The purpose of this activity is for students to connect values expressed using square roots with values expressed in decimal form by determining the length of a diagonal line segment on a grid. 

Monitor for students who use the following strategies to find the length of the segment, ordered to show students a reasoning strategy that produces an exact square root length followed by concrete measurement strategies that help students understand the value in a decimal form that they may be more familiar with:

• Draw a tilted square and find the area.
• Use tracing paper.
• Use a compass to make a circle.

The purpose of this discussion is for students to see multiple ways the length of a line segment can be represented. This helps students transition from thinking of square roots as side lengths to thinking about them as values that can be plotted on the number line.

Invite previously selected students to share how they determined the length of the line segment. Sequence the discussion of the methods in the order listed in the Activity Narrative. If possible, record and display their work for all to see.

Since the task did not specify whether students should find an exact or approximate side length, some students will draw a square and use the area to find the exact side length, which is a familiar strategy. Other students may use tracing paper to use the number line as a ruler. Students who use a compass are finding another way to use the number line as a ruler.

Connect the different responses to the learning goals by asking questions, such as:

• “Which of these methods gives the most accurate length of the line segment?” (Finding the area of the square and taking the square root will give an exact answer.)
• “What do the different methods tell us about the exact value of ?” (It is about 3.1.)
• "How could you use the compass method to approximate the side lengths of other squares?" (Use the compass to measure a side length, then set the point at and draw a circle, marking where it crosses the number line.)

In previous activities and lessons, students used the areas of squares with whole number side lengths to find an approximation for the square root of an integer. In this activity, students start with the square root of an integer and then use a drawn square to explain why a given approximation of the square root is reasonable or not (MP3). 

The goal of this discussion is to make sure students understand both a visual and an algebraic strategy for checking the value of square root approximations and to connect this thinking to the number line. 

Display the diagram from the Task Statement for all to see. Invite 1–2 students to share their reasoning for why 2.5 is not a very good approximation for . Emphasize that squaring a point on the number line can be visualized as the area of a literal square sitting on the number line. This can help us estimate the value of a square root. 

Ask students, “2.5 might be a good approximation for the square root of what number?”. After a brief quiet think time, invite students to share their values. If not mentioned by students, make sure these two strategies are brought up:

• Estimate the area of the square by decomposing and rearranging squares and rectangles to get an area of 6.25 square units.
• Find the exact value by squaring. Since , the square root of 6.25 is 2.5.