Students build on the Warm-up 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).

If students do not yet correctly find the area of Figure D, consider saying:

• “Can you explain how you found the area of the central square in the Warm-up.”

• “I know the area is less than 9. Can you see how I know that?”

The purpose of the discussion is for students to recognize the relationship between a value and its square root.

To discuss these connections, consider asking students,

• “If a square has side length units, what is its area?” (28 square units)
• “If a square has an area of 28 square units, what is the side length?” ( units)
• “What is a solution to the equation ?” (A solution is .)
• “What is the value of ?” (28)
• “What do all these questions have in common?” (All of these questions are about the relationship between and 28.)
• “If a square has side length , what is the area of the square?” ()
• “If a square has area , what is the length of one of its sides?” ()
• “Explain what means.” (It is the value that squares to make .)
   

Record students’ responses during the discussion for all to see, drawing squares where needed to help illustrate connections if students did not.

Students extend their work with squares and square roots to review cubes and cube roots. 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.

The purpose of the discussion is for students to see the connection between a value and its cube root.

To discuss these connections, consider asking students,

• “If a cube has edge length units, what is its volume?” (28 cubic units)
• “If a cube has a volume of 28 cubic units, what is the edge length?” ( units)
• “What is a solution to the equation ?” (A solution is .)
• “What is the value of ?” (28)
• “What do all these questions have in common?” (All of these questions are about the relationship between and 28.)
• “If a cube has edge length , what is the volume of the cube?” ()
• “If a cube has volume , then what is the length of one of its edges?” ()
• “Explain what means.” (It is the value that cubes to make .)

Record students’ responses during the discussion for all to see, drawing cubes where needed to help illustrate connections if students did not.