Beginning and intermediate algebra spend a considerable amount of time solving equations exactly without any reference to functions.

Very little time is spent in those courses visualizing equations or making estimates for solutions- though some time is spent on estimating roots or discussing the fact that technology often gives only estimates for numbers.

In more advance courses before calculus, functions and estimation become more important but still little is done to make sense visually of the solution of equations beyond the connection to intersections of curves with other curves or with the x-axis.

Estimates are visualized in almost haphazard fashion, along with discussion of limit concepts.

Estimating a solution to an equation (or the value of a function) does have some general numerical techniques. These are still evolving as technology improves to make them more effective and efficient. [See Root-finding algorithm, Wikipedia]

In this section we provide mapping diagrams that illustrate some important yet simple numerical algorithms for estimating function values and equation solutions.

Basic approaches: In most estimation techniques the key idea is to find a procedure that is applied to some initial estimate to arrive at a second estimate. Following a general principle that what's good for one is good for all, the procedure is then repeated on each result. Iteration of a procedure with some initial estimating information is possible as long as the initial conditions needed for the procedure continue to be satisfied.

Each of the following three techniques (bisection, false position, and Newton secant) has some positive and negative attributes, which are discussed in the comparison, where the issues of effectiveness and efficiency are illustrated with mapping diagrams.

**Example**AEF.NEB Numerical Estimation Bisection.**Example**AEF.NEFP Numerical Estimation False Position.**Example**AEF.NENS Numerical Estimation Newton Secant.**Example**AEF.NEC Numerical Estimation Comparison ("The More, The Merrier")