Course: 1.4 使用作业顺序评价数值和变量表达式 | The Way To Learn

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使用操作命令评价数值和变量表达式
Jeb enters his name into a draw to win a handheld game system from his local electronics megastore. A week later he visits the store for the draw and his name is picked! In order to claim the prize, he must correctly answer a skill testing question.
::Jeb为了赢得当地电子公司大型商店的手持游戏系统,将他的名字输入抽奖中。一周后,他前往商店抽奖,并被选中。为了获得奖项,他必须正确地回答技能测试问题。

The question is:
$\begin{aligned} \frac{4}{5} [30 - (4 \times 2 - 3)] \end{aligned}$

::问题是:45[30-(4x2-3)]

Jeb has to answer the skill testing question without the use of technology. How can Jeb answer this question correctly to claim the prize?
::Jeb必须在不使用技术的情况下回答技能测试问题。 Jeb怎样才能正确地回答这个问题才能获得奖项呢?

In this concept, you will learn to use the order of operations to solve numerical and variable expressions.
::在此概念中,您将学会使用操作顺序来解析数字和变量表达式。

Order of Operations
::行动命令令

In mathematics, you will often hear the word evaluate . Before you begin, it is important for you to understand what the word evaluate means. When you evaluate a mathematical sentence, you figure out the value of the number sentence. Often times you think of evaluating as solving, and it can be that, but more specifically, evaluating is figuring out the value of a sentence.
::在数学中,您通常会听到对词的评价。在开始之前,重要的是要理解对词的评价意味着什么。当对一个数学句进行评价时,您会发现数字句的值。通常,当你认为对词的评价是解决的时,可能,但更具体地说,对词的评价是确定一个句子的价值。

In mathematics, you can evaluate different types of number sentences. Sometimes you will be working with equations and other times you will be working with expressions. First you need to know the difference between an equation and an expression .
::在数学中,您可以评估不同的数字句类型。有时您会用方程式工作,有时您会用表达式工作。首先,您需要知道方程式和表达式之间的区别。

An equation is a statement that two mathematical expressions have the same value. An equation has an equal sign such that the quantity on the left side of the equal sign is equal to the quantity on the right side of the equal sign. This means both sides of the equation stand for the same number. $\begin{aligned} 5 x = 30 \end{aligned}$ is an equation. It states that $\begin{aligned} " 5 x " and " 30 " \end{aligned}$ have the same value.
::等式是两个数学表达式具有相同值的语句。等式有一个等号, 等号左侧的数量等于等号右侧的数量。这意味着等式两侧的数值相同。 5x=30是一个方程。它表示“ 5x” 和“ 30” 具有相同值。

An expression is a general term in mathematics for a group of numbers, symbols and variables representing numbers and operations . You evaluate an expression to figure out the value of the mathematical statement itself, you are not trying to make one side equal another, as with an equation. $\begin{aligned} 3 a - 2 b + 8 \end{aligned}$ is an expression. There is no equal sign.
::表达式是数学中代表数字和操作的一组数字、符号和变量的通用术语。您对一个表达式进行评价,以找出数学语句本身的值,您没有试图使一边对等,就像方程式一样。 3a-2b+8 是一个表达式。没有相等的符号。

Let’s start with an example of evaluating expressions.
::让我们先举一个评价表达式的例子。

Two eighth grade math students evaluated the expression $\begin{aligned} 2 + 3 \times 4 \div 2 \end{aligned}$ . Macy’s answer was ten and Cole’s answer was eight. The students you are asked to write their step by step solutions on the board.
::两位八年级数学学生评估了2+3×42. 梅西的回答是10分,科尔的回答是8分。学生们被要求在董事会上一步一步地写出他们的解决方案。

Macy‘s Solution:
::梅西的解决方案 :

$\begin{aligned} = 2 + 3 \times 4 \div 2 \\ = 5 \times 4 \div 2 \\ = 20 \div 2 \\ = 10 \end{aligned}$

Cole‘s Solution:
::Cole's解决方案:

$\begin{aligned} = 2 + 3 \times 4 \div 2 \\ = 2 + 12 \div 2 \\ = 2 + 6 \\ = 8 \end{aligned}$

It appears that each student performed the indicated operations in different orders. Of course, there cannot be two correct solutions for the same expression. There is actually a specific order in which operations must be performed.
::显然,每个学生都按不同顺序执行指定的操作。当然,对同一表达式不能有两种正确的解决方案。实际上,操作必须按具体顺序进行。

The order of operations is a rule that tells you which operation you need to perform and the order in which it must be done to achieve the correct answer. The order of operations is often called PEMDAS and each of the letters represents one part of the rule. P : parenthesis and grouping symbols ; E : exponents; M : multiplication ; D : division ; A : addition ; S : subtraction . MD (multiplication and division) are performed in the order they appear in the expression from left to right. AS (addition and subtraction) are also performed in the order they appear in the expression from left to right.
::操作的顺序是一条规则, 它告诉您您需要执行什么操作, 以及实现正确答案必须完成的顺序。操作的顺序通常被称为 PEMDAS , 字母中的每个字母代表了规则的一部分。 P: 括号和分组符号; E: 引号; M: 乘法; D: 分数; A: 添加; S: 减法; S: 倍数和分数) 的顺序是按从左到右表达式的顺序执行的。自动( 增减) 也按从左到右表达式的顺序执行。

Looking at the two solutions for evaluating the expression $\begin{aligned} 2 + 3 \times 4 \div 2, \end{aligned}$ who has the correct answer?
::查看评价2+3×42表达式的两种解决办法,谁有正确的答案?

Macy
::梅西

$\begin{aligned} First : 2 + 3 = 5 \\ Next : 5 \times 4 = 20 \\ Then : 20 \div 2 = 10 \end{aligned}$

::第一: 2+3=5 下一个: 5x4=20 然后: 20 @% 2=10

Cole
::科尔

$\begin{aligned} First : 3 \times 4 = 12 \\ Next : 12 \div 2 = 6 \\ Then : 6 + 2 = 8 \end{aligned}$

::第一: 3x4= 12Next: 12 @ @ @ 2= 6 然后: 6+2=8

Macy evaluated the expression by performing the addition first, followed by multiplication and division. Cole performed the multiplication and division first, then addition. Macy simply completed the operations as they appeared from left to right. Cole completed the multiplication and division as they appeared from left to right and then performed the addition as his final step. Cole used the order of operations rule, PEMDAS, and his answer is correct.
::梅西首先进行添加,然后进行乘法和分法。科尔先进行乘法和分法的乘法和分法,然后加法。梅西只是完成了从左向右出现的操作。科尔完成了从左向右出现的乘法和分法的乘法和分法,然后又完成了最后一步的加法。科尔使用了操作顺序规则,即PEMDAS,他的答复是正确的。

This is called evaluating or simplifying a numerical expression. A numerical expression is an expression made up only of numbers and operations.
::这被称为评价或简化数字表达式。数字表达式只由数字和操作组成。

In addition to numbers, expressions can also have letters. The letters in an expression are called variables . These variables represent an unknown quantity. When an expression is written with a variable in it, you call it a variable expression .
::除数字外,表达式也可以有字母。表达式中的字母被称为变量。这些变量代表一个未知的数量。当一个表达式用变量写入时,您将它称为变量表达式。

A variable expression is evaluated using the order of operations in the same way as a numerical expression is evaluated. In this C oncept there will be a value given for the variable and you will substitute it into the variable expression before evaluating the expression.
::以与数字表达式相同的方式使用操作顺序对变量表达式进行评价。在此概念中,将给变量指定一个值,并在评价表达式之前将其替换为变量表达式。

Evaluate the variable expression:
::评估变量表达式 :

$\begin{aligned} 60 \div 2 \cdot 2 a + 16 - 4 when a = 5. \end{aligned}$
::a=5时为6022a+16-4。

First, substitute the given value for $\begin{aligned} a \end{aligned}$ in the expression:
::首先,将给定值替换为:

$\begin{aligned} 60 \div 2 \cdot 2 (5) + 16 - 4 \end{aligned}$

Remember that, used this way, parenthesis mean you are multiplying 2 times (5).
::记住,用这种方式,括号表示你乘以2乘(5)。

Now apply the order of operations (PEMDAS) and continue to evaluate the expression.
::现在应用操作顺序( PEMDAS) 并继续评价表达式。

First , divide: $\begin{aligned} 60 \div 2 = 30 \end{aligned}$ , and write the new expression.
::首先,划分: 602=30, 并写下新表达式。

$\begin{aligned} 30 \cdot 2 (5) + 16 - 4 \end{aligned}$

Next, multiply: $\begin{aligned} 30 \cdot 2 = 60, \end{aligned}$ since this is the first multiplication from left to right.
::下一个乘以: 302=60, 因为这是从左向右的第一次乘法。

$\begin{aligned} 60 (5) + 16 - 4 \end{aligned}$

Next, multiply: $\begin{aligned} 60 (5) = 300 \end{aligned}$ and write the new expression.
::下一个乘以: 60(5)=300, 并写下新表达式。

$\begin{aligned} 300 + 16 - 4 \end{aligned}$

Next, add: $\begin{aligned} 300 + 16 = 316 \end{aligned}$ and write the new expression.
::下加: 300+16=316, 并写下新表达式。

$\begin{aligned} 316 - 4 \end{aligned}$

Then, subtract: $\begin{aligned} 316 - 4 = 312 \end{aligned}$
::然后减去:316-4=312

The answer is 312.
::答案是312

Now let’s add in the grouping symbols. The grouping symbols that you will be working with are brackets [ ] and parenthesis ( ). According to the order of operations (PEMDAS), you perform all operations inside the grouping symbols BEFORE any other operation in the list.
::现在让我们在分组符号中添加。您将要使用的分组符号是括号 [ ] 和括号。根据操作顺序( PEMDAS) , 您在列表中的任何其他操作之前在分组符号中执行全部操作。

Evaluate the numerical expression
::评估数字表达式

$\begin{aligned} 7 + 4 (15 \div 5) - 6. \end{aligned}$

First, perform the operation in the parenthesis: $\begin{aligned} 15 \div 5 = 3, \end{aligned}$ and write the new expression.
::首先,在括号中执行操作:155=3,并写新表达式。

$\begin{aligned} 7 + 4 (3) - 6 \end{aligned}$

Next, multiply $\begin{aligned} 4 (3) = 12 \end{aligned}$ to clear the parenthesis and write the new expression.
::下一步,乘以 4(3)=12来清除括号并写入新表达式。

$\begin{aligned} 7 + 12 - 6 \end{aligned}$
Next, add: $\begin{aligned} 7 + 12 = 19 \end{aligned}$ and write the new expression.
::7+12-6Next,加上:7+12=19,并写下新表达式。

$\begin{aligned} 19 - 6 \end{aligned}$

Then, subtract: $\begin{aligned} 19 - 6 = 13 \end{aligned}$
::然后减去:19-6=13

The answer is 13.
::答案是13岁

Brackets can be used to group more than one operation. When you see a set of brackets, remember that brackets are a way of grouping numbers and operations with other groups already in parenthesis. Always evaluate grouping symbols from the innermost to the outermost.
::括号可以用来组合多个操作。当您看到一组括号时,请记住括号是一种将数字和操作与其他括号中已经括号中的组组合在一起的方式。总是要从最内层到最外层来评估组合符号。

Evaluate the numerical expression.
::评估数字表达式。

$\begin{aligned} 6 + [5 + (4 \times 6)] - 17. \end{aligned}$

First, evaluate the innermost grouping symbols: the parenthesis.
::首先,评价最核心的分组符号:括号。

Multiply: $\begin{aligned} 4 \times 6 = 24 \end{aligned}$ and write the new expression.
::乘以:4x6=24,然后写新表达式。

$\begin{aligned} 6 + [5 + 24] - 17. \end{aligned}$

Next, perform the operation inside the brackets.
::下一步,在括号内执行操作。

Add: $\begin{aligned} 5 + 24 = 29 \end{aligned}$ and write the new expression.
::加:5+24=29,并写新表达式。

$\begin{aligned} 6 + 29 - 17 \end{aligned}$

Next, add: $\begin{aligned} 6 + 29 = 35 \end{aligned}$ and write the new expression.
::下加:6+29=35,并写新表达式。

$\begin{aligned} 35 - 17 \end{aligned}$
Then, subtract: $\begin{aligned} 35 - 17 = 18 \end{aligned}$
::35 - 17, 减去: 35 - 17=18

The answer is 18.
::答案是18岁

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Jeb and his (almost) prize.
::以前,你曾因耶卜和他的奖项而受辱。

Remember, w hen solving a problem like th is skill testing question, you need to complete the indicated operations in the order of PEMDAS.
::记住,当解决类似技能测试的问题时, 您需要按照 PEMDAS 的顺序完成指定的操作。

$\begin{aligned} \frac{4}{5} [30 - (4 \times 2 - 3)] \end{aligned}$

Notice there are parenthesis within the brackets. You need to evaluate from the inside out. So start with the parenthesis. According to PEMDAS multiplication comes before subtraction, and this is just as true within a group as without.
::括号中有括号括号括号。您需要从括号内向外进行评价。所以从括号开始。根据 PEMDAS 的乘法, 乘数在减法之前, 在一个组中和在不计组中一样正确。

First, multiply: $\begin{aligned} 4 \times 2 = 8 \end{aligned}$ and write the new expression.
::首先,乘以: 4x2=8, 并写入新表达式。

$\begin{aligned} \frac{4}{5} [30 - (8 - 3)] \end{aligned}$
Next, subtract: $\begin{aligned} 8 - 3 = 5 \end{aligned}$ and write the new expression.
::45[30-(8-3)] 下,减去:8-3=5,并写下新表达式。

$\begin{aligned} \frac{4}{5} [30 - 5] \end{aligned}$

Next, subtract: $\begin{aligned} 30 - 5 = 25 \end{aligned}$ and write the new expression.
::下一步,减去:30-5=25,并写下新表达式。

$\begin{aligned} \frac{4}{5} (25) \end{aligned}$

Next, multiply: $\begin{aligned} \frac{4}{5} \times \frac{25}{1} = \frac{100}{5} \end{aligned}$ and write the new expression.
::下一位, 乘以: 45x251=1005, 并写入新表达式。

$\begin{aligned} \frac{100}{5} \end{aligned}$

Then, divide: $\begin{aligned} \frac{100}{5} = 20 \end{aligned}$
::然后,分开:1005=20

The answer is 20.
::答案是20岁

Jeb needs to answer 20 to claim his prize.
::Jeb需要回答20 才能获得他的奖项

Example 2
::例2

Evaluate the expression $\begin{aligned} 3 + 9 \cdot 2 \div 3 + 8 \end{aligned}$
::评价3+9+2+3+8的表达式

First, multiply $\begin{aligned} 9 \cdot 2 = 18, \end{aligned}$ and write down the new expression.
::首先,乘以 92=18, 并写下新表达式。

$\begin{aligned} 3 + 18 \div 3 + 8 \end{aligned}$

Next, divide $\begin{aligned} 18 \div 3 = 6 \end{aligned}$ and write down the new expression.
::下一个分隔 183=6, 并写下新表达式。

$\begin{aligned} 3 + 6 + 8 \end{aligned}$

Next add $\begin{aligned} 3 + 6 = 9 \end{aligned}$ and write down the new expression.
::下一步添加 3+6=9 并写入新表达式。

$\begin{aligned} 9 + 8 \end{aligned}$
Then, add $\begin{aligned} 9 + 8 = 17. \end{aligned}$
::9+8,加上9+8=17。

The answer is 17.
::答案是17。

Example 3
::例3

Evaluate the variable expression $\begin{aligned} 6 y + 3 - (2 \times 4) when y = 15. \end{aligned}$
::y=15 时评价变量表达式 6y+3-( 2x4) 。

First, substitute $\begin{aligned} y = 15 \end{aligned}$ into the expression.
::首先,将 Y=15 替换为表达式。

$\begin{aligned} 6 (15) + 3 - (2 \times 4) \end{aligned}$

Next, perform the operation inside the parenthesis.
::接下来,在括号内执行操作。

Multiply: $\begin{aligned} 2 \times 4 = 8 \end{aligned}$ and write the new expression.
::乘以: 2x4=8 并写入新表达式。

$\begin{aligned} 6 (15) + 3 - 8 \end{aligned}$

Next, multiply: $\begin{aligned} 6 (15) = 90 \end{aligned}$ to clear the parenthesis and write the new expression.
::下一个, 乘: 6( 15) = 90 来清除括号并写入新表达式。

$\begin{aligned} 90 + 3 - 8 \end{aligned}$

Next add: $\begin{aligned} 90 + 3 = 93 \end{aligned}$ and write the new expression.
::下加:90+3=93,并写下新表达式。

$\begin{aligned} 93 - 8 \end{aligned}$
Then, subtract:
::93 - 8 然后减去:

$\begin{aligned} 93 - 8 = 85 \end{aligned}$
The answer is 85.
::93 -8=85 答案是85

Example 4
::例4

Evaluate the numerical expression.
::评估数字表达式。

$\begin{aligned} 7 + 4 [4 + (3 \times 2)] - 5 \end{aligned}$

Remember to evaluate the grouping symbols from the innermost to the outermost.
::记住要从最内层到最外层的分组符号。

First, perform the operation inside the parenthesis.
::首先,在括号内执行操作。

Multiply: $\begin{aligned} 3 \times 2 = 6 \end{aligned}$ and write the new expression.
::乘以: 3x2=6, 并写入新表达式。

$\begin{aligned} 7 + 4 [4 + 6] - 5 \end{aligned}$

Next, perform the operation inside the brackets.
::下一步,在括号内执行操作。

Add: $\begin{aligned} 4 + 6 = 10 \end{aligned}$ and write the new expression.
::加:4+6=10,并写新表达式。

$\begin{aligned} 7 + 4 [10] - 5 \end{aligned}$

Then multiply $\begin{aligned} 4 \times 10 = 40. \end{aligned}$
::然后乘以 4x10=40。

$\begin{aligned} 7 + 40 - 5 \end{aligned}$

Next, add: $\begin{aligned} 7 + 40 = 47 \end{aligned}$ and write the new expression.
::下加:7+40=47,并写下新表达式。

$\begin{aligned} 47 - 5 \end{aligned}$
Finally , subtract
::47-5 最后,减去

$\begin{aligned} 47 - 5 = 42 \end{aligned}$

The answer is 4 2.
::答案是42岁

Example 5
::例5

Evaluate the variable expression.
::评估变量表达式。

$\begin{aligned} 14 \times 2 \div 7 + 3 b - 4 when b = 12. \end{aligned}$

::b=12时为14x2+7+3b-4。

First substitute $\begin{aligned} b = 12 \end{aligned}$ into the expression.
::表示式中的第一个替代品 b=12。

$\begin{aligned} 14 \times 2 \div 7 + 3 (12) - 4 \end{aligned}$

First , multiply: $\begin{aligned} 14 \times 2 = 28 \end{aligned}$ and write the new expression.
::首先,乘以:14x2=28,然后写新表达式。

$\begin{aligned} 28 \div 7 + 3 (12) - 4 \end{aligned}$

Next, divide: $\begin{aligned} 28 \div 7 = 4 \end{aligned}$ and write the new expression.
::下一个分隔: 287=4, 并写入新表达式。

$\begin{aligned} 4 + 3 (12) - 4 \end{aligned}$

Then, multiply again: $\begin{aligned} 3 \times 12 = 36, \end{aligned}$ and write another new expression.
::然后,再乘以: 3x12=36, 并写新表达式。

$\begin{aligned} 4 + 36 - 4 \end{aligned}$

Next, add: $\begin{aligned} 4 + 36 = 40 \end{aligned}$ and write the new expression.
::下加:4+36=40,并写新表达式。

$\begin{aligned} 40 - 4 \end{aligned}$

Finally , subtract : $\begin{aligned} 40 - 4 = 36 \end{aligned}$
::最后,减去:40-4=36

The answer is 36.
::答案是36岁

Review
::回顾

Evaluate each numerical expression using the order of operations.
::使用操作顺序评估每个数字表达式。

1. $\begin{aligned} 4 + 5 \times 2 - 3 \end{aligned}$

2. $\begin{aligned} 6 + 6 \times 3 \div 2 - 7 \end{aligned}$

3. $\begin{aligned} 5 + 5 \times 8 \div 2 + 6 \end{aligned}$

4. $\begin{aligned} 13 - 3 \times 2 + 8 - 2 \end{aligned}$

5. $\begin{aligned} 17 - 5 \times 3 + 8 \div 2 \end{aligned}$

6. $\begin{aligned} 9 + 4 \times 2 + 7 - 1 \end{aligned}$

7. $\begin{aligned} 8 + 5 \times 6 + 2 \times 4 - 3 \end{aligned}$

8. $\begin{aligned} 19 + 2 \times 4 - 3 \cdot 2 + 10 \end{aligned}$

9. $\begin{aligned} 12 + 4 \times 4 \div 8 - 3 \end{aligned}$

10. $\begin{aligned} 12 \times 2 + 6 \div 2 - 12 \end{aligned}$

Evaluate each variable expression. Remember to use PEMDAS when necessary.
::评估每个变量表达式。请记住必要时使用 PEMDAS 。

1 1. $\begin{aligned} 4 y + 6 - 2, when y = 6 \end{aligned}$
::11.4y+6-2,y=6时

12. $\begin{aligned} 9 + 3 x - 5 + 2, when x = 8 \end{aligned}$
::12. 9+3x- 5+2, 当x=8时

13. $\begin{aligned} 6 y + 2 y - 5, when y = 3 \end{aligned}$
::13.6y+2y-5,y=3时

14. $\begin{aligned} 8 + 3 y - 5, when y = 4 \end{aligned}$
::14.8+3y-5,y=4时

15. $\begin{aligned} 7 x - 2 \times 3 \div 3 + 12, when x = 5 \end{aligned}$
::15. 7x-2x3+12, 当 x=5

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。

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