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  • 描述模式的方程

    lesson content

    Layla's school just won a contest! Ten lucky students will get to go to Disney World to participate in a computer animation program. In  order  to determine who will get to go on the trip, the principal assigns each of the 150 interested students a number between 1 and 150. Then, the principal starts listing off the numbers for the students who will be able to attend:
    ::莱拉的学校刚刚赢得了一场比赛! 十名幸运的学生将去迪士尼世界参加一个计算机动画程序。 为了决定谁能去旅行,校长为150名有兴趣的学生分配了1至150个号码。 然后,校长开始列出能够参加的学生的号码:

    12, 27, 42, 57, ...

    Layla realizes there is a  pattern  to the numbers being called. How is the principal choosing which numbers to call? If Layla has number 82 will she get to go on the trip?
    ::莱拉意识到被调用的数字有一个模式。 主选人如何选择要调用的数字? 如果莱拉有82个,她能去旅行吗?

    In this concept, you will learn to recognize and describe  numerical patterns  by finding a pattern rule.
    ::在此概念中,您将学会通过寻找模式规则来识别和描述数字模式。

    Pattern Rules
    ::规则规则

    A  numerical pattern  is a sequence of numbers that has been created based on a  formula  or rule called a  pattern rule . Pattern rules can use one or more mathematical  operations  to describe the relationship between  consecutive  numbers in the pattern.
    ::数字模式是一种数字序列,它是根据一种称为模式规则的公式或规则创建的。 模式规则可以使用一个或多个数学操作来描述模式中连续数字之间的关系。

    There are two primary categories of numerical patterns.
    ::数字模式有两大类。

    1. When numbers in a pattern get larger as the sequence continues, they are in an  ascending pattern . Ascending patterns often involve  multiplication  or  addition .
      ::当一个模式中的数字随着序列的继续而增大时,它们就呈上升趋势。 升序模式往往涉及乘法或加法。
    2. When numbers in a pattern get smaller as the sequence continues, they are in a  descending pattern . Descending patterns often involve  division  or  subtraction .
      ::当一个模式中的数字随着序列的继续而变小时,它们就是一种递减模式。 递减模式往往涉及分裂或减法。

    When given a pattern, you will often want to figure out the pattern rule that created the pattern. To figure out the pattern rule you must determine how the consecutive numbers are related.
    ::当给定一个模式时, 您通常会想要找出创建模式的模式规则。 要找出模式规则, 您必须确定连续数字的关联性 。

    Here is an example.
    ::举一个例子。

    Find the pattern rule for the sequence: 243, 81, 27, 9.
    ::查找序列模式规则: 243, 81, 27, 9。

    First, take an overview of the numbers. The numbers get smaller in value as the sequence continues, so this is a descending pattern. This means the rule likely involves division or subtraction.
    ::首先,对数字进行概览。随着序列的继续,数字的价值会越来越小,所以这是一个递减模式。这意味着规则可能涉及分割或减法。

    Look at the smaller numbers at the end of the sequence. Think: "What could you do to 27 to get 9?"
    ::看看序列结尾的较小数字。想想: “你能对27做什么才能拿到9?”

    • You could subtract 18.
      ::您可以减去 18 。
    • You could divide by 3.
      ::你可以除以3
    • You could do a combination of two or more operations.
      ::您可以将两个或两个以上的操作组合在一起。

    Next, check if any of these potential pattern rules work with the rest of the sequence.
    ::接下来,检查这些可能的模式规则是否与顺序的其余部分有效。

    Consider 81 and 27.
    ::考虑81和27。

    • If you subtract 18 from 81 you get 63, not 27. So the pattern rule is "not subtract 18."
      ::如果从81中减去18, 则得到63, 不是27, 那么模式规则是“ 不减去18 ” 。
    • If you divide 81 by 3 you get 27. So the pattern rule "divide by 3" seems to work.
      ::如果你将81除以3,你就会得到27 所以模式规则"除以3"似乎有效

    Now, make sure "divide by 3" works throughout the whole sequence.
    ::现在,确保"3分解" 在整个序列中有效

    "Divide by 3" works for the whole sequence.
    ::"3分而出" 整个序列都有效

    The answer is that the pattern rule is "divide by 3."
    ::答案是 模式规则是"3比3"

    Figuring out pattern rules can take some amount of guessing and checking. You will often have to come up with more than one potential pattern rule based on two of the numbers in the sequence and check which one works throughout the whole sequence.
    ::制定模式规则需要一定的猜测和检查。 您通常需要根据序列中的两个数字得出一个以上的潜在模式规则, 并检查一个序列在整个序列中有效。

    Here is another example.
    ::下面是另一个例子。

    Find the pattern rule for the sequence: 1, 3, 11, 43.
    ::查找序列的模式规则: 1, 3, 11, 43。

    First, take an overview of the numbers. The numbers get larger in value as the sequence continues, so this is an ascending pattern. This means the rule likely involves multiplication or addition.
    ::首先,对数字进行概览。随着序列的继续,数字的价值会增加,因此这是一个向上模式。这意味着规则可能涉及乘法或增加。

    Look at the smaller numbers at the beginning of the sequence. Think: "What could you do to 1 to get 3?"
    ::看看序列开始时的较小数字。 想想: “ 你能对 1 做什么才能得到 3 ? ”

    • You could add 2.
      ::你可以加上2个。
    • You could multiply by 3.
      ::你可以乘以3乘以3
    • You could do a combination of two or more operations.
      ::您可以将两个或两个以上的操作组合在一起。

    Next, check if any of these potential pattern rules work with the rest of the sequence.
    ::接下来,检查这些可能的模式规则是否与顺序的其余部分有效。

    Consider 3 and 11.
    ::考虑3和11。

    • If you add 2 to 3 you get 5, not 11. So the pattern rule is not "add 2."
      ::如果你加上2到3,你得到5,而不是11 所以模式规则不是"加2"
    • If you multiply 3 by 3 you get 9, not 11. So the pattern rule is not "multiply by 3."
      ::如果你乘以3乘以3,你就会得到9,而不是11 所以模式规则不是"乘以3乘以3"

    Since neither of those pattern rules work, the pattern rule must involve more than one  operation . Notice how the jump between the numbers increases each time as you move through the sequence. This means multiplication must be involved, but addition or subtraction will be involved as well.
    ::由于这两种模式规则都不起作用, 模式规则必须涉及不止一次操作 。 注意数字之间的跳动在您通过序列移动时会如何增加 。 这意味着必须包含乘法, 但也会包含增减 。

    Next, consider possible pattern rules that involve multiplication and addition or subtraction. Think: "What else can you do to 1 to get 3?"
    ::下一步, 考虑可能的模式规则, 包括乘数和加法或减法。 想想 : “ 您还可以对 1 做些什么来获得 3 ? ”

    • You could multiply by 2 and add 1.
      ::您可以乘以 2 并添加 1 。
    • You could multiply by 4 and subtract 1.
      ::您可以乘以 4 乘以 4 减去 1 。
    • You could do some other combination of two or more operations.
      ::你可以做其他两种或多种操作的组合。

    Now, look back at the rest of the sequence.
    ::现在,看看其余的顺序。

    Again consider 3 and 11.
    ::再次考虑3和11。

    • If you multiply 3 by 2 and add 1 you get 7, not 11. So the pattern rule is not "multiply by 2 and add 1."
      ::如果乘以 3 乘以 2 , 加上 1, 得到 7, 不是 11, 那么模式规则不是“ 乘以 2 再加 1 ” 。
    • If you multiply 3 by 4 and subtract 1 you get 11. So the pattern rule "multiply by 4 and subtract 1" seems to work.
      ::如果乘以 3 乘以 4 减去 1 , 就会得到 11 。 因此, 模式规则“ 乘以 4 乘以 4 减去 1 ” 似乎有效 。

    Now, make sure "multiply by 4 and subtract 1" works throughout the whole sequence.
    ::现在,确保“乘以4乘以4再减去1” 在整个序列中有效。

    "Multiply by 4 and subtract 1" works for the whole sequence.
    ::“以 4 乘以 4 减去 1 ” 在整个序列中有效 。

    The answer is that the pattern rule is "multiply by 4 and then subtract 1."
    ::答案是模式规则是"乘以4再减去1"

    Examples
    ::实例

    Example 1
    ::例1

    Earlier, you were given a problem about Layla and her trip to Disney World. 
    ::早些时候,你得到一个问题 关于Layla和她去迪士尼世界的旅行。

    Her school won a contest and gets to send 10 students to a computer animation program at Disney World. All interested students were assigned numbers and the principal has started reading off the numbers for the students who will attend: 12, 27, 42, 57, ... Layla realizes there is a pattern to these numbers. She has number 82 and wonders if she will be chosen to go on the trip this time.
    ::她的学校赢得了一场比赛,并派了10名学生去迪斯尼世界的电脑动画程序。所有感兴趣的学生都被分配了人数,校长已经开始为将要参加的学生朗读数字:12、27、42、57......莱拉意识到这些数字有一个模式。她有82个号码,并想知道她这次是否会被选中去旅行。

    You should first notice that this is an ascending pattern so the rule likely involves multiplication or addition.
    ::您应该首先注意,这是一个不断上升的模式,因此规则可能涉及乘法或加法。

    Look at the numbers at the beginning of the sequence. Think: "What could you do to 12 to get 27?"
    ::看看序列开始的时候的数字。想想: “你能对12做什么才能得到27?”

    • You could add 15.
      ::你可以加上15个
    • You could multiply by a mixed number.
      ::您可以乘以混合数字。
    • You could do a combination of two or more operations.
      ::您可以将两个或两个以上的操作组合在一起。

    Next, check if any of these potential pattern rules work with the rest of the sequence.
    ::接下来,检查这些可能的模式规则是否与顺序的其余部分有效。

    Consider 27 and 42.
    ::考虑27和42。

    • If you add 15 to 27 you get 42. So the pattern rule "add 15" seems to work.
      ::如果你加上15到27,你会得到42 所以模式规则"添加15"似乎有效

    Now, make sure "add 15" works throughout the whole sequence.
    ::现在,确保"加15"整个过程都有效

    "Add 15" works for the whole sequence.
    ::"Add 15"为整个序列工作。

    The first answer is that the pattern rule is "add 15."
    ::第一个答案是 模式规则是"加15"

    You can continue the pattern by continuing to add 15:
    ::您可以继续添加 15 来继续此模式 :

    The next two numbers to be chosen will be 72 and then 87, so unfortunately Layla's number will not be chosen.
    ::接下来要选择的两个数字将是72和87, 所以不幸的是,Layla的号码将不会被选中。

    The final answer is that number 82 will not be chosen so Layla won't be going on the trip this time.
    ::最后的答案是 82号不会被选中 所以Layla这次不会去旅行了

    Example 2
    ::例2

    Find the rule for the pattern: 4, 12, 36, 108.
    ::确定模式的规则: 4, 12, 36, 108。

    First, take an overview of the numbers. This is an ascending pattern so the rule likely involves multiplication or addition.
    ::首先,对数字进行概述。这是一个不断上升的模式,因此该规则可能涉及乘数或增加数。

    Look at the smaller numbers at the beginning of the sequence. Think: "What could you do to 4 to get 12?"
    ::看看序列开始时的较小数字。想想: “你能对4做什么才能得到12?”

    • You could add 8.
      ::你可以加上8个
    • You could multiply by 3.
      ::你可以乘以3乘以3
    • You could do a combination of two or more operations.
      ::您可以将两个或两个以上的操作组合在一起。

    Next, check if any of these potential pattern rules work with the rest of the sequence.
    ::接下来,检查这些可能的模式规则是否与顺序的其余部分有效。

    Consider 12 and 36.
    ::考虑12和36。

    • If you add 8 to 12 you get 20, not 36. So the pattern rule is not "add 8."
      ::如果你加上8到12,你得到20,而不是36 所以模式规则不是"加8"
    • If you multiply 12 by 3 you get 36. So the pattern rule "multiply by 3" seems to work.
      ::如果你乘以12乘以3,你就会得到36 所以模式规则"乘以3"似乎有效

    Now, make sure "multiply by 3" works throughout the whole sequence.
    ::现在,确保"乘以3" 在整个序列中起作用。

    "Multiply by 3" works for the whole sequence.
    ::整个序列都用"3"来进行"3"

    The answer is that the pattern rule is "multiply by 3."
    ::答案是模式规则是"乘以3乘以3"

    Example 3
    ::例3

    Find the rule for the pattern: 5, 8, 11, 14.
    ::查找模式的规则: 5, 8, 11, 14。

    First, take an overview of the numbers. This is an ascending pattern so the rule likely involves multiplication or addition.
    ::首先,对数字进行概述。这是一个不断上升的模式,因此该规则可能涉及乘数或增加数。

    Look at the smaller numbers at the beginning of the sequence. Think: "What could you do to 5 to get 8?"
    ::看看序列开始时的较小数字。想想: “你能对5做什么才能得到8?”

    • You could add 3.
      ::你可以加上3个
    • You could multiply by a mixed number.
      ::您可以乘以混合数字。
    • You could do a combination of two or more operations.
      ::您可以将两个或两个以上的操作组合在一起。

    Next, check if any of these potential pattern rules work with the rest of the sequence.
    ::接下来,检查这些可能的模式规则是否与顺序的其余部分有效。

    Consider 8 and 11.
    ::考虑8和11。

    • If you add 3 to 8 you get 11. So the pattern rule "add 3" seems to work.
      ::如果你加上3到8,你得到11。 所以模式规则“添加3”似乎有效。

    Now, make sure "add 3" works throughout the whole sequence.
    ::现在,确保"加3"整个过程都有效

    "Add 3" works for the whole sequence.
    ::"加3"为整个序列工作

    The answer is that the pattern rule is to "add 3."
    ::答案是模式规则是"加3"

    Example 4
    ::例4

    Find the rule for the pattern: 20, 10, 5, 2.5.
    ::查找模式的规则: 20, 10, 5, 5, 2.5。

    First, take an overview of the numbers. This is a descending pattern so the rule likely involves division or subtraction.
    ::首先,对数字进行概述。这是一个递减模式,因此规则可能涉及分割或减法。

    Look at the numbers at the beginning of the sequence. Think: "What could you do to 20 to get 10?"
    ::看看序列开始的时候的数字。想想: “你对20能做些什么才能得到10?”

    • You could subtract 10.
      ::你可以减10。
    • You could divide by 2.
      ::你可以除以2。
    • You could do a combination of two or more operations.
      ::您可以将两个或两个以上的操作组合在一起。

    Next, check if any of these potential pattern rules work with the rest of the sequence.
    ::接下来,检查这些可能的模式规则是否与顺序的其余部分有效。

    Consider 10 and 5.
    ::考虑10和5。

    • If you subtract 10 from 10 you get 0, not 5. So the pattern rule is not "subtract 10."
      ::如果从 10 中减去 10, 则得到 0, 不是 5, 那么模式规则不是“ 减10 ” 。
    • If you divide 10 by 2 you get 5. So the pattern rule "divide by 2" seems to work.
      ::如果将10除以2除以2, 就会得到5。 所以模式规则“ 将2除以2” 似乎有效 。

    Now, make sure "divide by 2" works throughout the whole sequence.
    ::现在,确保"分解 2" 在整个序列中有效。

    "Divide by 2" works for the whole sequence.
    ::"乘以2"为整个序列工作

    The answer is that the pattern rule is to "divide by 2."
    ::答案是模式规则是"二分之一"

    Example 5
    ::例5

    Find the rule for the pattern: 4, 7, 13, 25, 49.
    ::确定模式的规则:4、7、13、25、49。

    First, take an overview of the numbers. This is an ascending pattern. This means the rule likely involves multiplication or addition.
    ::首先,对数字进行概述。这是一个不断上升的模式。这意味着规则可能涉及乘数或增加。

    Next, look at how the numbers are related. The  difference  between the numbers increases as you move through the sequence. To get from 4 to 7 you have to add 3, but to get from 7 to 13 you have to add 6. This means multiplication is involved. Since there is no whole number you can multiply 4 by to get 7, the pattern rule is likely multiplication with either addition or subtraction.
    ::接下来,请看看数字是如何关联的。随着您在序列中移动,数字之间的差别会增加。要从 4 到 7, 您必须增加 3, 但从 7 到 13, 您必须增加 6 。 这意味着要进行乘法。 因为没有整数, 您可以乘以 4 乘以 7 , 模式规则可能会随着加法或减法而倍增 。

    Now, consider possible pattern rules that involve multiplication and addition or subtraction. Think: "What could you do to 4 to get 7?"
    ::现在,考虑一下可能涉及乘法和加法或减法的模式规则。想想: “你对4能做些什么才能得到7?”

    • You could multiply by 2 and subtract 1.
      ::您可以乘以 2 乘以 2 减去 1 。
    • You could multiply by 3 and subtract 5.
      ::您可以乘以 3 乘以 3 减去 5 。
    • You could do some other combination of two or more operations.
      ::你可以做其他两种或多种操作的组合。

    Next, look back at the rest of the sequence.
    ::接下来,看看其余的顺序。

    Consider 7 and 13.
    ::考虑7和13。

    • If you multiply 7 by 2 and subtract 1 you get 13. So the pattern rule "multiply by 2 and subtract 1" seems to work.
      ::如果乘以 7 乘以 2 , 减去 1 , 则得到 13 。 因此, 模式规则“ 乘以 2 乘以 2 , 减去 1 ” 似乎有效 。

    Now, make sure "multiply by 2 and subtract 1" works throughout the whole sequence.
    ::现在,确保“乘以2乘以2再减去1”在整个序列中有效。

    "Multiply by 2 and subtract 1" works for the whole sequence.
    ::“以 2 乘以 2 减去 1 ” 在整个序列中有效 。

    The answer is that the pattern rule is "multiply by 2 and then subtract 1."
    ::答案是模式规则是"乘以2乘以2再减去1"

    Review
    ::回顾

    Find the pattern rules for the following numerical patterns.
    ::为下列数字模式寻找模式规则。

    1. 1, 6, 21, 66
    2. 95, 80, 65, 50
    3. 3, 10, 17, 24
    4. 256, 64, 16, 4
    5. 3, 11, 43, 171
    6. 81, 27, 9, 3
    7. 4, 13, 40, 121
    8. 1, 6, 31, 156
    9. 3, 18, 108, 648
    10. 100, 90, 80, 70
    11. 2, 3, 5, 9
    12. 45, 15, 5
    13. 142, 70, 34, 16
    14. 5, 35, 245, 1715
    15. 900, 300, 100

    Review (Answers)
    ::回顾(答复)