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# Equations that Describe Patterns

## Describe numerical sequences by finding a rule.

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Equations that Describe Patterns

### Guidance

We see patterns all around us each and every day. We see them on soccer balls, in fabric, in the way different gardens are designed, even our day can be organized in a pattern. Recognizing patterns is second nature to us and feels very natural.

In primary grades, you learned to count by 2s,3s,4s,5s,\begin{align*}2s, 3s, 4s, 5s,\end{align*} etc. This type of counting involves counting in patterns. In fact, counting in this way was probably so natural that you didn’t even realize how simple it was when you were doing it.

This chapter is about algebraic thinking. We could say that algebraic thinking is about learning to think mathematically. One of the keys to algebraic thinking comes from recognizing and understanding patterns. As you learn more advanced mathematics, patterns are going to become more and more useful. We can investigate and work to understand two types of patterns: numerical patterns and geometric patterns.

What is a numerical pattern?

A numerical pattern is a sequence of numbers that uses a formula or rule to generate the sequence.

Numerical patterns can be organized in a couple of different ways.

1. When numbers in a pattern get larger as the sequence continues, they are in an ascending pattern.
2. When numbers in a pattern get smaller as the sequence continues, they are in a descending pattern.

Every pattern has a sequence that has been created based on a pattern rule. Examining the relationship between the numbers in a pattern can help us to determine the rule used by the pattern. Pattern rules can use one or more mathematical operations to describe the relationship.

Once you have figured out the relationship between the numbers, you can work on expressing this relationship in the form of a rule.

Find the pattern rule in the following sequence: 243, 81, 27, 9.

First we can take an overview of the numbers. All the numbers are odd and all have 9 as a factor. The numbers get smaller in value as the sequence continues, so this is a descending pattern.

Think about this. The pattern is descending-the numbers get smaller. We need to figure out which operation is involved in the pattern rule.

Which operations help us in making numbers smaller?

Subtraction and division help us to make numbers smaller. This tells us that subtraction or division is involved in the pattern rule.

Next we have to dive a little farther into the figuring out the rule.

What is the relationship between 27 and 9?

Nine is a factor of 27 because 9×3=27\begin{align*}9 \times 3 = 27\end{align*}.

But we are looking for division or subtraction here.

Since 9×3=27\begin{align*}9 \times 3 = 27\end{align*}, it is possible that 27÷3=9\begin{align*}27 \div 3 = 9\end{align*}

A possible pattern rule is ÷ 3\begin{align*}\div \ 3\end{align*}.

How can we test it out to be sure that this is the correct pattern rule?

To test it out, we have to be sure that it works for all of the other numbers in the pattern. Let’s look at the pattern again.

243,81,27,9243÷3=8181÷3=2727÷3=9\begin{align*}&243, 81, 27, 9\\ &243 \div 3 = 81\\ &81 \div 3 = 27\\ &27 \div 3 = 9\end{align*}

The pattern works! Our pattern rule is ÷ 3\begin{align*}\div \ 3\end{align*}.

How can we write a pattern rule?

The pattern rule can be described algebraically by writing an expression with a variable.

A variable is a letter or symbol used to represent a quantity that can vary.

For example, the letter x\begin{align*}x\end{align*} could represent any number in the pattern above.

y\begin{align*}y\end{align*} would represent the number directly following it.

Then the pattern rule describes the relationship between x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*}.

Because any number x\begin{align*}x\end{align*} in the pattern is divided by 3 to give the number y\begin{align*}y\end{align*}, we can write the pattern rule as y=x3\begin{align*}y= \frac{x}{3}\end{align*}.

That pattern was a descending pattern. Now let’s look at an ascending pattern.

Find the pattern rule in the following sequence: 1, 3, 11, 43.

First we take an overview of the numbers. All the numbers are odd and this is an ascending pattern. Therefore, addition and/or multiplication are operations involved in the pattern rule because that is the way we increase numbers in mathematics.

Now let’s compare the first and second number.

What is the relationship between 1 and 3?

1+2=3\begin{align*}1 + 2 = 3\end{align*}, so +2 is a possible pattern rule, but the jump between the second number (3) and the third number (11) is much larger than +2. Such a large jump indicates that multiplication must be one of the operations in the pattern rule.

How can we figure out how multiplication is used in the pattern rule?

To do this, we look at the relationship between the second and third number. The third number (11) is close to 9 and 12, both of which have relationships with the second number (3), 3×3=9\begin{align*}3 \times 3 = 9\end{align*} and 3×4=12\begin{align*}3 \times 4 = 12\end{align*}. This means that the pattern rule could either be (× 3+2)\begin{align*}(\times \ 3 + 2)\end{align*} or (× 41)\begin{align*}(\times \ 4 - 1)\end{align*}.

When we try out both rules with the list, we can see that × 41\begin{align*}\times \ 4 - 1\end{align*} is the correct pattern rule.

How do we write this rule algebraically?

The pattern rule can be described algebraically by the expression 4x1\begin{align*}4x - 1\end{align*}. Where x\begin{align*}x\end{align*} is any number in the pattern, and y\begin{align*}y\end{align*} is the number directly following it, y=4x1\begin{align*}y = 4x - 1\end{align*}.

Now it's time for you to try a few on your own. Practice finding the rule for each pattern.

#### Example A

5, 8, 11, 14

Solution:y=x+3\begin{align*}y = x + 3\end{align*}

#### Example B

20, 10, 5, 2.5

Solution: y=x÷2\begin{align*}y = x \div 2\end{align*}

#### Example C

4, 7, 13, 25, 49

Solution: y=2x1\begin{align*}y = 2x - 1\end{align*}

Now let's go back to the patterns that the girls were seeing in nature.

Now that you have learned all about patterns, it is time to revisit the original problem with Kelly and Sara and their nature patterns. Reread the problem and underline any important information.

Kelly is very excited for summer vacation. She has been accepted into a Teen Wilderness program and will be spending four weeks hiking, camping and learning wilderness skills in the White Mountains of New Hampshire. There are a bunch of other students who will be going too. Kelly hasn’t met them yet.

In two weeks there will be a pre-trip meeting. At the pre-trip meeting, Kelly will meet the other kids in the program and learn specifics about the summer. In the meantime, Kelly is so excited that she is counting the days. To help her pass the time, her Mom bought her a couple of books about hiking and nature.

One of the books talks all about nature and unique features about nature. One of the chapters in the books talks all about patterns in nature. Kelly is fascinating.

“Listen to this,” she tells her friend Sara. “There are patterns on leaves and trees and flowers. In fact, according to this, the way a tree divides into branches follows a specific pattern.”

“That is cool,” Sara says. “Do you think it is really true?”

“We could go and check it out,” Kelly suggests.

The girls decide to investigate. On a piece of paper Kelly writes the following pattern.

1, 1, 2, 3, 5, 8, 13....

When the girls arrive at the park, they sit down in front of one of the trees. Kelly reads the following information.

“This pattern in nature is called the Fibonacci pattern after an Italian mathematician. He discovered that many things in nature follow this pattern of numbers.”

“Very interesting, but how does it apply to trees?” Sara asked.

You just finished learning all about patterns. What is the rule for the Fibonacci pattern of numbers that Sara and Kelly are using?

1, 1, 2, 3, 5, 8, 13,

If you look you can see that the two previous numbers add together to equal the next number. This is the rule.

### Vocabulary

Pattern
a sequence of number or geometric figures that repeats according to a pattern unit or a rule.
Algebraic Thinking
thinking in a mathematical way
Numerical Patterns
number patterns that are organized in a sequence according to a rule.

### Guided Practice

Here is one for you to try on your own.

3, 9, 27, ....

First, we have to figure out what is happening to each given number to create the next value. If you look, you can see that each value is being multiplied by 3.

We can write the following equation.

y=3x\begin{align*}y = 3x\end{align*}

This means that if we multiply any x value by 3 that it will give us the next value in the sequence.

### Practice

Directions: 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. 144, 70, 34, 16

14. 5, 35, 245, 1715

15. 900, 300, 100

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