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# Common Core Standards: Math

### The Number System 8.NS.A.1

1. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

Kyle and his mom were having a "discussion" about his curfew. After Kyle insisted that it be extended to midnight and his mom refused, he yelled, "Mom! I can't believe you! You're being so irrational!"

As much as Kyle loves his mom, he was certainly not expressing that love when he called her irrational. Irrational people, like irrational numbers, behave in a difficult manner and are hard to work with.

Your students probably haven't encountered irrational numbers until now, so don't be surprised if they throw a few tantrums in the beginning. Remind them that you aren't inventing these numbers just to be cruel (and you certainly don't have any say concerning their curfews).

Students should know that irrational numbers are numbers that calculators don't like, such as π or √3. They can't be expressed either as a fraction or as a terminating or repeating decimal, so we give them their very own symbols.

Of course, that means students should know what terminating decimals are and how to convert them to and from fractions. For instance, one-half or ½ is easy to convert to 0.5—just divide the numerator by the denominator. The numbers after the decimal point terminate neatly (that is, they don't go on forever), so it's called a terminating decimal. Because it terminates.

On the other hand, students should be aware that repeating decimals require a bit more work. For example, to convert  to a decimal, we divide the numerator by the denominator. What we get is a pattern of numbers after the decimal point: 0.181818181818….

See how they repeat? That explains their name, doesn't it? Students should know to write such a decimal as 0.18 to show which digits repeat, and be able to convert these decimals into fractions and vice versa.

If students encounter a decimal that don't terminate or repeat, then they're in the presence of an irrational number. If they complain about these numbers being such a pain, remind them that it's one of the consequences of growing up. Along with later curfews, world deems these young Padawans strong and mature enough to deal with the crazy antics of these numbers.

#### Drills

1. Classify ¾ as terminating, repeating, or irrational.

Terminating, equal to 0.75

We already know that if the number can be written as a fraction, it's a rational number. That means we can eliminate (D). Three divided by four is 0.75, so our answer is (A). You can thank your fourth grade math teacher for that one.

2. Classify 73 as terminating, repeating, or irrational.

Repeating, equal to 2.3

When we divide 7 by 3, we get 2.3333333…, which is a non-terminating but repeating decimal. That makes it rational.

3. Classify 0.1001000100001000001… as terminating, repeating, or irrational.

Irrational

Just because we can see what the pattern is does NOT mean that it repeats itself. The pattern of adding a 0 in between the 1s can keep going until there are a million zeros and then some, but if it doesn't repeat or terminate, then it's irrational.

4. Classify π as rational or irrational and explain why.

Irrational because it isn't equal to any terminating or repeating decimal

While π does have its own symbol, that's not why it's irrational. It's irrational because if we write it out as a decimal, it looks like this: 3.14159265358979323846264338… There's no repetition and clearly no termination. That's why it's irrational.

5. Classify as rational or irrational and explain why.

Rational because it equals 41

This one was easy, right? Don't let the radical confuse you. The square root of 16 is 4. Whether you put that 4 over 1 or not doesn't change its value.

6. Classify -2.3145 as rational or irrational, and explain why.

Rational because it repeats

Any repeating decimal is rational, regardless of whether there are numbers or negative signs before it. Since this number repeats at the end, we know that (A) is the right answer.

7. Classify 0.12345 as rational or irrational and explain why.

Rational because it terminates

Look at that number! After the 5, there's nothing else. The number has ended and that's all there is to it. It's a rational number because it terminates.

8. Classify 0 as rational or irrational and explain why.

Rational because it terminates

This may seem like we're trying to trick you, but we're really not. Zero is zero, a rational, terminating number. Also, don't try to claim that 0.0000… makes it irrational or repeating. After all, we don't add zeros to 1 to make it 1.0000… do we? We just say it terminates. And 0 terminates, too.

9. Classify 2021 as rational or irrational and explain why.

Rational because it terminates

Hopefully you haven't forgotten that if a number can be written as a fraction, it's automatically rational. Whether it repeats or not is something for decimals to take care of. If we divide 20 by 21, we end up with 0.952380952380953280… or 0.952380, which repeats. Obviously.

10. Which of the following numbers is irrational?