4.10整数乘法

Section outline

• Select section General

  Collapse Expand

  General

  Collapse all Expand all

  • Select activity 新闻通告

    

    新闻通告 Forum
• Select section 整数乘法

  Collapse Expand

  整数乘法

  

  [Figure 1]

  播放段落

  Marybeth just helped give her baby brother a bath. As the water is draining out of the tub, Marybeth notices that the height of the water in the tub is decreasing by about 2 inches  per  minute. How could Marybeth describe the change in the height of the water in the tub after 3 minutes?
  ::玛丽贝丝只是帮她弟弟洗澡。 玛丽贝丝注意到浴缸的水位每分钟下降2英寸左右。 玛丽贝丝如何描述3分钟后浴缸水位变化?

  播放段落

  In this concept, you will learn how to multiply integers.
  ::在这个概念中,你会学会如何乘以整数。

  播放段落

  Multiplying Integers
  ::乘数整数

  播放段落

  Integers  are the set of  whole numbers  and their opposites.
  ::整组是整组数字和反面数字的组合。

  播放段落

  When you multiply integers, the numbers that you are multiplying are called  factors . Your answer after multiplying is called a  product .
  ::当乘以整数时,乘以的数字被称为因数。乘以后的答案被称为产品。

  播放段落

  Here is an example.
  ::举一个例子。

  $\begin{array}{r}3\times 7=21\end{array}$

  播放段落

  In this example, 3 and 7 are the factors. 21 is the product.
  ::在这个例子中,因素是3和7。 21是产品。

  播放段落

  Multiplying integers is very similar to multiplying whole numbers. The only  difference  is you will have to decide if your product is negative or  positive .
  ::乘数整数与整数的乘数非常相似。 唯一的区别是您必须决定您的产品是负还是正。

  播放段落

  To multiply two integers:
  ::乘以两个整数:

  播放段落
  1. First, multiply the absolute value of the factors.
     ::首先,乘以因素的绝对值。
  播放段落
  2. Next, determine the sign of your product according to the following rules:
     播放段落
     • A positive number times a positive number equals a positive number.
       ::正数乘以正数等于正数。
     播放段落
     • A positive number times a negative number equals a negative number.
       ::负数乘以负数等于负数。
     播放段落
     • A negative number times a negative number equals a positive number.
       ::负数乘以负数等于正数。
     播放段落
     • A negative number times a positive number equals a negative number.
       ::负数乘以正数等于负数。
     播放段落
     • Any integer times  zero  equals zero.
       ::任何整数乘以零等于零。
     
     ::接下来,根据以下规则确定您产品的标识:正乘以正数等于正数。正乘以负数等于负数。正乘以负数等于负数。负数乘以负数等于正数。负数乘以正数等于负数。负数乘以正数等于负数。任何整数乘以零等于零。
  播放段落
  3. Your product will be your result from step 1 with the sign from step 2.
     ::您的产品将来自第1步的结果 与第2步的标志。

  播放段落

  It might help to remember that if your original two integers have the same sign, then their product will be positive. If your original two integers have different signs, then their product will be negative.
  ::这或许有助于记住,如果原来的两个整数有相同的符号,那么它们的产品将是正数。如果原来的两个整数有不同的符号,那么它们的产品将是负数。

  播放段落

  Let's look at an example.
  ::让我们举个例子。

  播放段落

  Multiply  $\begin{array}{r}(-4)(-3)\end{array}$ .
  ::乘(-4) (-3) 。

  播放段落

  The first step is to multiply the absolute value of the factors. This means you are basically ignoring the negative signs for now.
  ::第一步是乘以因素的绝对值。 这意味着您现在基本上忽略了消极的迹象。

  $\begin{array}{r}(4)(3)=12\end{array}$

  播放段落

  The next step is to decide on the sign of your answer. Both factors were negative and a negative times a negative equals a positive. This means your product is positive.
  ::下一步是决定您回答的信号。 两种因素都是负的, 负的则等于正的。 这意味着您的产品是正的 。

  $\begin{array}{r}(-4)(-3)=12\end{array}$

  播放段落

  The answer is 12.
  ::答案是12岁

  播放段落

  Here's another example.
  ::下面是另一个例子。

  播放段落

  Multiply  $\begin{array}{r}-5\cdot 8\end{array}$ .
  ::乘以 - 5 8 。

  播放段落

  Again, the first step is to multiply the absolute value of the factors. Don't worry about the negative sign for now.
  ::第一步是将因素的绝对值乘以倍增。不要担心现在的负信号。

  $\begin{array}{r}5\cdot 8=40\end{array}$

  播放段落

  The next step is to decide on the sign of your answer. The two factors had different signs. A negative times a positive is a negative. This means your product is negative.
  ::下一步是决定您答案的符号。 这两个因素有不同的征兆。 负数正数是负数。 这意味着您的产品是负数 。

  $\begin{array}{r}-5\cdot 8=-40\end{array}$

  播放段落

  The answer is -40.
  ::答案是 -40

  播放段落

  You can use the same process to multiply more than two integers together. Just work from left to right and multiply two factors at a time.
  ::您可以使用相同的进程将两个整数相乘。只需从左到右工作,然后一次乘以两个系数。

  播放段落

  Here is an example.
  ::举一个例子。

  播放段落

  Multiply  $\begin{array}{r}(-8)(-3)(-2)\end{array}$ .
  ::乘(-8) (-3) (-2) 。

  播放段落

  First you will multiply  $\begin{array}{r}(-8)(-3)\end{array}$ . You know that  $\begin{array}{r}(8)(3)=24\end{array}$ . You also know that a negative times a negative equals a positive. So you have
  ::首先,你将乘以(-8) (- 3) 。你知道(8) (3) = 24。你也知道,负负数等于正数。因此,你已经知道,负数乘以负数等于正数。

  $\begin{array}{r}(-8)(-3)=24\end{array}$

  播放段落

  Next, take your product of 24 and multiply by the third factor, -2.
  ::接下来,取24的产物 乘以第三个因数 -2

  $\begin{array}{r}(24)(-2)\end{array}$

  播放段落

  You know that  $\begin{array}{r}(24)(2)=48\end{array}$ . You also know that a positive times a negative equals a negative.
  ::您知道( 24) (2) = 48。 您也知道正乘以负等于负。

  $\begin{array}{r}(24)(-2)=-48\end{array}$

  播放段落

  The final answer is  $\begin{array}{r}(-8)(-3)(-2)=-48\end{array}$ .
  ::最后一个答案是(-8) (-3) (-2) = 48。

  播放段落

  Examples
  ::实例

  播放段落

  Example 1
  ::例1

  播放段落

  Earlier, you were given a problem about Marybeth and the bathtub.
  ::之前,你得到一个问题 关于玛丽贝丝和浴缸。

  播放段落

  She noticed that the height of the water in her baby brother's bathtub was decreasing by 2 inches every minute. She wondered how she could represent the change in the height of the water after 3 minutes.
  ::她注意到她弟弟浴缸的水位每分钟减少2英寸,她想知道她如何能代表3分钟后水位变化。

  播放段落

  First, represent the decrease in the height as an integer.
  ::首先,以整数表示高度的下降。

  播放段落

  The height of the water is decreasing by 2 inches per minute. Use a negative integer, -2, to show the decrease.
  ::水的高度每分钟减少2英寸。 使用负整数 - 2 来显示水的下降 。

  播放段落

  To represent the decrease in the height of the water after 3 minutes, multiply the number of minutes by the integer representing the decrease.
  ::要表示3分钟后水高度的下降,请乘以分钟数乘以表示减少的整数。

  $\begin{array}{r}3\times (-2)=?\end{array}$

  播放段落

  Find the product to solve the problem.
  ::找到产品解决问题。

  播放段落

  Multiply the absolute value of the factors.
  ::乘以系数的绝对值。

  $\begin{array}{r}3\times 2=6\end{array}$ .

  播放段落

  Next, decide on the sign for your product. A positive times a negative equals a negative, so your product is a negative.
  ::下一步, 请决定您的产品标记。 负负值等于负值, 所以您的产品是负值 。

  $\begin{array}{r}3\times -2=-6\end{array}$

  播放段落

  The answer is -6.
  ::答案是 -6

  播放段落

  The change in the height of the water after 3 minutes can be represented by -6 inches.
  ::3分钟后水高度的变化可以用 -6英寸表示。

  播放段落

  Example 2
  ::例2

  播放段落

  The number of students voting in a school election has been decreasing at a  rate  of 15 students per year. Represent the change in the number of students voting over the last 3 years as an integer.
  ::参加学校选举投票的学生人数以每年15名学生的速度下降,代表了过去三年来参加投票的学生人数的整数变化。

  播放段落

  First, represent the decrease in the number of students voting as an integer.
  ::首先,是整数投票的学生人数的减少。

  播放段落

  The problem states that the number of students voting has been decreasing by 15 students each year. To show a decrease, use a negative integer -15.
  ::这个问题表明,每年参加投票的学生人数减少了15人,如果减少,则使用负整数-15。

  播放段落

  To represent the decrease in the number of students voting over the last 3 years, multiply the number of years by the integer representing the decrease.
  ::为了代表过去三年来参加投票的学生人数的减少,将年数乘以整数表示的减少。

  $\begin{array}{r}3\times (-15)=?\end{array}$

  播放段落

  Find the product to solve the problem.
  ::找到产品解决问题。

  播放段落

  Multiply the absolute value of the factors.
  ::乘以系数的绝对值。

  $\begin{array}{r}3\times 15=45\end{array}$

  播放段落

  Next, decide on the sign for your product. A positive times a negative equals a negative, so your product is a negative.
  ::下一步, 请决定您的产品标记。 负负值等于负值, 所以您的产品是负值 。

  $\begin{array}{r}3\times -15=-45\end{array}$

  播放段落

  The answer is -45.
  ::答案是 -45

  播放段落

  The change in the number of students voting over the last 3 years can be represented as -45.
  ::过去三年来参加投票的学生人数的变化可代表为-45人。

  播放段落

  Example 3
  ::例3

  播放段落

  Multiply  $\begin{array}{r}-9(-3)\end{array}$ .
  ::乘- 9 (- 3) 。

  播放段落

  First, multiply the absolute value of the factors.
  ::首先,乘以因素的绝对值。

  $\begin{array}{r}9(3)=27\end{array}$

  播放段落

  Next, decide on the sign for your product. A negative times a negative equals a positive, so your product is a positive. 
  ::下一步, 请决定您的产品标记。 负负值等于正值, 所以您的产品是正值 。

  $\begin{array}{r}-9(-3)=27\end{array}$

  播放段落

  The answer is 27.
  ::答案是27个

  播放段落

  Example 4
  ::例4

  播放段落

  Multiply  $\begin{array}{r}(-3)(12)\end{array}$ .
  ::乘(-3) (12) 。

  播放段落

  First, multiply the absolute value of the factors.
  ::首先,乘以因素的绝对值。

  $\begin{array}{r}(3)(12)=36\end{array}$ .

  播放段落

  Next, decide on the sign for your product. A negative times a positive equals a negative, so your product is a negative.
  ::下一步, 决定您产品的标志 。 负值为正值等于负值, 所以您的产品是负值 。

  $\begin{array}{r}(-3)(12)=-36\end{array}$

  播放段落

  The answer is -36.
  ::答案是 -36

  播放段落

  Example 5
  ::例5

  播放段落

  Multiply  $\begin{array}{r}(-4)(3)(-2)\end{array}$
  ::乘数 (-4) (3) (-2)

  播放段落

  First you will multiply  $\begin{array}{r}(-4)(3)\end{array}$ . You know that  $\begin{array}{r}(4)(3)=12\end{array}$ . You also know that a negative times a positive equals a negative. So you have
  ::首先,您会乘以( 4) (3) 乘以( 4) (3) 。 您知道( 4) (3) = 12 。 您也知道, 负乘以正值等于负值。 因此, 您知道, 负值乘以正值等于负值 。

  $\begin{array}{r}(-4)(3)=-12\end{array}$

  播放段落

  Next, take your product of -12 and multiply by the third factor, -2.
  ::接下来,拿你的 -12的产物乘以第三个因数 -2

  $\begin{array}{r}(-12)(-2)\end{array}$

  播放段落

  You know that  $\begin{array}{r}(12)(2)=24\end{array}$ . You also know that a negative times a negative equals a positive.
  ::您知道( 12) (2) = 24。 您也知道负乘负等于正。

  $\begin{array}{r}(-12)(-2)=24\end{array}$

  播放段落

  The final answer is  $\begin{array}{r}(-4)(3)(-2)=24\end{array}$ .
  ::最后一个答案是(-4) (3) (-2) = 24。

  播放段落

  Review
  ::回顾

  播放段落

  Use integer rules to find each product.
  ::使用整数规则查找每种产品。

  1. $\begin{array}{r}(-3)(9)\end{array}$  
  2. $\begin{array}{r}8(-9)\end{array}$
  3. $\begin{array}{r}12\times 8\end{array}$
  4. $\begin{array}{r}(-4)(7)\end{array}$
  5. $\begin{array}{r}(-2)\cdot (-11)\end{array}$
  6. $\begin{array}{r}3\cdot (-25)\end{array}$
  7. $\begin{array}{r}5\times (-6)\times (-1)\end{array}$
  8. $\begin{array}{r}(-8)(-7)\end{array}$
  9. $\begin{array}{r}(-2)(3)(-4)\end{array}$
  10. $\begin{array}{r}(9)(1)(-1)\end{array}$
  11. $\begin{array}{r}(-9)(2)(-1)\end{array}$

  播放段落

  Review (Answers)
  ::回顾(答复)

  播放段落

  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 。

  播放段落

  Resources
  ::资源