1.1: Describing Patterns
Introduction
Summer Hiking
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 found in nature. One of the chapters in the book talks about patterns in nature. Kelly is fascinated.
“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's 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.
As the girls investigate patterns in trees, it is your turn to learn about patterns in Algebra. Kelly and Sara are going to discover patterns found in nature. Patterns in nature are mathematical and there are many of them. In this Concept you will learn about mathematical patterns as you explore thinking algebraically. At the end of the Concept we will return to Kelly and Sara and discover what they learned about patterns in nature.
What You Will Learn
By the end of this lesson you will be able to demonstrate the following skills.
- Recognize and describe numerical patterns by finding a rule.
- Extend numerical patterns.
- Recognize and describe visual or geometric patterns.
- Extend a visual or geometric pattern.
- Interpolate missing elements of numerical, visual or geometric patterns.
Teaching Time
I. Recognize and Describe Numerical Patterns by Finding a Rule
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 to day lives can follow a pattern. Recognizing patterns is second nature to us and feels very natural.
In primary grades, you learned to count by \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. Let's discuss two types of patterns: numerical patterns and geometric patterns.
Numerical patterns use numbers and geometric patterns use visual shapes and figures. Both are useful and this lesson will teach you about both of them.
Let’s begin with numerical patterns.
What is a numerical pattern?
A numerical pattern is a sequence of numbers that uses a formula or rule to generate a sequence.
Numerical patterns can be organized in a couple of different ways.
- When numbers in a pattern get larger as the sequence continues, they are in an ascending pattern.
- 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.
Looking for pattern rules is a lot like being a detective! You have to use your detective skills to decipher the relationship between the numbers. Once you have figured out the relationship between the numbers, you can work on expressing this relationship in the form of a rule.
Let’s look at an example.
Example
Find the pattern rule in the following sequence: 243, 81, 27, 9.
Alright math detectives, get ready. We need to figure out how each of these numbers is related to the others. There is a way that they are related, we just have to figure out how.
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 likely involved in the pattern rule.
Next we have to dive a little farther into the process of figuring out the rule.
Let’s start with the smaller numbers.
What is the relationship between 27 and 9?
Nine is a factor of 27 because \begin{align*}9 \times 3 = 27\end{align*}.
But we are looking for division or subtraction here.
Since \begin{align*}9 \times 3 = 27\end{align*}, we know that \begin{align*}27 \div 3 = 9\end{align*}
A possible pattern rule is \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.
\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 \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 \begin{align*}x\end{align*} could represent any number in the pattern above.
\begin{align*}y\end{align*} would represent the number directly following it.
Then the pattern rule describes the relationship between \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.
Because any number \begin{align*}x\end{align*} in the pattern is divided by 3 to give the number \begin{align*}y\end{align*}, we can write the pattern rule as \begin{align*}y= \frac{x}{3}\end{align*}.
That pattern was a descending pattern. Now let’s look at an ascending pattern.
Example
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?
\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), \begin{align*}3 \times 3 = 9\end{align*} and \begin{align*}3 \times 4 = 12\end{align*}. This means that the pattern rule could either be \begin{align*}(\times \ 3 + 2)\end{align*} or \begin{align*}(\times \ 4 - 1)\end{align*}.
When we try out both rules with the list, we see that \begin{align*}\times \ 4 - 1\end{align*} is a correct pattern rule.
How do we write this rule algebraically?
The pattern rule can be described algebraically by the expression \begin{align*}4x - 1\end{align*}. Where \begin{align*}x\end{align*} is any number in the pattern, and \begin{align*}y\end{align*} is the number directly following it, \begin{align*}y = 4x - 1\end{align*}.
1A. Lesson Exercises
Practice figuring out pattern rules. Decipher each rule and write it algebraically.
- 5, 8, 11, 14
- 20, 10, 5, 2.5
- 4, 7, 13, 25, 49
Take a few minutes to check your work with a peer.
II. Extend Numerical Patterns
Once you have figured out a pattern rule it is easy to use that rule to extend the pattern. Extending a pattern involves writing the numbers that come next in the pattern according to the rule.
Let’s look at an example.
Example
Find the next term in the following pattern: 3, 6, 9, 12, ____
First, notice that this is an ascending pattern meaning that it will involve addition, multiplication or both.
What is the relationship between these numbers? How were they increased?
If you think about this question, you can see that each number was increased by adding three.
To extend the pattern, we simply add three to the last number in the sequence.
\begin{align*}12 + 3 = 15\end{align*}
Our answer is 15.
Sometimes we need to extend patterns in other ways too. Remember how we used a variable to create an expression which described the pattern rule? We can use these expressions to extend number patterns. Sometimes, a table called a function table can help us organize our values as we extend the pattern. A function is a set of ordered pairs in which each element of the domain \begin{align*}(x)\end{align*} has only one element associated with it in the range \begin{align*}(y)\end{align*}.
Let’s apply this information to one of the patterns from the first section. Our pattern was 1, 3, 11, 43 and the pattern rule was \begin{align*}4x - 1\end{align*}. If \begin{align*}y\end{align*} is the next number in the pattern, then we can write the expression \begin{align*}y = 4x - 1\end{align*}.
Let’s organize these values into a table.
\begin{align*}y = 4x - 1\end{align*}
position | \begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|---|
1st | 1 | 3 |
2nd | 3 | 11 |
3rd | 11 | 43 |
4th | 43 | 171 |
5th | 171 |
Notice how \begin{align*}x\end{align*} is the number in the pattern and \begin{align*}y\end{align*} is the next number in the pattern. Theoretically, any number could be \begin{align*}x\end{align*}, but for the purposes of extending a numerical pattern, \begin{align*}x\end{align*} has to be the last number in the pattern. When the number 43 is entered into the equation as the value for \begin{align*}x\end{align*}, the value of \begin{align*}y\end{align*} is 171. 171 is the 5th number in the pattern.
Sometimes using a table can be helpful and other times it can be confusing. As long as you understand the pattern rule, you can extend the pattern by applying the rule.
Sometimes, you need to extend rule by looking far out into the future. Let’s look at an example of this.
Example
What is the seventh number in the sequence: 1, 3, 9, 27, ____
First, let’s figure out the rule. This is an ascending sequence so it uses addition, multiplication or both. The rule in this case is \begin{align*}\times \ 3\end{align*}. Using the rule we can write the following expression \begin{align*}y=3x\end{align*}.
Next we can organize this information into a table.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
1 | 3 |
3 | 9 |
9 | 27 |
27 | 81 |
81 | 243 |
243 | 729 |
729 | 2187 |
Our answer is 2187.
1B. Lesson Exercises
Figure out the pattern rule for each pattern and then extend the pattern two terms.
- 9, 17, 33, ___, ___
- 3, 10, 31, ___, ___
Check your work. What rule did you write for each pattern? Did your partner write the same rule?
III. Recognize and Describe Visual or Geometric Patterns
Visual or geometric patterns are similar to numerical patterns in the sense that the follow a rule. Some geometric patterns are repeating patterns. Repeating patterns have a pattern unit which repeats. Other geometric patterns, much like numerical patterns, change position, increase, or decrease following a pattern rule or formula.
Example
Look at the pattern below. What is the pattern unit?
Let’s begin by naming the shapes. The shapes follow this order: circle, square, triangle, triangle, circle, square. To find the pattern unit, we have to find the unit which repeats.
Where do you begin to notice that the pattern is repeating?
In this case, the pattern unit is circle, square, triangle, triangle
You can see repeating patterns all around you. The tiling of a floor is a repeated pattern. Wallpaper is often designed in a repeating pattern as well.
What happens when a visual pattern doesn’t repeat? When it is created by some other rule?
When this happens, you have to pull out your detective skills once again. You will have to figure out the rule. How did the visual images change? What rule can you write to describe the change? Let’s look at an example.
Example
Look at the pattern below. What is the pattern rule?
We can see that this is not a repeating pattern because no two shapes repeat in the same position. Therefore, we need to look for a pattern rule. Like with numerical patterns, we compare the shapes to determine how they change. All the shapes are right triangles. In the first figure, the right angle is in the bottom left. In the second shape, the right angle has changed to the top left. In the third shape, the right angle has changed to the top right.
In this pattern, the right angle is moving clockwise by 90 degrees each time. This is our pattern rule.
1C. Lesson Exercises
Practice examining visual or geometric patterns. Explain how the pattern unit repeats or write a pattern rule for each pattern.
Take a few minutes to check your answers with a friend.
IV. Extend Visual or Geometric Patterns
Once you figure out how the pattern unit is repeating or what the rule is to the geometric pattern, you can extend the pattern according to this information.
Example
How many triangles will be in the next step of this pattern?
How does this geometric pattern change? To figure this out, we have to look at each figure as if it is a step in the pattern. The first step has one triangle. The second step has four triangles. The third step has nine triangles.
We can write the pattern as: 1, 4, 9
Because this pattern does not change in a systematic way, we can’t write a rule for it. We can see that each step increases by two more than the previous step.
The first step adds 3 to get the second step.
The second step adds 5 to get the third step.
The third step will add two more which is 7.
There are 9 triangles in the third step, \begin{align*}9 + 7 = 16\end{align*}. There will be 16 triangles in the fourth step.
Practice extending the following pattern. Express how the pattern is changing. Write the number of images in the next step of the pattern. Then draw the next step of the pattern.
Take a few minutes to check your work with a neighbor. This question should have three parts to the answer. Be sure that you have all three parts written down.
V. Interpolate Missing Elements of Numerical, Visual or Geometric Patterns
If we know how to describe and extend numerical, visual, or geometric patterns, we can also find missing elements in these patterns. The same tools we used to extend patterns—finding the pattern rule or pattern unit—will help us find the missing elements.
Example
What is the missing number in the pattern?
11, 23, 47, ___, 191
First, we have to find the pattern rule. In an increasing number pattern like this one, we start by looking for an addition and/or multiplication relationship between the first two numbers. \begin{align*}11 + 12 = 23\end{align*}, so +12 is a possible pattern rule, but it doesn’t apply to the second number (23) and third number (47), since \begin{align*}23 + 12 = 35\end{align*}, not 47. The third number is 47, which is one more than \begin{align*}23 \times 2\end{align*}, so \begin{align*}(\times \ 2 + 1)\end{align*} is a possible pattern rule.
Once you think you have found the pattern rule, you have to test it out. Going back to test the first two numbers, you can see that \begin{align*}(\times \ 2 +1)\end{align*} also describes that relationship. Written algebraically, \begin{align*}y = 2x + 1\end{align*} is a pattern rule that explains the progression.
To find the missing number (the fourth number), we need to plug 47 (the third number) into the value for \begin{align*}x\end{align*}.
\begin{align*}y &= 2x + 1\\ y &= 2(47) + 1\\ y &= 94 + 1\\ y &= 95\end{align*}
To check that we have the right number, we can plug our answer (the fourth number) into the equation to see if we get the fifth number in the pattern.
\begin{align*}y &= 2x + 1\\ y &= 2(95) + 1\\ y &= 190 + 1\\ y &= 191\end{align*}
191 is the fifth number, so our solution is correct!
Our answer is 95.
Example
Fill in the missing piece of the pattern.
First, we examine the figures in the pattern. The picture is a pattern of repeating images, so we can look at how the images repeat and figure out the missing image in the pattern.
This is the missing image. We can check our work by examining the pattern. The design faces to the left, then to the right, then to the left, then to the right, then to the left, then to the right.
This is the correct image to complete the pattern.
Real Life Example Completed
Summer Hiking
Now that you have learned 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 found in nature. One of the chapters in the book talks all about patterns in nature. Kelly is fascinated.
“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's 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 and sequences. 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. Given this information, what is the next number in the pattern?
\begin{align*}8 + 13 = 21\end{align*}
What is the next one after that?
\begin{align*}13 + 21 = 34\end{align*}
Now let’s apply the Fibonacci pattern to nature. First, think about leaves on a tree. Many leaves do contain Fibonacci numbers. Look at a maple leaf. There are five main sections to this leaf, thirteen major "points" (circled), and 1 stem that holds the leaf on the tree. 1, 5, and 13 are all Fibonacci numbers.
What about trees?
Trees also work with Fibonacci numbers. The way that the tree moves from the trunk and divides into branches is a pattern of Fibonacci numbers. Let’s look at an image and a table of values.
The numbers on the left represent the levels of the tree. The numbers on the right represent the number of branches at each level of the tree. You can see that when we get to the top of the tree that the numbers become challenging to count, however you can trust the Fibonacci pattern for the correct number of branches!
\begin{align*}1 \qquad 1\\ 2 \qquad 3\\ 3 \qquad 5\\ 4 \qquad 8\\ 5 \qquad 13\\ 6 \qquad 21\end{align*}
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Pattern
- a sequence of numbers 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.
- Geometric Patterns
- visual patterns of geometric figures that follow a rule or repeat according to a pattern unit.
- Ascending Pattern
- a pattern that increases
- Descending Pattern
- a pattern that decreases
- Variable
- a letter used to represent an unknown quantity
- Expression
- combines variables, numbers and operations but does not relate equal values with an "equals" sign
Technology Integration
Khan Academy, Patterns and Equations
James Sousa, Arithmetic Sequences
Other Videos:
- http://www.linkslearning.org/Kids/1_Math/2_Illustrated_Lessons/5_Patterns/index.html – This is a great video on patterns.
- http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html – This is a great website on Fibonacci numbers for students to explore.
Time to 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
Directions: Use the pattern rule \begin{align*}y = 4x - 2\end{align*} to complete each table.
6.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
1 | 2 |
2 | |
3 | |
4 | 14 |
7.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
2 | |
4 | 14 |
6 | 22 |
8 |
8.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
9 | 34 |
7 | |
5 | 18 |
3 |
9. Use the function table to find the sixth place in the following pattern.
1, 3, 13, 63
position | \begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|---|
1st | 1 | 3 |
2nd | 3 | 13 |
3rd | 13 | 63 |
4th | 63 | |
5th | ||
6th |
10. Use the function table to find the sixth place in the following pattern.
1000, 500, 250, 125
position | \begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|---|
1st | 1000 | 500 |
2nd | 500 | 250 |
3rd | 250 | 125 |
4th | 125 | |
5th | ||
6th |
Directions: Write a rule for each number pattern.
11. 4, 7, 13, 25
12. 216, 196, 176, 156
13. 1, 7, 19, 43
14. 10,000, 1,000, 100, 10
Directions: Write the next number in each pattern for 15 – 18.
15. 4, 7, 13, 25, ____
16. 216, 196, 176, 156, ____
17. 1, 7, 19, 43, ____
18. 10,000, 1,000, 100, 10, ____
Directions: Fill in the missing number in each pattern. Then answer the questions.
19. 1, 5, 17, ___, 161
20. 7776, ___, 216, 36, 6
21. What is the rule for the pattern in number 19?
22. What is the rule for the pattern in number 20?
Directions: Use the following problem to answer the questions about patterns.
A sandwich shop showed the following sales in its first four days of business: $2.50, $7.50, $22.50, $67.50.
23. If the pattern continues, what will be the shop’s sales on the next day of business?
24. What will the shop’s sales be on the seventh day of business?
25. What is the rule to this pattern of sales?
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