The purpose of this activity is for students to think about square roots in relation to the two whole number values they are closest to. Students are encouraged to use numerical approaches to explain their thinking (MP3), especially the fact that is a solution to the equation . 

If needed, students can draw a number line to help them reason about the magnitude of the given square roots, but this is not required.

The goal of this discussion is to make sure students understand how to reason about the value of a square root by comparing it to square roots that are whole numbers. Ask students:

• “What strategy did you use to figure out the two whole numbers?” (I made a list of perfect squares and then found which two the number was between.)
• “Did anyone use inequality symbols when writing their answers?” (Yes, for the first problem, I wrote .)

When students are in agreement about which two whole numbers each square root lies between, ask students to think more deeply about the relationship by asking them to figure out which of the two whole numbers it is closer to. Give 1–2 minutes of quiet work time, then invite students to share their thinking.

If not brought up by students, highlight that if a number is exactly halfway between two perfect squares, it is not true that the square root of that number is also halfway between the square root of the perfect squares. For example, students may think that is halfway between 4 and 6 since 26 is halfway between 16 and 36. It’s close since , but it’s slightly more than “halfway.” 

If time allows, remind students of the graph they made in an earlier lesson showing the relationship between the side length and area of a square. The graph showed a non-proportional relationship, so making proportional assumptions about relative sizes will not be accurate.

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 an equation of the form . This is also the first time that students are asked to consider negative square roots. 

The purpose of this discussion is to reinforce the idea that irrational numbers are still numbers on the number line, though their location cannot be found by subdividing the unit interval into parts and taking of them in the way that rational numbers written as can be. 

Display the number line from the activity for all to see. Select groups to share how they chose to place values, recording them on the number line as they share. After each placement, survey the class and ask if students used the same or different reasoning. Invite any groups that used different reasoning to share with the class.

Conclude the discussion by asking students to share how they placed . (I found the approximate location of and then placed the same distance from 0 to the left.)