Lesson 2
Square Roots and Cube Roots
2.1: It’s a Square (5 minutes)
Warmup
The purpose of this warmup is to review the meaning of square root in the geometric context of the area of a square.
Look for students who use the Pythagorean Theorem to find that the side length of the square \(ABCD\) is \(\sqrt{13}\). This will be an important discussion point.
Student Facing
Find the area of square \(ABCD\).
Student Response
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Activity Synthesis
Briefly discuss students’ responses. Here are some possible approaches:
 The four triangles each have area 3 square units, and the big square has area 25 square units. So square \(ABCD\) has area 13 square units, because \(25  4\boldcdot 3 = 13\).
 Let \(s\) be the side length of square \(ABCD\). Then \(2^2 + 3^2 = s^2\). That means \(s^2 = 13\), which is the area of the square.
Ask students with different solutions to explain their approach. If not mentioned by students, make explicit the connection that the side length of a square with an area of 13 square units is \(\sqrt{13}\) units.
2.2: Squares and Their Side Lengths (15 minutes)
Optional activity
This activity is optional because it revisits below gradelevel content. Students build on the warmup to find areas of various squares and express their side lengths exactly in terms of square roots. Through this repeated reasoning, students see patterns relating area, length, and solutions to equations involving squares and square roots (MP8).
Launch
Be prepared to help students understand what exact means if they are unsure.
Supports accessibility for: Memory; Conceptual processing
Student Facing
 Complete the table with the area of each square in square units, and its exact side length in units.
figure A B C D E area side length  This table includes areas in square units and side lengths in units of some more squares. Complete the table.
area 9 23 89 side length 4 6.4
Student Response
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Student Facing
Are you ready for more?
In the first question, all of the squares have vertices at grid points.
 Is there a square whose vertices are at grid points and whose area is 7 square units? Explain how you know.
 Is there a square whose vertices are at grid points and whose area is 10 square units? Explain how you know.
Student Response
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Anticipated Misconceptions
If students struggle with figure D, remind them of their work in the warmup.
Activity Synthesis
These ideas are all connected:
 If a square has side length \(s\), then the area is \(s^2\).
 If a square has area \(A\), then the side length is \(\sqrt{A}\).
 The number \(\sqrt{a}\) is the number that squares to make \(a\). In other words, \((\sqrt{a})^2 = a\).
 We can also think of \(\sqrt{a}\) as a solution to the equation \(x^2 = a\).
To discuss these connections, consider asking students,
 “If a square has side length \(\sqrt{28}\) units, what is its area?” (28 square units)
 “If a square has an area of 28 square units, what is the side length?” (\(\sqrt{28}\) units)
 “What is a solution to the equation \(x^2=28\)?” (A solution is \(x=\sqrt{28}\).)
 “What is the value of \((\sqrt{28})^2\)?” (28)
 “What do all these questions have in common?” (All of these questions are essentially the same question. They are all asking about the relationship between \(\sqrt{28}\) and 28.)
Record students’ responses during the discussion for all to see, drawing squares where needed to help illustrate connections if students did not.
Design Principle(s): Support sensemaking
2.3: Cube It (15 minutes)
Optional activity
This activity is optional because it revisits below gradelevel content. Students extend their work with squares and square roots to review cubes and cube roots. As in the previous activity, students repeat the same kinds of calculations in order to notice patterns (MP8). Students also cube numbers to figure out which whole numbers a cube root lies between. They will use this kind of thinking in an upcoming lesson to approximate bases raised to fractional powers using a graph.
Launch
Supports accessibility for: Language; Socialemotional skills
Student Facing
 A cube has edge length 3 units. What is the volume of the cube?
 A cube has edge length 4 units. What is the volume of the cube?
 A cube has volume 8 units. What is the edge length of the cube?
 A cube has volume 7 units. What is the edge length of the cube?

\(\sqrt[3]{1,\!200}\) is between 10 and 11 because \(10^3 = 1,\!000\) and \(11^3 = 1,\!331\). Determine the whole numbers that each of these cube roots lies between:
 \(\sqrt[3]{5}\)
 \(\sqrt[3]{10}\)
 \(\sqrt[3]{50}\)
 \(\sqrt[3]{100}\)
 \(\sqrt[3]{500}\)
between  1 and 2  2 and 3  3 and 4  4 and 5  5 and 6  6 and 7  7 and 8  8 and 9 

Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
These ideas are all connected:
 If a cube has edge length \(s\), then the volume is \(s^3\).
 If a cube has volume \(V\), then the edge length is \(\sqrt[3]{V}\).
 The number \(\sqrt[3]{a}\) is the number you cube to get \(a\). In other words, \((\sqrt[3]{a})^3 = a\).
 We can also think of \(\sqrt[3]{a}\) as the solution to the equation \(x^3 = a\).
To discuss these connections, consider asking students,
 “If a cube has edge length \(\sqrt[3]{28}\) units, what is its volume?” (28 cubic units)
 “If a cube has a volume of 28 cubic units, what is the edge length?” (\(\sqrt[3]{28}\) units)
 “What is a solution to the equation \(x^3=28\)?” (A solution is \(x=\sqrt[3]{28}\).)
 “What is the value of \((\sqrt[3]{28})^3\)?” (28)
 “What do all these questions have in common?” (All of these questions are essentially the same question. They are all asking about the relationship between \(\sqrt[3]{28}\) and 28.)
Record students’ responses during the discussion for all to see, drawing cubes where needed to help illustrate connections if students did not.
Design Principle(s): Support sensemaking
Lesson Synthesis
Lesson Synthesis
In this lesson, students have reviewed square and cube roots in geometric contexts. Here are some questions for discussion:
 “What is the difference between \(\sqrt{10}\) and \(\sqrt[3]{10}\)?” (The square root of 10 is a number that squares to make 10, while the cube root of 10 is a number that cubes to make 10. In other words, \(\left(\sqrt{10}\right)^2=10\) and \(\left(\sqrt[3]{10}\right)^3=10\).)
 “Which is larger, \(\sqrt{10}\) or \(\sqrt[3]{10}\)? How do you know?” (The square root of 10 is larger because it takes fewer repeated factors to make 10. This is similar to reasoning that \(\frac{10}{2}\) is larger than \(\frac{10}{3}\), because 10 is cut into fewer, larger pieces.)
 “What is a solution to the equation \(x^3=10\)?” (\(x=\sqrt[3]{10}\))
 “What would \(\sqrt[5]{10}\) mean?” (It would be a number that, when raised to the power of 5, produces 10. In other words, \(\left(\sqrt[5]{10}\right)^5=10\).)
2.4: Cooldown  What is a Square Root? (5 minutes)
CoolDown
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Student Lesson Summary
Student Facing
If a square has side length \(s\), then the area is \(s^2\). If a square has area \(A\), then the side length is \(\sqrt{A}\). For a positive number \(b\), the square root of \(b\) is defined as the positive number that squares to make \(b\), and it is written as \(\sqrt{b}\). In other words, \(\left(\sqrt{b}\right)^2 = b\). We can also think of \(\sqrt{b}\) as a solution to the equation \(x^2 = b\). This square has an area of \(b\) because its sides have length \(\sqrt{b}\):
Similarly, if a cube has edge length \(s\), then the volume is \(s^3\). If a cube has volume \(V\), then the edge length is \(\sqrt[3]{V}\). The number \(\sqrt[3]{a}\) is defined as the number that cubes to make \(a\). In other words, \(\left(\sqrt[3]{a}\right)^3 = a\). We can also think of \(\sqrt[3]{a}\) as a solution to the equation \(x^3 = a\). This cube has a volume of \(a\) because its sides have length \(\sqrt[3]{a}\):