Introduction:

This is a set of guiding questions and materials for creating your own lesson plan on introducing the basic notions of Euclidean geometry in an axiomatic yet exploratory way.

Common Core State Standards:

G-CO.1

Learning Objectives:

To explore knowledge of geometric notions and objects.
To understand what an axiomatic system is.
To understand the importance of definitions in an axiomatic system.
To understand the need for primitive or undefined terms in an axiomatic system.
To understand the meaning and purpose of the three basic components (primitive terms/definitions, postulates/axioms, and rules of inference) of an axiomatic system.
To appreciate how logical reasoning combined with the rules of inference lead to conclusions called theorems.
To appreciate the historical significance of Euclidean geometry in rational discourse and intellectual development.
To become aware of Euclid’s five axioms.

Guiding Questions:

What words or concepts come to mind when you think of “geometry”? Make a list.
What does geometry mean?
- Try to elicit a loose etymological definition: geo = earth and metry = measurement.
Why do you think it was given that name?
What uses do you think it had in antiquity?
Is it relevant today?
Do more concepts come to mind now that you are aware of the meaning of geometry?
How do the terms given relate to each other? Do you see any connections?
Can you define any of those terms?
Prompt for circle and square. Take their definitions and construct counterexamples. For instance, if they say that a square is a “shape with four sides,” then draw a rectangle or a clover.
What do we want to avoid in an axiomatic (or any deductive reasoning) system?
Are contradictions and paradoxes acceptable?
- Although they are not, contradictions and paradoxes have been a major force in the development and refinement of mathematics.
Why are right angles 90 degrees?
- Point out that there are several ways to measure angles and that the preferred method in other math courses will be a unit called the radian.

Notes for Teachers:

I have indented some suggestions in the questions above.
Give a short (5-10 min) lecture on the history of Euclidean geometry, its relevance in scientific and rational thinking, and about axiomatic systems in general (what they are and why they are important).
Present and discuss Euclid’s 23 definitions and five axioms. Use the parallel postulate version as the fifth axiom. Make sure to point out that there are hidden assumptions and that other mathematicians (most notably David Hilbert, more than 2,000 years after Euclid) saw the need to refine Euclid’s list of axioms. However, every conclusion drawn from the original five holds under Hilbert’s revision.
When explaining why a right angle is 90 degrees, talk about the importance of geometry for astronomy and how the 360 degree circle relates to the number of days in a year. It also makes for easier computations.

Video of the Day:

8.1.1 Axiomatic Systems and Primitive Terms

We explain why primitive terms are needed to avoid circular definitions and introduce the basic elements of an axiomatic system.

8.1.1 Axiomatic Systems.mp4

8.1.2 Nested Definitions

We explain the concept of nested definition with Venn diagrams.

8.1.2 Nested Definitions.mp4

Exercises and Problems:

Online Sources:

Euclid’s definitions and axioms

Khan Academy: Tools of Geometry

Additional Resource for Teachers:

If you wish to download the contents of this page as a printable pdf, click here: 8.1 Introduction to Euclidean Geometry.pdf