## Length

1 - 1.25 Hours

## Materials

## Curriculum Standards

CCSS.MATH.CONTENT.HSG.CO.A.1

Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

## Warmup and Instructions

- Inform your students they will be learning about the famous Greek mathematician Euclid, as well as his axiomatic geometry.
- Ask them if anyone is familiar with Euclid, or has even read about him in the past.
- Have them read the short introduction to the video lesson.
- Start the video lesson Euclid's Axiomatic Geometry: Developments & Postulates and pause for the first time at 2:18.
- Who was Euclid and what is he known for?
- What was the name of his book?
- What are Euclid's five axioms?

- Next, restart the video and pause this time at 3:31.
- How does the axiomatic system work?

- Now, restart the video and pause for the final time at 4:35.
- How do Euclidean and non-Euclidean geometry differ?

- Next, restart the video and watch the 'Lesson Summary' section.
- Finally, have your students take the lesson quiz to confirm their grasp of this newfound knowledge about Euclid's axiomatic geometry.

## Activity

- Let your students know they will be taking part in a fun activity related to Euclidean geometry.
- Tell them that today we are going to act out Euclid's five postulates so that we can visually see what they represent. Let us first try an example with one of the postulates.
- Pick out two volunteers. Place a yardstick on the floor. Have the two students stand at either end of the yardstick, with their backs facing each other. Now, have the two students wave their arms and point off into the distance while saying 'forever' out loud.
- Ask students what postulate does this represent? (any line segment can be extended into infinity in both directions.)
- Next, divide your students into small groups of 4-6.
- Instruct students to invent their own creative demonstrations of Euclid's other four postulates from the video lesson. If you wish, you can also ask them to create an alternate idea for the postulate just demonstrated.
- Give students 20 minutes to develop a way to act out each postulate.
- Have your students act out their demonstrations for the entire class.
- Alternatively, you can set this up like it is a game of charades and other students have to guess the postulate as a group acts it out. You could even award points for teams who guess correctly.
- Note to the teacher - Below are four helpful examples of demonstrations that you might offer to a group who is struggling to find ideas.
- The first student stands in a clear space while holding the string. The second student holds the other end of the string and gently pulls it taut. Now the second student walks very slowly and smoothly around the first student. Thus, the first student represents the center of the circle, the string represents the radius, and the second student walking represents the circumference. (a circle can be described with just a center point and radius.)
- Two students stand about five yards apart. Now, a third student walks very slowly and directly from the first student to the second student. (a line can be drawn between any two points.)
- Each student finds various right angles in your classroom (examples include corners of books, places where walls meet, and chalkboards.) As they point out each right angle, they announce out loud 'this right angle is equal to exactly ninety degrees.') It is okay if students talk at the same time, but tell them not to run, push, or shove. (a right angle is equal to all other right angles.)
- Place two parallel pieces of string on the floor, and then place a third piece of intersecting string across them. Now, two students stand on the inside angles where the lone string intersects the two strings. The first student announces out loud 'I am an acute angle', and then the second student announces out loud 'I am also an acute angle.' Next, two other students stand at the far end holding the two strings. Now, those two students walk very slowly toward each other until they meet. They drop the two strings on the floor so that they are touching at one point. They announce together 'these two lines meet because those two angles are acute.' (if one line intersects two other lines and forms angles less than 90 degrees on one side, then the two lines will intersect on that side.)

- Lastly, ask your students if they have any other final questions.

## Extension

- Hipparchus was another Greek mathematician and is often considered the founder of trigonometry. Write a one-page paper detailing some of his accomplishments.