Course: 3.20十进制运算中的属性

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十进制运算中的属性
Casey enjoys running. He plans to run 9.5 miles on Monday, 13.2 miles on Wednesday and 11.5 miles on Friday. The next week, he plans to run 11.5 miles on Monday, 13.2 miles on Wednesday and 9.5 miles on Friday. Since he runs different distances on different days, he's confused that he ends up running the exact same number of miles each week. Which property is represented by Casey's situation?
::凯西喜欢跑步。他计划在周一跑9.5英里,周三跑13.2英里,周五跑11.5英里。下周,他计划在周一跑11.5英里,周三跑13.2英里,周五跑9.5英里。由于他在不同的日子跑的距离不同,他感到困惑的是,他最后每星期跑同样的英里。凯西的情况代表了哪些财产?

In this concept, you will learn how to apply properties to decimals.

::在此概念中,您将学习如何将属性应用到小数点。

Properties of Decimal Addition or Multiplication
::十进制增加或乘数的属性

A property is a rule that remains true when applied to certain situations in mathematics.
::财产是一项规则,在适用于数学的某些情况时依然适用。

The commutative property means that you can switch the order of any of the numbers in an addition or multiplication problem and you will still reach the same answer.
::通量属性意味着您可以在附加或乘法问题中切换任何数字的顺序,您仍然会达到相同的答案。

4 + 5 + 9 = 18 is the same as 5 + 4 + 9 = 18
::4+ 5+ 5+ 9= 18等于 5+ 4+ 9= 18等于 5+ 4+ 9= 18

The order of the numbers being added does not change the sum of these numbers. This is an example of the commutative property.
::添加数字的顺序不改变这些数字的总和。这是通货财产的例子。

If you switch the order of the decimals in an addition problem, the sum does not change.
::如果在附加问题中切换小数点的顺序,总和不会改变。

4.5 + 3.2 = 7.7 is the same as 3.2 + 4.5 = 7.7
::4.5 + 3.2 = 7.7 等于 3.2 + 4.5 = 7.7

The associative property means that you can change the groupings of numbers being added or multiplied and it does not change the result. This applies to problems with and without decimals.
::连带属性意味着您可以改变数字的组合,即添加或乘数,不会改变结果。这适用于小数和小数的问题。

(1.3 + 2.8) + 2.7 = 6.8 is the same as 1.3 + (2.8 + 2.7) = 6.8
:1.3+2.8)+2.7=6.8与1.3+2.8+2.7=6.8相同。

Notice that " data-term="Parentheses" role="term" tabindex="0"> parentheses are used to help with the groupings.
::注意用括号帮助分组。

Sometimes, you will have a problem with a variable and a decimal in it. You can apply the commutative property and associative property here too.
::有时,您会遇到变量和小数点的问题。您也可以在这里应用通勤属性和关联属性。

Let's look at an example.
::让我们举个例子。

$\begin{aligned} x + 4.5 \end{aligned}$ is the same as $\begin{aligned} 4.5 + x \end{aligned}$
::x+4.5等于4.5+x等于4.5+x

$\begin{aligned} (x + 3.4) + 5.6 \end{aligned}$ is the same as $\begin{aligned} x + (3.4 + 5.6) \end{aligned}$
:x+3.4)+5.6与x+(3.4+5.6)相同。

The most important thing is that the order of the numbers and the groupings can change but the sum will remain the same.
::最重要的是,数字和分组的顺序可以改变,但总数将保持不变。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Casey and his weekly runs.
::早些时候,你得到一个问题凯西和他每周的赛跑。

He runs 9.5 miles, 13.2 miles then 11.5 miles the first week and 11.5 miles, 13.2 miles and 9.5 miles the next week. Which property is represented by Casey's two weeks of running?
::他运行9.5英里,13.2英里,第一周11.5英里, 第一周11.5英里, 第11.5英里, 第13.2英里, 第9.5英里。

$\begin{aligned} 9.5 + 13.2 + 11.5 = 11.5 + 13.2 + 9.5 \end{aligned}$

First, check that all of the numbers are the same on both sides of the equal sign.
::首先,检查所有数字在等号两侧都是一样的。

yes
::是是

Next, determine if the order of the numbers changes or the groupings of the numbers changes.
::下一步,确定数字的顺序是否改变或数字的分组是否改变。

the order changes
::顺序更改

Then, determine the property.
::然后,确定财产。

commutative property
::平价财产

The answer is the commutative property of addition . Casey's two weeks of running represent the commutative property.
::凯西的两周运行时间代表着通货财产

Example 2
::例2

Name the property illustrated below.
::说明下列财产。

$\begin{aligned} 3.2 + (x + y) + 5.6 = (3.2 + x) + y + 5.6 \end{aligned}$
::3.2+(x+y)+5.6=(3.2+x)+y+5.6

First, check that all of the numbers and variables are the same on both sides of the equal sign.
::首先,检查所有数字和变量在等号两侧都是一样的。

yes
::是是

Next, determine if the order of the numbers and variables change or the groupings of the numbers and variables change.
::下一步,确定数字和变量的顺序是否发生变化,或者数字和变量的组合是否发生变化。

the groupings change
::组合组合改变

Then, determine the property.
::然后,确定财产。

associative property
::组合财产

The answer is that the associative property of addition changes.
::答案是,加法的连带属性会发生变化。

Example 3
::例3

Look at the following example and name the property illustrated in the example.
::以下列示例为例,并列出示例中说明的财产名称。

$\begin{aligned} 3.4 + 7.8 + 1.2 = 7.8 + 1.2 + 3.4 \end{aligned}$

First, check that all of the numbers are the same on both sides of the equal sign.
::首先,检查所有数字在等号两侧都是一样的。

yes
::是是

Next, determine if the order of the numbers changes or the groupings of the numbers changes.
::下一步,确定数字的顺序是否改变或数字的分组是否改变。

the order changes
::顺序更改

Then, determine the property.
::然后,确定财产。

commutative property
::平价财产

The answer is the commutative property of addition.
::答案是增加的通货性能。

Example 4
::例4

Look at the following example and name the property illustrated in the example.
::以下列示例为例,并列出示例中说明的财产名称。

$\begin{aligned} (1.2 + 5.4) + 3.2 = 1.2 + (5.4 + 3.2) \end{aligned}$

First, check that all of the numbers are the same on both sides of the equal sign.
::首先,检查所有数字在等号两侧都是一样的。

yes
::是是

Next, determine if the order of the numbers changes or the groupings of the numbers changes.
::下一步,确定数字的顺序是否改变或数字的分组是否改变。

the groupings change
::组合组合改变

Then, determine the property.
::然后,确定财产。

associative property
::组合财产

The answer is the associative property of addition.
::答案是附加的连带属性。

Example 5
::例5

Look at the following example and name the property illustrated in the example.
::以下列示例为例,并列出示例中说明的财产名称。

$\begin{aligned} x + 5.6 + 3.1 = 3.1 + x + 5.6 \end{aligned}$
::x + 5.6 + 3.1 = 3.1 + x + 5.6 + 5.6

First, check that all of the numbers and variables are the same on both sides of the equal sign.
::首先,检查所有数字和变量在等号两侧都是一样的。

yes
::是是

Next, determine if the order of the numbers and variables change or the groupings of the numbers and variables change.
::下一步,确定数字和变量的顺序是否发生变化,或者数字和变量的组合是否发生变化。

the order changes
::顺序更改

The answer is the commutative property of addition.
::答案是增加的通货性能。

Review
::回顾

Identify the property illustrated in each number sentence.
::查明每一句数字中说明的财产。

$\begin{aligned} 4.5 + (x + y) + 2.6 = (4.5 + x) + y + 2.6 \end{aligned}$
::4.5+4.5+(x+y)+2.6=(4.5+x)+y+2.6

$\begin{aligned} 3.2 + x + y + 5.6 = x + 3.2 + y + 5.6 \end{aligned}$
::3.2+xx+y+y+5.6=x+3.2+y+5.6

$\begin{aligned} 1.5 + (2.3 + y) + 5.6 = (1.5 + 2.3) + y + 5.6 \end{aligned}$
::1.5+(2.3+y)+5.6=(1.5+2.3)+y+5.6

$\begin{aligned} 3.2 + 5.6 + 1.3 + 2.6 = 3.2 + 2.6 + 5.6 + 1.3 \end{aligned}$

$\begin{aligned} 4.5 + 15.6 = 15.6 + 4.5 \end{aligned}$

$\begin{aligned} (x + y) + 5.6 = x + (y + 5.6) \end{aligned}$
:x+y)+5.6=x+(y+5.6)

$\begin{aligned} 17.5 + 18.9 + 2 = 2 + 17.5 + 18.9 \end{aligned}$

$\begin{aligned} (x + y) + z = x + (y + z) \end{aligned}$
:x + y ) + z = x + (y + z )

$\begin{aligned} 5.4 + 5.6 = 5.6 + 5.4 \end{aligned}$

$\begin{aligned} 1.2 + 3.2 + 5.6 = 1.2 + 5.6 + 3.2 \end{aligned}$

$\begin{aligned} 3.2 + (x + y) + 5.6 = 3.2 + x + (y + 5.6) \end{aligned}$
::3.2+(x+y)+5.6=3.2+x+(y+5.6)

$\begin{aligned} 3.4 + x + y + .6 = .6 + y + x + 3.4 \end{aligned}$
::3.4 + x + y + y + 6 = 6 + y + x + 3.4

$\begin{aligned} 2.2 + 4.3 + 1.1 = 1.1 + 2.2 + 4.3 \end{aligned}$

$\begin{aligned} (1.2 + 3.4) + 7.6 = 1.2 + (3.4 + 7.6) \end{aligned}$

$\begin{aligned} 8.9 + 9.3 + 3.1 = 9.3 + 8.9 + 3.1 \end{aligned}$

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

General

十进制运算中的属性

Properties of Decimal Addition or Multiplication ::十进制增加或乘数的属性

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

Properties of Decimal Addition or Multiplication
::十进制增加或乘数的属性

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