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 VOLUME

Solve 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 STANDARDS

Primarily Focused on Plane Euclidean Geometry

Shapes 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 PERSPECTIVE

Rigid 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