Section outline

  • Babylon
    ::巴比伦

    Before ancient  Babylonian  times (around 1,700 BCE in ancient Babylon, modern-day Iraq), i t was believed that mathematics had predominantly been used for counting. However, we have an understanding of the ways Babylonian mathematicians used math because they wrote it down. Records were carved into clay tablets in a system of writing called cuneiform. One tablet, named Plimpton 322 and stored at Columbia University, shows that the Babylonians had an understanding of trigonometry. Another tablet, named BM92698 because it is stored in the British Museum, displays the first known list of perfect squares and perfect cubes. Babylonians used this list to perform quick calculations in matters of land allotment and architecture.
    ::在古代的巴比伦时代(古代巴比伦,现代伊拉克,约1,700 BCE),人们相信数学主要用于计算。然而,我们了解巴比伦数学家在写下来时使用数学的方式。记录被刻成粘土平板块,在一种称为Cuneiform的写作系统中。一个名为Plimpton 322并储存在哥伦比亚大学的平板块显示,巴比伦人对三角测量有了解。另一个名为BM92698的平板块,因为它存放在英国博物馆,展示了第一个已知的完美方块和完美立方体清单。巴比伦人用这个单子快速计算土地分配和建筑事项。

     

    Perfect Squares
    ::完美广场

    Y ou may not realize it, but you have already been exploring perfect squares. A perfect square is the square of a whole number. Recall that squaring a number means to multiply it by itself. The perfect squares can be found along the diagonal of the multiplication table when a whole number is multiplied by itself.
    ::您可能没有意识到它, 但是您已经探索过完美的方块。 一个完美的方块是整个数字的正方块。 提醒您注意, 将数字进行分解是自动乘以它的手段。 当一个整数本身乘以一个整数时, 在乘法表的对角上可以找到完美的方块 。

    lesson content
    Perfect Squares

    For example, the number 25 is a perfect square because  5 × 5 = 25 . Use the interactive below to explore perfect squares.
    ::例如,25号是一个完美的正方形, 因为 5x5=25。 使用下面的交互方式来探索完美的正方形 。

    INTERACTIVE
    Perfect Squares
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    • Move the blue point to change the value of  n .  
      ::移动蓝色点以更改 n 的值。
    • Notice the relationship between the value of  n  and the number of squares. 
      ::注意 n 值与方形数之间的关系。
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    Discussion Question
    ::讨论问题

    How many squares would be added to the 9th perfect square to get the 10th perfect square?
    ::在第九完美的广场上要加多少平方 才能得到第十完美的广场?

     Perfect Cubes
    ::完美立方体

    A perfect cube is the cube of a whole number, found by multiplying a whole number by itself three times. For example, the number 8 is a perfect cube because  2 × 2 × 2 = 8 . Use the interactive below to explore perfect cubes.
    ::一个完美的立方体是整个数字的立方体,它通过将整个数字乘以三次而发现。例如,数字8是一个完美的立方体,因为 2x2x2=8. 使用下面的交互作用来探索完美的立方体。

    INTERACTIVE
    Perfect Cubes
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    • Change the value of  n  by pressing the + and - buttons.
      ::按下 + 和 - 按钮, 更改 n 的值。
    • Observe how the blue cube is affected.
      ::观察蓝色立方体如何受到影响。
    • Press Expand to count the number of unit cubes that make up the big blue cube.
      ::按 展开以计算构成大蓝色立方体的单位立方体数 。
    • Drag the screen to change your point of view.
      ::拖曳屏幕以改变您的观点 。
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    Completing the Tablet
    ::正在完成目录

    Earlier, you learned about the Babylonian tablet known as BM92698, which listed the first 59 perfect squares and the first 32 perfect cubes. Use the interactive below and your knowledge of perfect squares and perfect cubes to recreate this historic feat.
    ::早些时候,你学到了称为BM92698的巴比伦平板,它列出了前59个完美的方块和前32个完美的立方块。 利用下面的互动以及你对完美的方块和完美的立方块的知识来重建这个历史性的伟大成就。

    INTERACTIVE
    Babylonian Tablet
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    • Fill out the stone tablet with the correct perfect squares and cubes.
      ::用完美的方块和立方体填满石板
    • Click "Switch" to change between filling out squares or cubes.
      ::单击“ 切换” 以在填充方块或立方体之间更改 。
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    Connection to Geometry
    ::连接到几何

    The lesson discussed how you could solve the problem n 2 = 16  by listing perfect squares until you found the number 4, which when squared is equal to 16.
    ::课程讨论了如何通过列出完美的方块来解决问题n2=16,直到你发现数字4,当平方等于16。

    This approach can also be used to find missing dimensions of squares and cubes. Suppose a cube needs to be constructed to contain a specific volume or a square needs to be constructed to contain a specific area. Use the interactive below to explore this idea.
    ::这种方法也可以用来查找方块和立方体缺失的维度。 假设需要构造立方体以包含一个特定体积, 或者需要构造一个方块以包含一个特定区域。 使用下面的交互功能来探索这个概念 。

    INTERACTIVE
    Connection to Geometry
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    • A random square or cube will be generated. Input the correct side length for the corresponding geometric figure.
      ::将生成随机方块或立方体。 输入相应的几何图的正确边长。
    • Click "New" once finished to generate a new problem.
      ::点击“ 新建” 完成后产生新问题 。
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     Summary
    ::摘要

    • Perfect squares are the squares of a whole number (when a number is multiplied by itself two times).
      ::完美的正方形是整数的正方形(当数字本身乘以两次时)。
    • Perfect cubes are the cubes of a whole number (when a number is multiplied by itself three times).
      ::完美的立方体是整个数字的立方体(当一个数字本身乘以三次时)。