Grade 8 Pythagorean Theorem

GRADE 8 PYTHAGOREAN THEOREM

Understand and apply the Pythagorean Theorem.  

Explain a proof of the Pythagorean Theorem and its converse.  

 Here is one ofmany proofs of the PythagoreanTheorem.

How does this prove the Pythagorean Theorem?

GRADE 8 PYTHAGOREAN THEOREM Apply the Pythagorean Theorem to determine unknown

side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

 FromKahnAcademy

GRADE 8 VOLUMESolve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.  

http://www.math.com

TURN AND TALK TO YOUR NEIGHBOR

What concepts and skills that HS Geometry have traditionally spent a lot of time on are now being introduced in middle school?

How does that change your ideas for focus in HS Geometry?

What concepts and skills do you predict will be areas of major focus in HS Geometry?

STRUCTURE OF THE HS GEOMETRY CONTENT STANDARDSCongruence (G-CO)Similarity, Right Triangles, and

Trigonometry (G-SRT)Circles (G-C)Expressing Geometric Properties with

Equations (G-GPE)Geometric Measurement and Dimension

(G-GMD)Modeling with Geometry (G-MG)

STRUCTURE OF THE HS GEOMETRY CONTENT STANDARDSCongruence (G-CO)• Experiment with transformations in the plane• Understand congruence in terms of rigid motions• Prove geometric theorems (required theorems

listed)• Theorems about Lines and Angles• Theorems about Triangles• Theorems about Parallelograms

Make geometric constructions (variety of tools and methods…by hand and using technology) (required constructions listed)

STRUCTURE OF THE HS GEOMETRY CONTENT STANDARDSSimilarity, Right Triangles, and Trigonometry

(G-SRT)• Understand Similarity in terms of similarity

transformations• Prove theorems involving similarity• Define trigonometric ratios and solve

problems involving right triangles• (+) Apply trigonometry to general triangles

• Law of Sines• Law of Cosines

STRUCTURE OF THE HS GEOMETRY CONTENT STANDARDSCircles (G-C)Understand and apply theorems about circles • All circle are similar• Identify and describe relationships among inscribed angles, radii, and chords.• Relationship between central, inscribed, and circumscribed angles• Inscribe angles on a diameter are right angles• The radius of a circle is perpendicular to the tangent where the radius intersects the circle

Find arc lengths and sectors of circles

STRUCTURE OF THE HS GEOMETRY CONTENT STANDARDSExpressing Geometric Properties with

Equations (G-GPE)• Translate between the geometric

description and the equation for a conic section

• Use coordinates to prove simple geometric theorems algebraically

STRUCTURE OF THE HS GEOMETRY CONTENT STANDARDSGeometric Measurement and Dimension (G-

GMD)• Explain volume formulas and use them to

solve problems• Visualize relationships between two-

dimensional and three-dimensional objects

Modeling with Geometry (G-MG)• Apply geometric concepts in modeling

situations

HS GEOMETRY CONTENT STANDARDSPrimarily Focused on Plane Euclidean GeometryShapes are studied Synthetically & Analytically• Synthetic Geometry is the branch of geometry

which makes use of axioms, theorems, and logical arguments to draw conclusions about shapes and solve problems

• Analytical Geometry places shapes on the coordinate plane, allowing shapes to defined by algebraic equations, which can be manipulated to draw conclusions about shapes and solve problems.

FORMAL DEFINITIONS AND PROOF

HS Students begin to formalize the experiences with geometric shapes introduced in K – 8 by

• Using more precise definitions• Developing careful proofs

When you hear the word “proof”, what do you envision?

INSTRUCTIONAL SHIFT: MORE FOCUS ON TRANSFORMATIONAL PERSPECTIVE

Congruence, Similarity, and Symmetry are understood

from the perspective of

Geometric Transformation

extending the work that was started in Grade 8

INSTRUCTIONAL SHIFT: MORE FOCUS ON TRANSFORMATIONAL PERSPECTIVERigid Transformations (translations, rotations,

reflections) preserve distance and angle and therefore result in images that are congruent to the original shape.

G-C0 Cluster Headings Revisited• Experiment with transformations in the plane• Understand congruence in terms of rigid

motions• Prove geometric theorems• Make geometric constructions

PROVING SIMILARITY VIA TRANSFORMATIONSDilation is a Non-Rigid Transformation that

preserves angle, but involves a scaling factor that affects the distance, which results in images that are similar to the original shape.

G-SRT Cluster Headings dealing with Similarity:

• Understand Similarity in terms of similarity transformations

• Prove theorems involving similarity

PROVING SIMILARITY VIA TRANSFORMATIONSFrom a transformational perspective…Two shapes are defined to be similar to

each other if there is a sequence of rigid motions followed by a non-rigid dilation that carries one onto the other.

A dilation formalizes the idea of scale factor studied in Middle School.

PROVE SIMILARITY BY TRANSFORMATIONSWhat non-rigid transformationproves that these trianglesare similar?What is the center of dilation?What is the scale factor of theDilation?

FIND SCALE FACTORS GIVEN A TRANSFORMATION

www.ck12.org Similarity Transformations Created by: Jacelyn O'Roark

CIRCLES IN ANALYTIC GEOMETRYG-GPE (Expressing Geometric Properties with Equations) Derive the equation of a circle given center (3,-2) and radius 6 using the

Pythagorean Theorem

Complete the square to find the center and radius of a circle with equation x2 + y2 – 6x – 2y = 26

Think of the time spent in Algebra I on factoringVersus completing the square to solve quadraticEquations. What % of quadratics can be solvedby factoring? What % of quadratics can be Solved by completing the square?Is completing the square using the area modelmore intuitive for students?

CONIC SECTIONS – CIRCLES AND PARABOLAS

• Translate between the geometric description and the equation for a conic section • Derive the equation of a parabola given a focus and directrix• Parabola – Note: completing the square to find the vertex of a parabola is in

the Functions Standards(+) Ellipses and Hyperbolas in Honors or Year 4

Sketch and derive the equation for the parabola withFocus at (0,2) and directrix at y = -2

Find the vertex of the parabola with equationY = x2 + 5x + 7

VISUALIZE RELATIONSHIPS BETWEEN 2-D AND 3-D OBJECTS

• Identify the shapes of 2-dimensional cross sections of 3-dimensional objects

VISUALIZE RELATIONSHIPS BETWEEN 2-D AND 3-D OBJECTS

• Identify 3-dimensional shapes generated by rotations of 2-dimensional objects

http://www.math.wpi.edu/Course_Materials/MA1022C11/volrev/node1.html

NORTH COUNTRY INSERVICE OUTLINE

• Review with Agreed Upon Expectations from 2-15-13 Inservice – Share Experiences

• Review of CCSSM Practice Standards – Share Experiences

• Presentation of How Geometry Unfolds over K – 12 in CCSSM

• Focus on Volume Standard in HS Geometry• Develop one unit focusing on HS Volume Standard

and Practice Standards

HS.GMD.A.1Give an informal argument for the

formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

INFORMAL ARGUMENT FOR AREA OF CIRCLEhttp://www.youtube.com/watch?v=7zoqL2iOpvo

Area of Circle GeoGebra Applethttp://www.geogebratube.org/student/m279?mobile=true

From Don Steward

HS.GMD.A.3Use volume formulas for cylinders,

pyramids, cones, and spheres to solve problems.★

Dan Meyer 3 Act: Popcorn PickerDan Meyer 3 Act: The Coffee CarrierDan Meyer 3 Act: You Pour, I ChooseAndrew Stadel 3 Act: Trashketball

MATHEMATICS ASSESSMENT PROJECTH.G-GMD: Geometric measurement and dimension Explain volume formulas and use them to solve

problems

Equations of Circles 2Evaluating

Statements About Enlargements (2D and 3D)

Calculating Volumes of Compound Objects

ILLUSTRATIVE MATH G-GMDG-GMD.3 CenterpieceG-GMD.3 Doctor’s Appointment

MATHEMATICS ASSESSMENT PROJECTVisualize relationships between two-dimensional and three-

dimensional objects 4: Identify the shapes of two-dimensional cross-sections of

three- dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Modeling: Rolling Cups 2D Representations of 3D Objects

VOLUME ANIMATIONS

Charles A. Dana Center Mathematics Unfolding: Volume

http://ccsstoolbox.agilemind.com/animations/standards_content_mathematics_volume.html

HS.GMD.B.4Identify the shapes of two-dimensional

cross-sections of three-dimensional objects generated by rotations of two-dimensional objects.

G-GMD.4 Tennis Balls in a Can

NORTH COUNTRY INSERVICE OUTLINE

• Review with Agreed Upon Expectations from 2-15-13 Inservice – Share Experiences

• Review of CCSSM Practice Standards – Share Experiences

• Presentation of How Geometry Unfolds over K – 12 in CCSSM

• Focus on Volume Standard in HS Geometry• Develop one unit focusing on HS Volume

Standard and Practice Standards