n this lesson, students begin to transition from understanding square roots simply as side lengths to recognizing that all square roots are specific points on the number line. This understanding takes time to develop because students have previously only worked with rational numbers, which can be found by dividing the segment between two numbers into equal intervals. In the first activity, they still find \(\sqrt{10}\) by relating it to the side length of a square of area 10 square units, but then are asked to approximate the value of \(\sqrt{10}\) to the nearest tenth. In the second activity, students find a decimal approximation for \(\sqrt{3}\) by looking at areas and also computing squares of numbers. This lesson shows students that irrational numbers are numbers—specific points on the number line—and we can find them by rotating a tilted square until it is sitting "flat." This is the conceptual foundation for the approximation work in the next lesson.
Lesson overview
 4.1 Warmup: Diagonals (5 minutes)
 4.2 Activity: Squaring Lines (10 minutes)

4.3 Activity: Square Root of 3 (10 minutes)
 Includes "Are you Ready for More?" extension problem
 Lesson Synthesis
 4.4 Cooldown: Approximating \(\sqrt{18}\) (5 minutes)
Learning goals:
 Calculate an approximate value of a square root to the nearest tenth, and represent the square root as a point on the number line.
 Determine the exact length of a line segment on a coordinate grid and express the length (in writing) using square root notation.
 Explain (orally) how to verify that a value is a close approximation of a square root.
Learning goals (student facing):
 Let’s explore square roots.
Learning targets (student facing):
 I can find a decimal approximation for square roots.
 I can plot square roots on the number line.
Required materials:
 compasses
 fourfunction calculators
 tracing paper
Glossary:
 Access the complete Grade 8 glossary.
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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