Content Area: Math

 

Index: 4.3C Grade12 CPI 3

 

Standard: 4.3 - Patterns and Algebra

 

Strand: C - Modeling

 

Cumulative Progress Indicator: 3 - The student will convert recursive formulas to linear or exponential functions (e.g., Tower of Hanoi and doubling).

 

Grade: 12

 

Sample Activities:

 

·        Students in all mathematics classes use functions, making explicit connections to what they have previously learned about functions. As students encounter a new use or meaning for functions, they relate it to their previous understandings.

 

·        Students use recursive definitions of functions in both geometry and algebra. For example, they define n! recursively as n! = n×(n-1)! They use recursion to generate fractals in studying geometry. They may use patterns such as spirolaterals, the Koch snowflake, the Monkey's Tree curve, the Chaos Game, or the Sierpinski triangle. They may use Logo or other software to iterate patterns, or they may use the graphing calculator. In studying algebra, students consider the equation y = .1x + .6, starting with an x-value of .6, and find the resulting y value. Using this y value as the new x value, they then calculate its corresponding y value, and so on. (The resulting values are .6, .66, .666, .6666, ... - providing closer and closer approximations to the decimal value of 2/3!) Students investigate the results of iterations which are other starting values for the same function; the results are surprising! They use other equations and repeat the procedure. They graph the results with a graphing calculator, adjusting the range values to permit viewing the resulting y-values. (See Fractals for the Classroom by H.-O. Peitgen, et al.)