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 Kayla (Koenig) Lambert. Her topic, from Pre-Algebra: solving proportions.

**B) Curriculum: How does this topic extend what your students have learned in previous courses?**

Multiplying and dividing rational expressions extends so many topics because the students have to use what they have learned up to multiplying and dividing the rational expressions. For example, this topic extends multiplying and dividing fractions. For multiplying and dividing fractions the students need to multiply across the numerators and multiply across the denominators and then simplify when possible (Multiplying Rational Expressions). Students also use factoring, which they should have learned before getting to this topic. When factoring, the students should remember different ways to factor. Some different ways are finding the greatest common factor, factoring by grouping, and finding the perfect square. They should also remember how to factor polynomials of different degrees.

The students also need to remember how to divide numerical fractions because they use the same method when dividing rational expressions; multiplying by the reciprocal. Another topic students should have previously learned is how to simplify rational expressions and how to multiply polynomials. Lastly, the students should also remember what a term, coefficient, constant, degree of a term, degree of a polynomial and should remember different types of polynomials (monomial, binomial, etc.). I could keep going with what topics are used when multiplying and dividing rational expressions all the way down to counting, addition, and subtraction. There are obviously so many different topics students have learned in the past that are extended when multiplying and dividing rational expressions.

**D) History: ****What are the contributions of various cultures to this topic?**

We can break multiplying and dividing rational expressions into many different mathematical subjects. In order to accomplish multiplying and dividing rational expressions, basic algebra and other basic mathematics had to come first. Methods of multiplication were documented by ancient Egyptian, Greek, and Chinese civilizations (Multiplication-Wikipedia). Around 1800 BC, Egyptians were the first known to use fractions. In 1600 BC, the Babylonians already knew solutions to quadratic equations and also solutions to equations to the third and fourth degree (Mathematics History). Egyptians used papyrus to make papers and used these to “calculate fractions” (Mathematics History).

The word polynomial comes from the Greek work “poly” meaning “many” and from the Latin word “binomium” meaning “binomial” and was introduced in Latin by a French mathematician, Franciscus Vieta (Polynomial-Wikipedia). The history of algebra goes back to ancient Egypt and Babylon where people learned to solve linear and quadratic equations. Also, Islamic mathematicians were able to multiply, divide and find the square root of polynomials. The Hindu-Arabic numerical system was first described by Brahmagupta who gave rules for addition, subtraction, multiplication and division. In orient mathematics, algebra “ultimately evolved from arithmetic” (Mathematics History). Nicole Oresme, from Normandy, was the first person to use fraction and exponents. Many cultures have contributed to multiplying and dividing rational expressions, but I would have to say that the Egyptians, Babylonians, Chinese, and Arabic have contributed the most.

**E) Technology: How can technology (YouTube, Geometers Sketchpad, graphing calculator, etc.) be used to efficiently engage students with this topic?**

Rational functions are used for many things including:

- Fields and forces in physics
- Spectroscopy in chemistry
- Enzyme kinetics in biochemistry
- Electronic circuitry
- Aerodynamics
- Medicine concentration
- Wave functions for atoms and molecules
- Optics and photography to improve image resolution
- Acoustics and sound

Since the above topics are a little too advanced, I could show the student a video on YouTube to introduce the topic and to show them what multiplying and dividing rational functions are used for in the real world. After this, I would explain to the students that many other careers use rational functions like architects, foresters, and chemists. After talking about the topic, I could them give them a problem like the one below and ask them to graph the rational function with their calculator and can use their calculator to set up tables of values for their rational function. This will make it easy for them to see the maximum and minimum of the function and to see how the function behaves.

Example 9 from PreCalculus:

A rectangular page is designed to contain 48 square inches of print. The margins at the top and bottom of the page are 1 inch deep. The margins on each side are 1 ½ inches wide. What should the dimensions of the page be so the least amount of paper is used?

Works Cited

Larson, Ron, and David C. Falvo. “Precalculus – Ron Larson, David C. Falvo – Google Books.” 7 Feb. 2012. http://books.google.com/books?id=JRzhE6yqeFcC&pg=PA125&dq=what+are+rational+functions+used+for&hl=en&sa=X&ei=1lo1T9zDN-GusQLcrpyuAg&ved=0CFwQ6AEwBQ#v=onepage&q=what%20are%20rational%20functions%20used%20for&f=false.

“Mathematics History.” *ThinkQuest : Library*. 7 Feb. 2012. http://library.thinkquest.org/22584.

“Multiplication – Wikipedia, the free encyclopedia.” *Wikipedia, the free encyclopedia*. 7 Feb. 2012. <http://en.wikipedia.org/wiki/Multiplication>.

“Multiplying Rational Expressions.” *Purplemath*. 7 Feb. 2012. http://purplemath.com/modules/rtnlmult.htm.

“Polynomial – Wikipedia, the free encyclopedia.” *Wikipedia, the free encyclopedia*. 7 Feb. 2012. http://en.wikipedia.org/wiki/Polynomial_Functions#Polynomial_functions.

“Who Created Fractions | Ask Kids Answers.” *AskKids Answers | AskKids.com*. 7 Feb. 2012. http://answers.askkids.com/Math/who_created_fractions.