Perimeter, Area, Volume, and All That: A Study of Measurement

“To a scholar, mathematics is music.”

- Amit Kalantri, Wealth of Words

Introduction

Many rising 8^th grade scholars around the country will be given the opportunity to take Algebra I; some skipping 8^th grade Math all together. That was my experience until last year. I advocated for scholars to receive a double block in Math to ensure those valuable concepts in 8^th grade math was introduced. My administration team agreed and we were able to adjust their schedules to assure they didn’t miss any valuable standards. But for many teachers around the county, that isn’t the case. Many find themselves teaching 2 years’ worth of math topics in 180 academic school days, and not surprisingly, there are just some students who can’t handle the work load. In her book Mathematical Mindsets Jo Boaler states,

Students were failing algebra not because algebra is so difficult, but because students don’t have number sense, which is the foundation for algebra.¹

This statement alone paints a picture of what numerous of Algebra teachers across the country experience time and time again. Moving from numerical concepts to algebraic ideas often opens the door for frustration and disconnection for many students. They simply do not understand or see how variables belong in math nor do they find much similarity between variables and numerical values. I intend on designing a curriculum unit that will highlight the relevance of applying some arithmetic concepts, especially rules for multiplication, to multiply polynomials, using area models and box models. I find that polynomials are the building blocks for algebraic thinking from the 8th grade and beyond. Being able to effectively and accurately manipulate polynomials will allow students to represent a variety of situations/scenarios algebraically. I plan on making significant connections between my previously written curriculum, Closing Deficits Exponentially: Addressing Base Ten & Small Numbers Using Exponents and the rules for multiplication, in such a way that students begin to transfer their elementary math understandings to more complex algebraic ideas.

The goal is to take these elementary structures and use them as stepping stones to introduce operations with polynomials. In a lot of curriculum materials, they miss the mark on making these basic associations when using the same or similar process in a more advanced manner. I am hoping to use this unit to illustrate the similarities and ease in using strategies learned in elementary grades to serve as a link to Algebra concepts. As we move into the less traditional and conventional ways of doing Math, I will demonstrate various visual models that will provide my students with concrete examples. These examples will serve as a bridge for understanding how basic mathematical ideas can be utilized to conquer problems with a higher level of difficulty. I will use an array model, often known as the box method, to allow students to have a visual representation of how polynomials can be manipulated. By using the box method, I’ve witnessed fewer errors, and increased ability to keep track of the transformation each polynomial makes when an operation is applied.

By teaching my students to understand the process of using the box method for multiplying multi-digit numbers, and then having them advance to working with monomials, binomials, and trinomials, I hope to strengthen their understanding of not just polynomial operations but, the laws and properties of exponents, the behavior of the degree of polynomials under multiplication, area models, and certain geometric notions. Also, I hope they will come to appreciate the value of the Rules of Arithmetic (aka, Properties of the Operations). We will again explore the basis for the 5 Stages of Place Value² to set up this practice, and use those general ideas to master the procedure.

Lastly, aside from the idea of doing the actual math, I hope to shift the mindsets of my scholars as they enter the last year of their middle school careers. I hope to get them excited and engaged about diving into algebraic thinking.