It’s Not the Same Old
Algebra and Geometry
Dr. Joyce Bernstein
East Williston UFSD
LIASCD
October 19, 2007
Investigate/Explore - Students will be given situations in which they will be
asked to look for patterns or relationships between elements within the setting.
Discover - Students will make note of possible patterns and generalizations
that result from investigation/exploration.
Conjecture - Students will make an overall statement, thought to be true, about
the new discovery.
Reasoning - Students will engage in a process that leads to knowing
something to be true or false.
Argument - Students will communicate, in verbal or written form, the reasoning
process that leads to a conclusion. A valid argument is the end result of the
conjecture/reasoning process.
Justify/Explain - Students will provide an argument for a mathematical
conjecture. It may be an intuitive argument or a set of examples that support
the conjecture. The argument may include, but is not limited to, a written
paragraph, measurement using appropriate tools, the use of dynamic software,
or a written proof.
Proof - Students will present a valid argument, expressed in written form,
justified by axioms, definitions, and theorems.
Apply - Students will use a theorem or concept to solve an algebraic or
numerical problem.
Math A
Math B
2006-07
X
X
2007-08
X
X
X
First admin.
in June 2008
2008-09
X
Last admin.
in January
2009
X
X
X
First admin.
in June 2009
X
Last admin.
in June 2010
X
X
2009-10
Algebra
Geometry
Algebra 2 and
Trigonometry
X
First admin.
in June
2010-11
X
X
X
2011-12
X
X
X
Important Websites:
Math Toolkit grades 9 – 12:
http://emsc32.nysed.gov/3-8/guidance912.htm
Sample Tasks
NYS Mathematics Core Curriculum
Curriculum Performance Indicators
Implementation Timeline
Glossary for Teachers by Grade Level
Regents Approved Commencement-Level Course Descriptions
Template for Analysis of Mathematics Program Series
Commencement Level Crosswalk
Powerpoint Overview
Test Samplers – Late October prior to each test
http://www.emsc.nysed.gov/osa
Test Specifications
http://www.emsc.nysed.gov/osa/mathre/testspecsalgebra.pdf
http://www.emsc.nysed.gov/osa/mathre/testspecs-geometry.pdf
http://www.emsc.nysed.gov/osa/mathre/testspecs-alger2trig.pdf
Association of Math Assistant Principals of NYC:
http://www.amaps.org/
Integrated Algebra
In implementing the Algebra process and content performance
indicators, it is expected that students will identify and justify
mathematical relationships. The intent of both the process
and content performance indicators is to provide a variety
of ways for students to acquire and demonstrate
mathematical reasoning ability when solving problems.
Local curriculum and local/state assessments must support
and allow students to use any mathematically correct
method when solving a problem.
Throughout this document the performance indicators use the
words investigate, explore, discover, conjecture, reasoning,
argument, justify, explain, proof, and apply. Each of these terms
is an important component in developing a student’s
mathematical reasoning ability.
Crosswalks To Algebra Other than from Math A
From Middle School:
A.N.5 - Solving Algebraic problems involving fractions,
decimals, percents (decrease/increase and discount), and
proportionality/direct variation
Math 8 does not use the term “direct variation”.
A.N.6 - Evaluating expressions involving factorial(s),
absolute value(s), and exponential expression(s)
Factorials and absolute value in grade 7
Math 8 handles expressions with exponents.
A.A.9 - Analyze and solve verbal problems that involve
exponential growth and decay Start from scratch!
A.A.21 - Verifying a value as a solution to a linear equation
or inequality in one variable
A.A.45 - Application of Pythagorean theorem (grade 7)
A.M.1 - Calculations of rate (grade 7)
Crosswalks To Algebra Other than from Math A
From Math B
A.G.3 - Determine when a relation is a function
A.G.5 - How coefficient change in a function effects its graph
(Associate with graphs of parabolas. Use the calculator.)
A.S.3 - Determine when collected data or display may be biased
A.S.4 - Compare and contrast the appropriateness of different
measures of central tendency for a given data set
A.S.8 - Line of best fit for a scatter plot and its equation
(The line drawn depends on the 2 points chosen to define
the line. Or.. Stat..calc…4:LinReg(ax + b))
A.S.15 - Identify and describe sources of bias and its effect,
drawing conclusion from data
A.S.17 - Use a reasonable line of best fit to make a prediction
involving interpolation or extrapolation
Crosswalks To Algebra- New to 2005 Standards
A.A.3 - Difference between an Algebraic expression and an
Algebraic equation
(Now a Middle School topic: Is there an equal sign?)
A.A.28 - Relation between roots and factors of a quadratic
equation (Not really new)
A.A.29 - Set builder notation and/or interval notation to represent
the elements of a set
A.A.30 - Complement of a set
A.A.31 – Intersection of sets
A.A.32 - Slope as a rate of change (not really new)
Highlights from the Specifications for
Integrated Algebra
Content Strand
Number Sense & Operations
%
6 – 10
Algebra
50 – 55
Geometry
14 – 19
Measurement
Probability & Statistics
3–8
14 – 19
Highlights from the Specifications for
Integrated Algebra
Question Type
Number of Questions
Multiple Choice (2 pt)
2-credit open ended
3-credit open ended
4-credit open ended
30
3
3
3
87 Points 60 points are Multiple Choice
Highlights from the Specifications for
Integrated Algebra
Reference Sheet:
•Trig Ratios
•Selected area, volume, surface area formulas
•Slope
Reminder about graphing calculators
Riverside Test Publishing has overall responsibility
for the Regents exams for the next few years. New
York State teachers are the question writers, trained
by Riverside. Riverside is responsible for final
format.
State Ed Update:
For standard-setting the new exams, the Department
will use the same processes that were used for the
Grades 3 – 8 Testing Program, including standard
setting AFTER the test has been administered using
operational testing data (rather than field testing
data).
Administration: Tuesday, June 17
Return with scaled scores: Thursday, June 26
Geometry
There is no other school mathematics course that offers
students the opportunity to act as mathematicians. Within this
course, students will have the opportunity to make conjectures
about geometric situations and prove in a variety of ways, both
formal and informal, that their conclusion follows logically from
their hypothesis.
The variety of approaches to verification and proof is what
gives curriculum developers and teachers the flexibility to
adapt strategies to address these performance indicators in a
manner that meets the diverse needs of our students. Local
curriculum and local/state assessments must support and
allow students to use any mathematically correct method
when solving a problem.
It is intended that students will use the traditional tools of compass
and straightedge as well as dynamic geometry software that
models these tools more efficiently and accurately, to assist in
these investigations.
Crosswalks to Geometry
Solid Geometry NOT Previously Addressed:
G.G.1 A line perpendicular to each of two intersecting
lines at their point of intersection 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.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 Two planes perpendicular to the same line are
parallel.
GG.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.12 The volume of a prism is the product of the area of
the base and the altitude
Concurrency Theorems NOT Previously Addressed:
G.G.21 Investigate and apply the concurrence of
medians, altitudes, angles bisectors, and perpendicular
bisectors of triangles.
That means centroids, orthocenters, incenters, and
circumcenters
Highlights from the Specifications for
Geometry
Content Band
%
Geometric Relationships
8 – 12
Constructions
3–7
Locus
4–8
Informal and Formal Proofs
41 – 47
Transformational Geometry
23 - 28
Highlights from the Specifications for
Geometry
Question Type
Number of Questions
Multiple Choice (2 pt)
2-credit open ended
4-credit open ended
6-credit open ended
28
6
3
1
86 Points
56 points are Multiple Choice
Algebra 2 and Trigonometry
This course is a continuation and extension of the two courses
that preceded it. While developing the algebraic techniques that
will be required of those students that continue their study of
mathematics, this course is also intended to continue
developing alternative solution strategies and algorithms.
For example, technology can provide to many students the
means to address a problem situation to which they might
not otherwise have access. In implementing the Algebra 2 and
Trigonometry process and content performance indicators, it is
expected that students will identify and justify mathematical
relationships, formally and informally. The intent of both the
process and content performance indicators is to provide a
variety of ways for students to acquire and demonstrate
mathematical reasoning ability when solving problems. Local
curriculum and local/state assessments must support and
allow students to use any mathematically correct method
when solving a problem
Crosswalks to Algebra 2 and Trigonometry
Topics NOT Previously Addressed
(my edited list)
A2.A.12 Evaluate exponential expressions, including
those with base e
A2.A.24 Know and apply the technique of completing
the square
A2.A.50 Approximate the solution to polynomial
equations of higher degree by inspecting the graph
A2.A.63 Restrict the domain of the sine, cosine, and
tangent functions to ensure the existence of an inverse
function
A2.A.65 Sketch the graph of the inverses of the sine,
cosine, and tangent functions
A2.A.70 Sketch and recognize one cycle of a function of
the form y = AsinBx or y = Acos Bx
A2.A.71 Sketch and recognize the graphs of the functions
y = sec(x), y = csc(x), y = tan(x), and y = cot(x)
Highlights from the Specifications for
Algebra 2/Trigonometry
Content Strand
Number Sense & Operations
Algebra
Measurement
Probability & Statistics
%
6 – 10
70 – 75
2–5
13 – 17
Highlights from the Specifications for
Algebra 2/Trigonometry
Question Type
Multiple Choice (2 pt)
2-credit open ended
4-credit open ended
6-credit open ended
Number of Questions
27
8
3
1
88 Points 54 points are Multiple Choice
Questions from Our Curriculum Leaders
Let’s Discuss……
1. In what grade are you starting Algebra?
2. How are you providing AIS?
3. How are you providing staff development?
4. How are you selecting textbooks?
5. In what order are you offering Integrated Algebra,
Geometry, and Algebra 2/Trigonometry?
6. Are Regents Examinations counted in final averages?