Section: 解决问题的计划，心理数学 | 1.18解决问题的计划，心理数学

Section outline

Sayber and Tuscany ride the city bus to school every weekday. The bus charges $2 per passenger, and, with the student discount they get, the kids pay another $3 each for the extended ride.
::Sayber和Tuscany每周天都坐公交车上学。公交车每个乘客收费2美元,学生折扣后,孩子们每人多付3美元。

If the kids’ parents give them enough money at the beginning of the week to cover bus fare, how much do they receive together each week?
::如果孩子的父母在周初给他们足够的钱支付公交车费,他们每星期一起领取多少?

In this concept, you will learn about using mental math to solve word problems.
::在这个概念中,你会学会使用心理数学解决字词问题。

Problem Solving
::解决问题

There are a number of common ways to solve word problems, and practicing each of them is a great idea, since a plan that works perfectly in one situation my not be so great for another. The Guess and Check and the Work Backward models are great problem solving plans, but sometimes it makes more sense to use mental math.
::有很多共同的方法可以解决字词问题,而实践其中的每一个都是一个好主意,因为一个计划在一种情况下运作完美,我对另一种计划并不怎么好。猜和检查和倒工模式是解决计划问题的大难题,但有时使用精神数学更有意义。

By applying mental math to a problem, you can often solve it as quickly (and more efficiently) as you could have by guessing. The trick to using mental math is to look for more and easier operations rather than fewer but more complex ones.
::通过将心理数学应用到问题中,你往往可以像你猜想的那样快速(和更有效地)解决问题。使用心理数学的技巧是寻找更多、更简单的操作,而不是更少、更复杂的操作。

For example, consider this multiplication problem:
::例如,考虑这一乘数问题:

Kayte has 14 boxes of popcorn, each with 7 packages inside. How many packages does she have in total?
::Kayte有14箱爆米花,每盒里面有7包。她总共有多少包?

You could solve this using vertical multiplication on paper, or even by guessing and then dividing to check, but even though you may not know what 14 times 7 is right away, you can break in into two more easily computed multiplication problems.
::你可以在纸面上进行垂直乘法, 或甚至通过猜想然后进行分解来解决这个问题, 但是即使你可能不知道什么是14乘以7, 你也可以破解成两个更容易计算的乘法问题。

$\begin{aligned} 14 is the same as 10 + 4 \\ 10 times 7 = 70 \\ 4 times 7 = 28 \\ 70 + 28 = 98 \\ 70 + 20 = 90 \\ 90 + 8 = 98 \end{aligned}$

Even though you end up with more steps by splitting up the 14 before you multiply, the math is easier this way and you are more likely to be able to quickly compute it in your head.
::即使你最终会有更多的步骤在你乘数之前把14分解出来, 数学也比较容易, 你更有可能快速地计算出你的脑中。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Sayber and Tuscany each receiving $2 to board the bus and $3 for the ride each day for 5 days.
::早些时候,你遇到一个问题赛伯和托斯卡纳每人得到2美元搭乘公交车,3美元每天搭车五天。

You can use mental math to calculate the total amount of money their parents give them.
::你可以用心理数学来计算他们父母给他们的钱的总额。

First, consider the problem.
::首先,考虑一下问题。

2 kids times ($2 + $3) times 5 days
::2个孩子2次(2+3美元)5天

Next, break the problem into smaller steps that are easier to calculate in your head. 2 + 3 is easy, that’s 5. Now you have $\begin{aligned} 2 \times 5 \times 5 \end{aligned}$ .
::下一步,将问题分成更小的一步,在你的脑海中更容易计算。 2+3很容易,也就是5。现在你有2×5×5×5。

$\begin{aligned} 2 \times 5 = 10 \end{aligned}$ , so that would leave $\begin{aligned} 10 \times 5 \end{aligned}$ .
::2 × 5 = 10 , 这样就会剩下 10 × 5 。

Remember, you can multiply by 10 by just adding a zero, so $\begin{aligned} 10 \times 5 = 50 \end{aligned}$ .
::记住,你可以乘以 10 乘以 10 只要加一个零, 所以10 × 5 = 50 。

Sayber and Tuscany should receive a total of $50 from their parents for bus fares each school week.
::Sayber和Tuscany每周从父母那里领取50美元的公共汽车票价。

Example 2
::例2

Ground beef for hamburgers is sold by the pound. If Delilah buys 12 pounds at $2.50 per pound, how much did she spend?
::Delilah买12磅,每磅2.50美元,她花了多少钱?

First, consider the problem and see if it can be split into easily-computed pieces.
::首先,考虑一下这个问题,看看是否可以将其分成容易分类的碎片。

The hamburger is $2.50 per pound, and Delilah is buying 12 pounds, so you need to multiply 12 by 2.5. You may not know what 12 times 2.5 is right away, but 12 is the same as 10 + 2, and multiplying by 10 is simple as you just add a zero. Then, you can just add 2.5 times 2 to the answer.
::汉堡是每磅2.50美元,Delilah正在购买12磅,所以你需要乘以12乘以2.5。你可能不知道12乘以2.5是什么,但12等于10+2,乘以10简单,因为你只需加零。然后,你可以在答案中加上2.5乘以2。

Next, compute each piece separately using mental math.
::接下来,用心理数学分别计算每块碎片。

$\begin{aligned} 2.5 \times 10 & = 25 (Multiply by ten easily by moving the decimal one place to the right.) \\ 2.5 \times 2 & = 5 \\ 25 + 5 & = 30 \end{aligned}$

The answer is Delilah will spend a total of $30 on 12 pounds of hamburger, at $2.50 per pound.
::答案是Delilah在12磅汉堡上将花费30美元,每磅2.50美元。

Example 3
::例3

Ravi lives in Florida and loves going to pick strawberries during strawberry season. He can fit 20 strawberries in one box. If he is given a barrel of 500 strawberries, how many boxes will it take for Ravi to sort the strawberries?

First, consider the problem to see if there is a way to break it into smaller pieces that you may be able to calculate quickly in your head.
::拉维住在佛罗里达,喜欢在草莓季节采摘草莓。他可以把20个草莓装在一个盒子里。如果给他一桶500个草莓,那么拉维要用多少个盒子来分拣草莓?首先,考虑一下问题,看有没有办法把它分成小块,你就可以快速地计算出来。

In this case, you need to divide 500 by 20. You may not know right away what 500 divided by 20 is, but dividing by 10 is always easy, you just remove a zero. Then you can just divide that answer in half, since there can only be half as many 20-strawberry boxes as 10-strawberry boxes.
::在此情况下,您需要将500除以20, 您可能不知道500除以20是什么, 但除以10总是容易的, 您只需删除一个零。然后您就可以将答案分成一半, 因为只有10草莓盒的20个草莓盒的一半。

Next, handle each mental calculation separately.
::接下来,分别处理每一项心理计算。

$\begin{aligned} 500 divided by 10 = 50 \\ 50 divided by 2 = 25 \end{aligned}$

The answer is Ravi will need 25 boxes of 20 strawberries each in order to pack 500 strawberries total.
::答案是拉维需要25箱每箱20只草莓才能把总共500只草莓打包起来

Example 4
::例4

If Ravi fills 40 boxes during his shift at work, how many strawberries did he start with?
::如果Ravi在值班时填满了40个箱子, 他从多少草莓开始?

First, consider the problem and see if it can be broken into smaller bits.
::首先,考虑一下问题,看看能否将其分成小部分。

Ravi puts 20 strawberries into each box, so if he fills 40 boxes, that is 20 times 40 strawberries total. You may not know what 20 times 40 is right away, but 10 times 40 is easy, and 20 times 40 is just twice that many.
::拉维把20个草莓放进每个盒子, 所以如果他填满40个盒子, 总共是20倍于40个草莓。你可能不知道什么是20乘以40, 但10乘以40很容易, 20乘以40只是两倍。

Next, use mental math to multiply each of the smaller parts of the overall problem.
::接下来,用心理数学来乘以整个问题中每一个较小的部分。

$\begin{aligned} 10 times 40 & = 400 \\ 400 times 2 & = 800 \\ If 4 times 2 = 8, then 400 times 2 & = 800 \end{aligned}$

The answer is if Ravi fills 40 boxes of 20 strawberries each, he packed 800 strawberries.
::答案是,如果Ravi装满40箱每箱20草莓,他装了800草莓。

Example 5
::例5

If Ravi works two shifts, at this rate , how many boxes will he fill?
::如果Ravi工作两班,按这个速度,他要填多少箱?

This one is easy, since it is only one step to start with!
::这很简单,因为它只是第一步!

40 boxes per shift times 2 shifts is 80 boxes total
::两班总共80箱

The answer is 80 boxes.
::答案是80箱。

Review
::回顾

Use mental math to solve each of the following problems.
::使用心理数学来解决以下每一个问题。

Josie helps label pints of strawberries. She labels them at $\begin{aligned} $ 2.00 \end{aligned}$ per pint. If she sells 100 pints, how much money has she made?
::Josie帮助给一品脱草莓贴标签,每品脱2美元。如果她卖100品脱,她赚了多少钱?

If Josie sells $400.00 worth of strawberries, how many pints has she sold?
::如果乔西卖了价值400.00美元的草莓她卖了多少品脱?

Josie also enjoys making strawberry milkshakes. If it takes 5 strawberries to make one milkshake, how many can she make with 20 strawberries?
::Josie也喜欢做草莓奶昔。如果需要5个草莓才能做一个奶昔,她能用20个草莓做多少?

If Josie makes 35 strawberry milkshakes in one day, how many strawberries does she need to accomplish this task?
::如果乔西一天做35个草莓奶昔她完成这项任务需要多少草莓?

If there are 25 strawberries in a pint, how many pints does Josie use to make her 35 milkshakes?
::如果一品脱有25个草莓, 乔西用多少品脱来做35个奶昔?

Carl loves to collect old vinyl records. He has a whole collection that he received from his Dad. If he has five different categories of records with twenty records in each category, how many records does Carl have altogether?
::卡尔喜欢收集旧的乙烯基记录。他有他从父亲那里收到的全部收藏。如果他有五类不同的记录,每类有20个记录,那么卡尔总共有多少记录?

Julie is a friend of Carl’s. She brought over her collection of records. Julie has 254 records. If she and Carl were to combine their collections, how many would they have altogether?
::朱莉是卡尔的朋友,她带来了她收集的记录。朱莉有254个记录。如果她和卡尔合并了这些记录,他们总共会有多少记录?

When Carl and his Mom went to a yard sale, Carl got a box of vinyl records for $25.00. He brought them home and looked in the box. Out of 30 records, five of them were broken or scratched. If he puts these new records with his collection, how many does he now have?
::当卡尔和他妈妈去场外销售时,卡尔得到了一盒25美元的黑乙烯唱片,他带他们回家看箱子。在30个记录中,有5个被打破或划伤。如果他把这些新记录与收藏放在一起,他现在有多少记录?

Carl’s sister borrowed ten records to show her friend. How many are left in Carl’s collection?
::卡尔的妹妹借了十张唱片给朋友看。卡尔的收藏中还剩下多少?

Mario is an outstanding skateboarder. He recently purchased a new skateboard. He wants to sell his old one. A friend wishes to buy it for $\begin{aligned} $ 45 \end{aligned}$ . If he gives Mario three twenty dollar bills, how much change should Mario give his friend?
::Mario是个优秀的滑板运动员。他最近买了一个新的滑板。他想卖掉他的旧滑板。一个朋友想用45美元买它。如果他给Mario320美元的钞票,Mario应该给朋友多少零钱?

If Mario buys a new skateboard for double the price that he sold his old one, how much did he pay for the new skateboard?
::如果马里奥买一个新的滑板价格是他卖掉旧滑板价格的两倍那他为新的滑板花了多少钱?

If Mario has $\begin{aligned} $ 100.00 \end{aligned}$ and he buys the skateboard for double the price that he sold his old one, does he have enough money to make the purchase?
::如果马里奥有100.00美元买滑板价格比他卖掉旧的要高一倍那么他有足够的钱来买吗?

Did he receive any change back at the skateboard shop? How much?
::他在滑板店收到零钱了吗?

Karen collects fairy figurines. She was given 3 for her birthday, 2 for Christmas, 4 from her grandmother and 3 she bought on her own. How many fairy figurines does she have in all?
::Karen收集了童话雕像。她的生日3个,圣诞节2个,祖母4个,她自己买3个。她一共有多少童话雕像?

Karen’s little sister loves one of the figurines. Karen has decided to give her the little fairy as a gift. After she does this, how many figurines will Karen have left?
::凯伦的妹妹爱一个雕像。凯伦决定把小仙女送给她作为礼物。在她这样做之后,卡伦将离开多少雕像?

At a yard sale, Karen bought 5 fairies for $\begin{aligned} $ 20.00 \end{aligned}$ . How much did she pay per fairy?
::Karen买了5个仙子花了20美元她给每个仙子付了多少钱?

How many fairies does Karen have now?
::现在凯伦有多少仙女了?

Karen’s friend Emily also collects fairies. If Emily has twice as many fairies as Karen, how many does she have?
::Karen的朋友Emily也收集仙子。如果Emily拥有的仙子是Karen的两倍,那么她有多少?

Jamie runs track at school. He is one of the fastest runners on the team and runs one mile in about 5 minutes. How long will it take Jamie to run 10 miles?
::杰米在学校跑赛道他是队里跑得最快的选手之一 5分钟内跑一英里

If Jamie runs a 3 mile race, about how much time will it take to run the 3 miles at his one mile pace?
::如果Jamie跑3英里的赛跑跑3英里一英里的速度要花多少时间呢?

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

Section outline

Problem Solving ::解决问题

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Resources ::资源

Problem Solving
::解决问题

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)

Resources
::资源