In this lesson, students learn about square roots. The warmup 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 noninteger side lengths.
Lesson overview
 2.1 Warmup: Notice and Wonder: Intersecting Circles (5 minutes)

2.2 Activity: One Square (15 minutes)
 Includes "Are you Ready for More?" extension problem

2.3 Activity: The Sides and Areas of Tilted Squares (15 minutes)
 There is a digital applet in this activity.
 Lesson Synthesis
 2.4 Cooldown: What Is the Side Length? (5 minutes)
Learning goals:
 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 noninteger side lengths.
 Determine the exact side length of a square and express it (in writing) using square root notation.
Learning goals (student facing):
 Let’s investigate some more squares.
Learning targets (student facing):
 I understand the meaning of expressions like \(\sqrt{25}\) and \(\sqrt3\).
 If I know the area of a square, I can express its side length using square root notation.
 I can explain what a square root is.
Required materials:
 fourfunction calculators
 geometry toolkits
Glossary:
 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.
 Access the complete Grade 8 glossary.
Standards:
 This lesson builds on the standard: CCSS.5.G.B MS.5.G
 This lesson builds towards the standards: CCSS.8.EE.A.2MS.8.EE.2CCSS.8.NS.A
IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 20172019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/mathcurriculum/.
Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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