Lesson 5Reasoning About Square Roots

Learning Goal

Let’s approximate square roots.

Learning Targets

When I have a square root, I can reason about which two whole numbers it is between.

Lesson Terms

square root

Warm Up: True or False: Squared

Decide if each statement is true or false.

${(\sqrt{5})}^{2} = 5$
${(\sqrt{9})}^{2} = 3$
$7 = {(\sqrt{7})}^{2}$
${(\sqrt{10})}^{2} = 100$
$(\sqrt{16}) = 2^{2}$

Activity 1: Square Root Values

What two whole numbers does each square root lie between? Be prepared to explain your reasoning.

$\sqrt{7}$
$\sqrt{23}$
$\sqrt{50}$
$\sqrt{98}$

Are you ready for more?

Can we do any better than “between 3 and 4” for $\sqrt{12}$ ? Explain a way to figure out if the value is closer to 3.1 or closer to 3.9.

Activity 2: Solutions on a Number Line

The numbers $x$ , $y$ , and $z$ are positive, and $x^{2} = 3$ , $y^{2} = 16$ , and $z^{2} = 30$ .

Plot $x$ , $y$ , and $z$ on the number line. Be prepared to share your reasoning with the class.
Plot $- \sqrt{2}$ on the number line.

A number line that shows the integers from negative 3 to 9

Lesson Summary

In general, we can approximate the values of square roots by observing the whole numbers around it, and remembering the relationship between square roots and squares. Here are some examples:

$\sqrt{65}$ is a little more than 8, because $\sqrt{65}$ is a little more than $\sqrt{64}$ and $\sqrt{64} = 8$ .
$\sqrt{80}$ is a little less than 9, because $\sqrt{80}$ is a little less than $\sqrt{81}$ and $\sqrt{81} = 9$ .
$\sqrt{75}$ is between 8 and 9 (it’s 8 point something), because 75 is between 64 and 81.
$\sqrt{75}$ is approximately 8.67, because ${8.67}^{2} = 75.1689$ .

A number line with the numbers 8 through 9, in increments of zero point 1, are indicated. The square root of 64 is indicated at 8. The square root of 65 is indicated between 8 and 8 point 1, where the square root of 65 is closer to 8 point 1. The square root of 75 is indicated between 8 point 6 and 8 point 7, the square root of 75 is closer to 8 point 7. The square root of 80 is indicated between 8 point 9 and 9, where the square root of 80 is closer to 8 point 9. The square root of 81 is indicated at 9.

If we want to find a square root between two whole numbers, we can work in the other direction. For example, since $22^{2} = 484$ and $23^{2} = 529$ , then we know that $\sqrt{500}$ (to pick one possibility) is between 22 and 23.

Many calculators have a square root command, which makes it simple to find an approximate value of a square root.

Lesson 5Reasoning About Square Roots

Learning Goal

Learning Targets

Lesson Terms

Warm Up: True or False: Squared

Problem 1

Activity 1: Square Root Values

Problem 1

Are you ready for more?

Problem 1

Activity 2: Solutions on a Number Line

Problem 1

Lesson Summary