Lesson 20

Which number line accurately plots the value of $\sqrt{21}$? Explain your reasoning.

A

B

C

D

Rational numbers are fractions and their opposites.

1. All of these numbers are rational numbers. Show that they are rational by writing them in the form $\frac{a}{b}$ or $\text{-}\frac{a}{b}$ for integers $a$ and $b$.
   1. 6.28
   2. $\text{-}\sqrt{81}$
   3. $\sqrt{\frac{4}{121}}$
   4. -7.1234
   5. $0.\overline{3}$
   6. $\frac{1.1}{13}$
2. All rational numbers have decimal representations, too. Find the decimal representation of each of these rational numbers.
   1. $\frac{47}{1,000}$
   2. $\text{-}\frac{12}{5}$
   3. $\frac{\sqrt{9}}{6}$
   4. $\frac{53}{9}$
   5. $\frac{1}{7}$
3. What do you notice about the decimal representations of rational numbers?

Although $\sqrt{2}$ is irrational, we can approximate its value by considering values near it.

1. How can we know that $\sqrt{2}$ is between 1 and 2?
2. How can we know that $\sqrt{2}$ is between 1.4 and 1.5?
3. Approximate the next decimal place for $\sqrt{2}$.
4. Use a similar process to approximate the $\sqrt{5}$ to the thousandths place.