Looking at the New Algebra 2 / Trigonometry Course New York City Department of Education Department of Mathematics

Agenda Content and Process Strands Topics New to Algebra 2 / Trigonometry Algebra 2 Course Topics Looking at the New Regents Exam Note to presenter: Distribute file #04 ‘Table of Contents – Mathematics Toolkit’. This document from the NY State Education Department website lists many of the resources available there for downloading. Particularly useful are the glossaries and sample tasks for all of the high school courses. Several of the documents that they will be receiving during today’s presentation come from this website. Since there is a lot of material in this presentation, our advice would be to spend relatively little time on the first segment ‘Content and Process Strands’, since some of your audience has probably already been exposed to this information. Should there be questions on any of the information presented in slides 3 through 8, refer participants to the NYSED website (URL is listed on file # 11 and on the last slide in this presentation). New York City Department of Education Department of Mathematics

Algebra 2 and Trigonometry 2006-07 X 2007-08 New Mathematics Regents Implementation / Transition Timeline     Math A Math B Algebra Geometry Algebra 2 and Trigonometry 2006-07 X School curricular and instructional alignment and SED item writing and pre-testing 2007-08 First admin. in June 2008, Post-equate 2008-09 Last admin. in January 2009 First admin. in June 2009, Post-equate 2009-10 Last admin. in June 2010 X First admin. in June 2010, Post-equate 2010-11 2011-12 Note to presenter: Distribute files #01 ‘Course Descriptions HS’, #02: ‘New Mathematics Regents Implementation/Transition Timeline’, and #03 ‘ATS Codes Alg + Geometry’ As you see, we will soon be entering year three of the transition to new high school math standards. As background let’s take a brief look at those new standards.

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. According to the New York State Education Department, the goal of math education is to strengthen our students’ abilities in these areas. The SED classifies Regents exam questions into three categories; Procedural, Conceptual and Problem Solving. Procedural A procedural question is designed to test the student’s ability to perform mathematical computation. Example: The product of 4x2y and 2xy3 is (1) 8x2 y3 (2) 8x3 y3 (3) 8x3 y4 (4) 8x2 y4 Conceptual A conceptual question tests knowledge or/and understanding of a mathematical concept. Typical examples are questions involving definitions or interpretations of definitions. This type of question involves little or no computation. Which set is closed under division? (1) {1} (3) integers (2) counting numbers (4) whole numbers Problem Solving A Problem Solving question requires critical thinking. All problem-solving questions contain procedural and conceptual elements. Typical examples of this type of questions are word problems and math puzzles. Two trains leave the same station at the same time and travel in opposite directions. One train travels at 80 kilometers per hour and the other at 100 kilometers per hour. In how many hours will they be 900 kilometers apart?

Content and Process Strands 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 Again, the 2005 standards are organized into five Content Strands and five Process Strands. The content strands tell us the “WHAT” while the process strands tell us the “HOW”. Within each content strand there are bands which organize topics more specifically, and finally a performance indicator specific to each individual topic in the curriculum.

Note to presenter: This grid is a graphical representation of the way that the process and content strands are interrelated. Process strands are woven throughout the content strands. For example, Problem Solving is a process used in every one of the content strands. Please note that the bars rest on a “mat” which underlines the importance of conceptual understanding, procedural proficiency and problem solving. This graphic implicitly defines Mathematical Proficiency as a balance among the three of them. Students will become successful in mathematics only if they see mathematics as a whole, not as isolated skills and facts. As we develop our instructional plans and assessment techniques, we must pay attention to the integration of process and content. Otherwise we risk producing students who have temporary knowledge but who are unable to apply mathematics in realistic settings. Curriculum, instruction, and assessment are intricately related and must be designed with this in mind. All three of them must address conceptual understanding, procedural fluency, and problem solving. If this is accomplished, we will produce students who will (1) have mathematical knowledge, (2) have an understanding of mathematical concepts, and (3) be able to apply mathematics in the solution of problems.

Number of Performance Indicators for Each Course Content Strand Integrated Algebra Geometry Algebra 2 and Trigonometry Total Number Sense and Operations 8 10 18 Algebra 45 77 122 74 84 Measurement 3 2 5 Statistics and Probability 23 16 39 TOTAL 89 105 268 This table helps us to see at a simple glance the distribution of the Performance Indicators within the Content Strands. Ask participants to consider in their groups: “In what way does the Algebra 2 / Trigonometry course differ from the other two high school math courses?” [Many more topics.]

Some Useful Websites Related to Algebra 2 Eric Schlytter’s Web Page http://classrooms.tacoma.k12.wa.us/stadium/eschlytter/index.php Illuminations http://illuminations.nctm.org/ Math Forum at Drexel http://mathforum.org/ National Library of Virtual Manipulatives http://nlvm.usu.edu/en/nav/vlibrary.html Jefferson Math Project http://www.jmap.org/ Regents Prep http://www.regentsprep.org/ Department of Mathematics (Educator Resources) http://schools.nyc.gov/Academics/Mathematics/EducatorResources/default.htm

Which topics have not been addressed in previous high school courses?

Performance Topics Indicators Equations and Inequalities A2.A.5 Use direct and inverse variation to solve for unknown values A2.A.23 Solve rational equations and inequalities A2.A.24 Know and apply the technique of completing the square These slides present an overview. For much more detailed information, please visit the NY State Education Department’s website. (We’ve listed the URL on the last slide in this presentation).

A2.A.5a Use direct or inverse variation to solve for the unknown values: If p varies directly as q, and p = 7 when q = 9, find p when q = 12. If m varies inversely as t, and m = 5 when t = 6, find t when m = 10.

A2.A.23c Solve the inequality:

Performance Topics Indicators A2.A.26 Find the solution to polynomial equations of higher degree that can be solved using factoring and/or the quadratic formula A2.A.50 Approximate the solution to polynomial equations of higher degree by inspecting the graph

Polynomial equations of higher degree A2.A.26a Solve the following equations. Express any irrational solutions in simplest radical form. = 0

A2.A.50a The function is graphed below. Use the graph to approximate the solutions to the equation

Performance Topics Indicators A2.A.27 Solve exponential equations with and without common bases A2.A.28 Solve a logarithmic equation by rewriting as an exponential equation

A2.A.28a Solve the following equations:

Performance Topics Indicators Patterns, Relations, and Functions A2.A.29 Identify an arithmetic or geometric sequence and find the formula for its nth term A2.A.30 Determine the common difference in an arithmetic sequence A2.A.31 Determine the common ratio in a geometric sequence

A2.A.29a Maya has decided to train for a marathon (26 miles) and has set up a practice schedule to build her stamina. When she began she was able to run 3 miles, but she intends to train every day and increase her run by 2 miles each week. Find a pattern and write a formula that will give the number of miles Maya can run in week n. Using the formula, how many weeks will Maya need to train in order to be ready for the marathon?

A2.A.30a What is the common difference in the following arithmetic sequences? 5, 9, 13, 17,… and A2.A.31a What is the common ratio in the following geometric sequence?

Performance Topics Indicators A2.A.32 Determine a specified term of an arithmetic or geometric sequence A2.A.33 Specify terms of a sequence, given its recursive definition

A2.A.32a Find the specified term of this arithmetic sequence. t20: A2.A.32b Find the specified term of this geometric sequence.

A2.A.33a Use the recursive rule given to write the first four terms of each sequence.

Performance Topics Indicators A2.A.34 Represent the sum of a series, using sigma notation A2.A.35 Determine the sum of the first n terms of an arithmetic or geometric series

A2.A.34a Use sigma notation to represent the sum of the following series. for the first 33 terms. for the first 50 terms. for n terms.

A2.A.35c Given the sequence , Paul notices a pattern and finds a formula he believes will find the sum of the first n terms. His formula is Show that Paul’s formula is correct.

Performance Topics Indicators A2.A.43 Determine if a function is one-to-one, onto, or both A2.A.45 Determine the inverse of a function and use composition to justify the result

A2.A.43a For each of the following functions, state whether the function is one-to-one, onto, neither, or both:

A2.A.45b Demonstrate that and are inverses using at least two different strategies (numeric, graphic or algebraic).

Performance Topic Indicator A2.A.46 Perform transformations with functions and relations: f(x + a), f(x) + a, f(–x), –f(x), af(x),

A2.A.46d Given the graph of the function f(x), sketch the graphs of

Performance Topic Indicator Coordinate Geometry A2.A.47 Determine the center-radius form for the equation of a circle in standard form

A2.A.47a Use the technique of completing the square to convert the equation into center-radius form. What is the center and what is the radius of this equation?

Performance Topics Indicators Trigonometric Functions 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.65a Sketch the graph of over the interval Reflect the graph over the line How would you restrict the domain to make the image a function?

Performance Topics Indicators A2.A.69 Determine amplitude, period, frequency, and phase shift, given the graph or equation of a periodic function A2.A.71 Sketch and recognize the graphs of the functions y = sec(x), y = csc(x), y = tan(x), and y = cot(x)

A2.A.69a A pet store clerk noticed that the population in the gerbil habitat varied sinusoidally with respect to time, in days. He carefully collected data and graphed his resulting equation. From the graph, determine amplitude, period, frequency, and phase shift.

A2.A.71a Sketch one cycle of each of the following equations. Carefully label each graph. Graph and at the same time on your calculator with a window of . What conclusions can you make? Describe the similarities between the 2 functions. Now, graph and at the same time on your calculator with a window of . What conclusions can you make? Describe the similarities between the 2 functions.

Performance Topic Indicator A2.A.72 Write the trigonometric function that is represented by a given periodic graph

A2.A.72a Write a trigonometric function that matches each of the following graphs. Check your answers with a partner. If different equations have been obtained, confirm by graph or table, the accuracy of each equation.

Performance Topics Indicators Collection of Data A2.S.1 Understand the differences among various kinds of studies (e.g., survey, observation, controlled experiment) A2.S.2 Determine factors which may affect the outcome of a survey Organization and Display of Data A2.S.4 Calculate measures of dispersion (range, quartiles, interquartile range, standard deviation, variance) for both samples and populations

Looking at the new Regents exam

Exam credit aligned with each content strand % of Total Credits 1) Number Sense 6-10% 2) Algebra 70-75% 4) Measurement 2-5% 5) Probability and Statistics 13-17%

Specifications for the Regents Examination in Algebra 2 Question Type Number of Questions Point Value Multiple choice 27 54 2-credit open-ended 8 16 4-credit open-ended 3 12 6-credit open-ended 1 6 Total 39 88

Calculators Schools must make a graphing calculator available for the exclusive use of each student while that student takes the Regents examination in Algebra 2 / Trigonometry.

Reference Sheet The Regents Examination in Algebra 2 will include a reference sheet containing the formulas specified below.

The reference sheet will also include a bell curve.