Lesson 2

Side Lengths and Areas

Lesson Narrative

In this lesson, students learn about square roots. The warm-up helps them see a single line segment as it relates to two different figures: as a side length of a triangle and as a radius of a circle. In the next activity, they use this insight to estimate the side length of a square via a geometric construction that relates the side length of the square to a point on the number line, and verify their estimate using techniques from the previous lesson. Once students locate the side length of the square as a point on the number line, they are formally introduced to square roots and square root notation:

\(\sqrt{a}\) is the length of a side of a square whose area is \(a\) square units.

In the final activity, students use the graph of the function \(y = x^2\) to estimate side lengths of squares with integer areas but non-integer side lengths.

Learning Goals

Teacher Facing

  • Comprehend the term “square root of $a$” (in spoken language) and the notation $\sqrt{a}$ (in written language) to mean the side length of a square whose area is $a$ square units.
  • Create a table and graph that represents the relationship between side length and area of a square, and use the graph to estimate the side lengths of squares with non-integer side lengths.
  • Determine the exact side length of a square and express it (in writing) using square root notation.

Student Facing

Let’s investigate some more squares.

Learning Targets

Student Facing

  • I can explain what a square root is.
  • If I know the area of a square, I can express its side length using square root notation.
  • I understand the meaning of expressions like $\sqrt{25}$ and $\sqrt{3}$.

CCSS Standards

Building On

Addressing

Building Towards

Glossary Entries

  • square root

    The square root of a positive number \(n\) is the positive number whose square is \(n\). It is also the the side length of a square whose area is \(n\). We write the square root of \(n\) as \(\sqrt{n}\).

    For example, the square root of 16, written as \(\sqrt{16}\), is 4 because \(4^2\) is 16.  

    \(\sqrt{16}\) is also the side length of a square that has an area of 16.