Lesson 6

Squares and Square Roots

Lesson Narrative

The purpose of this lesson is for students to reason from the meaning of a square and the meaning of a square root to solve equations of the form \(x^2 = a\) and \(\sqrt{x} = a\) where \(a\) is positive. They use graphs to understand that positive numbers always have two square roots, one positive and one negative. They also consider why we might want \(\sqrt{x}\) to be a function rather than giving us both values that square to make \(x\). Then they use graphs to understand why equations like \(x=\sqrt{5}\) only have one solution. There is no focus on equation-solving procedures in this lesson, which is something students will turn their attention to and develop techniques for in the following lessons.

Note that there are certain claims like, “The equation \(x^2=\text- 1\) has no solution,” which would be more precisely stated as “The equation \(x^2=\text- 1\) has no real solution.” However, students don’t know about any numbers other than real numbers, so it does not make sense to make this distinction at this time. In upcoming lessons, students will expand their concept of number to include imaginary and complex numbers.

Students attend to precision when they reason about solutions to equations involving squares and square roots from the meaning of the \(\sqrt{}\) symbol (MP6).

Learning Goals

Teacher Facing

  • Comprehend that the symbol $\sqrt{x}$ denotes only the positive square root of $x$.
  • Explain why equations of the form $x^2 = a$ can have two solutions, while equations of the form $\sqrt{x} = a$ cannot have more than one solution.

Student Facing

  • Let’s compare equations with squares and square roots.

Learning Targets

Student Facing

  • I understand that the square root symbol means the positive square root.

CCSS Standards

Addressing