# Common Core Standards: Math

### The Number System 8.NS.A.2

2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

A long, long, time ago in a place called Ancient Greece, the life of the people was very different from ours. There were no electronics: no DVDs, no Xboxes, no cell phones—not even email. There wasn't all that much to do other than invent the Olympics and think about math. So, once the Olympics were over, lots of people sat around and talked about math.

One of those ancient Greeks was named Zeno. He came up with the following notion:

"Before I can get to the pie on the other side of the room, I must first walk halfway there." (Sounds reasonable, right?) "But before I can do that," he said, "I must go a quarter of the way." (Still good with us.) "But first," Zeno said, "I must go an eighth of the way, and before that, a sixteenth."

Okay, where's he going with this? Well, nowhere, really. According to Zeno's Paradox, it's impossible to ever move at all, since one must always complete half the trip before he or she can go the rest of the way. That renders Zeno immobile and pie-less. How sad.

Similarly, it might be difficult for your students to see exactly what those pesky irrational numbers are equal to. Well, here's where Zeno's Paradox actually gets us places.

Let's say we want to approximate the square root of 2. We know it's between 1 (which is the square root of 1) and 2 (which is the square root of 4.) So it's got to be one-point-something.

If we square 1.5, we get 2.25. So the square root of 2 is a little less than 1.5. But if we square 1.4, we get 1.96. Since 2 is between 1.96 and 2.25, the square root of 2 is between 1.4 and 1.5. We could keep doing this all day, getting closer each time to the true value of the square root of 2, which is an irrational number.

Students should be able to come up with similar approximations for just about any irrational number they might encounter. The goal here is for students to know more or less where irrational numbers lie on the number line, not to have them find the exact location of π. (It's probably still across the room from Zeno. It's not like he moved it or anything.)

#### Drills

1. What is the approximate value of ?

2.8

The square root of 9 is 3, so we know that the square root of 8 is slightly less than 3. We can also use our knowledge that the square root of 2 is between 1.4 and 1.5, and is equal to . Either way, none of the other answers make sense.

2. What is the approximate value of ?

3.1

The square root of 10 should be just a little more than the square root of 9, which equals 3. We also know that 42 = 16, so the square root of 10 should be a little above 3 and definitely not above 4. The only answer that makes sense is (B).

3. What is the approximate value of ?

4.5

The square root of 20 should fall between , which is 4, and , which is 5. The only choice that falls in between these two numbers is (C).

4. What is the approximate value of ?

5.9

It only takes 52 = 25 to know that the square root of 35 will be greater than 5. In fact, it's so much greater than 5 that it's almost 6. We know that because 62 = 36, so we can estimate  to be about 5.9, or (D).

5. What is the approximate value of ?

8.1

The square root of 65 should be just a little more than the square root of 64. So the answer should be just a little more than 8.

6. Which is the best approximation of the square root of 51?

7.2

Knowing that 72 = 49 is helpful because we can eliminate at least (C) and (D). If we actually compute the other two, we get 7.12 = 50.41 and 7.22 = 51.84. Since 7.1 is closer, that is the best approximation.

7. Which is the best approximation for the square root of 75?

8.8

We know that the answer should be between 8 and 9 because 82 = 64 and 92 = 81. Since 75 is closer to 81 than 64, we can also assume that it'll be a little closer to 9 than 8. If we actually calculate the values, 8.72 = 75.69, while all the other choices are further from 75.

8. Which is the best approximation for π2?

1.6

If we use 3.14 to estimate π, we can calculate 3.14 ÷ 2 = 1.57, which is closest to 1.6. We could also estimate π using 22 ÷ 7 and then say that π2 is the same as 22 ÷ 14, but that isn't necessarily easier. Whatever floats your boat, though.

9. Which of the following is the best approximation for π2?

9.9

Again, we can use either 3.14 to estimate π. Since 32 = 9, we know it has to be above 9—and probably by more than 0.2. Knowing that  should be enough to choose (D).

10. Is there a limit to the number of decimal places you can approximate an irrational number?

No, that's why they're called irrational numbers