课程： 11.12整数乘法 | The Way To Learn

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整数乘法
Sarah, Benjie, and Sipple are biking up to the ski basin. They want to carry as little stuff as they can on their bikes because the weight makes it harder to bike up the mountain. They are trying to decide if they need to bring jackets, which each weigh a few pounds. It is $\begin{aligned} 70^{\circ} \end{aligned}$ in town. They decide they don't need jackets unless it drops below 65°. In town, it is about 7,000 ft. It is 12,000 ft at the ski basin. They know the temperature drops about 2° for every thousand feet they climb. Will they need their jackets?
::Sarah, Benjie, Sipple 正在骑自行车到滑雪盆地。他们想在自行车上尽可能地携带小东西, 因为重量使骑车上山更加困难。他们试图决定是否要带夹克, 每件重量只有几磅。在镇上是70磅。他们决定他们不需要夹克, 除非它降到65度以下。在镇上, 它大约是7000英尺。在滑雪盆地是12,000英尺。他们知道每爬上一千英尺,温度会下降2°左右。他们需要夹克吗?

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。

Multiplying Integers
::乘数整数

An integer is the set of whole numbers and their opposites.
::整数是整数及其反面的一组。

A product is the answer of a multiplication problem. Multiplication is repeated addition. When a value is being multiplied the multiplication symbol is a shorthand for this repeated addition.
::产品是乘法问题的答案。乘法是重复增加。当值乘法是乘法时,乘法符号是重复增加的简称。

Commutative means a group of quantities connected by a mathematical symbol that gives the same result whatever the order of the quantities involved. Multiplication is commutative.
::通量是指用数学符号连接的一组数量,无论所涉数量的顺序如何,结果都是一样的。乘法是通量的。

Here is an example of a multiplication problem.
::以下是一个乘法问题的例子。

3 x 4 = ____
::3 x 4 = __

First, write out the multiplication as four 3s added together.
::首先,写出乘法,四三加在一起。

3 + 3 + 3 + 3 = 12

Next, because multiplication is commutative, write out the multiplication as three 4s added together.
::其次,由于乘法具有通融性,因此写出乘法,将三四加在一起。

4 + 4 + 4 = 12

Then, verify that the two answers are the same.
::那么,核实这两个答案是相同的。

Here is another example.
::下面是另一个例子。

6 x (-5) = ____
::6 x (5) = __

First, write out the multiplication as six -5s added together.
::首先,写出乘法,将乘法加起来6 -5。

-5 + -5 + -5 + -5 + -5 + -5 = -30

The answer is -30.
::答案是 -30

If you know the multiplication table, you can simplify the process without converting to addition every time. Then, you just need to worry about the signs. Here are the sign rules for multiplication:
::如果您知道乘法表, 您可以简化程序, 而不是每次转换为加法。然后, 您只需要担心符号。以下是乘法的符号规则 :

Positive $\begin{aligned} \times \end{aligned}$ positive = positive
::正x正=正

Negative $\begin{aligned} \times \end{aligned}$ positive = negative(-->remember, because multiplication is commutative, this is the same as Positive X Negative)
::负x正=负( - - - > remember, 因为乘法是平流的, 这与正 X负相同)

Negative $\begin{aligned} \times \end{aligned}$ negative = positive
::负x负=正

Another way to think of it is: if there are an odd number of negatives in a multiplication problem, the answer is negative. If there are an even number, the answer is positive.
::另一种思考方式是:如果一个乘法问题有奇数的负数,答案是负数。如果数字是偶数,答案是正数。

Here is another example.
::下面是另一个例子。

(-5)(-2) = ____

First, multiply the integers ignoring their signs.
::首先,乘以整数,忽略其符号。

5 x 2 = 10
::5 x 2 = 10

Next, use the sign rules for multiplication to determine the sign of the answer.
::其次,使用乘法的标记规则来确定答案的标记。

In this case, there was a negative times a negative. That equals a positive.
::在这种情况下,负数为负数,等于正数。

The answer is 10.
::答案是10个

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Sarah, Sipple, and Benjie.
::早些时候,你遇到了一个问题关于莎拉, 锡普尔, 和本杰。

Their ride starts in town at 7,000 ft, where it is 70°, and ends at 12,000 ft. The temperature drops 2° for every thousand feet they climb. They don't need their jackets unless it drops below 65°. Do they need to bring them?
::他们骑马从7000英尺的小镇开始, 70英尺, 12000英尺结束。温度每爬上一千英尺就下降2°。他们不需要外套, 除非衣服降到65英尺以下。他们需要带吗 ?

First, calculate how many feet they climb in total.
::首先,计算他们总共爬了多少英尺。

12,000 - 7,000 = 5,000 ft. That is 5 thousand ft., or 5 x 1,000.
::12 000 - 7 000 = 5 000英尺,即5 000英尺,或5 x 1 000英尺。

Next, they know that the temperature drops 2° for every thousand feet. So they can write a multiplication for the final temperature. Note that a "drop" in temperature is a decrease, or a negative.
::其次,他们知道温度每千英尺下降2°。因此他们可以写出最后温度的乘数。请注意, 温度中的“ 滴数” 是下降, 或者负值。

5 x (-2) =
::5x(-2)=

Then, they multiply the integers.
::然后,它们会乘以整数。

5 x 2 = 10
::5 x 2 = 10

Then, they check the sign. A positive times a negative is a negative.
::然后,他们检查了信号,正数负数是负数。

The temperature up at the ski basin is -10° from the temperature in town. That means, it is 10° colder.
::滑雪盆地的温度比镇上的温度高10度, 也就是说更冷10度。

Finally, they take this temperature from the initial temperature.
::最后,他们从最初的温度开始就开始摄取这种温度。

70° - 10° = 60°

It will be $\begin{aligned} 60^{\circ} \end{aligned}$ up at the ski basin.
::在滑雪盆地会是60块

They decided they need to bring their jackets if it is under 65°. Since 60° is colder than 65°, they must bring their jackets.
::他们决定,如果夹克在65度以下,他们必须带上夹克。由于60度较冷,超过65度,他们必须带上夹克。

Example 2
::例2

Find the product.
::找到产品。

(9)(-6)(5) = _____

First, multiply the first two integers ignoring their sign (because multiplication is commutative you can do the multiplications in any order).
::首先,乘以前两个整数,忽略其符号(因为乘法是混合的, 您可以按任何顺序进行乘法)。

9 x 6 = 54
::9x6=54

Next, use the rule of signs to determine the sign.
::接下来,使用标志规则来确定标志。

In this case, it was a positive times a negative. That equals a negative.
::在这种情况下,正值是负值,等于负值。

So, the first product is -54.
::所以,第一种产品是 -54。

Then, multiply that answer by the final integer, ignoring the signs.
::然后,将回答乘以最后整数,忽略符号。

54 x 5 = 270
::54 x 5 = 270

Then, use the rule of signs to figure out the final sign.
::然后,使用标志规则找出最后的标志。

This was a positive times a negative, too. So, the final answer is negative.
::这是正时负时,也是负时。所以,最后的答案是负时。

The answer is -270.
::答案是 -270

Example 3
::例3

Find the product.
::找到产品。

-9(-8) = ____

First, multiply the integers together ignoring their sign.
::首先,将整数乘在一起,忽略其符号。

9 x 8 = 72
::9 x 8 = 72

Next, check the number of negative signs.
::接下来,检查负信号的数目。

In this problem, there are two, which is even. When there are an even number of negative signs, the overall answer is positive.
::在这个问题上,有两个是平衡的,当有甚至几个消极迹象时,总体答案是肯定的。

The answer is +72.
::答案是+72。

Example 4
::例4

Find the product.
::找到产品。

(4)(-12) = ____

First, multiply the integers together ignoring their sign.
::首先,将整数乘在一起,忽略其符号。

4 x 12 = 48
::4 x 12 = 48

Next, check the number of negative signs.
::接下来,检查负信号的数目。

In this problem, there is one, which is odd. When there is an odd number of negative signs, the overall answer is negative.
::在这个问题上,有一个问题,这是奇怪的。当有奇数的负面迹象时,总体答案是否定的。

The answer is -48.
::答案是 -48

Example 5
::例5

Find the product.
::找到产品。

(5)(13) = ____

First, multiply the integers together ignoring their sign.
::首先,将整数乘在一起,忽略其符号。

5 x 13 = 65
::5 x 13 = 65

Next, check the number of negative signs.
::接下来,检查负信号的数目。

In this problem, there are no negative signs, which means the answer is automatically positive.
::在这个问题上,没有消极的迹象,这意味着答案自动是肯定的。

The answer is +65.

::答案是+65。

Review
::回顾

Multiply each pair of integers to find a product.
::乘以每对整数寻找产品。

(-7)(-8)

-3(4)

5(8)

(-3)(-9)

6(12)

-9(-9)

8(-4)

-7(-2)

-7(-3)

15(-2)

-15(2)

-2(-15)

12(-5)

(-11)(-7)

(-4)(-5)

(-8)(-11)

(2)(-3)

-5(7)

-13(-2)

14(2)

Evaluate each numerical expression.
::评估每个数字表达式。

(-9)(2)(-1)

(-3)(2)(-4)

(-5)(9)(-1)

(8)(-9)(-2)

(2)(-3)(-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。

Resources
::资源

章节大纲

常规

整数乘法

Multiplying Integers ::乘数整数

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

Multiplying Integers
::乘数整数

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