Postulates
Objective: Students will be able to identify Euclid’s postulates.
Activity-Tie a piece of chalk to a string. Draw a dot on the board. Give a student a specific measurement for the length of the string. Holding the string at the specified length on the dot, the student draws a circle around the dot. Repeat with varying lengths. This activity will provide kinesthetic reinforcement of Euclid’s Postulate 3.
Activity-Have students describe a job or career in which some or all of the work is based on a set of postulates or assumptions. For example, the work of research scientists is almost entirely based on scientific postulates and/or theorems. Computer scientists, medical practitioners, and even musicians base much of their work on a given set of postulates and theorems. Instruct students to identify individuals in such a career and discuss with them the use of postulates in their work. Encourage students to find out how old the postulate or theorem is, who is responsible for it, and how it is applied in the given field of work.
Objective: Students will be able to identify the postulates used in given constructions. Students will make constructions using Euclid’s postulates.
Activity-Have students investigate how a draftsperson, graphic artist, or landscape architect makes use of constructions that are based on Euclid’s five postulates. Ask them to find specific constructions for each postulate and describe what the figures may represent in that given line of work. Encourage students to use sources such as encyclopedias, how-to manuals, the Internet, or interviews with someone working in the field the student has chosen. If possible, display graphic or landscape architectural drawings.
Activity-Ask each group of three to four students to create a drawing that combines figures based on all of Euclid’s postulates. Explain that their final product should be a construction that incorporates all of the postulates. However, they should also produce a progressive “slide show” of the stages leading up to their final drawing. The “slide show” would start with a single construction using one postulate, then show a second construction based on a different postulate added on to the first construction, and so on. The end product should show a series of six drawings, each labeled with the postulate that was used to make each new construction in the series of drawings. Encourage students to be creative in their design of constructions, use of color, and layout. Display the “slide shows” in your room or school.
Activity-Ask students to look around home for objects whose designs reflect, at least partially, one or more of Euclid’s five postulates. For example, Postulate 4 allows the corners of walls or ceilings to be equal, and Postulate 1 allows a fence to be built or a flag to be raised up a flagpole.
Objective: Students will be able to identify the postulates, axioms, and theorems that justify the statements in a proof.
Activity: Have students in groups of four or five write a proof for the theorem that states all right angles are congruent. The groups should write up a formal proof that includes statements and reasons columns, and all statements and reasons should be clearly, completely, and accurately stated. The groups should prepare posters as final products for display in the classroom.