Teaching High School Geometry

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Teaching High School Geometry
New York City Department of Education Department
of Mathematics
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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 Education Department
of Mathematics
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New Mathematics Regents Implementation /
Transition Timeline  
  Math A Math B Algebra 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   X First 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 X Last admin. in January 2009 X X   X First 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
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• Standard 3
• The 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.

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Performance Indicator Organization
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 Number Sense and Operations Algebra Geometry Measurement 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.
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• Standard 3
• Content and Process Strands

The Five Content Strands The Five Process Strands
Number Sense and Operations Problem Solving
Algebra Reasoning and Proof
Geometry Communication
Measurement Connections
Statistics and Probability Representation
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Comparison of 1998 Seven Key Ideas and 2005
Process and Content Strands
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.
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Number of Performance Indicators for Each Course Number of Performance Indicators for Each Course Number of Performance Indicators for Each Course Number of Performance Indicators for Each Course Number of Performance Indicators for Each Course
Content Strand Integrated Algebra Geometry Algebra 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
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Geometry Bands
Shapes Geometric Relationships Constructions
Locus Informal Proofs Formal
Proofs  Transformational Geometry Coordinate
Geometry
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Which topics are in the new geometry course?
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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

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Volume and Surface Areaof Rectangular Prism
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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

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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

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Area Without Numbers
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Performance TopicsIndicators

• 30 48 Investigate, justify and apply
  theorems (angles and polygons)
• Sum of angle measures (triangles and
  polygons) interior and exterior
• Isosceles triangle
• Geometric inequalities
• Triangle inequality theorem
• Largest angle, longest side
• Transversals and parallel lines

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Performance TopicsIndicators

• 30 48 Investigate, justify and apply
  theorems (angles and polygons)
• Parallelograms (including special cases),
  trapezoids
• Line segment joining midpoints, line
  parallel to side (proportional)
• Centroid
• Similar triangles (AA, SAS, SSS)
• Mean proportional
• Pythagorean theorem, converse

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Exhibit Semantic Feature Analysis Matrix
Terms Features Properties






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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

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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.
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Performance TopicsIndicators

• 54 61 Transformations
• Isometries (rotations, reflections,
  translations, glide reflections)
• Use to justify geometric relationships
• Similarities (dilations)
• Properties that remain invariant

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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.
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Venn Symmetry
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Performance TopicsIndicators

• 62 68 Coordinate geometry Distance,
  midpoint, slope formulas to find equations
  of lines perpendicular, parallel,
  and perpendicular bisector

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Performance TopicsIndicators

• 69 Coordinate geometry Properties of
  triangles and quadrilaterals
• 70 Coordinate geometry Linear- quadratic
  systems

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

• 71 74 Coordinate geometry Circles
  equations, graphs (centered on and off
  origin)

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About 20 of the topics in the new Geometry
course have not been addressed in previous high
school courses.
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Which topics have not been addressed in previous
high school courses?
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centroid circumcenter
incenter of a triangle
orthocenter
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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.
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isometry symmetry
plane
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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.
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A Symmetry Plane
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Geometric Relationships 1Theorems and Postulates

• G.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

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G.G.1b Study 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.
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G.G.3a Examine the diagram of a right triangular
prism.
Describe how a plane and the prism could
intersect so that the intersection is a line
parallel to one of the triangular bases a line
perpendicular to the triangular bases a
triangle a rectangle a trapezoid
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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.
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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

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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.
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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.
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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

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G.G.11a Examine 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.
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Locus

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

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G.G.21a Using 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.
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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 21

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G.G.43a The 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.
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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

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G.G.46a In ?ABC , 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.
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Transformational Geometry

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

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G.G.60a In the accompanying figure, ?ABC is an
equilateral triangle. If ?ADE is similar to ?ABC,
describe the isometry and the dilation whose
composition is the similarity that will transform
?ABC onto ?ADE.
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G.G.60b Harry claims that ?PMN is the image of
?NOP under a reflection over PN.. How would you
convince him that he is incorrect? Under what
isometry would ?PMN be the image of ?NOP?
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Looking at the new Regents exam
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Content Band of Total Credits
Geometric Relationships 812
Constructions 37
Locus 48
Informal and Formal Proofs 4147
Transformational Geometry 813
Coordinate Geometry 2328
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Specifications for the Regents Examination in
Geometry
Question Type Number 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
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Calculators
Schools must make a graphing calculator available
for the exclusive use of each student while that
student takes the Regents examination in Geometry.
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Reference Sheet The Regents Examination in
Geometry will include a reference sheet
containing the formulas specified below.
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Core Curriculum, Sample Tasks, Glossary, Course
Descriptions, Crosswalks and Other
Resourceshttp//www.emsc.nysed.gov/3-8/guidance9
12.htm
New York City Department of Education Department
of Mathematics