# Document ```New Standards in High School Mathematics,
New York State
Introduction to the Integrated Algebra Course
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
Standard 3
The Three Components
•Conceptual Understanding consists of those
relationships constructed internally and connected to
•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 3
Content and Process Strands
The Five Content Strands
The Five Process Strands
Number Sense and
Operations
Problem Solving
Algebra
Geometry
Measurement
Statistics and
Probability
Reasoning and Proof
Communication
Connections
Representation
Work with two other students to solve
the following problem:
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
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
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.
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.
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.
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.
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.
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.
The New Courses:
•Integrated Algebra
•Geometry
•Algebra 2 and Trigonometry
Number of Performance Indicators for Each Course
Content Strand
Integrated
Algebra
Geometry
Algebra 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 Regents
Implementation / Transition Timeline
200607
200708
200809
200910
Math A
Math
B
Algebra
Geometry
Algebra 2 and
Trigonometry
X
X
School curricular and
instructional alignment and
SED item writing and pretesting
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
School curricular and instructional
alignment and SED item writing and
pre-testing
X
X
January 2009
X
X
June 2008, Postequate
X
X
X
in June
2010
X
X
2009, Post-equate
X
School curricular and instructional
alignment and SED item writing and
pre-testing
X
Post-equate
201011
X
X
X
201112
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.
When A= {3,4,5} and B = {4,5,6,7},
find: AB and AB
A
B
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
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.
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
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
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-andwhisker graph below:
What is the
What is the
What is the
What is the
percentile?
What is the
minimum number of e-mails sent?
number at the 25th percentile?
number at the 50th percentile?
number of e-mails sent at the 75th
maximum number sent?
Other New Topics
Determine if the graph of each of the relations is a
Determine if each relation is a function. Justify your
x
3
7
9
-1
y
7
11
13
3
x
0
1
1
2
y
2
3
-3
4
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?
• 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
• 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 Challenge
of Communication
• Math Vocabulary
Definitions
Linear function
Correlation: negative, positive
Permutation
Vertex, axis of symmetry
Slopes of parallel lines
Undefined
Qualitative, quantitative
Questions
1
5
6
11
14
17
19
Definitions
Linear function
Correlation: negative, positive
Permutation
Vertex, axis of symmetry
Slopes of parallel lines
Undefined
Qualitative, quantitative
Definitions with
minimal application
Bias
, , , 
Cumulative frequency
Questions
3
21
22
Definitions with
minimal application
Bias
, , , 
Cumulative frequency
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:
[email protected]
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
[email protected]
New York City Department of Education
Department of Mathematics
Contact Information
• Miguel Cordero
High School Math Instructional Specialist
[email protected]
• Ronald Schwarz
High School Math Instructional Specialist
[email protected]
• Elaine Carman
Middle School Math Instructional Specialist
[email protected]
New York City Department of Education
Department of Mathematics
```