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For thousands of years the pyramids of Ancient Egypt have been a source of fascination for people of all ages. Archaeologists have offered a variety of theories as to how the Egyptians actually constructed these impressive structures at a time when tools and technology were still quite primitive by today's standards. One area of particular interest has been the methods employed to place layers and layers of stone blocks, weighing over two thousand pounds each, onto the pyramid structure. In this lesson students explore one theory that some archaeologists have offered, which involves the pulling of the stone on a sled up a single ramp. Students construct a model of a pyramid and use a variety of problem solving strategies to determine the length of a ramp needed to complete the building of a pyramid. Based on their mathematical explorations, they will evaluate the theory that just one ramp was built to move the blocks of stone.
Students have the opportunity to:
- Construct a geometric model of a pyramid
- Measure to the nearest inch
- Use a protractor to measure the angle of a ramp
- Use a variety of problem solving strategies, such as making a model and looking for a pattern to solve a problem
- Use a three dimensional model to estimate the length of a ramp
- Read to perform a task
- VCR and TV
- "Black Kingdoms of the Nile" video
- 21 inch square of poster board or tagboard (per group of four students)
- Ruler, customary (per group of four students)
- Protractor (per group of four students)
- Tape (per group of four students)
- Heavy cardboard, approximately 18" x 36" (per group of four students)
- Scissors (per group of four students)
- Lesson 1 Student Activity Sheets: Constructing a Pyramid and Building the Ramps
Approximately 1 ½ hours
Relevant National Standards
NCTM Standards 2000
- Understand numbers, ways of representing numbers, relationships among numbers, and number systems;
- Use computational tools and strategies fluently and estimate appropriately.
- Understand various types of patterns and functional relationships
- Use symbolic forms to represent and analyze mathematical situations and structures
- Use mathematical models and analyze change in both real and abstract contexts.
- Analyze characteristics and properties of two- and three- dimensional geometric objects
- Use visualization and spatial reasoning to solve problems both within and outside mathematics.
- Understand attributes, units, and systems of measurement
- Apply a variety of techniques, tools, and formulas for determining measurements.
- Build new mathematical knowledge though their work with problems
- Develop a disposition to formulate, represent, abstract, and generalize in situations within and outside mathematics
- Apply a wide variety of strategies to solve problems and adapt the strategies to new situations
- Monitor and reflect on their mathematical thinking in solving problems
- Organize and consolidate their mathematical thinking to communicate with others
- Express mathematical ideas coherently and clearly to peers, teachers, and others
- Extend their mathematical knowledge by considering the thinking and strategies of others
- Use language of mathematics as a precise means of mathematical expression.
- Recognize and use connections among different mathematical ideas
- Understand how mathematical ideas build on one anther to produce a coherent whole
- Recognize, use, and learn about mathematics in contexts outside mathematics.
- Create and use representations to organize, record, and communicate mathematical ideas
- Develop a repertoire of mathematical representations that can be used purposefully, flexibly, and appropriately
- Use representations to mode and interpret physical, social, and mathematical phenomena.
- View the "Black Kingdoms of the Nile" video. Take special note of segments that refer to the construction of the ancient pyramids, temples and monuments. Pertinent moments include:
Offer students a purpose for viewing by encouraging them to look at the size, shapes, and construction of the pyramids.
2:17 to 2:57-verbal cue: "I've wanted to see the pyramids since I was child."
7:37 to 10:24-verbal cue: "My search starts here at the Temple of Abu Simbel."
21:7 to 24:10-verbal cue: "These pyramids are spectacular."
- Initiate discussion about the constructions of the pyramids. Questions to ask may include:
- What building materials did the ancient Egyptians use to build their pyramids, temples, and monuments?
- What geometric shapes were used in the constructions?
- What types of tools were available for construction in that time period?
- How were the Egyptians able to transport and lift heavy stone blocks?
- Divide students into groups of approximately 3-4 students.
- Challenge students to make a model of a pyramid. You may give each group a set of directions (refer to handout 1, "Constructing a pyramid") or list the steps on the chalkboard or an overhead projector.
- Supply students with background information concerning the different theories of pyramid construction. Some archaeologists believe that the Egyptians used a single ramp made out of mud to help them move the heavy blocks of stone onto the pyramid structure. As the layers of stone were added, the ramp would have to be extended to keep the angle small enough for people to pull the sled carrying the stone. Other theories suggest that multiple ramps were built from step to step and then later removed. The use of levers to move the heavy stone has also been explored. Inform students that they will be involved in investigating the theory that one ramp was built to help move the stone blocks into place.
- Have students cut a ramp measuring 1½ inches by 5 inches from a piece of heavy cardboard. On approximately the middle of one side of the pyramid, tape the ramp one inch up from the base of the pyramid. Using a protractor, student should then measure an angle of the ramp. (You may need to remind students the angle that they are measuring is the angle formed where the ramp touches the ground. They may need to slide their model to the edge of a desk or table to make it easier to measure.) Students should record the height and length of the ramp and the measurement of the angle on handout 2, "Building the Ramp."
- Students should remove the first ramp from the structure. Challenge students to make a second ramp, one layer higher on the pyramid. (This ramp would measure two inches from the base.) The ramp must equal the same angle measurement as the first ramp. Students will need to make the ramp longer to have an even gradient. Before students begin the task, have them estimate the length of the second ramp and record this on the worksheet. (Explain to the students that a second ramp would not actually be built, but rather would be built higher and extended.) Have students continue to explore the length of the ramps needed by making ramps one inch higher each time. The same angle measurement should be used each time. Encourage students to look for patterns and to estimate how long the ramp on the top layer would be. Depending on the amount of time available, the teacher can determine how many ramps the students should construct in order to make the estimation. Encourage students to compare the length of the ramp with the actual measurements of the pyramid (ratio).
- Using their constructions and investigations, have students evaluate the theory of using one ramp to move the stone. Students should communicate their findings and theories in both oral and written format. (Many archaeologists do not believe that one ramp was used because the ramp would have to be extremely long to keep the same gradient. Also, they have not found physical evidence to indicate that such a ramp was built.)
Students may be evaluated using the following assessment techniques:
- Group and class participation.
- Performance-based product: constructing pyramid and ramps according to written directions.
- Written and oral evaluation of investigation based on mathematical findings.
- Students could use the pyramid models to explore other engineering theories such as the use of multiple ramps. Students could use the pyramid model to make a series of ramps that completely surround the pyramid. These ramps would be attached parallel to the sides of the pyramid. Students could also use sugar cubes to make a three dimensional model to explore the multiple ramp theory, since some observers suggest that the ramps had to be removed as the final blocks on each level were set into place.
- There are many curious and interesting mathematical aspects involving the Great Pyramid and its relationship to the size of the earth. Students can explore this on the internet at the Di-Soft web site, Facts About the Egyptian Pyramids, http://www.di-soft.demon.co.uk/pyramids.htm. The material found at this site could be used for a real world connection when studying proportion and ratio. The web site also has information on the concept of pi and the Great Pyramid. Historian believe that pi had not been discovered at the time of the building of the Great Pyramid, yet interesting mathematical calculations found at this web site could cause students to question this theory. Students could use calculators to perform and check computations presented on this interesting site.
- Students could investigate the engineering feats used to move the temple at Abu Simbel in order to save it from destruction when the Aswan Dam was built. (See the list of recommended web sites below for resources on this topic.) Students might present their information in a multimedia format (such as PowerPoint or Hyperstudio).
- Students could create a web site to share information they have acquired in this lesson. An example of such a project can be found at http://www.geocities.com/EnchantedForest/Meadow/1934/.
Recommended Web Sites
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Black Kingdoms of the Nile
The Swahili Coast
The Holy Land
Road to Timbuktu
Lost Cities of the South