# Engaging students: Synthetic Division

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Cire Jauregui. Her topic, from Precalculus: synthetic division.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

The website IXL has a series of Algebra 2 learning topics where students can do practice problems. It presents students with a problem and tracks how long it takes them to solve the question. It also gives them a score out of 100. This site also has examples students can use to help them learn. The “Learn with an example” page walks students through the process step by step so that they can learn the process. If a student answers correctly, they are congratulated, given points, and then given a new problem to solve. If a student answers the question incorrectly, they are given a full explanation with the steps to solve the problem written out so students can check where they messed up. There are so many problems this program can come up with and provide students with many examples of all kinds.

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Paolo Ruffini developed Ruffini’s rule which is now known most commonly as synthetic division. Ruffini was an Italian mathematician in the late 1700s. In 1796, Napoleon Bonaparte and his troops signed agreements with the duke of Modena where Ruffini was studying and teaching. Here Napoleon set up the Cisalpine Republic where Ruffini was appointed to be a representative for the Junior Council of the Cisalpine Republic. He did not wish to take the position, so he left to return to his studies at the University of Modena in 1798. However, when he was required to swear an oath to the Republic, Ruffini refused due to his religious grounds and was removed from his teaching position at the university and told he could not teach again.

How does this topic extend what your students should have learned in previous courses?

This topic extends on a student’s ability to do long division and also polynomial long division. Polynomial long division works exactly how students would expect dividing a polynomial would work. The polynomial dividend is under the bracket, the leading term (not just the coefficient) of the divisor is used as the primary divisor which determines what should be on top of the bracket. This process continues until the divisor cannot divide into the dividend and then is used as a remainder where the “leftover” part is put over the divisor and left as a fraction. Synthetic division simplifies this process by focusing on the coefficients of the polynomial being divided. By focusing on the coefficients, it can remove some of the confusion students face when trying to do polynomial division.

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