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

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

A Brief History

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

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

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

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

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

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

Intermediate Algebra: A FUNctional Approach Relationships between variables:

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

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

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

Intermediate Algebra: A FUNctional Approach

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

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?

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

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

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

Intermediate Algebra: A FUNctional Approach

Intermediate Algebra: A FUNctional Approach

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

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

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

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

Intermediate Algebra: A FUNctional Approach Sample activities and assessments: https://ctalgebra2.wikispaces.com/

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

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

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

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

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

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

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

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

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

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

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

Intermediate Algebra: A FUNctional Approach So what?

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

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

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

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

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

Contact Info & Resources Marlene Lovanio Director of STEM and Literacy Programs, CSDE marlene.lovanio@ct.gov Algebra 2 Wiki https://ctalgebra2.wikispaces.com/ Secondary School Reform – the CT Plan National Math Panel Report