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# Patterns

### Written by tutor Jeff S.

Understanding the words group and pattern will help us understand Algebraic and Geometric patterns.

A group is a number of things that we can see and touch that are related. They can be related based on any number of reasons. For example, the items may be in a group because they are the same color or size.

It’s important to remember that groups are made up of things we can touch and see. For example, we can’t have a group of “blue”; it doesn’t make sense! But, we can have a group of blue cars together in a parking lot.

The word pattern describes groups of things with characteristics that repeat in a predictable way. For example, our brain uses patterns to help us quickly make sense of the world around us. Imagine being shocked each time you saw green grass in front a house in a suburban neighborhood! Our brain makes things easy by recognizing that (almost) all grass in front of suburban homes is green.

Patterns help us to quickly learn new skills, too. Piano students soon learn to recognize the pattern of white and black keys on the keyboard. We can describe the pattern of the piano keys like this: “white, black, white, black, white, black, white, white”. We can also shorten this by using letters to represent each word: W, B, W, B, W, B, W, W.

Another word for pattern is sequence. We will use both words to learn about Algebraic and Geometric patterns.

## Algebraic Patterns

Algebraic patterns are number patterns with sequences based on addition or subtraction. In other words, we can use addition or subtraction to predict the next few numbers in the pattern, as long as two or more numbers are already given to us. Let’s look at an example:

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

We can use addition to figure out the next two numbers in this pattern. In the example, 1 + 2 = 3 and 2 + 3 = 5. We could say that the rule for this Algebraic pattern is "add the previous two numbers in the pattern together to find the next number."

So, we add 8 + 13 and get 21. Then, we add 13 + 21 and get 34. Our finished pattern looks like this:

1, 2, 3, 5, 8, 13, 21, 34.

Exercise 1: Look at each number sequence below. Use the numbers given to find the next numbers in the sequences.

4, 8, _____, 16, 20, 24, ______. What numbers complete this pattern? (Separate the two numbers with a comma.)

Answer: 12 and 28. We see that the first two numbers increase by 4. We can check out work by adding 4 to each of the numbers in the middle of the pattern to see if our answers match. The problems look like this:
16 + 4 = 20
20 + 4 = 24
Since 20 and 24 match the numbers in the middle of our pattern, we know we have found the right answers.

{12, 28|12,28|12 28|1228|12 and 28}
47, 43, 40, 38, 37, 33, ______, _______. What numbers complete this pattern? (Separate the two numbers with a comma.)

Answer: 30 and 28. This one is tricky! We know we will need to subtract because the numbers get smaller as we read from left to right. We find the answer by subtracting the number on the left from the number to its right. The problems in this pattern look like this:
47 – 43 = 4
43 – 40 = 3
40 – 38 = 2
38 – 37 = 1
37 – 33 = 4
Now we know that the rule for this pattern is to start by subtracting the first number by four, the next by three, the third by two, and the last by 1. Since we subtracted the last set of numbers by 4, we know that we need to subtract 3 and 2 to find the last two numbers in the pattern. The problems and answers look like this:
33 – 3 = 30
30 – 2 = 28

{30, 28|30,28|30 28|3028|30 and 28}

## Geometric Patterns

Geometric patterns are sequences of numbers with patterns that are based on multiplication and division. In other words, as long as we know two or more numbers in the pattern, we can use either multiplication or division to find missing numbers. Here’s an example:

128, 64, 32, ___, ___.

We know we will need to use division because the numbers get smaller as we read from left to right. Since the numbers are even, we begin by using the lowest even number (not including zero) we can think of - 2 - and dividing by that number to see if our answers match the numbers in the sequence. The problems look like this:

128/2 = 64
64/2 = 32

We found our rule! We divide by two to find the next number. When we divide 32 by 2 we get 16. When we divide 16 by 2 we get 8. Our answer would look like this:

128, 64, 32, 16, 8.

What would we do if our answers didn’t match? We would try dividing by another number - like 3 or 4 - until the answers matched the numbers in the pattern.

Exercise 2. Look at each number sequence below. Use the numbers given to find the next number in the sequences.

___, 27, 81, 243, 729. What number completes this pattern?

Answer: 9. We know that we will need to multiply to find the answer because the numbers get larger as we read from left to right. We can use algebra to help us find the missing number. The problem looks like this:
a*27=81
First, we divide both sides by 27 to isolate the variable. The problem looks like this:
27a/27 = 81/27

a = 3

So, we know we need to multiply each number by 3 to get the answer. Our multiplication tables tell us that 9 x 3 = 27. So, the answer is 9.

{9|nine}
30, 15, 7.5, 3.75, ____. What number completes this pattern?

Answer: 1.875. The numbers get smaller as we read from left to right. This tells us we will need to divide to find the rule for this pattern. Since the first number is even, we begin by dividing each number by the lowest even number we can think of (not including zero) - 2 - to see if we can find the answer. The problem looks like this:
30/2 = 15
15/2 = 7.5
7.5/2 = 3.75

We found our rule! The rule for this pattern is “divide each number by 2 to find the next number in the sequence”. To find our answer, we divide 3.75 by 2. The problem looks like this:
3.75/2 = 1.875

{1.875}

Exercise 3. Look at the group of numbers below. Use what you have learned about patterns to determine whether the sequence of numbers is an Algebraic or Geometric pattern. Type your answer in the box below. Be careful! This is a tricky one!

12, ___, 72, 216, 432, 1296. What number completes this pattern? What kind of pattern is this (algebraic or geometric)? Separate your answers with a comma.

Answer: 36 and geometric. The numbers get larger as we read from left to right. That tells us we’re going to use either addition or multiplication to find the rule for this pattern. Let’s check addition first.

We can subtract the numbers we know in the set that to see if addition will work. The problems look like this:
216 – 72 = 144
432 – 216 = 216
1,296 – 432 = 864

Since there are no consistent numbers or patterns of numbers we can add to the numbers we already see in our set, we move on to multiplication.

We can set up an algebra problem to find the answer. The problem looks like this:
a72 = 216
First, divide both sides by 72 to isolate the variable. The problem looks like this:
72a/72 = 216/72

a = 3

Check your work by multiplying 72 by 3. When we do that, we get 216! We’re on the right track! Let’s make sure that 3 works for the whole set.

Check your work by multiplying 216 by 3. When we do that, we get 648?? Uh oh . . . 3 doesn’t work.

Set up another Algebra problem to find the rest of the pattern. The problem looks like this:
a216=432
First, divide both sides by 216 to isolate the variable. The problem looks like this:
216a/216 = 432/216

a = 2

Check your work by multiplying 216 by 2. When we do that, we get 432! We found it! The rule for the pattern “multiply the first number by 3 and the second number by 2”.

Check your work by trying out the pattern on the last number in the sequence. If we’re correct, we should be able to multiply 432 by 3 and get the last number. When we do, we see that 432 x 3 = 1,296!

We learned that Algebraic patterns use addition and subtraction. Geometric patterns use multiplication and division. That means that this pattern is Geometric.

{36, geometric|geometric, 36|36 geometric|geometric 36}