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ICAS Mathematics - Paper D: Test Prep & Practice8 chapters | 64 lessons

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Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

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.

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!

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

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.

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

The first pattern is one of whole numbers. Notice that each consecutive term in the pattern doubles the term that came before it. That is our rule! To get to the next term, we multiply by 2. Therefore, the next term in the sequence would be 16 ⋅ 2 = 32.

The second pattern is a pattern of fractions. This pattern is a little different, because it is a pattern within the sequence. Notice that the denominator of each consecutive fraction goes up by 1. There's our rule! To continue the pattern, we simply add 1 to the denominator of 1/5 to get 1/6.

Lastly, the third pattern is a pattern of decimals, and once again, the pattern is within the sequence. It looks like each consecutive term extends the decimal one more digit as if we were counting, so the next term in the pattern would be 0.12345.

**Whole numbers** are the counting numbers along with the number 0. **Fractions** and **decimals** are parts of whole numbers, where fractions have the form *a*/*b*, where *a* and *b* are whole numbers, and decimals contain a decimal point with digits extending beyond it.

A **pattern** of whole numbers, fractions, or decimals is a sequence of these numbers that follows a specific rule or set of rules. When we are able to identify the rule that a given pattern follows, we can continue the pattern using that rule. This is extremely useful in the study of these types of numbers, as well as when these types of patterns show up in our daily lives, such as in the pizza examples. Has all of this pizza talk made you hungry? Let's celebrate our newly acquired knowledge with a fraction or decimal of a whole pizza!

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ICAS Mathematics - Paper D: Test Prep & Practice8 chapters | 64 lessons

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