Functional thinking methodology can illuminate most of the problem solving students will encounter in the digital age.

(under development)

Numberlines and Discrete Variables

Numberlines of all kinds with parameters initial value x0 and increment Δx by using rules to explore and order numbers.

Operations and Functions

Functions with input and output columns to transform x into f(x) using rules and explore patterns with all 4 operations and constants stored in the parameter table.

Tables: Hundreds & Times

Tables (100’s tables and multiplication) using a vertical (column) and horizontal (row) numberlines to explore addition and multiplication patterns. (Multiplication table numberlines should grow up and right to match standard graphs.)

Ordered Pairs

Ordered pair tables (x, f(x)) to explore discrete variables and functions that can be displayed with line graphs, linked to tables, and changed by adjusting the parameters.


Ratios (x, f(x)) and division tables to visualize order, conversion, and equivalence of fractions, decimals, rates, percents, interest, averages, proportions, and linear functions.


Build multiplication tables for the integers in 4 quadrants using x0 and Δx.

Inverse Variation

Use x0 and Δx to manipulate the hyperbola graphs as well as concrete tables like multiplication tables.

Classic Story Problems

Classic story problems (work problems, motion problems, mixture problems, etc.) using (x, f(x)) tables adjusting x0 and Δx parameters to find solution values or just increase accuracy.

Solving Equations

Solve equations as Judah Schwartz has taught by setting two functions equal to each other and changing the parameters to set up the functions.

Linear Functions

Linear functions with slope (m) and y-intercept (b) as parameter values along with x0 and Δx to generate discrete tables of values and graphs.

Exponential Functions

What do they look like? What happens when you change the starting value of x? What happens when you change the base (common ratio)? What does the graph look like when the base is negative?

Quadratic Functions

With (a, b, c) as parameters along with x0 and Δx in both graphs and tables. Ask questions like what does changing b do to the graph?

Families of Functions

Use the parameter table and easily adjusting and changing domains with x0 and Δx to find maxima, minima, and points of inflection.

Operations on Functions

Find inverses, absolute values using x0 and Δx to picture how the shapes of these graphs change when these parameters change.

Visualizing Derivatives

Zoom into a function by shrinking Δx, at different location on a graph using x0. Students see the function becoming more and more linear to represent the slope at each point

Visualizing Integrals

Visualize integrals as Riemann sums using bar graphs and shrinking Δx.