Common Core Standards: Math
- The Standard
- Sample Assignments
- Practice Questions
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.