# Continuing Patterns With Fractions, Decimals & Whole Numbers

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

This lesson will have a quick review of the definitions of whole numbers, fractions, and decimals. Then, we will look at patterns of these types of numbers and the process of continuing these patterns based on identifying rules that a pattern follows.

## Fractions, Decimals, and Whole Numbers

Mathematicians love pizza! Not only because of how delicious it is, but also because pizza is a great tool for studying numbers. You see, if we let a whole pizza represent the number 1, then parts of that pizza can be thought of as parts of the number 1. In mathematics, these values represent three different types of numbers, and those are whole numbers, fractions, and decimals.

• Whole Numbers: The counting numbers with the number 0; 0, 1, 2, 3, …
• Fractions: These numbers are part of a whole number; a/b, where a and b are whole numbers.
• Decimals: These numbers are part of a whole number. They contain a decimal point with digits extending past the decimal point.

For example, 3 whole pizzas represent the whole number 3, or if a pizza is sliced in 8 equal-sized slices, then one slice represents the fraction 1/8, or the decimal 0.125.

Pretty neat, wouldn't you agree? Well, if you think that's neat, you'll be happy to hear that the excitement of relating pizzas and numbers doesn't stop there! Pizzas are also great for studying patterns with these numbers!

## Patterns with Fractions, Decimals, and Whole Numbers

A pattern of numbers is an arrangement of numbers that follows a specific rule or set of rules. Suppose that a certain pizza parlor can put four pizzas in the oven at a time. Therefore, after each batch of pizzas come out of the oven, the number of pizzas that the pizza parlor has made that day goes up by four. This is a pattern that continues throughout the day.

Notice that the running total number of pizzas made can be written as a sequence, or list, of whole numbers.

• 0, 4, 8, 12, 16, …

We see this is a pattern of whole numbers! Do you see a rule that the pattern follows? If you are thinking that each consecutive number goes up by 4, then you are correct!

Based on this, what do we think would be the next number in the sequence after 16? Hmm…well, the next number should be 4 more than 16.

• 16 + 4 = 20

It looks like 20 would be the next number in the pattern. Yum! That's a lot of pizza!

Now, let's suppose that you have a group of friends over, so you order a pizza to eat that is sliced into 10 equal-sized pieces. In other words, one slice of pizza represents the fraction 1/10, and we can represent the whole pizza as 10/10, or 1.

Each time somebody eats a slice of pizza, the amount of pizza goes down by 1/10, so the amount of pizza left after each slice is eaten, one by one, can be put into a sequence, or pattern, of fractions.

• 10/10, 9/10, 8/10, 7/10, …

Ah! We've got a pattern of fractions! Do you see the rule that this pattern follows, and can you use it to continue the pattern? I bet you can! We see that each term in the pattern goes down by 1/10, so to find the next term in the sequence, we simply subtract 1/10 from 7/10.

• 7/10 - 1/10 = 6/10

Then to get the next term, we subtract 1/10 from 6/10, and we continue this pattern until there are no more pieces of pizza left.

• 10/10, 9/10, 8/10, 7/10, 6/10, 5/10, 4/10, 3/10, 2/10, 1/10, 0/10

We can also look at this pattern of fractions as a pattern of decimals. If we write the fractions as decimals, we have the following sequence:

• 1.0, 0.9, 0.8, 0.7, …

Notice this still follows the pattern of each term going down by 1/10. It's just displayed in decimal form, so each term goes down by 0.1, and that is the rule that we can use to continue this pattern of decimal numbers.

• 1.0, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2, 0.1, 0.0

## More Examples

Let's just look at a few more examples. Consider the following patterns, and let's figure out what number comes next in each of them.

1. 2, 4, 8, 16, …
2. 1/1, 1/2, 1/3, 1/4, 1/5, …
3. 0.1, 0.12, 0.123, 0.1234, …

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