There is no royal road to Geometry.
Welcome to Integrated Math II with Mr. Jones!
What is integrated math?
Euclid (pictured above) is considered to be the father, not just of geometry, but of our current axiomatic system upon which all of mathematics is erected. In his historic textbook, Euclid's Elements, written 300 BC, we find the seeds for the geometry, algebra, number theory, logic, set theory, and calculus that we study today.
Many people go through high school believing that geometry, algebra, and calculus are a collection of unrelated topics that have little to do with each other (other than the fact that they are all math subjects studied in school). Most people don't know that the properties of algebra, such as the reflexive and transitive properties, are actually postulates of the real numbers, not unlike how the postulates of geometry are the postulates that give meaning to our three-dimensional space. And taking the right approach, the postulates of one system can be proven as theorems of the other. In Integrated Math II at University High School, we emphasize that integrated math is not a disintegrated collection of math topics such as algebra and geometry thrown together into one class. Rather, we emphasize that the algebra is taught through the geometry and vice versa.
When Euclid wrote The Elements, he intended it to not just to be a book about geometry, but rather a study of all the mathematical knowledge that he had available to him in his day. The properties of algebra, such as the transitive property, were either stated as axioms (Two things equal to the same thing are equal to each other, Book I, Common Notion I) or, as in the case of the distributive property (Book II, Proposition I), were proven as theorems. Thus Euclid, by taking this approach, was engaged in an exercise in the integration of algebra and geometry. This is the approach we take in this class as well.
Although Euclid's Elements was a historic milestone, by today's standards of rigor it is grossly outdated and is in need of a revision. We have attempted to do that here by making the use of set theory to fill in the missing gaps in The Elements. This is a work in progress, but it is one of the core resources that will be used in this class, along with the Integrated Math II Pearson textbook.
As you will see, there is a lot of history behind the math that we study today, and I intend to highlight these gems of knowledge as we embark on our journey together on the topics of integrated math. Each unit of this website begins with a painting or photo that highlights a historically significant aspect about the respective topic of study, along with an explanation of the same. Hopefully this will aid in the students' understanding of the context in which each topic that we study was discovered.
I look forward to working with you this year and hope that you will enjoy our journey through topics in integrated math!