Download Teaching High School Geometry - New York City Department of

Teaching High School Geometry
New York City Department of Education
Department 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 Education
Department of Mathematics
New Mathematics Regents
Implementation / Transition Timeline
200607
200708
200809
200910
Math A
Math
B
Algebra
Geometry
Algebra 2 and
Trigonometry
X
X
School curricular and
instructional alignment and
SED item writing and pretesting
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
School curricular and instructional
alignment and SED item writing and
pre-testing
X
X
Last admin. in
January 2009
X
X
First admin. in
June 2008, Postequate
X
X
X
Last admin.
in June
2010
X
X
First admin. in June
2009, Post-equate
X
School curricular and instructional
alignment and SED item writing and
pre-testing
X
First admin. in June 2010,
Post-equate
201011
X
X
X
201112
X
X
X
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.
Performance Indicator Organization
1996 Mathematics Standard and
1998 Core Curriculum
2005 Mathematics Standard and
2005 Core Curriculum
1996 Mathematics Standard
2005 Mathematics Standard
Seven Key Ideas
 Mathematical Reasoning
 Number and Numeration
 Operations
 Modeling/Multiple Representation
 Measurement
 Uncertainty
 Patterns/Functions
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 indicators are organized under the
seven key ideas and contain an includes (testing
years) or may include (non-testing years) columns
for further clarification.
Standard 3
Content and Process Strands
The Five Content Strands
The Five Process Strands
Number Sense and
Operations
Problem Solving
Algebra
Geometry
Measurement
Statistics and
Probability
Reasoning and Proof
Communication
Connections
Representation
Comparison of 1998 Seven Key Ideas
and 2005 Process and Content Strands
1998 Key Ideas
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.
2005 Process and Content Strands
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.
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
Geometry Bands
•Shapes
•Geometric Relationships
•Constructions
•Locus
•Informal Proofs
•Formal Proofs
•Transformational Geometry
•Coordinate Geometry
Which topics are in
the new geometry
course?
Performance
Indicators
Topics
1–9
Perpendicular lines and planes
10 – 16
Properties and volumes of threedimensional figures, including
prism, regular pyramid, cylinder,
right circular cone, sphere
Volume and Surface Area
of Rectangular Prism
Performance
Indicators
Topics
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
Indicators
24 – 27
28, 29
Topics
Logic and proof: negation, truth
value, conjunction, disjunction,
conditional, biconditional, inverse,
converse, contrapositive;
hypothesis → conclusion
Triangle congruence (SSS,
SAS,ASA, AAS, HL) and
corresponding parts
Area
Without
Numbers
Performance
Indicators
Topics
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
Performance
Indicators
Topics
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
Exhibit: Semantic Feature Analysis Matrix
Terms
Features
Properties
Performance
Indicators
49 – 53
Topics
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 Circle
Find 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
Indicators
54 – 61
Topics
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.
Venn Symmetry
Reflective Symmetry
Translational Symmetry
Rotational Symmetry
Performance
Indicators
62 – 68
Topics
Coordinate geometry: Distance,
midpoint, slope formulas to
find equations of lines
perpendicular, parallel, and
perpendicular bisector
Performance
Indicators
69
70
Topics
Coordinate geometry: Properties
of triangles and quadrilaterals
Coordinate geometry: Linearquadratic systems
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.
14
12
10
8
6
4
2
-10
-5
5
-2
10
Performance
Indicators
71 – 74
Topics
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 1
Theorems 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
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.
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
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 2
More 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 3
Prisms
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.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.
Locus
G.G.21 Concurrence of medians, altitudes,
angle bisectors, and perpendicular
bisectors of triangles
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.
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.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.
Informal and Formal Proofs 2
Similarity
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.46a
In 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.60a
In 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.60b
Harry 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%
Specifications for the Regents Examination in Geometry
Question Type
Number of
Questions
Point Value
Multiple
choice
28
56
2-credit openended
6
12
4-credit openended
3
12
6-credit openended
1
6
Total
38
86
Calculators
Schools must make a graphing
calculator available for the
exclusive use of each student while
that student takes the Regents
examination in Geometry.
Reference Sheet
The 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/38/guidance912.htm
New York City Department of Education
Department of Mathematics