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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.
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
Sets•Set-Builder Notation and Interval
Notation•Complement of a Subset of a Given Set•Intersection and/or Union of Sets
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?
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
Which of the new topics we’ve looked at were assessed on the June 2008 Integrated Algebra
Regents exam?
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
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
New York City Department of EducationDepartment of Mathematics
Contact Information
• Miguel CorderoHigh School Math Instructional Specialist
• Ronald SchwarzHigh School Math Instructional Specialist
• Elaine CarmanMiddle School Math Instructional Specialist
New York City Department of EducationDepartment of Mathematics