New Standards in High School Mathematics, New York State Introduction to the Integrated Algebra Course New York City Department of Education Department.

New Standards in High School Mathematics,New York State

Introduction to the Integrated Algebra Course

New York City Department of EducationDepartment of Mathematics

Session Objectives:Session Objectives:

• Content and Process StrandsContent and Process Strands

• Performance IndicatorsPerformance Indicators

• New CoursesNew Courses

• Looking at Integrated AlgebraLooking at Integrated Algebra

• The New Regents ExamThe New Regents Exam

• For More Information For More Information

Standard 3Standard 3

The Three ComponentsThe 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.

Standard 3Standard 3

Content and Process StrandsContent and Process Strands

The Five Content Strands The Five Process Strands

Number Sense and Operations

Problem Solving

Algebra Reasoning and Proof

Geometry Communication

MeasurementConnections

Statistics and Probability

Representation

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?

The Five Content StrandsThe 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

Number Sense and Operations Number Sense and Operations StrandStrand

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.

Algebra StrandAlgebra 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.

Geometry StrandGeometry 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.

Measurement StrandMeasurement 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.

Statistics and Probability StrandStatistics 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.

The Five Process StrandsThe 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 StrandProblem 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 StrandReasoning 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 StrandCommunication 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 StrandConnections 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 StrandRepresentation 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.

The New Courses:

•Integrated Algebra

•Geometry

•Algebra 2 and Trigonometry

Number of Performance Indicators for Each Course

Content StrandIntegrated Algebra

GeometryAlgebra 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

New Mathematics RegentsImplementation / Transition

Timeline 

  Math AMath

BAlgebra Geometry

Algebra 2 and Trigonometry

2006-07

X X 

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

2007-08

X X

XFirst admin. in

June 2008, Post-equate

 School curricular and instructional

alignment and SED item writing and pre-testing

 School curricular and instructional

alignment and SED item writing and pre-testing

2008-09

XLast admin. in January 2009

X X

XFirst admin. in June 2009, Post-equate

 School curricular and instructional

alignment and SED item writing and pre-testing

2009-10

 X

Last admin. in June

2010

X 

X

XFirst admin. in June 2010,

Post-equate

2010-11

X X X

2011-12

    X X X

Looking at Integrated Algebra

Some Major Topics in Algebra

Not in Math A

Sets•Set-Builder Notation and Interval

Notation•Complement of a Subset of a Given Set•Intersection and/or Union of Sets

Given that U={1,2,3,4,5} and A={3,4,5} list the elements in the complement of set A, Ā.

A B

When A= {3,4,5} and B = {4,5,6,7}, find: AB and AB

Data:

•Qualitative or Quantitative•Univariate or Bivariate•Bias, Including Sources•Evaluation of Reports or Graphs

Experimental DesignAppropriateness of Data

AnalysisSoundness of Conclusions

(more…)

Data (continued):

•Percentile Rank of Item in Data SetFirst, Second, Third Quartiles

•Variables: Correlation But Not Causation•Linear Transformations Affect Mean,

Median, Mode•Scatter Plots, Line of Best Fit

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.

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.

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.

Gasoline Milk

March 12, 2006 2.36 2.30

March 19, 2006 2.50 2.35

March 26, 2006 2.49 2.33

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

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.

The retail price of various diamonds by size was recorded at a local jewelry store, as seen in the graph below.

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?

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?

Other New Topics

Determine if the graph of each of the relations is a function. Justify your answer.

x y

3 7

7 11

9 13

-1 3

x y

0 2

1 3

1 -3

2 4

Determine if each relation is a function. Justify your answer.

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?

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

Standard Curriculum

Integrated Algebra Regents Exam

Format of the Integrated Algebra Exam

Topics on the Integrated Algebra Regents

Which of the new topics we’ve looked at were assessed on the June 2008 Integrated Algebra

Regents exam?

The Challengeof Communication

• Academic Language

• Math Vocabulary

Definitions

Linear function

Correlation: negative, positive

Permutation

Vertex, axis of symmetry

Slopes of parallel lines

Undefined

Qualitative, quantitative

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 withminimal application

Bias

, , , Cumulative frequency

Questions Definitions withminimal application

3 Bias

21 , , , 22 Cumulative frequency

NY State Education DepartmentNY 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: [email protected]

New York City Department of EducationDepartment of Mathematics

Department of MathematicsNew York City Department of Education

Contact Information:Linda Curtis-Bey, Director of Mathematics

[email protected]

New York City Department of EducationDepartment of Mathematics

Contact Information

• Miguel CorderoHigh School Math Instructional Specialist

[email protected]

• Ronald SchwarzHigh School Math Instructional Specialist

[email protected]

• Elaine CarmanMiddle School Math Instructional Specialist

[email protected]

New York City Department of EducationDepartment of Mathematics