Automated Generation of Region Based Geometric Questions

Abstract

We extend our previously proposed framework that combines a combinatorial approach, pattern matching and automated deduction to generate geometry questions which, directly or indirectly, require finding the congruent regions formed by the intersection of geometric objects. The extension involves proposing a knowledge representation for regions and a rule-based algorithm for generation of region-based knowledge representation. In addition, several algorithms such as circle/arc projection to straight line (s) are proposed to avoid numerical reasoning for proving congruent regions, making the solution eligible for high school geometry domain. Furthermore, we propose the integration of this framework with our previously proposed framework to generate questions involving both implicit construction and congruent regions. The system is able to generate the solution (s) of the questions for their validation. Such a system would help teachers to quickly generate large numbers of questions based on several properties of geometric objects such as length, angle, area and perimeter. Students can explore, revise and master specific topics covered in classes and textbooks based on generated questions. This system may also help standardize tests such as Primary School Leaving Exam (PSLE), GMAT and SAT. Our methodology uses (i) a combinatorial approach for generating geometric figures (ii) Pattern matching and rule-based approach for region generation (iii) automated deduction for checking equality of properties of geometric objects (iv) linear equation solver to generate new questions and solutions. By combining these methods, we are able to generate questions involving finding or proving congruence relationships between the regions generated by the geometric objects based on a various specifications such as objects and concepts. Experimental results show that a large number of questions can be generated in a short time. A survey shows that the generated questions and the solutions are useful and fulfills the high school criteria.