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 Algebra Geometry Algebra 2 and Trigonometry

2006-07 X X   

   

 

   

2007-08 X X XFirst admin.

in June 2008  

       

2008-09 X Last admin. in January 2009

X X XFirst admin. in June 2009

   

2009-10   X Last admin. in June 2010

X X 

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.htmSample Tasks

NYS Mathematics Core Curriculum Curriculum Performance Indicators

Implementation TimelineGlossary for Teachers by Grade Level

Regents Approved Commencement-Level Course DescriptionsTemplate for Analysis of Mathematics Program Series

Commencement Level CrosswalkPowerpoint 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.pdfhttp://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 AFrom Middle School:

A.N.5 - Solving Algebraic problems involving fractions, decimals, percents (decrease/increase and discount), and proportionality/direct variationMath 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 7Math 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 variableA.A.45 - Application of Pythagorean theorem (grade 7)A.M.1 - Calculations of rate (grade 7)

Crosswalks To Algebra Other than from Math AFrom Math B

A.G.3 - Determine when a relation is a functionA.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 biasedA.S.4 - Compare and contrast the appropriateness of different measures of central tendency for a given data setA.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 dataA.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 setA.A.30 - Complement of a setA.A.31 – Intersection of setsA.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 3 – 8

Probability & Statistics 14 – 19

Highlights from the Specifications for Integrated Algebra

Question Type Number of Questions

Multiple Choice (2 pt) 302-credit open ended 33-credit open ended 34-credit open ended 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

GeometryThere 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 planeG.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) 282-credit open ended 64-credit open ended 36-credit open ended 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 6 – 10

Algebra 70 – 75

Measurement 2 – 5

Probability & Statistics 13 – 17

Highlights from the Specifications for Algebra 2/Trigonometry

Question Type Number of Questions

Multiple Choice (2 pt) 272-credit open ended 84-credit open ended 36-credit open ended 1

88 Points 54 points are Multiple Choice

Questions from Our Curriculum LeadersLet’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?