使用代数模型

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Bailey is planting an outdoor garden. She wants the length of the planting area to be twice as large as the width. If she has 128 square feet of soil to work with, what dimensions must the garden be?
::贝利正在种植户外花园。 她希望种植面积的长度是宽度的两倍。 如果她有128平方英尺的土壤可以工作,那么花园的尺寸应该有多大?

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Algebraic Models
::代数模型

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Word problems are some of the hardest types of problems for students to grasp. There are a few steps to solving any word problem :
::语言问题是学生最难面对的问题。

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Step 1: Read the problem at least twice.
::步骤1:至少两次阅读问题。

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Step 2: Cross out any unnecessary words, circle any numbers or words that represent mathematical operators, or translate words into mathematical expressions.
::第2步:将任何不必要的单词、任何数字或代表数学运算符的单词、或将单词转换为数学表达式的单词划开来。

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Step 3: Write an equation and solve.
::第三步 写一个方程和解析。

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To help you with Steps 2 and 3, generate a list of words that represent: add, subtract, multiply, divide, equal, etc. Here are a few to get you started.
::为了帮助您处理步骤2和步骤3, 生成一个代表单词列表 : 添加、 减、 乘、 除、 等等 。 这里有一些可以让您启动 。

Operator	words			Operator	words
add	sum, plus, and	increase, more (than)		multiply	times, of, product	double (x2), triple (x3)
subtract	difference , minus	decrease, less (than)		equal	is, total, to	made/make, spend/spent
divide	quotient, half, ÷2	split, share		variable	播放段落 how many __ ::有多少___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 播放段落 how much __ ::多少钱... ? 播放段落 what amount (of) __ ::数额多少? ___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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See if you can add anything to these lists.
::看看能不能在这些清单中添加任何内容。

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Let's use  this chart to help you with decoding the following word problems.
::让我们用这个图表来帮助您解码下面的单词问题 。

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1. Two consecutive numbers add up to 55. What are the two numbers?
   ::连续两个数字加起来等于55个数字,这两个数字是什么?

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First, translate the statement. “Consecutive” means numbers that are one after the other. So if the first number is $\begin{align*}x\end{align*}$ , then the second number will be $\begin{align*}x + 1\end{align*}$ , and they add up to (equal)  55. The equation is: $\begin{align*}x+(x+1)=55.\end{align*}$
::首先,翻译语句。 “ concective” 表示数字是一对一。 所以, 如果第一个数字是 x, 那么第二个数字将是 x+1, 然后它们加起来等于( 相等) 55。 方程式是: x+( x+1)=55 。

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We put $\begin{align*}x + 1\end{align*}$ in parenthesis to show that it is a separate value . Solve the equation.
::我们在括号中插入 x+1 以显示它是单独的值。 解析方程 。

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$$\begin{align*}x+(x+1) &=55\\ 2x+1 &= 55\\ 2x &= 54\\ x &= 27\end{align*}$$
::x+(x+1)=552x+1=552x=552x=54x=27

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The smaller number is 27, and the larger number will be 28, since  $\begin{align*}27 + \color{red}28 \color{black}= 55 \end{align*}$
::人数少为27人,人数多为28人,因为27+28=55

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Sometime you may encounter problems with “consecutive even numbers” or “consecutive odd numbers.” All even numbers are divisible by 2, so the smallest should be $\begin{align*}2x\end{align*}$ , then the next even number would be $\begin{align*}2x + 2\end{align*}$ . For consecutive odd numbers, they will always be an even number plus 1, 3, 5, etc. So, the smaller odd number would be $\begin{align*}2x + 1\end{align*}$ and the larger odd number would be $\begin{align*}2x + 3\end{align*}$ .
::有时候,你可能会遇到“连续偶数”或“连续奇数”的问题。 所有偶数都可以除以 2 。 因此,最小的数字应该是 2x, 下一个偶数应该是 2x+2 。 对于连续的奇数,它们永远是一个偶数+ 1, 3, 5 等。 因此,较小的奇数将是 2x+1 , 较大的奇数将是 2x+3 。

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2. Over the Winter Break, you worked at a clothing store and made $9.00 an hour. During  the two weeks you worked 65 hours at  regular pay and 10 hours of overtime ( 1.5 times regular pay, or 'time and a half' ). How much money did you make?
   ::在冬季休息期间,你在一个服装店工作,每小时挣9.00美元,在这两周内,你按正常工资工作65小时,加班10小时(正常工资1.5倍,或“时间半”)。你挣了多少钱?

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First, we need to figure out how much you make for overtime. 'Time and a half' would be $\begin{align*}\$ 9.00 + \$ 4.50 = \$ 13.50\end{align*}$   per hour. So, you made:
::首先,我们需要弄清楚你加班挣多少钱。“时间半”是9.00+4.50美元=每小时13.50美元。所以,你做了:

$$\begin{align*}& \$ 9.00(65)+ \$ 13.50(10) \\ & = \$ 585.00+ \$ 135.00 \\ & = \$ 720.00\end{align*}$$

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3. Elise is taking piano lessons. The first lesson is twice as expensive as each additional lesson. Her mom spends $270 for 8 lessons. How much was the first lesson?
   ::伊莉斯在学钢琴,第一课比其他课贵两倍,她妈妈花了270美元8课。第一课多少钱?

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Translate each statement.
::翻译每个声明。

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Call the regularly priced lessons $\begin{align*}l\end{align*}$ . Then, the first lesson will be $\begin{align*}2l\end{align*}$ .
::然后,第一堂课将是21号课。

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"Mom spends $270 for 8 lessons" $\begin{align*} \rightarrow 2l + 7l = \$ 270\end{align*}$
::"妈妈花270美元8次上课"... ...2l+7l=270美元

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Solve:
::解决 :

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$$\begin{align*}2l+7l &= 270\\ 9l &= 270\\ l &= 30\end{align*}$$
::2l+7l=270l=30

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The regularly priced lessons are $30. The first lesson will be $60.
::定期的课价是30美元 第一课是60美元

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Examples
::实例

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Example 1
::例1

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Earlier, you were asked to find the dimensions the garden must be if Bailey wants the length to be twice as long as the width.
::早些时候,有人要求你 找到花园的尺寸 如果贝利想要长度是宽度的两倍

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If the width is $\begin{align*}w\end{align*}$ , then the length will be $\begin{align*}2w\end{align*}$ . The area of the soil is 128 square feet and the formula for area is $\begin{align*}A=l \cdot w\end{align*}$ .
::如果宽度为 w,则长度为 2w。土壤面积为128平方英尺,区域公式为A=lw。

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$$\begin{align*} 128&=2w \cdot w \\ 128 &= 2w^2 \\ 64 &= w^2 \\ 8 &= w \end{align*}$$
::128=2wwww128=2w264=2w28=ww

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Technically, the answer could also be $\begin{align*}\text{-}8, \text{ since }\sqrt{64}= \pm 8,\end{align*}$  but because we are talking about length, $\begin{align*}w\end{align*}$  cannot be negative. Therefore , the width is 8 feet and the length is $\begin{align*}2w=16 \text{ feet}.\end{align*}$
::从技术上讲,答案也可能是 - 8, 因为648, 但由于我们谈论的是长度, 所以不可能是负的。 因此宽度是8英尺, 长度是2w=16英尺。

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Example 2
::例2

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Bob is twice as old as his daughter. In that same year, his granddaughter is one-tenth his daughter's age. His granddaugther is 3. How old is Bob?
::同年,他的孙女是女儿的十分之一,他的外孙女是女儿的3岁。

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Rewrite each statement as a mathematical one.
::将每个语句重写为数学语句 。

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$$\begin{align*}b &= 2d\\ 10g&=d\\\end{align*}$$
::b=2d10g=d =2d10g=d

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Now, plug in the granddaughter's age and work through each equation.
::现在,插上孙女的年龄 并且通过每一个方程工作。

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$$\begin{align*}10\cdot 3&=30\\ b&=2\cdot 30\end{align*}$$
::103=30b=230

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Bob is 60 years old.
::鲍勃60岁了

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Example 3
::例3

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Javier needs to get a tank of gas. Gas costs $3.79 per gallon. How much money does Javier need to fill up his 16 gallon tank?
::汽油每加仑3.79美元 哈维尔需要多少钱来装满他16加仑的油箱?

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You need  to know how much money Javier needs to fill up his gas tank. Gas costs $3.79 per gallon and he needs 16 gallons of gas. It will cost $\begin{align*}\$ 3.79 \cdot 16= \$ 60.64\end{align*}$ to fill up his tank.
::你需要知道Javier需要多少钱来填充他的油箱。汽油每加仑3.79美元,他需要16加仑的汽油。装满油箱需要3.7916美元=60.64美元。

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Review
::回顾

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Answer each question to the best of your ability.
::尽你最大能力回答每个问题

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1. The average speed on highway 101 is 65 miles per hour (mph). Assuming you drive the speed limit, how long will it take you to drive 350 miles? Use the formula $\begin{align*}distance = rate \cdot time\end{align*}$ . Round your answer to two decimal places.
   ::101号高速公路的平均速度为每小时65英里。 假设您驾驶速度限制, 您需要多久才能驾驶350英里? 使用公式的距离=时间。 您的回答四舍五入到小数点后两位位数 。
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2. Using the information in #1, how many miles did you drive on highway 101 if you drove for 2.5 hours?
   ::使用1号公路的信息,如果开了2.5小时,你在101号高速公路上开了多少英里?
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3. The sum of two consecutive numbers is 79. Find the two numbers.
   ::连续两个数字的总和是79 找出这两个数字。
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4. The sum of two consecutive odd numbers is 44. Find the two odd numbers.
   ::连续两个奇数的总和是44 找出这两个奇数
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5. You borrowed $350 from your parents for a new Wii and games. They are not going to charge you interest, but you need to pay them back as quickly as possible. If you pay them $15 per week, how long will it take you to pay them back?
   ::你从父母那里借了350美元买新Wii和游戏。他们不会收取利息,但你必须尽快还清。如果你每周支付15美元,你还要花多久才能还清?
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6. George is building a rectangular, fenced-in, dog run. He has 120 feet of fencing and wants the length to be 20 feet greater than the width. If you use all the fencing, find the length and width of the dog run.
   ::George正在建造长方形、围栏内、狗跑。他有120英尺的栅栏,希望长度大于宽度20英尺。如果你使用所有栅栏,请找到狗跑的长度和宽度。
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7. Cynthia is selling chocolate bars for a fundraiser for school. Each bar costs $1.50. If she needs to raise $225, how many chocolate bars does she need to sell?
   ::辛西娅为学校募捐卖巧克力棒,每个巧克力棒要1.50美元。 如果她需要筹到225美元,她要卖多少巧克力棒?
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8. Harriet bakes and sells cookies to local stores. Her cost to bake one dozen cookies is $2.75 and she sells them to stores for $7.00 per dozen. How many dozen cookies does she need to make to earn $500? Round to the nearest dozen.
   ::哈丽特烤面包和向当地商店出售饼干。她烤十几个饼干的成本是2.75美元,每打售7.00美元。要挣500美元,她需要挣多少个饼干?圆到最近的十几个。
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9. A football field is a rectangle where the length is 100 yards. If the total perimeter is 1040 feet, what is the width of a football field? Leave your answer in feet.
   ::足球场是一个长100码的矩形。 如果总周界是1040英尺,那么足球场的宽度是多少? 请不要回答。
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10. Challenge The sum of three consecutive even numbers is 138. What are the three numbers?
    ::连续3个偶数的总和是138个 这三个数字是什么?

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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 。