Lesson 5
Reasoning About Square Roots
5.1: True or False: Squared (5 minutes)
Warmup
The purpose of this warmup is for students to analyze symbolic statements about square roots and decide if they are true or not based on the meaning of the square root symbol.
Launch
Display one problem at a time. Tell students to give a signal when they have an answer and a strategy. After each problem, give students 1 minute of quiet think time and follow with a wholeclass discussion.
Student Facing
Decide if each statement is true or false.
\(\left( \sqrt{5} \right)^2=5\)
\(\left(\sqrt{9}\right)^2 = 3\)
\(7 = \left(\sqrt{7}\right)^2\)
\(\left(\sqrt{10}\right)^2 = 100\)
\(\left(\sqrt{16}\right)= 2^2\)
Student Response
Teachers with a valid work email address can click here to register or sign in for free access to Student Response.
Activity Synthesis
Poll students on their responses for each problem. Record and display their responses for all to see. If all students agree, ask 1 or 2 students to share their reasoning. If there is disagreement, ask students to share their reasoning until an agreement is reached.
5.2: Square Root Values (10 minutes)
Activity
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, especially the fact that \(\sqrt{a}\) is a solution to the equation \(x^2=a\), rather than less efficient geometric methods (which may not even work). Students can draw a number line if that helps them reason about the magnitude of the given square roots, but this is not required. However the reason, students must construct a viable argument (MP3).
Launch
Do not give students access to calculators. Students in groups of 2. 2 minutes of quiet work time followed by a partner then a wholeclass discussion.
Supports accessibility for: Visualspatial processing; Organization
Student Facing
What two whole numbers does each square root lie between? Be prepared to explain your reasoning.

\(\sqrt{7}\)

\(\sqrt{23}\)

\(\sqrt{50}\)

\(\sqrt{98}\)
Student Response
Teachers with a valid work email address can click here to register or sign in for free access to Student Response.
Student Facing
Are you ready for more?
Can we do any better than “between 3 and 4” for \(\sqrt{12}\)? Explain a way to figure out if the value is closer to 3.1 or closer to 3.9.
Student Response
Teachers with a valid work email address can click here to register or sign in for free access to Extension Student Response.
Activity Synthesis
Discuss:
 “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 \(2<\sqrt5<3\).)
Once the class is satisfied with which two whole numbers the square roots lie between, ask students to think more deeply about their relationship. Give 1–2 minutes for students to pick one of the last two square roots and figure out which whole number the square root is closest to and to be ready to explain how they know. One possible misconception that could be covered here is that if a number is exactly halfway between two perfect squares, then the square root of that number is also halfway between the square root of the perfect squares. For example, students may think that \(\sqrt{26}\) is halfway between 4 and 6 since 26 is halfway between 16 and 36. It’s close, since \(\sqrt{26}\approx5.099\), but it’s slightly larger than “halfway.”
This is a good opportunity to remind students of the graph they made earlier showing the relationship between the side length and area of a square. The graph showed a nonproportional relationship, so making proportional assumptions about relative sizes will not be accurate.
Design Principles(s): Cultivate conversation; Maximize metaawareness
5.3: 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 wholeclass discussion.
Supports accessibility for: Conceptual processing
Design Principles(s): Optimize output (for explanation); Maximize metaawareness
Student Facing
The numbers \(x\), \(y\), and \(z\) are positive, and \(x^2 = 3\), \(y^2 = 16\), and \(z^2 = 30\).
 Plot \(x\), \(y\), and \(z\) on the number line. Be prepared to share your reasoning with the class.
 Plot \(\text \sqrt{2}\) on the number line.
Student Response
Teachers with a valid work email address can click here to register or sign in for free access to Student Response.
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
To approximate a square root, start by finding the whole numbers it lies between, and then try to get more accurate approximations.
 “How can we find the whole numbers that a square root lies between?” (Look at the squares of whole numbers whose squares are greater than and less than the number inside the square root symbol, like 121 and 144 for \(\sqrt{130}\).)
 “How can we get a better approximation?” (Test values between those two whole numbers.)
5.4: Cooldown  Betweens (5 minutes)
CoolDown
Teachers with a valid work email address can click here to register or sign in for free access to CoolDowns.
Student Lesson Summary
Student Facing
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 \(8.67^2=75.1689\).
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.