Numerical Thinking:
Is It Part of Your Algebra Curriculum?
SCCTM Conference
Hilton Head, SC
19 November 1999
Ed Dickey
University of South Carolina
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
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.
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 912
II. Numerical and Algebraic Concepts and Operations
B. Use tables and graphs as tools to interpret expressions, equations, and inequalities, using technology whenever appropriate.
D. Develop an understanding of and facility in manipulating algebraic expressions, performing elementary operations on matrices, and solving equations and inequalities.
III. Patterns, Relationships, and Functions
D. Translate among tabular, symbolic, and graphical representations of functions, using technology whenever appropriate.
A Numerical Approach
x^{2} – x – 3 = 0
x^{2} – x = 3
x(x1) = 3
x 
x – 1 
x(x – 1) 
3 
Iteration and Interpolation
x^{2} + 3x – 6 = 0
x^{2} + 3x + 6 = 0
2x^{2}  5x – 8 = 0
Symbolic 
Graphic 
Numeric 

Robust 

Easy to Remember 

Easy to Use 
Bisection Method
x^{2} – x – 3 = 0
x 
x^{2} – x  3 
Program for TI calculator (73, 81, 82, or 83)
PrgmBISCTION
:Disp "LEFTPT="
:Input L
:Disp "RIGHTPT="
:Input R
:Lbl 1
: (L+R)/2M
:L^2L3ÆX
:M^2M3ÆY
:R^2R3ÆZ
: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^2x3 
3 
9 
2 
3 
1 
1 
0 
3 
1 
3 
2 
1 
3 
3 
4 
9 
5 
17 
x 
g(x)=x^2x3 
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
x^{2} – 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