2.9 多标准乘数和分部涉及合理数字的问题-interactive
章节大纲
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Properties of Multiplication
::乘法属性Just like addition , multiplication has a commutative and associative property . The Commutative Property of Multiplication says that the order in which you multiply any two rational numbers, order does not matter . So, for any two rational numbers a and b ,
::就像附加一样,乘法有一个通和和连带属性。乘法的公交属性表示,您乘以任何两个合理数字的顺序,顺序并不重要。因此,对于任何两个合理数字a和b,a × b = b × a a b = b a a ⋅ b = b ⋅ a
::axb=bxa ab=ba ab=bxa =bxa =bxa =bxb=bxa =bxb=bxaNotice the three different ways to write the multiplication of two numbers. In addition, you could put parenthes e s around a and b, ( a ) ( b ) = ( b ) ( a ) . It is important to note that if there is no sign between two variables (letters) or a variable and number, you can assume that the two numbers are being multiplied.
::注意写两个数字乘法的三种不同方式。此外,您可以在a和b(a)(b)=(b)(a)前后加括号。必须指出,如果两个变量(字母)或变量和数字之间没有符号,您可以假设这两个数字是乘法。The Associative Property of Multiplication states that for any three rational numbers, you may pick the two that you want to multiply together first. You do not have to multiply them in the order in which they appear in the problem. So, in this case, for any three rational numbers : a , b and c ,
::乘法联系属性表示,对于任意三个合理数字,您可以先选择要一起乘的两种数字。您不必按问题出现时的顺序来乘。因此,在这种情况下,对于任何三个合理数字:a、b和c,a × b × c = a × ( b × c ) = ( a × b ) × c a b c = a ( b c ) = ( a b ) c a ⋅ b ⋅ c = a ⋅ ( b ⋅ c ) = ( a ⋅ b ) ⋅ c
::abxxc=axxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxJust like with the Associative Property of Addition, the Associative Property of Multiplication can use any of the grouping symbols , including absolute value . This property also works for more than three rational numbers.
::与“附加的联营财产”一样,“乘以的联营财产”可以使用任何组合符号,包括绝对值。该属性还可以使用三个以上的合理数字。
Switch the Order
::切换顺序How many different ways can you multiply - 8 ⋅ 2 ⋅ - 5 ⋅ 2 ?
::您可以乘以多少种不同的方式 - 8252?Discussion Question
::讨论问题Compare − 8 ÷ 2 ÷ ( − 5 ) ÷ 2 and − 8 ⋅ 1 2 ⋅ ( − 1 5 ) ⋅ 1 2 . Discuss the similarities and differences with your classmates. Can you apply the Commutative or Associative Properties to one and not the other? Why or why not?
::比较 - 82( - 5) 2 和 -812( - 15) 12。 与您的同学讨论相似和不同之处。 您能否将交流或联合属性应用到一个而不是另一个? 为什么或为什么不应用?T he Commutative and Associative Properties work for multiplication like they do for addition. There are two more properties to explore here : Identity and Inverse .
::computition and Associal Property 工作用于乘法, 和它们用来加法的乘法一样。 这里还有两个特性要探索: 身份和反向 。Recall that for addition the additive identity was 0 because any number you add to 0 is still that number. What number can you multiply by any number and the answer is still that number?
::提醒大家注意,加注身份为 0, 因为您加到 0 的任何数字仍然是这个数字。 您可以乘以哪个数字, 而答案仍然是那个数字 ?a ⋅ 1 = a
::a =1=aThe Identity Property of Multiplication states that for any number, a , when multiplied by 1, the answer is still a .
::乘以特性表示,对于任何数字,a乘以1,答案仍然是a。The Inverse Property of Multiplication is similar. W hat number times a is equal to 1?
::反乘属性相似。 乘以多少乘以等于 1 ?You can figure this out by working backwards from the identity property. Remember how when dividing by a fraction , you can flip the fraction and multiply by the reciprocal instead? Well, for any number a , you can multiply by its reciprocal to obtain 1.
::您可以从身份属性中反向工作来解决这个问题。 记住, 当分数除以一个分数时, 您可以翻转分数, 乘以对等法 。 那么, 对于任何一个数字 a , 您可以乘以对等法来获得 1 。a ⋅ 1 a = 1
::a1a=1It is important to note that the sign of the reciprocal of a number is the same as the sign of the number. Reciprocals are not opposites .
::必须指出,数字对等的标志与数字对等的标志相同,对等不是相反的。
Trail Mix
::轨迹混合Kylie and her friends are going on a hike and she is making trail mix to take. The recipe calls for 2 cups of peanuts, 1 cup of chocolate chips, 1 cup of dried cranberries, and 1 1 2 cups of pretzels. If she and three friends are going, how much of each ingredient will they get?
::凯莉和她的朋友们正在远足,她正在做杂食。 食谱需要两杯花生、一杯巧克力薯片、一杯咖啡干果和112杯薄饼。 如果她和三个朋友去,他们能得到多少?Move the slider to 4 to determine how much of each ingredient is in each bag if the mix is distributed evenly.
::将滑块移动到 4 , 以确定每袋中每个成分的份量, 如果混合成分分布均匀的话 。Recall that the Associative and Commutative Properties do not work with division . For the last question in the interactive, you need to perform the indicated operations in order from left to right. Additionally , this interactive used multiplying by an inverse interchangeably with dividing by the number. Familiarize yourself with using both multiplying by fractions and dividing by a whole number. Depending on the problem, one might be easier than the other.
::提醒注意“ 联合属性 ” 和“ 交流属性” 与 分区不起作用。 对于互动中最后一个问题, 您需要从左向右执行指定的操作。 此外, 此交互性用数字的除法以反相交法乘以反相交法。 既使用分数乘以分数,又使用整数, 使自己适应。 根据问题的不同, 一个比另一个容易。Summary
::摘要-
Commutative Property of Multiplication: for two rational numbers
a
and
b
,
a
⋅
b
=
b
⋅
a
.
Order does not matter for multiplication.
::乘法的通量属性:两个合理的数a和b, ab=ba。顺序与乘法无关。 -
Associative Property of Multiplication: for two rational numbers
a
and
b
,
a
×
b
×
c
=
a
×
(
b
×
c
)
=
(
a
×
b
)
×
c
.
It does not matter which two numbers you group together to multiply first.
::乘法的关联属性:对于两个合理数字a和b, axbxc=axx(bxc)=(axb)xc。您将两个数字组合在一起首先乘以两个数字并不重要。 -
Identity Property of Multiplication: for a rational number
a
,
a
×
1
=
a
.
Any number times one is that number.
::乘法属性:对于一个合理数字 a, ax1=a。任何数字乘以一就是该数字。 -
Inverse Property of Multiplication: for a rational number
a
,
a
×
1
a
=
1
.
Any number times its inverse (reciprocal) is always one.
::乘法的逆等值:对于一个合理的数 a, ax1a=1. 其反等值( 互惠) 的任意次数总是一个 。 -
Commutative and Associative Properties are not true for division.
::划分时不采用交流和共同属性。
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Commutative Property of Multiplication: for two rational numbers
a
and
b
,
a
⋅
b
=
b
⋅
a
.
Order does not matter for multiplication.