## Reflection: Vertical Alignment Origins of the Geometric Universe - Section 2: Concept Development

So during this geometry lesson, which took place the first week of the school year, I explained to students that we were actually exploring some fundamental Calculus concepts. Some students found this to be exciting, saying things like "Cool...I want to learn Calculus!" Others weren't so thrilled. The typical line from that camp was, "This is Geometry. Why are we learning Calculus?" And of course there were those in between who chose to wait and see how things unfolded.

My rationale for explicitly introducing calculus concepts in a geometry course is twofold: 1) I believe that students should see mathematics as a coherent body of knowledge, not as compartmentalized courses 2) For students who will eventually take calculus, I want them to have a strong conceptual foundation for what they will learn in calculus. When they get to calculus and they are learning the formal definitions of limits and integrals, I want them to have that 'Aha!' moment when they recall this experience in their 9th or 10th grade year and understand how it connects to what their calculus teacher is teaching them.

There are two main connections to calculus that I am trying to make in this lesson. First, is the concept of the limit. We are dealing with the concept of the point, which textbooks say has "no size". I've come to have a problem with this "undefinition". It reminds me, in a way, of how our elementary teaching has children believing that a rectangle is a four sided figure with a pair of shorter congruent sides and a pair of longer congruent sides...i.e., a square is not a rectangle. This is an example of how we fail to consider vertical alignment when we teach math at the lower levels and this puts students in the position of having to go through a conceptual change when they get to higher mathematics. But in any case, I want students to understand that a point must have size, but that size is not something we can actually quantify. It is not zero, but it is infinitely close to zero. This, I explain to students, brings us into the realm of the imaginary. Things we must envision with our minds. And I tell them that this skill of reasoning about things that don't exist in the real world is a major characteristic of the mathematician's mind.

The second connection I try to make is to the idea of the integral. More specifically, how things that are infinitely small (points in this case) can accumulate when infinitely many of them are summed. This is the idea of making a line out of points. To say that a point has no size, but somehow can make a line, leaves us with a quandary. But by introducing the idea of a limit, it makes more sense.

Later in this course, this idea will re-emerge. For example, when we explain why the formula for the volume of a prism is Bh, we can discuss how the area of an infinitely thin cross section of the prism is B and how these infinitely thin slices "accumulate" over the height of the prism. Hence V=Bh.

So even though this lesson topic is probably among the more bland in the geometry course, I see it as an opportunity to discuss weighty calculus concepts, to have students grapple with the concept of infinity, and to start training their brains to think abstractly.

Connecting with Calculus Concepts
Vertical Alignment: Connecting with Calculus Concepts

# Origins of the Geometric Universe

Unit 2: Geometry Foundations
Lesson 1 of 14

## Big Idea: Let's take a look back in time to see how the geometric universe was created...In this lesson, students imagine the undefined terms upon which synthetic geometry is founded.

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60 minutes

### Anthony Carruthers

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