New Standards in High School Mathematics, New York State Introduction to the Integrated Algebra Course Note to presenter: Printed Material: Distribute “20 Agenda Integrated Algebra” New York City Department of Education Department of Mathematics

Session Objectives: Content and Process Strands Performance Indicators New Courses Looking at Integrated Algebra The New Regents Exam For More Information Note to the Presenter: This is an overview of the topics covered in this PowerPoint presentation.

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. Note to the presenter: 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?

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 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 the presenter: This grid is a “creative” 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. We’ll be looking at the Content and Process Strands in more detail shortly. 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.

Work with two other students to solve the following problem: Cameron received a set of four grades. If the average of the first two grades is 50, the average of the second and third grades is 75, and the average of the third and fourth grades is 70, then what is the average of the first and fourth grades? Printed Material: Distribute “01 Table of Contents – Mathematics Toolkit” Sample Task PS7a Important Note: This problem is selected from the Integrated Algebra sample tasks published by the NYSED. (The links to download this and all other documents referred to in this PowerPoint are found on the sheet you’ve just distributed and on a slide at the end of this presentation). Time permitting, having participants work on it will make your presentation more interactive and will give them a better understanding of the new standards. For example, after working on the problem participants might, in groups, consider which of the content and process strands it is an example of. The Sample Task initials above identify which strand it is taken from. Instructions to the presenter: Allow participants to work together on this problem for about five minutes. (Seating arrangements should facilitate interaction among participants) While they work on the problem, walk around and visit all groups. It’s important that you become aware of how the different groups are solving the problem, so that you can strategically choose one or two groups to present their solutions to the whole group. Choose groups that have a correct approach. Different groups should be allowed to present if they have a different strategy.

The Five Content Strands Performance Indicators which: define a broad range of content knowledge that students must master are taught in an integrated manner engage students in construction of knowledge integrate conceptual understanding and problem solving should not be viewed as a checklist of skills void of understanding and application Printed Material: Distribute “02 The Five Content Strands/ The Five Process Strands” Note to the presenter: After participants have had an opportunity to read and discuss both sides of the sheet, pose this task: “working in your group, pick one strand (alternatively you may assign each group a different strand) and try to describe the strand in your own words”. Paraphrasing is a powerful way of learning new material.

Number Sense and Operations Strand Students will: •understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems; •understand meanings of operations and procedures, and how they relate to one another; •compute accurately and make reasonable estimates. Note to the presenter: Here we have the goals of the first of the content strands. Keep in mind that these goals are within specific Bands in Number Sense and Operations: Number Systems Number Theory Operations Estimation

Algebra Strand Students will: •represent and analyze algebraically a wide variety of problem solving situations; •perform algebraic procedures accurately; •recognize, use, and represent algebraically patterns, relations, and functions.   Note to the presenter: There are a total of 89 Performance Indicators in the Integrated Algebra course and 45 of them fall in this strand making this 51% of the total course. Bands within the Algebra Strand Variables and Expressions Equations and Inequalities Patterns, Relations, and Functions Coordinate Geometry Trigonometric Functions

Geometry Strand Students will: •use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes; •identify and justify geometric relationships, formally and informally; •apply transformations and symmetry to analyze problem solving situations; •apply coordinate geometry to analyze problem solving situations. Note to the presenter: Of the ten Performance Indicators in the Geometry Strand nine are topics of geometry that are closely tied to algebra such as: coordinate geometry, analyzing functions and using formulas. Only one P.I. is strictly about geometry (finding area and perimeter). Bands within the Geometry Strand Shapes Geometric Relationships Transformational Geometry Coordinate Geometry Constructions Locus Informal Proofs Formal Proofs

Measurement Strand Students will: •determine what can be measured and how, using appropriate methods and formulas; •use units to give meaning to measurements; •understand that all measurement contains error and be able to determine its significance; •develop strategies for estimating measurements. Note to the presenter: There are only three Performance Indicators in this strand; however one of the P.I.s is a topic new to 9th grade math (relative error in measuring). Bands within the Measurement Strand Units of Measurement Tools and Methods Units Error and Magnitude Estimation

Statistics and Probability Strand Students will: •collect, organize, display, and analyze data; •make predictions that are based upon data analysis; •understand and apply concepts of probability. Note to the presenter: This strand contains 22 PIs making it second only to the Algebra Strand in size. New topics to 9th grade math are also introduced through this strand (univariate / bivariate, quartiles with measures of central tendency, as well as data bias). Bands within this Strand Collection of Data Organization and Display of Data Analysis of Data Predictions from Data Probability

The Five Process Strands Performance Indicators which: highlight ways of acquiring and using content knowledge give meaning to mathematics as a discipline rather than a set of isolated skills engage students in mathematical content as they solve problems, reason mathematically, prove mathematical relationships, participate in mathematical connections, and model and represent mathematical ideas

Problem Solving Strand Students will: •build new mathematical knowledge through problem solving; •solve problems that arise in mathematics and in other contexts; •apply and adapt a variety of appropriate strategies to solve problems; •monitor and reflect on the process of mathematical problem solving.

Reasoning and Proof Strand Students will: •recognize reasoning and proof as fundamental aspects of mathematics; •make and investigate mathematical conjectures; •develop and evaluate mathematical arguments and proofs; •select and use various types of reasoning and methods of proof.

Communication Strand Students will: •organize and consolidate their mathematical thinking through communication; •communicate their mathematical thinking coherently and clearly to peers, teachers, and others; •analyze and evaluate the mathematical thinking and strategies of others; •use the language of mathematics to express mathematical ideas precisely.

Connections Strand Students will: •recognize and use connections among mathematical ideas; •understand how mathematical ideas interconnect and build on one another to produce a coherent whole; •recognize and apply mathematics in contexts outside of mathematics.

Representation Strand Students will: •create and use representations to organize, record, and communicate mathematical ideas; •select, apply, and translate among mathematical representations to solve problems; •use representations to model and interpret physical, social, and mathematical phenomena.

Integrated Algebra Geometry Algebra 2 and Trigonometry The New Courses: Integrated Algebra Geometry Algebra 2 and Trigonometry Printed Material: Distribute “03 Regents-Approved Commencement Level Mathematics Course Descriptions” and “04 ATS Codes”

These are the new ATS codes for the algebra and geometry courses These are the new ATS codes for the algebra and geometry courses. Please make sure that your students are being programmed properly.

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 Printed Material: “05 Content Performance Indicators Chart”. Ask participants to consider in their groups: “Why do you think that the Algebra course is called Integrated ?” And “What’s unusual about the Geometry course?” Note to the presenter: This table helps us to see at a simple glance the distribution of the Performance Indicators within the Content Strands. Although grade 8 is not included in this table it might be worth mentioning that the majority of 8th-grade Performance Indicators are in the Algebra and Geometry Strands (19 and 21 respectively). Also, as in 9th grade Integrated Algebra, most of the geometry consists of coordinate geometry.

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 Printed Material: Distribute “06 New Mathematic Regents Implementation/Transition Timeline” Note to the presenter: After letting participants study and discuss this timeline in their groups, ask them, “How many times will the Math A and the Integrated Algebra Regents be offered simultaneously?” and “How many times will the Math B and the Algebra 2 and Trigonometry Regents be offered simultaneously?”

This chart

Looking at Integrated Algebra Printed Material: distribute “07 Algebra Performance Indicators – Content” Have participants take a look at the 89 performance indicators, or topics, of the course. Printed Material: Distribute “08 Algebra Strand Trace Pre-K to Grade 8”. One of the key changes brought by the new state standards has been the introduction of algebraic concepts as early as elementary school. This document shows where such concepts and problems were first introduced. Printed Material: Distribute “09 Algebra Crosswalk”. This document shows that many of the algebra topics that had previously been taught in high school (in Math A) are now taught in eighth grade.

Some Major Topics in Algebra Not in Math A Printed Material: Distribute “ 10 Crosswalk Integrated Algebra Course” (16-page landscape-formatted document) After participants have had an opportunity to read and briefly comment on the Crosswalk in their groups, ask them, “Which topics in Integrated Algebra are NOT taken from Math A?” Allow a few minutes for discussion within their groups. Emphasize however the continuity between the old and new, which is what three-quarters of the new course consists of. Printed Material: Distribute “11 Algebra Performance Indicators Not in Math A”. This will be the focus in the next segment.

Sets Set-Builder Notation and Interval Notation Complement of a Subset of a Given Set Intersection and/or Union of Sets The following slides consist of sample tasks on some topics new to Integrated Algebra.

Given that U={1,2,3,4,5} and A={3,4,5} list the elements in the complement of set A, Ā. Sample Task A30a Note: This sample task is intended as an illustration of new topics that have been added to the curriculum, not as one of the problems for participants to work on. It may also serve as a stimulus for discussion of dealing with the changes in the curriculum.

When A= {3,4,5} and B = {4,5,6,7}, find: AB and AB B A Sample Task A31a Note: This sample task is intended as an illustration of new topics that have been added to the curriculum, not as one of the problems for participants to work on. It may also serve as a stimulus for discussion of dealing with the changes in the curriculum.

Data: Qualitative or Quantitative Univariate or Bivariate Bias, Including Sources Evaluation of Reports or Graphs Experimental Design Appropriateness of Data Analysis Soundness of Conclusions (more…)

Data (continued): Percentile Rank of Item in Data Set First, Second, Third Quartiles Variables: Correlation But Not Causation Linear Transformations Affect Mean, Median, Mode Scatter Plots, Line of Best Fit

Presidents and their places of birth. Identify the following data sets as either qualitative or quantitative: Presidents and their places of birth. Percent of persons living in poverty. Number of votes cast in the 2004 presidential election. Favorite places for vacation. Baseball players and the position they play. Sample Task S1a Note: This sample task is intended as an illustration of new topics that have been added to the curriculum, not as one of the problems for participants to work on. It may also serve as a stimulus for discussion of dealing with the changes in the curriculum.

State if the following data sets are univariate or bivariate: Three-year rate of return for various mutual funds. Relationship between per capita gross domestic product and the life expectancy of residents of a country. Gestation period of an animal and the animal’s life expectancy. The pulse rate of eight randomly selected individuals after jogging for one minute. Sample Task S2a Note: This sample task is intended as an illustration of new topics that have been added to the curriculum, not as one of the problems for participants to work on. It may also serve as a stimulus for discussion of dealing with the changes in the curriculum.

An inability to reach a person in 53% of the homes called. A research company wanted to obtain data on what is watched on television by community members who are 18 years old and older. Their research company made random telephone calls to homes in the community. The telephone calls resulted in: An inability to reach a person in 53% of the homes called. The exclusion of non-telephone homes in the community. Those surveyed were 72% male and 28% females. Explain how each of the three factors above could create a bias in the survey results. Sample Task S3a Note: This sample task is intended as an illustration of new topics that have been added to the curriculum, not as one of the problems for participants to work on. It may also serve as a stimulus for discussion of dealing with the changes in the curriculum.

Gasoline Milk March 12, 2006 2.36 2.30 March 19, 2006 2.50 2.35 The chart below shows the prices of gasoline and milk at a local convenience store, over a 3-week period. Price of Gasoline and Milk in March 2006 Gasoline Milk March 12, 2006 2.36 2.30 March 19, 2006 2.50 2.35 March 26, 2006 2.49 2.33 Sample Task S13a Note: This sample task is intended as an illustration of new topics that have been added to the curriculum, not as one of the problems for participants to work on. It may also serve as a stimulus for discussion of dealing with the changes in the curriculum. What type of correlation, if any, during this three week period existed between the price of gasoline and the price of milk? Could either of these events cause the other? Explain your answer.

On the graph determine the line of best fit. The retail price of various diamonds by size was recorded at a local jewelry store, as seen in the graph below. Sample Task S17a Note: This sample task is intended as an illustration of new topics that have been added to the curriculum, not as one of the problems for participants to work on. It may also serve as a stimulus for discussion of dealing with the changes in the curriculum. On the graph determine the line of best fit. Which is the best estimate of the price of a diamond that is 0.31 carats?

What is the minimum number of e-mails sent? The number of e-mails 20 different students sent in a week varied from 35 to 90, as seen in the box-and-whisker graph below: What is the minimum number of e-mails sent? What is the number at the 25th percentile? What is the number at the 50th percentile? What is the number of e-mails sent at the 75th percentile? What is the maximum number sent? Sample Task PS8c Note: This sample task is intended as an illustration of new topics that have been added to the curriculum, not as one of the problems for participants to work on. It may also serve as a stimulus for discussion of dealing with the changes in the curriculum.

Other New Topics

Determine if the graph of each of the relations is a function Determine if the graph of each of the relations is a function. Justify your answer. Sample Task G3a Note: This sample task is intended as an illustration of new topics that have been added to the curriculum, not as one of the problems for participants to work on. It may also serve as a stimulus for discussion of dealing with the changes in the curriculum.

Determine if each relation is a function. Justify your answer. x y 3 7 11 9 13 -1 x y 2 1 3 -3 4 Sample Task G3b Note: This sample task is intended as an illustration of new topics that have been added to the curriculum, not as one of the problems for participants to work on. It may also serve as a stimulus for discussion of dealing with the changes in the curriculum.

A ruler is accurate to 0. 1 of a centimeter A ruler is accurate to 0.1 of a centimeter. A rectangle is measured as 19.4 cm by 11.2 cm. What is the relative error, expressed as a decimal, in calculating the area? What is the percent error, to the nearest tenth of a percent, in calculating the area? Sample Task M3a Note: This sample task is intended as an illustration of new topics that have been added to the curriculum, not as one of the problems for participants to work on. It may also serve as a stimulus for discussion of dealing with the changes in the curriculum.

Some Additional New Topics Difference between an algebraic expression and an algebraic equation Verbal problems with exponential growth and decay Slope as a rate of change Equation of a line given two points Graphing linear inequalities Graphing solutions of systems of linear and quadratic equations How coefficient change of equation affects its graph Again referring to the previous group work, here are some additional new topics.

Standard Curriculum Printed Material: Distribute “15 Topics New to Integrated Algebra Keyed to Prentice Hall”. This is the standard curriculum for the Integrated Algebra course; we will be looking at it in depth shortly.

This is a sample page of the daily Pacing and Planning Guide.

Integrated Algebra Regents Exam

Format of the Integrated Algebra Exam Printed Material: Distribute “12 Specifications for the Regents Examination in Integrated Algebra”, refer to Table 2 Note to the Presenter: Emphasize that the 87 points of raw score on the exam will still be subject to a sliding scale to convert to the percent score that students finally receive on their records. To meet the exam requirement for a Regents Diploma, students must pass at least one math Regents exam. Students who first enter grade nine in September 2008, and thereafter, must attain a score of 65 or above on all required Regents examinations in order to earn a Regents diploma or a Regents diploma with advanced designation. Below is the formula used by the State Ed Dept to calculate how long it will take students to answer a question of each type: Multiple Choice question: 1 min. 2-credit open-ended: 5 mins. 3-credit open-ended: 10 mins. 4-credit open-ended: 15 mins.

Topics on the Integrated Algebra Regents Printed Material: “12 Specifications for the Regents Examination in Integrated Algebra”; refer to Table 1 Point out to participants that even some of the other Content Strands contain topics that have elements of algebra within them. For example, in the Geometry Strand most of the performance indicators are about Coordinate Geometry which is often considered to be an algebra topic. Printed Material: “14 Calculator Use Policy” Please emphasize that graphing calculator use is mandated on this exam, as it will be on the Geometry and Algebra 2 / Trigonometry Regents exams.

Printed Material: Distribute “13 June 2008 Integrated Algebra Exam” Printed Material: Distribute “13 June 2008 Integrated Algebra Exam”. This is the first of the new math Regents exams and we’re going to be taking a look at it now.

Which of the new topics we’ve looked at were assessed on the June 2008 Integrated Algebra Regents exam? Some examples: 1, 3, 5, 18, 19, 30, 33, 38

The Challenge of Communication Academic Language Math Vocabulary This coming school year will see a major effort to promote student use of academic language and to build their math vocabularies. Let’s take a look at the impact of math vocabulary on the Integrated Algebra Regents exam.

Definitions Linear function Correlation: negative, positive Permutation Vertex, axis of symmetry Slopes of parallel lines Undefined Qualitative, quantitative Understanding the definitions of these terms would be enough for a student to answer seven Part I questions correctly. Find the questions.

Questions Definitions 1 Linear function 5 Correlation: negative, positive 6 Permutation 11 Vertex, axis of symmetry 14 Slopes of parallel lines 17 Undefined 19 Qualitative, quantitative

Definitions with minimal application Bias , , ,  Cumulative frequency Likewise with these three terms or symbols, understanding the definition would make the correct answer choice apparent.

Questions Definitions with minimal application 3 Bias 21 , , ,  22 Cumulative frequency This makes a total of ten questions which can be answered with no computation or problem solving required, which represents two-thirds of the 30 points raw score needed for a passing mark of 65% on the exam! Here are the questions.

NY State Education Department Core Curriculum, Sample Tasks, Glossary, Crosswalks and Other Resources: http://www.emsc.nysed.gov/3-8/guidance912.htm Format of Integrated Algebra Regents Exam: http://www.emsc.nysed.gov/osa/mathre/testspecsalgebra.pdf Office of State Assessment www.emsc.nysed.gov/osa/ Testing Questions can be sent to: emscassessinfo@mail.nysed.gov Most of the material participants were given today can be downloaded from the SED’s website. New York City Department of Education Department of Mathematics

Department of Mathematics New York City Department of Education Contact Information: Linda Curtis-Bey, Director of Mathematics lcurtis@schools.nyc.gov For further information… New York City Department of Education Department of Mathematics

Contact Information Miguel Cordero High School Math Instructional Specialist mcordero@schools.nyc.gov Ronald Schwarz rschwarz@schools.nyc.gov Elaine Carman Middle School Math Instructional Specialist ecarman@schools.nyc.gov New York City Department of Education Department of Mathematics