章节大纲

  • How many branches are there on the fractal tree?
    ::分形树上有多少树枝?

    Fractals are mathematical patterns that can make beautiful artwork.  Try building your own fractal tree in the interactive below.  Throughout this lesson, you will use with other operations to describe patterns and predict outcomes.
    ::分形是数学图案, 可以制作美丽的艺术作品。 请尝试在下面的交互部分构建您自己的分形树。 在整个课程中, 您将与其他操作一起描述图案和预测结果 。

    INTERACTIVE
    How Many Branches Are on the Fractal Tree?
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    • Drag the slider to see the number of branches added to the tree. 
      ::拖动滑块以查看树上添加的树枝数 。
    • Fill in the table with the correct number of branches added on each step.
      ::填写表格,填写每一步骤增加的正确数目。
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    How do you find the acceleration of a bobsled?
    ::你是怎么找到一个振动加速度的?

    Experiment with the acceleration of a bobsled going around a turn using the CK-12 Simulation below.
    ::用下面的CK-12模拟法 实验一个圆转的波浪加速加速

    The turn radius is how tight that turn is. A sharp turn has a radius with a low number.  A wide turn has a radius with a high number
    ::转折半径是该转弯的紧度。 尖转弯的半径是一个低数的半径。 宽转的半径是一个高数的半径

    Can you build a fractal snowflake?
    ::你能造一个碎形雪花吗?

    Snowflakes have very detailed repeating patterns. Try building a snowflake with the interactive below and  find  the pattern in the numbers of branches added each time.
    ::雪花有非常详细的重复模式。 尝试用下面的交互模式构建雪花, 并找到每次增加的树枝数的图案 。

    INTERACTIVE
    Can You Build a Fractal Snowflake?
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    • Fill in the table with the number of new branches on each step and watch the snowflake grow!
      ::把每一步的新树枝数填满在桌子上, 看着雪花长起来!
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    INTERACTIVE
    Snowflake Table
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    • Complete the table based on what you learned from the snowflake interactive.
      ::根据您从雪花互动中学到的知识完成表格 。
    • Make sure to use the "^" format when filling in the exponential form. For example, 2^1 represents 2 1 .
      ::在填充指数表时,一定要使用“ ” 格式。例如, 21 代表 21 。
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    Discussion Questions
    ::讨论问题 讨论问题

    1. What expression would you use to find the number of snowflake branches added in the 10th step? Find the number of branches added. 
      ::您要使用什么表达式来查找第十步中添加的雪花树枝数量? 查找添加的树枝数量 。
    2. What is the general pattern you can use to find the number of snowflake branches added for any step? Pick any step number and find the number of branches added.
      ::您可以使用什么一般模式来查找为任何步骤添加的雪花树枝数 ? 选择任何步数并查找所添加的树枝数 。
    3. How did you figure out what to write in the top right of the table? What is the result of raising a natural number (like 1, 2, 3, etc.) to a zero power?
      ::您如何知道在表格右上方写什么? 将自然数字( 如 1, 2, 3 等) 提高到零功率的结果是什么 ?

    Extension Activity
    ::扩展活动活动

    Try drawing your own fractal snowflakes. The number of trunks to start with is the GCF multiplier. The number of times that the lines branch out each time is the base. Draw the branches up to a given number of steps. You may make a table to track the steps and numbers of branches added per step. Come up with an exponential expression for the number of branches added for each step. Using the pattern, calculate the number of branches added in a bigger step, like step 10. Share your snowflake and math work with another student, if you can.
    ::尝试绘制您自己的折形雪花。 要开始的树干数是 GEF 乘数。 每次线条分支的次数是基数。 将树枝绘制到一个给定的步数。 您可以绘制一个表格来跟踪每步添加的树枝的步数和数目。 以指数表示每步添加的树枝数。 使用这个模式, 计算在更大步数中添加的树枝数, 如第10步 。 分享您的雪花和数学, 如果可以的话, 与另一个学生一起工作 。

    Remember that no matter what number you choose to multiply, if it is raised to the power of zero, it is equal to 1!
    ::记住 无论您选择乘以多少数字, 如果升到零, 它等于 1 !

    Can you make a light-up, ugly Christmas sweater?
    ::你可以做一件轻轻的,丑陋的圣诞毛衣吗?

    lesson content
    Ugly Christmas Sweater

    Here is how you can make a light-up tree on an ugly Christmas sweater.
    ::这样你就可以在一件丑陋的圣诞毛衣上 做一棵光树了

    What you need:
    ::您需要的是:

    • Ugly knit sweater with Christmas tree on it
      ::上面有圣诞树的丑毛毛衣
    • Strings of LED lights with attached battery packs
      ::LED灯的字符串和挂着的电池盒
    • Safety pins
      ::安全针

    What to do:
    ::怎么办:

    • Turn the ugly sweater inside out.
      ::把丑陋的毛衣 里面了出来。
    • Poke lights through the knitting so they show up on the other side.
      ::编织时用灯泡亮 所以他们出现在另一边
    • Carefully f asten the lights in place with safety pins.
      ::谨慎地用安全针把灯关上
    • Turn sweater right side out and put it on!
      ::把毛衣右转出来穿上!

    Small strings of LED lights often come with 10 lights per string.
    ::LED灯的小字符串通常每字符串有10个灯。

    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.
    ::利用下面的交互效果 来弄清楚多少根LED灯线 你需要把圣诞树装饰在 一件丑陋的圣诞毛衣上

    INTERACTIVE
    Ugly Sweater Lights
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    • Enter the total number of lights on the tree including the unlit lights in the current row.
      ::输入树上的灯光总数,包括当前行的无光灯。
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    There is 1 light at the top of the tree for the star.  Try adding that light on separately and see if you can find a pattern in the total number of lights up to a given row on the tree.
    ::恒星树顶部有一光。 尝试将光分开添加, 并查看您能否在树上找到到给定行的灯光总数中的图案 。

    Complete the table to find the pattern.
    ::填写表格以查找图案。

    Remember : The result of 0 raised to a natural number is always 0.
    ::记住:0升到自然数的结果总是0。

    INTERACTIVE
    Star Table
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    • Fill in the table to describe how many stars there are in total when each new row is added.
      ::填入表格以描述每增加一行新列时总共有多少颗恒星。
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    Extension Activity : Make sketches of your own designs for light-up Christmas sweaters. Add extra lights for the star or along the outside, if you like. You can also try using a different exponent , but you may find your trees are getting very wide and using a lot of lights. Notice that no matter which counting number is used as an exponent, if 0 is the base, the result is 0. 
    ::扩展活动 : 绘制您自己设计的轻松圣诞毛衣的草图。 如果您愿意的话, 可以为恒星或外观添加额外的灯光。 您也可以尝试使用不同的提示, 但是您可能会发现您的树正在变得非常宽, 并且使用很多灯光。 请注意, 无论数字用什么计数作为提示, 如果 0 是底部, 结果为 0 。

    CK-12 PLIX Interactive: Building Blocks
    ::CK-12 PLIX 交互式互动:建筑块

    In this PLIX, you can  break down groups of blocks to solve problems with exponents.
    ::在这个PLIX中, 您可以拆分街区组, 以解决引言人的问题 。

      Summary
    • Exponents represent repeated multiplication. For example:  7 3 = 7 ⋅ 7 ⋅ 7  
      ::指数表示重复乘法。 例如: 73=7=7+7+7