1. Intermediate Algebra: A FUNctional Approach
MATYCONN Meeting
October 24, 2008
Marlene M. Lovanio
Connecticut State Department of Education

2. Intermediate Algebra: A FUNctional Approach
Today, our goals include:
Fostering a functional approach
Sharing ideas with colleagues
Considering future change

3. A Brief History

4. A Brief History The concern:
Increasing number of students needing remedial math.
The culprit:
Placement Tests, Algebra 2, Not Enough Math?

5. Intermediate Algebra Subcommittee Focus:
A Brief History
Intermediate Algebra Subcommittee Focus:
Determine skills and concepts necessary to succeed in high school and college.
At the same time, the state pulled together the P-16 council focused on Data and Curriculum and work was being done on secondary school reform. After a false start, a P-20 Council is being formed under the direction of the new leadership at SDE and DHE

6. Intermediate Algebra: A FUNctional Approach
Activity
What are the five most important concepts you would teach in Intermediate Algebra?

7. Intermediate Algebra: A FUNctional Approach
The results, so far…

8. Intermediate Algebra: A FUNctional Approach
Overview of the documents:
Philosophy
Pedagogical considerations
Minimal core topics
Sample activities and assessments

9. Intermediate Algebra: A FUNctional Approach
Philosophy:
The fundamental question is not “simplify this expression” or “find the value of x that makes this equation true,” but “how does this quantity change as it relates to some other quantity

10. Intermediate Algebra: A FUNctional Approach
Relationships between variables:

11. After this course, students should know and be able to:
function as a relationship and as an object
functions represented graphically, in context, symbolically, in tables or in data sets
identify the function family
understand the characteristics of function families

12. Intermediate Algebra: A FUNctional Approach
Key features for teachers:
Represent functions contextually, graphically, symbolically, and with tables/data sets
Expect students to identify families of functions from different representations
Build on benchmark functions for each family
Find a balance: depth vs. breadth and avoiding or including complex algebraic manipulation

13. Intermediate Algebra: A FUNctional Approach
Examples:
Represent functions contextually, symbolically, graphically, and with tables/data sets

14. Intermediate Algebra: A FUNctional Approach

15. Intermediate Algebra: A FUNctional Approach
A single bacterium begins to double after the first hour in the incubation lab.

16. Intermediate Algebra: A FUNctional Approach
f(x) = 16*0.75x ; when is f(x) < 1
AND
You take 16 mg. of a controlled substance
at 8 a.m. Your body metabolizes 25%
of the substance every hour. Will you
pass a 4 p.m. drug test the a requires
a level of less than 1 mg? At what time
could you first pass the test?

17. Intermediate Algebra: A FUNctional Approach
Key features:
Be able to recognize families of functions from different representations

18. Intermediate Algebra: A FUNctional Approach
Philosophy:
Build on benchmark functions for each family
Y = x^2 y = x^3

19. Intermediate Algebra: A FUNctional Approach
Key features:
Finding a balance: depth vs. breadth and avoiding or including complex algebraic manipulation

20. Intermediate Algebra: A FUNctional Approach

21. Intermediate Algebra: A FUNctional Approach

22. Intermediate Algebra: A FUNctional Approach
Key features:
Finding a balance: depth vs. breadth and avoiding or including complex algebraic manipulation.

23. Intermediate Algebra: A FUNctional Approach
Pedagogical Considerations:
Minimal Core Topics + more
Emphasis on modeling
Utilize “real” data
Explore important features of families of functions

24. Intermediate Algebra: A FUNctional Approach
Pedagogical Considerations:
Balance skill sets development with conceptual understanding
Balance avoidance of complex expressions with getting bogged down
Balance depth with breadth

25. Intermediate Algebra: A FUNctional Approach
Minimal Core Topics:
Linear functions and systems
Quadratic functions
Polynomial functions
Exponential functions
Rational functions
Radical functions
Logarithmic functions

26. Intermediate Algebra: A FUNctional Approach
Sample activities and assessments:

27. Intermediate Algebra: A FUNctional Approach
Example Activity:
The Martian Activity
Introduction to Quadratic Functions 27

28. The Martian Activity uses physics of gravity and motion to compare and contrast linear and quadratic functions.
The functional approach philosophy is used throughout this rich and engaging activity.
Students explore functions by plotting points, finding rates of change, critically thinking about constants as well as creating tables, charts, and graphs. 28

29. Real data – The data below compares the force of gravity on the surface of the various planets with gravity at the Earth’s surface as being 1. Since Martian gravity is only 38% that of the Earth, astronauts on the Martian surface can easily jump over craters and boulders. They can jump approximately 47 feet (15 meters).
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
0.39
0.91 1
0.38
2.6
1.1
.88
1.14 29

30. Height above surface (m)
The table of values below describes the position of an astronaut jumping up and forward over a small crater on the surface of Mars.
Vertical motion (height) involves gravity.
Horizontal motion (forward) is only affected by minimal air resistance.
Time (Sec)
Height above surface (m)
Forward motion (m) 1 8 3 2 12 6
2.5
12.5
7.5 9 4 5 15
What is the average rate of change in the number of meters forward per second? 30

31. Students will first explore forward (horizontal) motion by completing a table. Key Ideas: Constant Rate of Change Linear Function Relate horizontal distance with time by writing and plotting the function. 31

32. Height above surface (m)
The table of values below describes the position of an astronaut jumping up and forward over a small crater on the surface of Mars.
Vertical motion (height) involves gravity.
Horizontal motion (forward) is only affected by minimal air resistance.
Time (Sec)
Height above surface (m)
Forward motion (m) 1 8 3 2 12 6
2.5
12.5
7.5 9 4 5 15
Describe if the
∆output/ ∆input
is constant for vertical motion? 32

33. Average Rate of Change is not constant
Height (vertical) motion will be explored by completing a table and plotting points.
Key Ideas:
Average Rate of Change is not constant
Introduce Quadratic Function
Paths of objects moving under gravitational force traces a curve (parabola) Relate vertical distance with time by writing and plotting the function 33

34. Changes in the initial conditions (velocity) provides concrete understanding how the shape is affected by parameter changes. 34

35. By analyzing the data and graphs students
will be able to answer the following questions:
How far can the astronaut jump horizontally when their initial vertical velocity if 5 m/sec? 20 m/sec?
Using the plots find the value(s) of t for which g(t) = 0
If gravity on Mars was weaker, for example, the coefficient of was -1 how would the vertical jump be affected? 35

36. The Martian Activity engages students in:
analyzing the relationship between time and distance;
exploring how distance (output) changes related to time (input); and
investigating maximums and minimums of functions in realistic context.
This mathematical model activity is one of many resources being provided that uses the functional approach philosophy. 36

37. Intermediate Algebra: A FUNctional Approach
Discussion:
How is this approach similar/different to what you already do?

38. Intermediate Algebra: A FUNctional Approach
So what?

39. National Math Panel Report Curricular Content
An Authentic Algebra Course
All school districts:
Should ensure that all prepared students have access to an authentic algebra course, and
Should prepare more students than at present to enroll in such a course by Grade 8. 39 39

40. National Math Panel Report Curricular Content
The Major Topics of School Algebra
Covering all of school algebra traditionally extending over two courses, Algebra I and Algebra II
Symbols and Expressions
Linear Equations
Quadratic Equations
Functions
Algebra of Polynomials
Combinatorics and Finite Probability 40 40

41. Secondary School Reform Proposal
Engagement
21st Century Skills
Rigor
All students take Algebra, Geometry, Algebra 2 or Prob/Stat and a 4th course.
Model curriculum including formative assessments
Final exams for Algebra 1 and Geometry
P-20 Council

42. If this is the future, what are the next steps in developing courses in a P-20 system?

43. Final Thoughts
Never be afraid to try something new. Remember, amateurs built the ark; professionals built the Titanic.
- Unknown

44. Contact Info & Resources
Marlene Lovanio
Director of STEM and Literacy Programs, CSDE
Algebra 2 Wiki
Secondary School Reform – the CT Plan
National Math Panel Report