章节大纲

  • Pythagorean Screen Size
    ::毕达哥里屏幕大小

    Not only has the functionality of modern technology been heavily influenced by the Pythagorean Theorem , but the design has as well. Companies hire mathematicians to perform studies to determine the optimal size of phones, tablets, and televisions. When a phone is described as having a 5.5-inch screen or a television is described as having a 62-inch screen this refers to the diagonal of the screen.
    ::现代技术的功能不仅受到Pytagoren Theorem的严重影响,而且设计也受到了很大影响。 公司雇用数学家进行研究以确定电话、平板电脑和电视的最佳尺寸。 当电话被描述为拥有5.5英寸的屏幕或电视被描述为拥有62英寸的屏幕时,这是指屏幕的对角。

    lesson content
    Diagonal of a Phone

    The width and length of the screen can vary as long as the diagonal is the desired length. 
    ::只要对角线是理想的长度,屏幕的宽度和长度可以变化。

    Use the interactive below to explore this concept.  N otice that as you move the red dot to change the length of the sides (also the legs of the triangle) the diagonal of the screen (the hypotenuse of the triangle) stays at 5.5 inches.
    ::使用下面的交互作用来探索这个概念。 请注意, 当您移动红色点以改变边的长度( 也包括三角形的腿) 时, 屏幕的对角( 三角形的假设值) 将维持在5. 5 英寸 。

     

    INTERACTIVE
    Pythagorean Screen Size
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    • Drag the red dot to see what kinds of dimensions result in a 5.5 inch display.
      ::拖曳红色点以查看何种尺寸导致5.5英寸显示。
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    Proving the Pythagorean Theorem
    ::证明毕达哥里定理

    You  have been exploring the Pythagorean Theorem but how do  you know whether it's always true? Use the interactive below to prove the theorem visually.
    ::您一直在探索毕达哥里安理论, 但您如何知道它是否总是真实的呢? 使用下面的交互功能来以视觉来证明理论。

     

    INTERACTIVE
    Proving the Pythagorean Theorem
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    • Press the button to see how the Pythagorean Theorem works.
      ::按下按钮,看看毕达哥伦神话是如何运作的。
    • Drag each of the shapes into the empty square below so that they fill it completely.
      ::将每个形状拖到下面的空方形中,以便完全填满。
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    Proving the Pythagorean Theorem Continued
    ::证明毕达哥里定理

    The proof above shows that this relationship is true for certain triangles. However, is it true for every triangle ? Use the interactive below to prove that the Pythagorean Theorem applies to all right triangles , even triangles with irrational sides.
    ::上述证据表明,对于某些三角形来说,这种关系是真实的。 但是, 对于每一个三角形来说, 这种关系是真实的吗? 使用下面的互动来证明 Pytagoren Theorem 适用于所有正确的三角形, 甚至适用于有非理性边的三角形 。

     

    INTERACTIVE
    Proving the Pythagorean Theorem Continued
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    • Use the sliders to resize the two squares to their desired sizes and press "Prove".
      ::使用滑动器将两个方形的大小调整到它们想要的大小, 按“ 证明 ” 键 。
    • Notice how the two squares' combined shape is split and press "Next".
      ::注意两个方形的组合形状是如何分裂的, 按下“ 下一步 ” 键 。
    • Drag the two triangles to create one big tilted square.
      ::拖曳两个三角形以创建一个大的倾斜方形。
    • Experiment with other values of a   and  b .  
      ::用a和b的其他值进行实验。
    • Your answer is rounded to the nearest hundredths place.
      ::您的答案四舍五入到最近的一百个位置 。

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    Pythagorean Screen Size Continued
    ::Pydagorean 屏幕屏幕大小继续

    Know that you know that the Pythagorean Theorem applies to right triangles with irrational side lengths, look at an example. Use the interactive below to explore an example of how this theorem is used in cell phone design.
    ::您知道 Pythagorean Theorem 适用于右三角形, 外长不合理, 请举一个例子。 使用下面的互动来探索一个示例, 说明该定理是如何在手机设计中使用的 。

     

    INTERACTIVE
    Pythagorean Screen Size Continued
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    • Drag the red and blue points to resize the display size of the phone.
      ::拖曳红色和蓝色点以调整手机显示大小 。
    • The length of the green diagonal is rounded to the nearest hundredths place.
      ::绿色对角线的长度四舍五入到最接近的百位。
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    Diagonal of a Present
    ::当前对数

    Katerina bought a magician's wand as a gift.  The wand is  10 inches across. She has a cube-shaped box that  has a side length of   9 inches.
    ::卡捷琳娜买了魔术师的魔杖作为礼物。魔杖横跨10英寸。她有一个立方体形状的盒子,侧长9英寸。

     

    INTERACTIVE
    Diagonal of a Cube
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    Katerina's wand needs to fit her wand in this cubic box. If the wand is 10 inches long will it fit?
    ::卡捷琳娜的魔杖需要把魔杖放进这个立方盒子里 如果魔杖长10英寸,它会合身吗?

    • Tap the red diagonal line to show right triangles needed to find the length of the diagonal.
      ::拖放红色对角线以显示右三角以查找对角线长度。
    • Drag the screen to change your point of view.  Use two fingers on your trackpad or touch screen or scroll mouse wheel to zoom.
      ::拖曳屏幕以改变您的观点。 在您的轨迹板上使用两根手指或触摸屏幕或滚动鼠标方向盘以缩放 。
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    Discussion Question
    ::讨论问题

    1. Will the wand fit inside the box?
      ::魔杖能放进盒子里吗?
    2. What is the smallest box that could contain the wand?
      ::什么是最小的盒子 可以包含魔杖?
    3. The diagonal of a cube that runs from corner to opposite corner through the center of the cube is called the axial diagonal. Through repeatedly attempting to find the axial diagonal using different side lengths in the previous question. Develop an algorithm for finding the diagonal of a cube that runs from corner to opposite corner?
      ::从角到对角穿过立方体中心的一个立方体的对角,称为轴对角。通过反复尝试,在前一个问题的不同侧长下找到轴对角。开发一个算法,以找到从角到对角运行的一个立方体的对角?
      Summary
    • The Pythagorean Theorem states that  a 2 + b 2 = c 2 .  
      ::《毕达哥里人理论论》指出,a2+b2=c2。