Number and Number Sense, Computation and Estimation 

Measurement and Geometry 

Probability, Statistics, Patterns, Functions, and Algebra 

SOL Review Materials 




Geometry
Focus: Problem Solving with 2 and 3Dimensional Figures
The student will
a) verify the Pythagorean Theorem; and
b) apply the Pythagorean Theorem. 


Essential Knowledge and Skills 
 Identify the parts of a right triangle (the hypotenuse and the legs).
 Verify a triangle is a right triangle given the measures of its three sides.
 Verify the Pythagorean Theorem, using diagrams, concrete materials, and measurement.
 Find the measure of a side of a right triangle, given the measures of the other two sides.
 Solve practical problems involving right triangles by using the Pythagorean Theorem.

Essential Understandings 
How can the area of squares generated by the legs and the hypotenuse of a right triangle be used to verify the Pythagorean Theorem? For a right triangle, the area of a square with one side equal to the measure of the hypotenuse equals the sum of the areas of the squares with one side each equal to the measures of the legs of the triangle. 

Vertical Articulation 
 SOL 6.12  determine congruence of segments/angles/polygons
 SOL 7.6  determine similarity of plane figures and
write proportions to express relationships
between similar quads and triangles

Instructional Materials 
 VDOE Enhanced Scope and Sequence Lesson: Pythagorean Theorem (PDF)  Working with the Pythagorean Theorem (Word)
 Notes: Pythagorean Theorem (doc)
 Notes: Pythagorean Theorem Constructivist Lesson Plan and Practice (pdf)
 Activity: Exploring the Pythagorean Theorem  Stations (doc)

Real World Lessons 
 Dan Meyer: Dollar Wall  How much cash is on the walls?
 Mathalicious: 51Foot Ladder  How high can you (safely) climb with a ladder?

Practice 
 Worksheet: Pythagorean Theorem Practice (doc)
 Worksheet: Pythagorean Theorem in the NFL (pdf)
 Quiz: Pythagorean Theorem and Perfect Squares (pdf)
 Worksheet: Pythagorean Theorem Applications (pdf)

SelfAssessments 
 SOL 8.10 MiniQuiz: Pythagorean Theorem

Skills 
 Brain Teaser  Same Hypotenuse from Illuminations.com
 Mathalicious  Viewmongus: Televisions. Smart phones. iPads. We’re surrounded by screens. We can’t escape them! But how much do we really know about these screens? In particular, how big are they?
 Mathematics Assessment Project:
 A05:Hopewell Geometry  Apprentice  The Hopewell people were Native Americans whose culture flourished in the central Ohio Valley about 2000 years ago. They constructed earthworks using right triangles. In this task, you will look at some of the geometrical properties of a Hopewell earthwork.
 A13:Temple Geometry  Apprentice  During the Edo period (16031867) of Japanese history, geometrical puzzles were hung in the holy temples as offerings to the gods and as challenges to worshippers. Here is one such problem for you to investigate.
 A15:Jane's TV  Apprentice  In this task, you will need to work out the actual dimensions of TV screens, which are sold according to their diagonal measurements.
 E04:Proofs Of The Pythagorean Theorem?  Expert  In this task, you will look at three different attempts to prove the Pythagorean theorem. Which is the best ‘proof’?
 E08:Pythagorean Triples  Expert  (3, 4, 5), (5, 12, 13), (7, 24, 25) and (9, 40, 41) are called Pythagorean Triples because they satisfy the condition c^2 = a^2 + b^2. In this task, you will investigate these, and other Pythagorean Triples
 E15:Circles and Squares  Expert  In this task, you must solve a problem about circles inscribed in squares.
 Pythagorean Theorem  Park Development (doc)

Videos 
 The Science of NFL Football: Pythagorean Theorem  NBC's Lester Holt explores the path a defender must take in order to tackle a ball carrier, and how this distancecalled the "angle of pursuit"can be calculated by using the Pythagorean Theorem to find the hypotenuse of a right triangle, and the distance of a defender's angle of pursuit.
 Rap Video: Pythagorean Theorem (YouTube)
 LearnAlberta: Exploring Square Roots Video (Video Interactive)
 LearnAlberta: Exploring the Pythagorean Theorem (Video Interactive)
 PH: Using the Pythagorean Theorem to find the hypotenuse
 PH: Using the Pythagorean Theorem to find the leg

Explore
Learning 
 Distance Formula  Activity A: Explore the distance formula as an application of the Pythagorean theorem. Learn to see any two points as the endpoints of the hypotenuse of a right triangle. Drag those points and examine changes to the triangle and the distance calculation.
 Geoboard: The Pythagorean Theorem: Build right triangles in an interactive geoboard and build squares on the sides of the triangles to discover the Pythagorean Theorem.
 Pythagorean Theorem  Activity A: Explore the Pythagorean Theorem using a dynamic right triangle. Examine a visual, geometric application of the Pythagorean Theorem, using the areas of squares on the sides of the triangle.
 Pythagorean Theorem  Activity B: Explore the Pythagorean Theorem using a dynamic right triangle. Examine a visual, geometric application of the Pythagorean Theorem, using the areas of squares on the sides of the triangle.

Websites 

VDOE Interventions 
 VDOE Algebra Readiness Curriculum Companion: Working with TwoDimensional Measurement  The
lessons focus on understanding and applying the Pythagorean Theorem. (pdf)

Common Core
Standards 

