# Lesson 4

Square Roots on the Number Line

### Lesson Narrative

In this lesson, students begin to transition from understanding square roots simply as side lengths to recognizing that all square roots are specific points on the number line. This understanding takes time to develop because students have previously only worked with rational numbers, which can be found by dividing the segment between two numbers into equal intervals. In the first activity, they still find $$\sqrt{10}$$ by relating it to the side length of a square of area 10 square units, but then are asked to approximate the value of $$\sqrt{10}$$ to the nearest tenth. In the second activity, students find a decimal approximation for $$\sqrt{3}$$ by looking at areas and also computing squares of numbers. This lesson shows students that irrational numbers are numbers—specific points on the number line—and we can find them by rotating a tilted square until it is sitting "flat." This is the conceptual foundation for the approximation work in the next lesson.

### Learning Goals

Teacher Facing

• Calculate an approximate value of a square root to the nearest tenth, and represent the square root as a point on the number line.
• Determine the exact length of a line segment on a coordinate grid and express the length (in writing) using square root notation.
• Explain (orally) how to verify that a value is a close approximation of a square root.

### Student Facing

Let’s explore square roots.

### Student Facing

• I can find a decimal approximation for square roots.
• I can plot square roots on the number line.

### Glossary Entries

• irrational number

An irrational number is a number that is not a fraction or the opposite of a fraction.

Pi ($$\pi$$) and $$\sqrt2$$ are examples of irrational numbers.

• rational number

A rational number is a fraction or the opposite of a fraction.

Some examples of rational numbers are: $$\frac74,0,\frac63,0.2,\text-\frac13,\text-5,\sqrt9$$

• square root

The square root of a positive number $$n$$ is the positive number whose square is $$n$$. It is also the the side length of a square whose area is $$n$$. We write the square root of $$n$$ as $$\sqrt{n}$$.

For example, the square root of 16, written as $$\sqrt{16}$$, is 4 because $$4^2$$ is 16.

$$\sqrt{16}$$ is also the side length of a square that has an area of 16.