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.
Required Materials
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 $\sqrt{3}$.
Glossary Entries
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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.
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