Pythagorean Theorem

Grade Level: 6-8

Objective: The goal of the activity is to explain that scientific investigations must use standardized units of measurement to maintain accuracy. A grid is one form of standardized measurement used by archaeologists. A grid helps archaeologists maintain context by documenting where artifacts are found on a site during excavation.

Learning Outcomes: Learners will understand that math is important in all disciplines, including archaeology. Archaeologists use the Pythagorean Theorem to accurately measure out a square for excavation.

STEM: Math

Materials: Outdoors: 4 nails per “unit,” hammers, two tape measures per group, string, scissors; Indoors: painter’s tape, two tape measures per group, 4 small paper circles (no larger than 4 cm in diameter) per group.

Time: 35 minutes

Overview: In this case, the standardized measurement for excavation is a 1 meter x 1 meter square (although units can be larger or smaller, rectangular or square). To make a perfectly square unit (with four sides measuring exactly 1 meter each), we use a basic geometry concept – the Pythagorean Theorem (named for the ancient Greek thinker Pythagoras). Each unit is placed on the site at specific coordinates on the established grid. The grid helps archaeologists maintain and record context – where on a site an artifact is recovered or a feature is identified during site excavation. Each square unit is given a number and a coordinate (North/South and East/West) on the grid to identify its location on the site. A map of the site, with the location of each square unit, is made for reference (see example below). This is done either by hand or with an engineer’s surveying tool called a Total Station. Maps can then be input into computer mapping/analysis programs, such as geographic information system (GIS).

Pythagorean Theorem: the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other sides. To calculate the value of the hypotenuse in a right triangle use this formula: a2+b2=c2, where c is the length of the hypotenuse and a and b are the lengths of the other two sides of the right triangle.

If the lengths of both a and b are known (such as in a 1 meter square unit), then c can be calculated as:

Vocabulary: archaeological unit, Pythagorean Theorem, right triangle, hypotenuse

Procedure: Discuss why a grid is important for archaeological excavation. Have the learners try to lay out a 1 x 1 m square. When they are done, ask them what they know about the Pythagorean Theorem? How can this theorem be used to set up an excavation unit? Discuss what the Pythagorean Theorem is and how it can be used to set up a perfectly square excavation unit. What are the values of a and b if we are trying to form a 1 m2 unit? Have learners work with you to figure out what the hypotenuse of a 1 m square is. Once this is done, and if the weather is good, take learners outside where there will already be two roofing nails/spikes laid out 1 m apart to represent two points of a 1 m2 unit. Several of these partial units will be set up in the grass. Learners will use the tape measures to place the other two nails of each unit making a perfect square, 1 m on a side (see photo below). If indoors, take learners to a large room, such as a gym or cafeteria, where the “nails” will be circles of paper taped to a floor. The learners will each receive two circles to serve as the other two points of the square. One person holds the beginning end of a tape measure at one nail and the beginning of the other tape measure at the other nail. The second person crosses the two tape measures so that the side of the triangle measures 1 m and the hypotenuse measures 1.41 m. Where they cross is where the nail is placed (see figure below). Have learners check their work by measuring the length of each side of their square; each should measure 1 m in length. If there is time, you can have students wrap string around the nails to see what an excavation unit looks like before digging has begun.

Ask learners why it is important that the unit be precisely 1 m2. Discuss why it is important. Ask learners why an archaeologist would use a 1 m2 unit vs. 0.5 m2, 2 m2, 4 m2, etc. Some of it has to do with the type of site or with your research questions, but it helps maintain the context of the artifacts you find because you can record their location within a relatively small area and the unit is not so small that excavation of the site takes a longer time.

Assessment Activities: Each student will use the Pythagorean Theorem to place one or two nails for their unit. Each group will check the sides of the square to be sure they measure 1 m in length.

Wrap up: Ask students why the Pythagorean Theorem was used to lay out the units vs. not using the Theorem. Ask them what other math concepts they use in their daily lives.