Essential Questions: What are the applications of the Pythagorean Theorem and its converse? What is the relationship among the lengths of the sides of a right triangle? What are the properties of special right triangles and how are these properties used? How can the Pythagorean Theorem be used to solve problems in life? Lesson Plans and Seeds Lesson Plan B.7: Apply the Pythagorean Theorem Lesson Seed B.7: Pythagorean Proof Download Seeds, Plans, and Resources (zip) Unit Overview Content Emphasis By Clusters in Grade 8 Progressions from Common Core State Standards in Mathematics Send Feedback to MSDE’s Mathematics Team
Lesson Plan B.7: Apply the Pythagorean Theorem
Lesson Seed B.7: Pythagorean Proof
Unit Overview
Content Emphasis By Clusters in Grade 8
Progressions from Common Core State Standards in Mathematics
Send Feedback to MSDE’s Mathematics Team
Lesson seeds are ideas that can be used to build a lesson aligned to the CCSS. Lesson seeds are not meant to be all-inclusive, nor are they substitutes for instruction. When developing lessons from these seeds, teachers must consider the needs of all learners. It is also important to build checkpoints into the lessons where appropriate formative assessment will inform a teachers instructional pacing and delivery.
For an in-depth discussion of the overarching, “big picture” perspective on student learning of content related to this unit, see:
The Common Core Standards Writing Team (10 September 2011). Progressions for the Common Core State Standards in Mathematics (draft), accessed at: http://ime.math.arizona.edu/progressions/
Vertical Alignment: Vertical curriculum alignment provides two pieces of information: (1) a description of prior learning that should support the learning of the concepts in this unit, and (2) a description of how the concepts studied in this unit will support the learning of additional mathematics.
Possible Organization of Unit Standards: This table identifies additional grade-level standards within a given cluster that support the over-arching unit standards from within the same cluster. The table also provides instructional connections to grade-level standards from outside the cluster.
Overarching Unit Standards
Supporting Standards within the Domain
Instructional Connections outside the Cluster
8.G.B.6: Explain a proof of the Pythagorean Theorem and its converse.
N/A
8.NS.A.1: Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
8.G.B.7:Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8.NS.A.2: Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., Π2).
8.EE.A.2: Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
8.EE.C.7b:Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational
Connections to the Standards for Mathematical Practice: This section provides examples of learning experiences for this unit that support the development of the proficiencies described in the Standards for Mathematical Practice. These proficiencies correspond to those developed through the Literacy Standards. The statements provided offer a few examples of connections between the Standards for Mathematical Practice and the Content Standards of this unit. The list is not exhaustive and will hopefully prompt further reflection and discussion.
In this unit, educators should consider implementing learning experiences which provide opportunities for students to:
Content Standards with Essential Skills and Knowledge Statements and Clarifications: The Content Standards and Essential Skills and Knowledge statements shown in this section come directly from the Maryland State Common Core Curriculum Frameworks. Clarifications were added as needed. Educators should be cautioned against perceiving this as a checklist. All information added is intended to help the reader gain a better understanding of the standards.
Standard
Essential Skills and Knowledge
Clarification
8.F.A.1: Explain a proof of the Pythagorean Theorem and its converse.
See the skills and knowledge that are stated in the Standard
proof of the Pythagorean Theorem and its converse: In any right triangle, the sum of the squares of the legs equals the square of the hypotenuse (leg2 + leg2 = hypotenuse2). The figure below shows the parts of a right triangle.
8.G.B.7: Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8.G.B.8:Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
The distance d between the points A = (x1, y1) and B = (x2, y2) is given by the formula:
The distance formula can be obtained by creating a triangle and using the Pythagorean Theorem to find the length of the hypotenuse. The hypotenuse of the triangle will be the distance between the two points.
This formula is an application of the Pythagorean Theorem for right triangles:
