Teaching High School Geometry

New York City Department of EducationDepartment of Mathematics

Agenda Content and Process Strands

Geometry Course Topics and Activities

Topics New to High School Geometry

Looking at the New Regents Exam

New York City Department of EducationDepartment of Mathematics

New Mathematics RegentsImplementation / Transition

Timeline 

  Math AMath

BAlgebra Geometry

Algebra 2 and Trigonometry

2006-07

X X 

School curricular and instructional alignment and SED item writing and pre-

testing

 School curricular and instructional

alignment and SED item writing and pre-testing

 School curricular and instructional

alignment and SED item writing and pre-testing

2007-08

X X

 

XFirst admin. in

June 2008, Post-equate

 School curricular and instructional

alignment and SED item writing and pre-testing

 School curricular and instructional

alignment and SED item writing and pre-testing

2008-09

XLast admin. in January 2009

X X

 

XFirst admin. in June 2009, Post-equate

 School curricular and instructional

alignment and SED item writing and pre-testing

2009-10

 X

Last admin. in June

2010

X 

X

 

XFirst admin. in June 2010,

Post-equate

2010-11

    

X X X

2011-12

    X X X

Standard 3Standard 3

The Three ComponentsThe Three Components

•Conceptual Understanding consists of those relationships constructed internally and connected to already existing ideas.

•Procedural Fluency is the skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.

•Problem Solving is the ability to formulate, represent, and solve mathematical problems.

1996 Mathematics Standard and 1998 Core Curriculum

2005 Mathematics Standard and 2005 Core Curriculum

1996 Mathematics Standard

Seven Key Ideas Mathematical Reasoning Number and Numeration Operations Modeling/Multiple Representation Measurement Uncertainty Patterns/Functions

 

  Performance indicators are organized under the seven key ideas and contain an includes (testing years) or may include (non-testing years) columns for further clarification.

2005 Mathematics Standard

Five Process Strands Problem Solving Reasoning and Proof Communication Connections Representation

Five Content Strands oNumber Sense and Operations oAlgebra oGeometry oMeasurement Statistics and Probability

Performance indicators are organized under major understandings within the content and process strands and content performance indicators are separated into bands within each of the content strands.

Performance Indicator Organization

Standard 3Standard 3

Content and Process StrandsContent and Process Strands

The Five Content Strands The Five Process Strands

Number Sense and Operations

Problem Solving

Algebra Reasoning and Proof

Geometry Communication

MeasurementConnections

Statistics and Probability

Representation

1998 Key Ideas 2005 Process and Content Strands

Broad in scope and transcend the various branches of mathematics (arithmetic, number theory, algebra, geometry, etc.)

Lack of specificity in the may include column for each performance indicators

Difference between the may include and includes columns for performance indicators is not clearly indicated

Processes of mathematics (problem solving, communication, etc.) are, for the most part, included in the narrative of the document.

Process and Content Strands are aligned to the National Council of Teachers of Mathematics Standards The processes of mathematics as well as the content of mathematics have performance indicators

Performance indicators are clearly delineated and more specific.

Comparison of 1998 Seven Key Ideas and 2005 Process and Content Strands

Number of Performance Indicators for Each Course

Content StrandIntegrated Algebra

GeometryAlgebra 2 and Trigonometry

Total

Number Sense and Operations

8 0 10 18

Algebra 45 0 77 122

Geometry 10 74 0 84

Measurement 3 0 2 5

Statistics and Probability 23 0 16 39

TOTAL 89 74 105 268

Geometry BandsGeometry Bands

•Shapes

•Geometric Relationships

•Constructions

•Locus

•Informal Proofs

•Formal Proofs 

•Transformational Geometry

•Coordinate Geometry

Which topics are in the new geometry

course?

Performance TopicsIndicators

1 – 9 Perpendicular lines and planes

10 – 16 Properties and volumes of three- dimensional figures, including prism, regular pyramid, cylinder, right circular cone, sphere

Volume and Surface Areaof Rectangular Prism

Performance TopicsIndicators

17 – 21 Constructions: angle bisector, perpendicular bisector, parallel through a point, equilateral triangle;

22, 23 Locus: concurrence of median, altitude, angle bisector, perpendicular bisector;

compound loci

Performance TopicsIndicators

24 – 27 Logic and proof: negation, truth value, conjunction, disjunction, conditional, biconditional,

inverse, converse, contrapositive; hypothesis → conclusion

28, 29 Triangle congruence (SSS, SAS,ASA, AAS, HL) and corresponding parts

Area Without Numbers

Performance TopicsIndicators

30 – 48 Investigate, justify and apply theorems (angles and polygons):Sum of angle measures

(triangles and polygons): interior and exterior

Isosceles triangleGeometric inequalitiesTriangle inequality theoremLargest angle, longest sideTransversals and parallel lines

Performance TopicsIndicators

30 – 48 Investigate, justify and apply theorems (angles and polygons):Parallelograms (including special

cases), trapezoidsLine segment joining midpoints, line parallel to side (proportional)CentroidSimilar triangles (AA, SAS, SSS)Mean proportionalPythagorean theorem, converse

Exhibit: Semantic Feature Analysis Matrix

Terms

FeaturesProperties

Performance TopicsIndicators

49 – 53 Investigate, justify and apply theorems (circles):

Chords: perpendicular bisector. relative lengths

Tangent lines

Arcs, rays (lines intersecting on, inside, outside)

Segments intersected by circle along tangents, secants

Center of a CircleFind different ways, as many as you can, to determine the center of a circle. Imagine that you have access to tools such as compass, ruler, square corner, protractor, etc.

Be able to justify that you have found the center.

Performance TopicsIndicators

54 – 61 Transformations

Isometries (rotations, reflections, translations,

glide reflections)

Use to justify geometric relationships

Similarities (dilations)

Properties that remain invariant

Fold and Punch

Take a square piece of paper. Fold it and make one punch so that you will have one of the following patterns when you open it.

Reflective Symmetry

Translational Symmetry Rotational Symmetry

Venn Symmetry

Performance TopicsIndicators

62 – 68 Coordinate geometry: Distance, midpoint, slope formulas to find equations of linesperpendicular, parallel, andperpendicular bisector

Performance TopicsIndicators

69 Coordinate geometry: Properties of triangles and quadrilaterals

70 Coordinate geometry: Linear-quadratic systems

-10 -5 5 10

14

12

10

8

6

4

2

-2

Area of a Triangle on a Coordinate PlaneTwo vertices of a triangle are located at (0,6) and (0,12).The area of the triangle is 12 units2.

Performance TopicsIndicators

71 – 74 Coordinate geometry: Circles: equations, graphs

(centered on and off origin)

About 20% of the topics in the new Geometry course have not been addressed in previous high school courses.

Which topics have not been addressed in

previous high school courses?

centroid

circumcenter

incenter of a triangle

orthocenter

centroid (G) The point of concurrency of the medians of a triangle; the center of gravity in a triangle.

circumcenter (G) The center of the circle circumscribed about a polygon; the point that is equidistant from the vertices of any polygon.

incenter of a triangle (G) The center of the circle that is inscribed in a triangle; the point of concurrence of the three angle bisectors of the triangle which is equidistant from the sides of the triangle.

orthocenter (G) The point of concurrence of the three altitudes of a triangle.

isometry

symmetry plane

isometry (G) A transformation of the plane that preserves distance. If P′ is the image of P, and Q′ is the image of Q, then the distance from P′ to Q′ is the same as the distance from P to Q.

symmetry plane (G) A plane that intersects a three-dimensional figure such that one half is the reflected image of the other half.

A Symmetry Plane

Geometric Relationships 1Theorems and PostulatesG.G.1 If a line is perpendicular to each of two intersecting

lines at their point of intersection, then the line is perpendicular to the plane determined by them

G.G.2 Through a given point there passes one and only one plane perpendicular to a given line

G.G.3 Through a given point there passes one and only one line perpendicular to a given plane

G.G.4 Two lines perpendicular to the same plane are coplanar

G.G.5 Two planes are perpendicular to each other if and only if one plane contains a line perpendicular to the second plane

G.G.1bStudy the drawing below of a pyramid whose base is quadrilateral ABCD. John claims that line segment EF is the altitude of the pyramid. Explain what John must do to prove that he is correct.

Describe how a plane and the prism could intersect so that the intersection is:a line parallel to one of the triangular basesa line perpendicular to the triangular basesa trianglea rectanglea trapezoid

G.G.3aExamine the diagram of a right triangular prism.

G.G.4b The figure below in three-dimensional space, where AB is perpendicular to BC and DC is perpendicular to BC, illustrates that two lines perpendicular to the same line are not necessarily parallel. Must two lines perpendicular to the same plane be parallel? Discuss this problem with a partner.

Geometric Relationships 2More Theorems and Postulates

G.G.6 If a line is perpendicular to a plane, then any line perpendicular to the given line at its point of intersection with the given plane is in the given plane

G.G.7 If a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane

G.G.8 If a plane intersects two parallel planes, then the intersection is two parallel lines

G.G.9 If two planes are perpendicular to the same line, they are parallel

G.G.7a Examine the four figures below:

Each figure has how many symmetry planes?Describe the location of all the symmetry planes for each figure.

G.G.9a The figure below shows a right hexagonal prism.

A plane that intersects a three-dimensional figure such that one half is the reflected image of the other half is called a symmetry plane. On a copy of the figure sketch a symmetry plane. Then write a description of the symmetry plane that uses the word parallel.On a copy of the figure sketch another symmetry plane. Then write a description that uses the word perpendicular.

Geometric Relationships 3Prisms

G.G.10 The lateral edges of a prism are congruent and parallel

G.G.11 Two prisms have equal volumes if their bases have equal areas and their altitudes are equal

G.G.11aExamine the prisms below. Calculate the volume of each of the prisms. Observe your results and make a mathematical conjecture. Share your conjecture with several other students and formulate a conjecture that the entire group can agree on. Write a paragraph that proves your conjecture.

Locus

G.G.21 Concurrence of medians, altitudes, angle bisectors, and perpendicular bisectors of triangles

G.G.21aUsing dynamic geometry software locate the circumcenter, incenter, orthocenter, and centroid of a given triangle. Use your sketch to answer the following questions:Do any of the four centers always remain inside the circle?If a center is moved outside the triangle, under what circumstances will it happen?Are the four centers ever collinear? If so, under what circumstances?Describe what happens to the centers if the triangle is a right triangle.

Informal and Formal Proofs 1

G.G.43 Theorems about the centroid of a triangle, dividing each median into segments whose lengths are in the ratio 2:1

G.G.43aThe vertices of a triangle ABC are A(4,5), B(6,1), and C(8,9). Determine the coordinates of the centroid of triangle ABC and investigate the lengths of the segments of the medians. Make a conjecture.

Informal and Formal Proofs 2Similarity

G.G.46 Theorems about proportional relationships among the segments of the sides of the triangle, given one or more lines parallel to one side of a triangle and intersecting the other two sides of the triangle

G.G.46aIn ΔABC , DE is drawn parallel to AC . Model this drawing using dynamic geometry software. Using the measuring tool, determine the lengths AD, DB, CE, EB, DE, and AC. Use these lengths to form ratios and to determine if there is a relationship between any of the ratios. Drag the vertices of the original triangle to see if any of the ratios remain the same. Write a proof to establish your work.

Transformational Geometry

G.G.60 Similarities: observing orientation, numbers of invariant points, and/or parallelism

G.G.60aIn the accompanying figure, ΔABC is an equilateral triangle. If ΔADE is similar to ΔABC, describe the isometry and the dilation whose composition is the similarity that will transform ΔABC onto ΔADE.

G.G.60bHarry claims that ΔPMN is the image of ΔNOP under a reflection over PN.. How would you convince him that he is incorrect? Under what isometry would ΔPMN be the image of ΔNOP?

Looking at the new Regents exam

Content Band % of Total

Credits

Geometric Relationships 8–12%

Constructions 3–7%

Locus 4–8%

Informal and Formal Proofs 41–47%

Transformational Geometry 8–13%

Coordinate Geometry 23–28%

Question TypeNumber of Questions

Point Value

Multiple choice

28 56

2-credit open-ended

6 12

4-credit open-ended

3 12

6-credit open-ended

1 6

Total 38 86

Specifications for the Regents Examination in Geometry

Schools must make a graphing calculator available for theexclusive use of each student while that student takes the Regents examination in Geometry.

Calculators

Measurement Three-Dimensional Figure Formula

Cylinder

where is the area of the base

Pyramid where is the area of the

base

Right Circular Cone where is the area of the

base

Volume

Sphere

Right Circular Cylinder rh Lateral Area ( )

Right Circular Cone rl where l is the slant height

Surface Area Sphere r2

Reference SheetThe Regents Examination in Geometry will include a reference sheet containing the

formulas specified below.

Core Curriculum, Sample Tasks, Glossary, Course

Descriptions, Crosswalks and Other Resources:

http://www.emsc.nysed.gov/3-8/guidance912.htm

New York City Department of EducationDepartment of Mathematics