# Soitirios Chasapis “Mathematical modelling on the falling ladder paradox”

**Please read the success story**

There is no better alternative than real life examples to make children understand mathematical problems.

My name is Sotirios Chasapis, I‘m teaching Mathematics in Evangeliki Model School of Smyrna, Greece, since 2013.

I am a mathematician who graduated from the National and Kapodistrian University of Athens, Greece with a Master’s degree in Pure Mathematics,

An I am a deputy principal in Evangeliki Model School of Smyrna, first founded in 1733 in Smyrna of Asia Minor. In 1922, after the war in Asia Minor, it stopped functioning and it was re-founded in Nea Smyrni in 1934. In 1971, it became a Model School.

I am interested in interdisciplinary collaborations to make the teaching of mathematics more intriguing and directly applicable. The school is open in new methods in teaching and the activity described below was organized jointly, with the school principal, physicist Dr. Christos Fanidis.

The challenge was identified during geometry math classes when the students had shown real difficulty in visualizing the mathematical problem. Shapes and drawings have always been used in mathematical problems, but in examples like a falling-ladder, the students could not actually link the description of the problem to an actual visualization. Therefore, a lot of students failed to comprehend the problem because they could not link it to a practical situation. There is no better alternative than real life examples to make children understand mathematical problems.

The falling ladder paradox is a classic paradigm in calculus. A ladder m meters long leans against a vertical wall and while it’s base moves outward at a constant rate, students calculate how fast the tip of the ladder is moving downward.

While the students were using standard methods of mathematical analysis and geometry to relate given facts and wanted, they were guided to calculate the velocity that the top of the ladder will hit the floor. Proceeding a little lightheartedly, they were astonished as the velocity calculated approached light speed! Students who were not interested in math but more in biology, they observed the paradox, and this sparked their interest.

The students decided to use a yardstick, 1 meter long, which leans against a wall in the class, in order to identify what is really happening. But the experiment was running fast in time, making experimental observations difficult and the conclusions inaccessible. How about using an interactive physical simulator?

After that, students, used the software Interactive Physics, an interactive physical simulator, in order to set the experiment on a computer. Additional programs like that were used, such as the STEP simulator, a free licensed interactive software simulator, accessible here: https://edu.kde.org/step/. The key of the experiment is that the ladder’s tip leaves the wall at some point in its descent near the ground. A fact that was closely observed, but not for sure, on the real experiment.

After using the software, the students could find the mathematical error in their calculations, as no actual triangle exists in the time that the ladder’s tip leaves the wall, so they realized they cannot use the Pythagorean Theorem. The successful results of the use of an interactive physical simulator were obvious at the time of the experiment, and later on. The students had apparently understood the mathematical problem during the time of the experiment, and the less interested students were a lot more eager to discuss mathematics, even requesting to use the simulator computer program.

A real experiment is not always possible to be used in the classroom and even if this happens, the observatory can’t always ‘catch’ the moment of impact. Model simulators on a computer, can make the experiment more interactive, accessible and observable, while a discussion with a physicist in the same time gives more hints for the students, more intriguing. Additionally, the use of digital tools by the students is encouraged as an added value which leads to an efficient engagement of them with mathematics, robotics and computer science competitions.

- Why can practical examples in mathematics be more easily depicted in simulations?
- How does identifying an error affect the interest of the less interested student?
- Can you think of mathematical examples that are ideal to be explained with model simulators?