Lesson 6: Finding Side Lengths of Triangles
About this lesson
This is the first of three lessons in which students investigate relationships between the side lengths of right and nonright triangles leading to the Pythagorean Theorem.
In the warmup for this lesson, students notice and wonder about 4 triangles. While there is a lot to notice, one important aspect is whether the triangle is a right triangle or not. This primes them to notice patterns of right and nonright triangles in the other activities in the lesson. In the next two activities, students systematically look at the side lengths of right and nonright triangles for patterns (MP8). By the end of this lesson, they see that for right triangles with legs \(a\) and \(b\) and hypotenuse \(c\), the side lengths are related by \(a^2+b^2=c^2\). In the next lesson they will prove the Pythagorean Theorem.
Lesson overview
 6.1 Warmup: Which One Doesn’t Belong: Triangles (5 minutes)
 6.2 Activity: A Table of Triangles (15 minutes)

6.3 Activity: Meet the Pythagorean Theorem (10 minutes)
 Includes "Are you Ready for More?" extension problem
 Lesson Synthesis
 6.4 Cooldown: Does a Squared Plus b Squared Equal c Squared? (5 minutes)
Learning goals:
 Comprehend the term “Pythagorean Theorem” (in written and spoken language) as the equation \(a^2+b^2=c^2\) where \(a\) and \(b\) are the lengths of the legs and \(c\) is the length of the hypotenuse of a right triangle.
 Describe (orally) patterns in the relationships between the side lengths of triangles.
 Determine the exact side lengths of a triangle in a coordinate grid and express them (in writing) using square root notation.
Learning goals (student facing):

Let’s find triangle side lengths.
Learning targets (student facing):
 I can explain what the Pythagorean Theorem says.
Glossary:
 hypotenuse  The hypotenuse is the side of a right triangle that is opposite the right angle. It is the longest side of a right triangle. Here are some right triangles. Each hypotenuse is labeled.
 legs  The legs of a right triangle are the sides that make the right angle. The legs of a right triangle are the sides that make the right angle. Here are some right triangles. Each leg is labeled.
 Pythagorean Theorem  The Pythagorean Theorem describes the relationship between the side lengths of right triangles.The diagram shows a right triangle with squares built on each side. If we add the areas of the two small squares, we get the area of the larger square. The square of the hypotenuse is equal to the sum of the squares of the legs. This is written as \(a^2+b^2=c^2\).
 Access the complete Grade 8 glossary.
Standards:
 This lesson builds on the standards:CCSS.5.G.B.4MS.5.G.4 CCSS.7.G.ACCSS.8.EE.A.2MS.8.EE.2
 This lesson builds towards the standards:CCSS.8.G.BCCSS.8.G.B.6MS.8.G.6 MS.8.G
M 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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