果第4天有10个人去动物园... ...第3天有诗歌... ...第4天有多少人去动物园?

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