Lesson 4Square Roots on the Number Line

Learning Goal

Let’s explore square roots.

Learning Targets

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

Lesson Terms

rational number
square root

Warm Up: Diagonals

What do you notice? What do you wonder?

A circle on a grid centered at (0,0) with a radius that has a point at (1,1).

Activity 1: Squaring Lines

Problem 1

A grid with a horizontal number line showing 0 and 1. A line extends up from 0.

Estimate the length of the line segment to the nearest tenth of a unit (each grid square is 1 square unit).

Problem 2

Find the exact length of the segment.

Activity 2: Square Root of 3

Diego said that he thinks that $\sqrt{3} \approx 2.5$ .

A grid with x axis 0 to 3 and y axis also 0 to 3. The is a square drawn in starting at (0,0) with a vertex at (2.5, 0).

Use the square to explain why 2.5 is not a very good approximation for $\sqrt{3}$ . Find a point on the number line that is closer to $\sqrt{3}$ . Draw a new square on the axes and use it to explain how you know the point you plotted is a good approximation for $\sqrt{3}$ .
Use the fact that $\sqrt{3}$ is a solution to the equation $x^{2} = 3$ to find a decimal approximation of $\sqrt{3}$ whose square is between 2.9 and 3.1.

Are you ready for more?

A farmer has a grassy patch of land enclosed by a fence in the shape of a square with a side length of 4 meters. To make it a suitable home for some animals, the farmer would like to carve out a smaller square to be filled with water, as in the figure.

What should the side length of the smaller square be so that half of the area is grass and half is water?

A green square with side lengths of 4 and a smaller blue square inside drawn from lower left corner vertex.

Lesson Summary

Here is a line segment on a grid. What is the length of this line segment?

A line segment slanted down from left to right. The right endpoint is 1 unit down and 2 units right from the left endpoint.

By drawing some circles, we can tell that it’s longer than 2 units, but shorter than 3 units.

Two circles that have the same center are drawn on a square grid with radii 2 and 3. A line segment slanted upward and to the left is drawn such that the bottom endpoint is the center of the two circles and is 1 unit down and 2 units right of the top endpoint of the line segment.

To find an exact value for the length of the segment, we can build a square on it, using the segment as one of the sides of the square.

The area of this square is 5 square units. (Can you see why?) That means the exact value of the length of its side is $\sqrt{5}$ units.

A square on a grid with side lengths equal to the hypotenuse of triangle with side lengths of 1 and 2 units. The square has an area of 5 square units.

Notice that 5 is greater than 4, but less than 9. That means that $\sqrt{5}$ is greater than 2, but less than 3. This makes sense because we already saw that the length of the segment is in between 2 and 3.

With some arithmetic, we can get an even more precise idea of where $\sqrt{5}$ is on the number line. The image with the circles shows that $\sqrt{5}$ is closer to 2 than 3, so let’s find the value of ${2.1}^{2}$ and ${2.2}^{2}$ and see how close they are to 5. It turns out that ${2.1}^{2} = 4.41$ and ${2.2}^{2} = 4.84$ , so we need to try a larger number. If we increase our search by a tenth, we find that ${2.3}^{2} = 5.29$ . This means that $\sqrt{5}$ is greater than 2.2, but less than 2.3. If we wanted to keep going, we could try ${2.25}^{2}$ and eventually narrow the value of $\sqrt{5}$ to the hundredths place. Calculators do this same process to many decimal places, giving an approximation like $\sqrt{5} \approx 2.2360679775$ . Even though this is a lot of decimal places, it is still not exact because $\sqrt{5}$ is irrational.