New Standards in High School Mathematics, New York State Introduction to the Integrated Algebra Course

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

2. Session Objectives: • Content and Process Strands • Performance Indicators • New Courses • Looking at Integrated Algebra • The New Regents Exam • For More Information

3. Standard 3 The Three Components • Conceptual Understanding consists of those relationships constructed internally and connected to already existing ideas. • Procedural Fluencyis the skill in carrying out procedures flexibly, accurately, efficiently, and appropriately. • Problem Solving is the ability to formulate, represent, and solve mathematical problems.

4. Standard 3 Content and Process Strands

6. 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?

7. 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

8. 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.

9. 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.

10. 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.

11. 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.

12. 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.

13. 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

14. 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.

15. 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.

16. 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.

17. 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.

18. 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.

19. The New Courses: Integrated Algebra Geometry Algebra 2 and Trigonometry

22. New Mathematics Regents Implementation / Transition Timeline

24. Looking at Integrated Algebra

25. Some Major Topics in Algebra Not in Math A

26. Sets • Set-Builder Notation and Interval Notation • Complement of a Subset of a Given Set • Intersection and/or Union of Sets

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

28. When A= {3,4,5} and B = {4,5,6,7}, find: AB and AB B A

29. 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…)

30. 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

31. 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.

32. 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.

33. 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.

34. 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.

35. 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?

36. 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?

37. Other New Topics

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

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

40. 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?

41. 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 Some Additional New Topics

42. Standard Curriculum

44. Integrated Algebra Regents Exam

45. Format of the Integrated Algebra Exam

46. Topics on the Integrated Algebra Regents

48. Which of the new topics we’ve looked at were assessed on the June 2008 Integrated Algebra Regents exam?

49. The Challengeof Communication • Academic Language • Math Vocabulary

50. Definitions Linear function Correlation: negative, positive Permutation Vertex, axis of symmetry Slopes of parallel lines Undefined Qualitative, quantitative