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 Enrique Alegria. His topic, from Algebra: multiplying polynomials.

**How can this topic be used in your students’ future courses in mathematics and science?**

This topic can be used in students’ future courses in mathematics by simplifying expressions of increasing degree. In Algebra II students are expected to simplifying polynomials of varying degrees as they move on to multiplying and dividing polynomials. From there determining the factors of a polynomial of degree three and degree four. Real-world problems can be solved through the simplification of several like terms. Each term representing a specific part of the problem. We can even compare the addition and subtraction of polynomials to runtime analysis in Computer Science. Measuring the change in the degree and how that affects the output. In a way, this can translate to the runtime of a program. For example, a chain of commands with a constant time is run. A loop is nested in another loop that is placed after the first expressions. This has changed the overall runtime of the program from constant time to quadratic because of the degree of the nested loops. The overall time would be the addition of the expressions and their corresponding times.

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

This topic extends from the early concept, ‘Combining Like Terms.’ Starting with adding and subtracting items of similar groupings such as 8 apples and 4 apples altogether are 12 apples. Bringing students to place value such as adding 3 ones and 2 ones to adding multi-digit numbers. We then leap towards Algebra introducing expressions and equations. Learning about linear and quadratic equations and graphing them. Students should have learned about monomials in correspondence with coefficients and exponents. From there, students are familiar with algebraic terms. Those are the building blocks that we are going to be expanding upon. Once students familiarize themselves with several terms in an expression, they will focus on adding or subtracting like terms by focusing on both the coefficient, term, and exponents on the variables. Shortly after the students can continue to be challenged by using terms such as or to focus on the terms and confirm if they are ‘like’ to be combined or just notice the fact that they have some common variables with the same exponents but with a slight difference other than the coefficient, the expression cannot be simplified as one may think.

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

Adding and subtracting polynomials can be engaging to students with the help of Brilliant. This site starts with helping students identifying polynomials and their degrees to help students understand how to describe them. Then moving to the arithmetic of polynomials performing addition and subtraction operations on the polynomial numbers. This source goes through polynomials through challenging and insightful exercises. For example, a quadrilateral of sides such as , , , , and from there simplifying the expression. Students would be able to substitute values and determine if a specific quadrilateral has been made. I can have students go through a few exercises as a class or on their own and then they can come up with a problem on their own that would be posted to the ‘public’ (which would be only their class) so that the students will be able to have classroom interaction and grow as they challenge each other. Students can apply this concept by creating a large polynomial expression and then simplifying it and lastly graphing the equation.

**References:**

Polynomials. *Brilliant.org*., from https://brilliant.org/wiki/polynomials/

Simplifying Expressions. *Brilliant.org*., from https://brilliant.org/wiki/simplifying-expressions/