Lesson 4

Square Roots on the Number Line

4.1: Notice and Wonder: Diagonals (5 minutes)

Warm-up

This warm-up transitions from work in previous lessons and prepares students to locate square roots on a number line in this lesson. Students must use the structure of the circle to relate the length of the segment to a point on the number line (MP7).

Launch

Arrange students in groups of 2. Tell students that they will look at an image, and their job is to think of at least one thing they notice and at least one thing they wonder. Display the image for all to see. Ask students to signal when they have noticed or wondered about something. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Student Facing

What do you notice? What do you wonder?

unit circle graphed on coordinate grid with center at origin. line segment from origin to the point 1 comma 1 on circle drawn.

Student Response

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Activity Synthesis

Ask students what the exact length is (it should be familiar to them from earlier lessons). The focus of the discussion is how you can see the decimal approximation from the diagram by looking at where the circle intersects an axis.

4.2: Squaring Lines (10 minutes)

Activity

In this activity, students determine the length of a “diagonal” line segment on a grid. Students can give an exact value for the length of the line segment by finding the area of a square and writing the side length using square root notation. The goal of this activity is for students to connect values expressed using square roots with values expressed in decimal form—a form they are more familiar with.

Monitor for students who draw a tilted square for the first problem during the first two minutes of work time. Then monitor for students who use the following strategies to find the length of the segment:

  • drawing a square and finding the area
  • using tracing paper
  • using a compass to make a circle

Launch

For this activity, it is best if students do not have access to a calculator with a square root button. If student calculators do have a square root button that students are familiar with, tell students that their explanations about their answers to the second problem need to dig deeper than pressing a button. In later lessons, however, they will be able to use it.

Before students begin, remind students that “exact length” means it can’t just be an approximation, so if it is not a rational number, we should write it with square root notation. For example, a square with area 17 has a side length of exactly \(\sqrt{17}\), which is a little larger than 4, since \(4^2=16\).

Begin by displaying the diagram for all to see. Ask students how this diagram is similar and how it is different from the diagram in the warm-up. Then 2–3 minutes after students begin working, pause the class and select a previously identified student who drew a square on the grid to share what they did and why. Give 2–3 minutes work time to finish the problems followed by a whole-class discussion.

Representation: Internalize Comprehension. Activate or supply background knowledge. Provide students with access to tracing paper, straight edges, and compasses to support information processing in estimating the length of the line segment.
Supports accessibility for: Visual-spatial processing; Organization

Student Facing

number line on grid, 0 and 1 plotted. line segment = square root 10 drawn.
  1. Estimate the length of the line segment to the nearest tenth of a unit (each grid square is 1 square unit).
  2. Find the exact length of the segment.

Student Response

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Activity Synthesis

Select students to present in this sequence:

  • Someone who drew a square and used the area to find the exact side length.
  • Someone who used tracing paper. This is essentially like the number line as a ruler. Ask students what this tells us about the exact value we found with the square. (\(\sqrt{17}\) is about 3.1.)
  • Someone who used a compass to find the approximate side length. This is a more formal geometric construction, but it is just another way to use the number line as a ruler.
Speaking, Listening: MLR8 Discussion Supports. As students share their methods for finding the exact length of the line segment, press for details in students’ reasoning by asking how they know the figure they drew in the coordinate plane is a square. Listen for and amplify the language students use to describe the important features of the square (e.g., opposite sides are equal, opposite sides are parallel, each angle is \(90^\circ\)). Then ask students to explain why the side length of the tilted square must be \(\sqrt{10}\). This will support rich and inclusive discussion about strategies for finding the exact length of a line segment in the coordinate plane.
Design Principle(s): Support sense-making

4.3: Square Root of 3 (10 minutes)

Activity

In previous activities and lessons, students found the exact area of a square in order to find an approximation for the square root of an integer. In this activity, students start with a square root of an integer, and draw a square to verify that a given approximation of the square root is reasonable. This is the first time students have seen or drawn squares that do not have vertices at the intersection of grid lines, so it may take them a few minutes to make sense of the new orientation.

Teacher Notes for IM 6–8 Math Accelerated
After arranging students in groups of 2, display for all to see \(\sqrt{7}\) and \(\sqrt{50}\) and ask, “What two whole numbers does each square root lie between? Be prepared to explain your reasoning.” After a brief quiet think time, invite students to share their reasoning. For example, students may make a list of perfect squares and then find which two the number was between. Once 2–3 strategies are shared, have students continue with the activity as indicated in the launch.

Launch

Give students access to four-function calculators. Display the diagram for all to see. Ask students what is the same and what is different about this diagram and diagrams they have seen in earlier activities. Arrange students in groups of 2. Give students 2–3 minutes of quiet work time followed by partner and whole-class discussions.

Action and Expression: Internalize Executive Functions. Provide students with a graphic organizer for data collection and organizing information.
Supports accessibility for: Language; Organization
Writing, Speaking, Listening: MLR1 Stronger and Clearer Each Time. After students have had time to identify a point on the number line that is closer to \(\sqrt{3}\), ask them to write a brief explanation of their reasoning. Give students time to meet with 2–3 partners, to share and get feedback on their writing. Display prompts that students can ask each other that will help students strengthen their ideas and clarify their language. For example, “How do you know that \(\sqrt{3}\) is between 1.5 and 2?” and “How did you determine the area of the square that you drew?” Students can borrow ideas and language from each partner to refine and clarify their original explanation. This will help students refine both their ideas and their verbal and written output.
Design Principles(s): Optimize output (for explanation); Maximize meta-awareness

Student Facing

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

quadrant 1, x axis, 0 to 3 by 1's. y axis, 0 to 3, by 1's. square with side length = 2 point 5 graphed.
  1. 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 \(\sqrt{3}\).
  2. 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.

Student Response

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Student Facing

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, side length = 4. Blue square in bottom left corner of green square, side length = 2.5.

Student Response

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Activity Synthesis

Invite one or two students to share their squares. Then tell students, “The square of a point on the number line can be visualized as the area of a literal square. This can help us estimate the value of a square root. Simply squaring the number can as well. Let’s check the squares of some numbers that are potential approximations of \(\sqrt{3}\).”

Then ask students to suggest decimal approximations, and check together as a class by finding their squares. Students should be using each guess to make a better guess next. For example, if they try 1.5, then the square is 2.25, which is too low. This suggests trying bigger. Because we know that 2 is too big (because \(2^2 = 4\), it should be somewhere in between 1.5 and 2.) For example, students might suggest this order:

\(1^2 = 1\) and \(2^2=4\)

\(1.5^2 = 2.25\)

\(1.8^2 = 3.24\)

\(1.7^2 = 2.89\)

\(1.72^2 = 2.9584\)

\(1.73^2 = 2.9929\)

So 1.73 is a pretty good approximation of \(\sqrt{3}\).

4.4: Solutions on a Number Line (10 minutes)

Activity

The purpose of this activity is for students to use rational approximations of irrational numbers to place both rational and irrational numbers on a number line and to reinforce the definition of a square root as a solution to the equation of the form \(x^2=a\). This is also the first time that students have thought about negative square roots. 

Launch

Do not provide students with access to calculators. Students in groups of 2. 2 minutes of quiet work time followed by a partner, then a whole-class discussion.

Representation: Develop Language and Symbols. Use virtual or concrete manipulatives to connect symbols to concrete objects or values. For example, use a kinesthetic representation of the number line on a clothesline. Students can place and adjust numbers on folder paper or cardstock on the clothesline in a hands-on manner.
Supports accessibility for: Conceptual processing
Writing, Speaking, Listening: MLR1 Stronger and Clearer Each Time. After students have had time to plot \(x\), \(y\), and \(z\) on the number line, ask them to write a brief explanation of their reasoning for each number on their paper. Ask each student to meet with 2–3 other partners in a row for feedback. Provide students with prompts for feedback that will help them strengthen their ideas and clarify their language (e.g., “How do you know that \(z = \sqrt{30}\)?” and “How do you know that \(\sqrt{30}\) is between 5 and 6?”, etc.). Students can borrow ideas and language from each partner to refine and clarify their original explanation. This will help students revise and refine both their ideas and their verbal and written output.
Design Principles(s): Optimize output (for explanation); Maximize meta-awareness

Student Facing

The numbers \(x\), \(y\), and \(z\) are positive, and \(x^2 = 3\), \(y^2 = 16\), and \(z^2 = 30\).

A numbre line that shows the integers from negative 3 to 9
  1. Plot \(x\), \(y\), and \(z\) on the number line. Be prepared to share your reasoning with the class.
  2. Plot \(\text- \sqrt{2}\) on the number line.

Student Response

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Activity Synthesis

Display the number line from the activity for all to see. Select groups to share how they chose to place values onto the number line. Place the values on the displayed number line as groups share, and after each placement poll the class to ask if students used the same reasoning or different reasoning. If any students used different reasoning, invite them to share with the class.

Conclude the discussion by asking students to share how they placed -\(\sqrt{2}\) and why.

Lesson Synthesis

Lesson Synthesis

The goal of this discussion is to check that students know how to approximate square roots. In the previous lesson, students learned that some square roots, \(\sqrt{2}\) in particular, are not rational. But they are still numbers, and we can reason about their approximate value using more familiar numbers. Here are some possible questions for discussion:

  • “How can you approximate the value of \(\sqrt{130}\)?” (\(\sqrt{130}\) is somewhere between 11 and 12 because \(11^2=121\) and \(12^2=144\).)
  • “So we know \(\sqrt{130}\) is somewhere between 11 and 12. Can we get more accurate than that? How?” (We could try squaring numbers from 11 to 12 like 11.1, 11.2, and so on to find the one closest to 130.)

4.5: Cool-down - Betweens (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

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.

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.

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.

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 \(\sqrt5\) units.

Notice that 5 is greater than 4, but less than 9. That means that \(\sqrt5\) 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.

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. 

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.