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2.1.1: Writing Basic Equations

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    4344
  • Equations That Describe Patterns

    Figure \(\PageIndex{1}\)

    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:

    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?

    In this concept, you will learn to recognize and describe numerical patterns by finding a pattern rule.

    Pattern Rules

    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.

    There are two primary categories of numerical patterns.

    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.
    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.

    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.

    Here is an example.

    Find the pattern rule for the sequence: 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.

    Look at the smaller numbers at the end of the sequence. Think: "What could you do to 27 to get 9?"

    • You could subtract 18.
    • You could divide by 3.
    • You could do a combination of two or more operations.

    Next, check if any of these potential pattern rules work with the rest of the sequence.

    Consider 81 and 27.

    • If you subtract 18 from 81 you get 63, not 27. So the pattern rule is "not subtract 18."
    • If you divide 81 by 3 you get 27. So the pattern rule "divide by 3" seems to work.

    Now, make sure "divide by 3" works throughout the whole sequence.

    "Divide by 3" works for the whole sequence.

    The answer is that the pattern rule is "divide by 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.

    Here is another example.

    Find the pattern rule for the sequence: 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.

    Look at the smaller numbers at the beginning of the sequence. Think: "What could you do to 1 to get 3?"

    • You could add 2.
    • You could multiply by 3.
    • You could do a combination of two or more operations.

    Next, check if any of these potential pattern rules work with the rest of the sequence.

    Consider 3 and 11.

    • If you add 2 to 3 you get 5, not 11. So the pattern rule is not "add 2."
    • If you multiply 3 by 3 you get 9, not 11. So the pattern rule is not "multiply by 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.

    Next, consider possible pattern rules that involve multiplication and addition or subtraction. Think: "What else can you do to 1 to get 3?"

    • You could multiply by 2 and add 1.
    • You could multiply by 4 and subtract 1.
    • You could do some other combination of two or more operations.

    Now, look back at the rest of the sequence.

    Again consider 3 and 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."
    • 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.

    Now, make sure "multiply by 4 and subtract 1" works throughout the whole sequence.

    "Multiply by 4 and subtract 1" works for the whole sequence.

    The answer is that the pattern rule is "multiply by 4 and then subtract 1."

    Examples

    Example 1.1.1

    Earlier, you were given a problem about Layla and her trip to Disney World.

    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.

    Solution

    You should first notice that this is an ascending pattern so the rule likely involves multiplication or addition.

    Look at the numbers at the beginning of the sequence. Think: "What could you do to 12 to get 27?"

    • You could add 15.
    • You could multiply by a mixed number.
    • You could do a combination of two or more operations.

    Next, check if any of these potential pattern rules work with the rest of the sequence.

    Consider 27 and 42.

    • If you add 15 to 27 you get 42. So the pattern rule "add 15" seems to work.

    Now, make sure "add 15" works throughout the whole sequence.

    "Add 15" works for the whole sequence.

    The first answer is that the pattern rule is "add 15."

    You can continue the pattern by continuing to add 15:

    The next two numbers to be chosen will be 72 and then 87, so unfortunately Layla's number will not be chosen.

    The final answer is that number 82 will not be chosen so Layla won't be going on the trip this time.

    Example 1.1.2

    Find the rule for the pattern: 4, 12, 36, 108.

    Solution

    First, take an overview of the numbers. This is an ascending pattern so the rule likely involves multiplication or addition.

    Look at the smaller numbers at the beginning of the sequence. Think: "What could you do to 4 to get 12?"

    • You could add 8.
    • You could multiply by 3.
    • You could do a combination of two or more operations.

    Next, check if any of these potential pattern rules work with the rest of the sequence.

    Consider 12 and 36.

    • If you add 8 to 12 you get 20, not 36. So the pattern rule is not "add 8."
    • If you multiply 12 by 3 you get 36. So the pattern rule "multiply by 3" seems to work.

    Now, make sure "multiply by 3" works throughout the whole sequence.

    "Multiply by 3" works for the whole sequence.

    The answer is that the pattern rule is "multiply by 3."

    Example 1.1.3

    Find the rule for the pattern: 5, 8, 11, 14.

    Solution

    First, take an overview of the numbers. This is an ascending pattern so the rule likely involves multiplication or addition.

    Look at the smaller numbers at the beginning of the sequence. Think: "What could you do to 5 to get 8?"

    • You could add 3.
    • You could multiply by a mixed number.
    • You could do a combination of two or more operations.

    Next, check if any of these potential pattern rules work with the rest of the sequence.

    Consider 8 and 11.

    • If you add 3 to 8 you get 11. So the pattern rule "add 3" seems to work.

    Now, make sure "add 3" works throughout the whole sequence.

    "Add 3" works for the whole sequence.

    The answer is that the pattern rule is to "add 3."

    Example 1.1.4

    Find the rule for the pattern: 20, 10, 5, 2.5.

    Solution

    First, take an overview of the numbers. This is a descending pattern so the rule likely involves division or subtraction.

    Look at the numbers at the beginning of the sequence. Think: "What could you do to 20 to get 10?"

    • You could subtract 10.
    • You could divide by 2.
    • You could do a combination of two or more operations.

    Next, check if any of these potential pattern rules work with the rest of the sequence.

    Consider 10 and 5.

    • If you subtract 10 from 10 you get 0, not 5. So the pattern rule is not "subtract 10."
    • If you divide 10 by 2 you get 5. So the pattern rule "divide by 2" seems to work.

    Now, make sure "divide by 2" works throughout the whole sequence.

    "Divide by 2" works for the whole sequence.

    The answer is that the pattern rule is to "divide by 2."

    Example 1.1.5

    Find the rule for the pattern: 4, 7, 13, 25, 49.

    Solution

    First, take an overview of the numbers. This is an ascending pattern. This means the rule likely involves multiplication or addition.

    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.

    Now, consider possible pattern rules that involve multiplication and addition or subtraction. Think: "What could you do to 4 to get 7?"

    • You could multiply by 2 and subtract 1.
    • You could multiply by 3 and subtract 5.
    • You could do some other combination of two or more operations.

    Next, look back at the rest of the sequence.

    Consider 7 and 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.

    Now, make sure "multiply by 2 and subtract 1" works throughout the whole sequence.

    "Multiply by 2 and subtract 1" works for the whole sequence.

    The answer is that the pattern rule is "multiply by 2 and then subtract 1."

    Review

    Find the pattern rules for the following numerical patterns.

    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)

    To see the Review answers, open this PDF file and look for section 1.1.

    Vocabulary

    Term Definition
    Algebraic Thinking Algebraic thinking is thinking in a mathematical way.
    Numerical Patterns Numerical patterns are number patterns that are organized in a sequence according to a rule.
    Pattern A pattern is a series of pictures, numbers or other symbols that repeat in some way according to a rule.

    Additional Resources

    Video:

    PLIX: Play, Learn, Interact, eXpore: Finding Patterns: Stars and Moons

    Activity: Equations that Describe Patterns Discussion Questions

    Practice: Writing Basic Equations

    Real World Application: Cheez Its

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