Conic Sections Problem Generator

Bachelor's thesis: a Mathematica tool that generates ellipse problems with step-by-step solutions.

Generated ellipse problem with its solution steps and a plot of the ellipse

About the project

For my bachelor's thesis I built a generator of analytical geometry problems about ellipses for students practising for exams. It covers ten problem types, from rewriting an equation into its general form to finding the tangents through a point, each in an easy, medium and hard variant.

Every problem is random but chosen to have clean numbers, and comes with a worked solution written the way a student would solve it on paper, a plot and the answer. Teachers can put together a whole test and export it with an answer key or full solutions.

Highlights

10 problem types in 3 difficulty levels
Step-by-step solutions recorded while the problem is solved
Student practice mode and teacher test builder

Under the hood

Each problem type is a self-contained Mathematica module that returns the statement, the solution steps, a plot and the answer
Parameters are kept readable: semi-axes are clamped so the ellipse is neither a near-circle nor too flat, and the general equation is reduced by its greatest common divisor
Some problem types are generated backwards: start from a random answer, derive the problem, then reverse the recorded steps and invert each operation
Ellipses with many integer points were found by a brute-force search and stored, so intersections and touching points can be computed by hand
Tested with final-year secondary school students for ease of use and how well the generated solutions matched their own
The interface and test export were adapted from two fellow students' bachelor's theses and connected to the generators

When: 2021 – 2022
Type: University
Supervision: Supervised by Mgr. Milada Omachelová, PhD. et PhD.
Versions: Mathematica 12.3
Stack: Wolfram Mathematica

Gallery

Screen recording of the generator building a test and solving a tangent problem step by step — A 50-second walkthrough: building a test in teacher mode, then a generated tangent problem with its solution steps and plot.

Three plots of an ellipse with two tangents through point M: original, mirrored across x = y, and shifted — Tangent problems come from stored cases with integer results. Mirroring across x = y and shifting by an integer vector turn each case into many different problems.

List of ten ellipse problem types coloured green, yellow and red — Student mode: the ten problem types, coloured from simpler (green) to harder (red).

Form with a number of problems for every problem type and difficulty — Teacher mode: choose how many problems of each type and difficulty the test should have, then generate it with an answer key or full solutions.