6.4 行动顺序-interactive

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  行动命令令

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  Can you make an ugly, light-up Christmas sweater?
  ::你能做一件难看的圣诞毛衣吗?

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  Small strings of LED lights often come with 10 lights per string. Use the interactive below to figure out how many strings of LED lights you would need to decorate the Christmas tree on an ugly Christmas sweater.
  ::LED灯的小字符串通常每字符串有10个光线。 使用下面的互动来决定需要多少个LED灯的字符串才能在一件丑陋的圣诞毛衣上装饰圣诞树。

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  T he pattern in the total number of lights on the tree as you complete each row is one plus the row number squared. To find the total number of lights do you add 1 to the row number and then take the square? Or do you square the row number and then add 1?
  ::您完成每行时树上灯光总数中的图案是 1+行号方形。 要找到灯光总数, 您会在行号中添加1, 然后选择方形 。 还是将行号平方, 然后添加1 ?

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  How many squares and unit cubes are there in an exploded cube?
  ::在爆炸的立方体中有多少方形和单位立方体?

  

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  In the interactive below, you can find the   volume of a cube by breaking one big cube into smaller unit cubes and counting them out. You can find the of a cube  in the interactive by counting the numbers of little painted squares on the outside of the cube.  
  ::在下面的互动中,您可以通过将一个大立方体破碎成一个小的立方体并将其数出来来找到一个立方体的体积。您可以通过计算立方体外的画面小方块数来找到互动中的立方体的体积。

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  Discussion Questions:
  ::讨论问题:

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  1. What is the relationship between the side length of the cube and the number of cubes?
     ::立方体的侧长与立方体数量之间的关系是什么?
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  2. How would you find the number of cubes with a side length of 10? With a side length of $\begin{align*}\frac12?\end{align*}$
     ::副长度为 10 的立方体数量? 副长度为 12 的立方体数量如何 ?
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  3. What is the relationship between the side length of the cube and the number of painted squares on one side of the cube?
     ::立方体的侧长与立方体一侧油漆的方形数之间的关系是什么?
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  4. How would you use the number of painted squares on one side of the cube to find the total number of painted squares on the cube?
     ::您如何使用立方体一侧的油漆方块数来找到立方体上的油漆方块总数 ?
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  5. How many total painted squares would there be on a cube with a side length of 10? How many would there be on a cube with a side length of $\begin{align*}\frac12?\end{align*}$
     ::边长为 10 的立方体上将有多少漆成的广场?边长为 12 的立方体上将有多少画成的广场?

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  T he total number of painted squares on the surface of a cube is called the surface area. You found the surface area by first squaring and then multiplying by 6. Would you get the same answers if you multiplied first and then squared? Complete the table below to try it out.
  ::立方体表面的油漆方块总数被称为表面区域。您通过先对齐然后乘以6 发现了表面区域。 如果您先乘以乘以乘以6, 你会得到相同的答案吗 ? 填写下面的表格来尝试它 。

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  The first 2 rows have been done for you.
  ::前两排已经为你做了。

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  N otice that the answers in the table do not match up when the is switched. T he order that the operations are performed does matter when getting to the final answer. When a side length is squared before finding the product,  you get the same number of squares that  you can count in the interactive. When multiplication happens before squaring, the number does not match the number of painted squares that  you can observe in the interactive.
  ::请注意, 表格中的答案在被调换时不匹配。 执行操作的顺序在获得最后答案时很重要。 当找到产品前的侧边长度是正方形时, 您得到的方形数与在互动中可以计算的方形数相同。 当乘法在交配前发生时, 数字与您在互动中可以观察到的油漆方形数不符 。

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  Which is the most powerful operation?
  ::最强大的行动是什么?

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  Explore the interactive below to figure out which of the operations (+, -, x, ÷, or exponent) increases or decreases the number most quickly.
  ::在下面探索互动关系,以找出哪些操作(+、 -、 x、 _ 或 exponent)增加或减少速度最快。

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  Some operations are equally powerful but perform inverse calculations. Inverse operations undo each other.
  ::有些操作同样强大, 但进行反向计算。 反向操作互相抵消 。

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  What is the inverse of each operation ?
  ::每项行动的反面是什么?

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  • 5 + 2 = 7
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    • The inverse of addition is subtraction .
      ::增加的逆数是减法。
    • 7 - 2 = 5
    
    ::5 + 2 = 7 + 5 + 2 = 7 增加数的反数是减去。 7-2 = 5
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  • 8 ÷ 4 = 2
    
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    • The inverse of division is multiplication.
      ::分裂的逆向是乘法。
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    • 4 x 2 = 8
      ::4 x 2 = 8
    
    ::8 4 = 2 分数的逆向是乘数 。 4 x 2 = 8
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  • 7 - 3 = 4
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    • The inverse of subtraction is addition.
      ::加上减法的逆数。
    • 3 + 4 = 7
    
    ::7 - 3 = 4 减去的反数是增加值 3+ 4 = 7
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  • 6 x 5 = 30
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    • The inverse of multiplication is division.
      ::乘法的逆数是除法。
    • 30 ÷ 5 = 6
    
    ::6 x 5 = 30 乘法的逆数是除法 30 = 5 = 6

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  Discussion Question
  ::讨论问题

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  What do you suppose is  the inverse of an exponent ?  Are you familiar with  taking the square root of a square or a cube root of a cube?  
  ::你是否熟悉用方块的平方根还是立方块根?

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  The most powerful operations in a mathematical expression get calculated before anything else .  The least powerful operations get performed last.  If you have equally powerful operations (like + & - or x & ÷) solve these from left to right, just like you are reading this sentence from left to right.
  ::数学表达式中最强大的操作会比其他任何操作都要先计算。 最弱的操作会最后完成。 如果您有同样强大的操作( 如 + & - 或 x & & & ) 从左到右解决这些操作, 就像您正在读从左到右的句子一样 。

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  Wh at is the order of operations for each of these expressions?
  ::每个表达式的操作顺序如何?

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  • 6 ²  - 7 x 3
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    • 1st exponent, 2nd multiplication, 3rd subtraction
      ::第一引号、第二乘法、第三减法
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    • 6 ² - 7 x 3 = 15
      ::62 - 7 x 3 = 15
    
    ::62 - 7 x 3 3 1 第1个引号, 第二次乘法, 第三次减法 62 - 7 x 3 = 15
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  • 4 ÷ 2 + 1 x 9
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    • 1st division, 2nd multiplication, 3rd addition
      ::第1师,第2乘数,第3加
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    • 4 ÷ 2 + 1 x 9 = 11
      ::4 2 + 1 x 9 = 11
    
    ::4++2+1x9 1 分区, 2乘数, 3加4 +2+1 x9 = 11
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  • 5 ³  ÷ 5 ² -  3
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    • 1st , 2nd division, 3rd subtraction
      ::第1次,第2次,第3次减
    • 5 ³  ÷ 5 ² - 3 = 2
    
    ::53 52 - 3 1 , 2 项, 3 减 53 52 - 3 = 2
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  • 7 + 6 - 5 x 4 ÷ 3 ²
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    • 1st exponent, 2nd multiplication, 3rd division, 4th addition, 5th subtraction
      ::第一引号、第二乘数、第三师、第4加、第5减
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    • 7 + 6 - 5 x 4 ÷ 3 ² =  $\begin{align*}10\frac{7}{9}\end{align*}$  
      ::7 + 6 - 5 x 4 32 = 1079
    
    ::7 + 6 - 5 x 4 + 32 1 演示文, 第二次乘数, 3 分, 4 添加, 5 减 7 + 6 - 5 x 4 + 32 = 1079

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  Extension
  ::延长

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  Come up with a different order of operations. Attempt to solve problems using this new order and observe the results. Share what works and what does not work so well with the system that you developed. When you change the order of operations, you may find that your answers get very large. Perhaps this system is beneficial for people in careers that often use large numbers (such as astronomers who calculate numbers of stars and the distances between them). What is beneficial about the standard order of operations for the average 6th grader? How can it be improved?
  ::产生不同的操作顺序 。 试图用这个新顺序来解决问题并观察结果 。 分享什么有效, 哪些不有效 与您开发的系统一样。 当你改变操作的顺序时, 你可能会发现答案非常大 。 也许这个系统对职业中经常使用大数量的人( 比如计算恒星数和星际距离的天文学家)有利 。 平均六年级的人的标准操作顺序有什么好处? 如何改进它呢 ?

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  Why is it important that everyone observe the same standard order of operations?
  ::为什么每个人都必须遵守同样的标准行动顺序?

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  What  picture does  connect the dots make ?
  ::将点连接到什么图画?

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  Show the correct order of operations for each of the questions in the interactive to connect the dots and  reveal a picture!
  ::显示互动中每个问题的正确操作顺序, 以连接点并显示图片 !

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  What is the correct order of operations for this problem? Find the answer.
  ::这个问题的正确操作顺序是什么? 找到答案 。

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  • $\begin{align*}{9}^{2}\div {3}^{2}\times {2}^{2} \\ \\\end{align*}$  
    
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    • 1st exponents, 2nd divide, 3rd multiply
      ::一号代表,二号分隔,三号乘以
    • $\begin{align*}\\ {9}^{2}\div {3}^{2}\times {2}^{2} \\ 81 \div 9 \times 4 \\ 9 \times 4 \\ 36 \\ \\\end{align*}$  
    
    ::9232x22 1 前列、 2 分、 3 乘 92_ 32x2281_ 9x49x436

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  In the problem you just solved, try solving again by first finding the exponents and then solving the other operations from right to left.
  ::在你刚解决的问题中, 尝试再解决一次, 首先找到出手者, 然后解决其他操作 从右到左。

  • $\begin{align*}{9}^{2}\div {3}^{2}\times {2}^{2} \\ \\\end{align*}$  
    
    • $\begin{align*}\\ {9}^{2}\div {3}^{2}\times {2}^{2} \\ 81 \div 9 \times 4 \\ 81 \div 36 \\ \frac{81\div 9}{36\div 9} \\ \frac{9}{4}=2\frac{1}{4} \\ \\\end{align*}$  

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  Discussion Question
  ::讨论问题

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  Compare  the answers that came from solving the same problem in different orders . What can you conclude about the order of operations for operations with the same level of power ( addition/subtraction  and multiplication/division )?
  ::比较从以不同顺序解决同一问题中得到的答案。 您能够得出什么结论来判断以相同水平的功率( 增加/ 减法和倍增/ 分离) 操作的操作顺序 ?

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  The answers are different when the operations are solved from right to left instead of left to right.  When finding the answer to problems with the same level of power, it is important to solve from left to right to get the correct answer.
  ::当操作从右到左解决,而不是从左到右解决时,答案是不同的。 当找到对权力水平相同的问题的答案时,必须从左到右解决,才能得到正确的答案。

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  CK-12 PLIX Interactive: Barrels of Candy
  ::CK-12 PLIX 互动:糖果桶

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  You can u se the order of operations in this PLIX interactive to find the total cost when buying different kinds of candies. One question mentions " data-term="Parentheses" role="term" tabindex="0"> parentheses which come before exponents in the order of operations. The very last question is a challenge! You need to find the order of operations with the formula for compound interest using letters as variables in place of numbers.
  ::您可以使用此 PLIX 互动中的操作顺序来寻找购买不同种类糖果的总成本。 有一个问题提到了在操作顺序中引言者面前的括号。 最后一个问题是挑战 。 您需要用字母作为变量来代替数字, 找到使用复利公式的操作顺序 。

  Summary

  播放段落 Exponents represent repeated multiplication. For example:  $\begin{align*}7^3=7\cdot7\cdot7\end{align*}$   ::指数表示重复乘法。 例如: 73=7=7+7+7 播放段落 When simplifying expressions with exponents: exponents come first, then multiplication and division, and then addition and subtraction. ::当用引言简化表达式时:引言先出现,然后乘法和除法,然后加法和减法。