Numerical Thinking:

Is It Part of Your Algebra Curriculum?

SCCTM Conference

Hilton Head, SC

19 November 1999

 

 

Ed Dickey

University of South Carolina

 

Ed.Dickey@SC.edu

www.ite.sc.edu/dickey.html

 

 

Solve 3x – 2 = x + 4

 

Symbolic:

3x – 2 = x + 4

2x = 6

x = 3

 

Graphic:

 

Numeric:

 

 

 

 

Symbolic

Graphic

Numeric

Traditional

(no technology)

      

Now

(with technology)

      

Ideal

     

 

 

 

Course Standards for South Carolina PACT

 

Algebra I

The student will solve quadratic equations in one variable both algebraically and graphically. Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions.

The student will estimate square roots at least to the nearest tenth and use a calculator to compute decimal approximations of radicals.

 

Precalculus

The student will recognize multiple representations of functions (linear, quadratic, absolute value, step, and exponential functions) and convert between a graph, a table, and symbolic form. A transformational approach to graphing will be employed through the use of graphing calculators.

 

 

Standards for Grades 9-12

II. Numerical and Algebraic Concepts and Operations

B. Use tables and graphs as tools to interpret expressions, equations, and inequalities, using technology whenever appropriate.

  • The student will represent linear equations or inequalities as tables or graphs.

    • D. Develop an understanding of and facility in manipulating algebraic expressions, performing elementary operations on matrices, and solving equations and inequalities.

  • The student will solve equations and inequalities using a variety of methods to include graphing, spreadsheets, and symbol manipulation, explaining procedures used.

    •  

    III. Patterns, Relationships, and Functions

    D. Translate among tabular, symbolic, and graphical representations of functions, using technology whenever appropriate.

  • The student will gather and plot data, fit a graph to plotted points, use the graph to illustrate the relationship between variables, predict outcomes, and form a generalized equation in a real-world context.
  • The student will recognize multiple representations of functions (linear, quadratic, absolute value, step, and exponential functions) and convert between a graph, a table, and symbolic form. A transformational approach to graphing will be employed through the use of graphing calculators.

    A Numerical Approach

    •

    x2 – x – 3 = 0

     

    x2 – x = 3

    x(x-1) = 3

     

    x

    x – 1

    x(x – 1)

    3
           
           
           
           
           
           

     

     

     

    Iteration and Interpolation

     

    x2 + 3x – 6 = 0

    x2 + 3x + 6 = 0

    2x2 - 5x – 8 = 0

     

     

     

     

    Symbolic

    Graphic

    Numeric

    Robust

    		      

    Easy to Remember

    		      

    Easy to Use

    		      

     

     

    Bisection Method

    x2 – x – 3 = 0

     

    x

    x2 – x - 3
       
       
       
       
       
       

     

    1. Find an interval where there is a sign change.
    2. Find the midpoint of that interval.
    3. If the midpoint is a root, we’re done.
    4. Of the two intervals .. left point to midpoint or midpoint to right point… choose the one in which there is a sign change.
    5. Go back to step 2.

     

    Program for TI calculator (73, 81, 82, or 83)

    PrgmBISCTION

    :Disp "LEFTPT="

    :Input L

    :Disp "RIGHTPT="

    :Input R

    :Lbl 1

    : (L+R)/2-M

    :L^2-L-3ÆX

    :M^2-M-3ÆY

    :R^2-R-3ÆZ

    :If abs(Y)_.0001

    :Goto 3

    :If Y/X _ 0

    :Goto 2

    :MÆR

    :Goto 1

    :Lbl 2

    :MÆL

    :Goto 1

    :Lbl 3

    :Disp "ROOT ="

    :Disp M

    :Pause

     

     

    Spreadsheet (Excel)

    x

    g(x)=x^2-x-3

    -3

    9

    -2

    3

    -1

    -1

    0

    -3

    1

    -3

    2

    -1

    3

    3

    4

    9

    5

    17

     

    x

    g(x)=x^2-x-3

    2

    -1

    2.1

    -0.69

    2.2

    -0.36

    2.3

    -0.01

    2.4

    0.36

    2.5

    0.75

    2.6

    1.16

    2.7

    1.59

    2.8

    2.04

    2.9

    2.51

    3

    3

     

    Fixed Point Iteration

    x2 – x – 3 = 0

    x

    		 g(x)
       
       
       
       
       
       

     

     

     

     

    x

    g(x)=sqrt(x+3)

    0

    1.732050808

    1.7320508

    2.175327747

    2.1753277

    2.274934669

    2.2749347

    2.296722593

    2.2967226

    2.301460969

    2.301461

    2.302490167

    2.3024902

    2.302713653

    2.3027137

    2.302762179

    2.3027622

    2.302772715

     

    References

  • Day, Roger P. "Solution Revolution." Mathematics Teacher, 86(1), (January, 1993), pp. 15-22.
  • Dickey, Edwin M. "The Golden Ratio: A Golden Opportunity to Investigate Multiple Representations of a Problem." Mathematics Teacher, 86(7), (October, 1993), pp. 554-557.
    •
  • Lovitt, Charles and Doug Clarke. Chapter 13: Iteration: Numerical Methods. Activity Band Volume 2, Mathematics Curriculum and Teaching Project, NCTM, 1988.
    •
  • Niess, Margaret. "Using Computer Spreadsheets to Solve Equations." Learning and Leading with Technology, 26(3), (November, 1998), pp. 22-27.
    •
  • Prichard, Mary Kim. "Mathematical Iteration through Computer Programming." Mathematics Teacher, 86(2), (February, 1993), pp. 150-156.
    •
  • Waits, Bert, and James Schultz. "An Interative Method for Computing Solutions to Equations Using a Calculator." Mathematics Teacher, 72(9), (December, 1979), pp. 685-689.
    •