Lesson 2

The purpose of this warm-up 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.

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.

This activity is optional because it revisits below grade-level content. 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).

Be prepared to help students understand what exact means if they are unsure.

If students struggle with figure D, remind them of their work in the warm-up.

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.

This activity is optional because it revisits below grade-level 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.

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.