节: 评估产品的心算 | 4.6评估产品的心算

章节大纲

Mr. Thompson is preparing supplies for an art lesson. Each student will be given a bag with 15 color pencils and markers. There are total of 25 students in the class. Mr. Thompson doesn’t have a calculator or a pencil with him. How can Mr. Thompson find the total number of drawing utensils he will need for the class?
::Thompson先生正在准备艺术课的用品。每个学生都会得到一个15个彩色铅笔和标记的袋子。班里共有25名学生。 Thompson先生没有计算器或铅笔。 Thompson先生如何找到班里需要的绘画用具的总数?

In this concept, you will use mental math to evaluate products using the distributive property .
::在此概念中,您将使用精神数学来评估使用分配属性的产品。

Mental Math
::心理数学

Some multiplication problems can be solved by doing the calculations in your head. That is called mental math. Some multiplication problems may seem too large to do in your head. But remember, a large number can be broken down into the sum of two smaller numbers.
::一些乘法问题可以通过在你的脑中进行计算来解决。这叫做精神数学。有些乘法问题在你的脑中似乎太大,无法解决。但记住,很多乘法问题可以分为两个小数的总和。

$\begin{aligned} 109 = 100 + 9 \end{aligned}$

The distributive property can help you evaluate the product of larger number. Remember that the distributive property is a property that allows you to multiply a number and a sum by distributing the multiplier outside the parentheses with each numbers inside the parentheses.
::分配性财产可以帮助您评估较大数字的产物。记住分配性财产是一种允许您通过在括号内分配括号外的乘数和乘数来乘以数字和数字的财产。

$\begin{aligned} a (b + c) = a b + a c \end{aligned}$

Here is a multiplication problem.
::这是一个乘法问题。

$\begin{aligned} 11 \times 109 = \underline{} \end{aligned}$

Think of 109 as the sum of 100 plus 9.
::把109当成100加9的总和

$\begin{aligned} 11 \times 109 = 11 (100 + 9) \end{aligned}$

Instead of multiplying 11 times 109, use the distributive property to distribute the multiplication to equal the sum of 11 times 100 and 11 times 9.
::使用分配财产分配乘以11乘以109,而不是乘以11乘以109,乘以11乘以100,乘以11乘以9。

$\begin{aligned} 11 (100 + 9) = 11 (100) + 11 (9) \end{aligned}$

Try to find the products using mental math. Multiply 11 times 100 in your head. Remember than when you multiply a number by a multiple of 10, ignore the zeros and place it into the product at the end.
::尝试用心理数学找到产品。将头部乘以 11 乘以 100 。记住, 数字乘以 10 乘以 10 , 忽略零, 将其放入产品末端。

$\begin{aligned} 11 \times 100 \to 11 \times 100 = 1100 \end{aligned}$

Think 11 times 9 is 99. Now find the sum of the products.
::想想11乘以9是99 现在找到产品的总和

$\begin{aligned} \begin{array}{rcl} 11 (100) + 11 (9) \\ 1, 100 + 99 = 1, 199 \end{array} \end{aligned}$

You have found the product of 11 times 109.
::你找到了11乘以109的产物

$\begin{aligned} 11 \times 109 = 1, 199 \end{aligned}$

Now, try it yourself on a few practice problems! Work the math in your head, then type in the answer.
::现在,你自己去尝试一些练习上的问题! 在你的脑子里算算,然后输入答案。

Here is another multiplication problem.
::还有一个乘法问题。

$\begin{aligned} 12 \times 53 = \underline{} \end{aligned}$

This problem would be difficult to do in your head. However, think of 53 as the sum of 50 plus 3.
::这个问题在你看来很难解决,不过,把53加50加3的总数算作50加3。

$\begin{aligned} 12 \times 53 = 12 (50 + 3) \end{aligned}$

Use the distributive property to distribute the multiplication of 12 between 50 and 3.
::利用分配财产将12乘数在50和3之间分配。

$\begin{aligned} 12 (50 + 3) = 12 (50) + 12 (3) \end{aligned}$

Multiply 12 times 50 in your head. 12 times 5 is 60, so 12 times 50 is 600. Multiply 12 times 3 in your head. 12 times 3 is 36. Now, add the sum of the products.
::12乘以12乘以50 头部。 12乘以5等于60, 12乘以50等于60。 12乘以50等于600。 12乘以12乘以3乘以3。 12乘以3乘以36。现在, 加上产品的总和。

$\begin{aligned} \begin{array}{rcl} 12 (50) + 12 (3) \\ 600 + 36 = 636 \end{array} \end{aligned}$

The product of 12 times 53 is 636.
::12乘以53等于636的产物是636。

$\begin{aligned} 12 \times 53 = 636 \end{aligned}$

Example 1
::例1

Earlier, you were given a problem about Mr. Thompson’s art lesson.
::先前,Thompson先生的艺术课给您带来了一个问题。

He needs 15 color pencils and markers for 25 students. Multiply the numbers using mental math to find out how many drawing utensils Mr. Thompson needs.
::他需要25个学生的15个彩色铅笔和标记。乘以使用心理数学的数字来找出Thompson先生需要多少绘画用具。

First, change one of the numbers to the sum of two smaller numbers. 15 is 10 plus 5. Note that the larger number does not always have to be the number broken down into two smaller numbers. Finding multiplies of 25 is similar to finding multiples of a quarter.
::首先,将数字之一修改为两个小数字的总和。 15是 10 + 5 。请注意, 较大数字不一定总是以数字细分为两个小数字。查找25 的倍数与查找四分之一的倍数相似。

Then, use the distributive property to find the product of a number and a sum.
::然后,使用分配财产找到一个数字和一个数额的产物。

$\begin{aligned} 25 (10 + 5) = 25 (10) + 25 (5) \end{aligned}$

Next, calculate the products using mental math and add the sum. 25 times 10 is 250. 25 times 5 is 125. (5 quarters is equal to $1.25 or 125 cents.
::接下来,使用心理数学计算产品,并加上总和。 25乘以10等于250。 25乘以5等于125( 5个季度等于1. 25或125美分)。

$\begin{aligned} \begin{array}{rcl} 25 (10) + 25 (5) \\ 250 + 125 = 375 \end{array} \end{aligned}$

Mr. Thompson will need a total of 375 drawing utensils.
::Thompson先生总共需要375个绘图用具。

Example 2
::例2

Use mental math to solve this problem.
::使用心理数学来解决这个问题。

$\begin{aligned} 9 \times 81 = \underline{} \end{aligned}$

First, change one of the numbers to the sum of two smaller numbers. 81 is $\begin{aligned} 80 + 1 \end{aligned}$ .
::首先,将数字之一改为两个小数字之和。 81是80+1。

Then, use the distributive property to find the product of a number and a sum.
::然后,使用分配财产找到一个数字和一个数额的产物。

$\begin{aligned} 9 (80 + 1) = 9 (80) + 9 (1) \end{aligned}$

Next, calculate the products using mental math. Multiply 9 times 80. Think 9 times 80 is 9 times 8, then add a zero , 720. 9 times 1 is 9.
::接下来,使用心理数学来计算产品。乘以9乘以80。乘以9乘以80, 9乘以80乘以9乘以8, 然后加上0, 720。 9乘以1乘以9等于9。

$\begin{aligned} 9 (80) + 9 (1) = 720 + 9 \end{aligned}$

Finally, add the sum of the products.
::最后,加上产品的总和。

$\begin{aligned} 720 + 9 = 729 \end{aligned}$

The product of 9 times 81 is 729.
::9乘81等于729的产物为729。

Example 3
::例3

Use mental math to evaluate the product.
::使用心理数学来评估产品。

$\begin{aligned} 5 (99) \end{aligned}$

First, change one of the numbers to the sum of two smaller numbers. 99 is 90 plus 9.
::首先,将一个数字改为两个小数字的总和。 99是90加9。

Then, use the distributive property to find the product of a number and a sum.
::然后,使用分配财产找到一个数字和一个数额的产物。

$\begin{aligned} 5 (90 + 9) = 5 (90) + 5 (9) \end{aligned}$

Next, calculate the products using mental math and add the sum. 5 times 90 is 450. 5 times 9 is 45.
::接下来,用心理数学计算出产品,再加一个总和。 5乘以90等于450。 5乘以9等于45。

$\begin{aligned} \begin{array}{rcl} 5 (90) + 5 (9) \\ 450 + 45 = 495 \end{array} \end{aligned}$

The product of 5 times 99 is 495.
::5乘以99等于495的产物是495。

Example 4
::例4

Use mental math to evaluate the product.
::使用心理数学来评估产品。

$\begin{aligned} 4 (65) \end{aligned}$

First, change one of the numbers to the sum of two smaller numbers. 65 is 60 plus 5.
::首先,将一个数字改为两个小数字的总和。 65是60+5。

Then, use the distributive property to find the product of a number and a sum.
::然后,使用分配财产找到一个数字和一个数额的产物。

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

Next, calculate the products using mental math and add the sum. 4 times 60 is 240. 4 times 5 is 20.
::接下来,用心理数学计算出产品,然后加上总和。 4乘60等于240。 4乘5等于20。

$\begin{aligned} \begin{array}{rcl} 4 (60) + 4 (5) \\ 240 + 20 = 260 \end{array} \end{aligned}$

The product of 4 times 65 is 260.
::4乘以65的产值是260。

Example 5
::例5

Use mental math to evaluate the product.
::使用心理数学来评估产品。

$\begin{aligned} 3 (140) \end{aligned}$

First, change one of the numbers to the sum of two smaller numbers. 120 is 100 plus 40.
::首先,将一个数字改为两个小数字的总和。 120是100加40。

Then, use the distributive property to find the product of a number and a sum.
::然后,使用分配财产找到一个数字和一个数额的产物。

$\begin{aligned} 3 (100 + 40) = 3 (100) + 3 (40) \end{aligned}$

Next, calculate the products using mental math and add the sum. 3 times 100 is 300. 3 times 40 is 120.
::接下来,用心理数学计算出产品,再加一个总和。 3乘以100等于300。 3乘以40等于120。

$\begin{aligned} \begin{array}{rcl} 3 (100) + 3 (20) \\ 300 + 120 = 420 \end{array} \end{aligned}$

The product of 3 times 140 is 420.
::3乘140的产物是420。

Review
::回顾

Use mental math to evaluate the following expressions.
::使用心理数学来评价以下表达式。

$\begin{aligned} 4 (68) \end{aligned}$

$\begin{aligned} 8 (45) \end{aligned}$

$\begin{aligned} 3 (61) \end{aligned}$

$\begin{aligned} 2 (53) \end{aligned}$

$\begin{aligned} 9 (22) \end{aligned}$

$\begin{aligned} 6 (44) \end{aligned}$

$\begin{aligned} 5 (120) \end{aligned}$

$\begin{aligned} 12 (12) \end{aligned}$

$\begin{aligned} 11 (18) \end{aligned}$

$\begin{aligned} 13 (12) \end{aligned}$

$\begin{aligned} 15 (22) \end{aligned}$

$\begin{aligned} 20 (106) \end{aligned}$

$\begin{aligned} 12 (310) \end{aligned}$

$\begin{aligned} 25 (16) \end{aligned}$

$\begin{aligned} 50 (720) \end{aligned}$

$\begin{aligned} 100 (210) \end{aligned}$

$\begin{aligned} 110 (180) \end{aligned}$

$\begin{aligned} 1, 200 (14) \end{aligned}$

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

章节大纲

Mental Math ::心理数学

Now, try it yourself on a few practice problems! Work the math in your head, then type in the answer. ::现在,你自己去尝试一些练习上的问题! 在你的脑子里算算,然后输入答案。

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

Mental Math
::心理数学

Now, try it yourself on a few practice problems! Work the math in your head, then type in the answer.
::现在,你自己去尝试一些练习上的问题! 在你的脑子里算算,然后输入答案。

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