Source: Wiggins, G., & McTighe, J. Understanding by Design.
Merrill Prentice Hall: 1998.
For further information about Backward Design refer to http://www.ubdexchange.org/
Title: The Volume of Ancient Egyptian Pyramids Subject/Course: Math
Topic: Geometry Grade(s):
7 Designer(s): Pam Joseph  
Stage
1 – Desired Results  
Established
Goal(s) (enter
goals here) M7G. Geometry Students will further develop and apply their understanding of plane and solid geometrical figures.
M7G2. Students will consider geometrical figures through various manipulations to deepen understanding of figures in space. a. Describe solid geometric figures formed by movement of plane figures through space.
b. Sketch/model
and describe various cross sections of cones, cylinders, pyramids
and prisms.  
Understanding(s)
Students will understand...
1. the social context in which the pyramids were built 2. the two formulas discovered by the Egyptians regarding the volume of a square pyramid and the volume of a frustum U
frustum: big pyramidlittle pyramid (off of the top) 
Essential
Question(s) 1. What is a pyramid? 2. What is volume? 3. What is a frustum?
Q

 
Students will know... 2. the importance of the Nile River and its annual flooding 3. the general idea of how and why the pyramids were built K

Students will be able to... 1. construct a square pyramid out of poster board 2. explain the formulas for the volume of a pyramid and frustum 3. use these formulas to obtain the theoretical volume of a given pyramid or frustum 4. determine the volume of a form by measuring how much sand it holds S

 
Stage
2 – Assessment Evidence 
 
Performance
Task(s) Summary in G.R.A.S.P.S.
form 1.
Students test the veracity of the formulas. Students do the following
worksheet from The Saga of Mathematics: A Brief History Lewinter &
Widulski website: http://math.widulski.net/worksheets/EgyptianGeometry.pdf 2. Students will work in groups of four to do the following experiment: (but each student must fill out her/his own worksheet)
Students will use equilateral triangles to draw four adjacent triangles
on poster board. They should mark the midpoint of each triangle and
connect them with dark lines. Then they measure the side of each
triangle (side of the base of the pyramid) and side of each segment that
connects midpoint (side of base of "halfpyramid"). They cut, fold and
tape these triangles together to form a square pyramid. Invert and
measure height of pyramid. (Assume height of "halfpyramid" is half this
measurement, which can be roughly confirmed with a ruler.) Measure how
much sand fits in the whole pyramid. Subtract values to find volume of
frustum. Compute theoretical values using the formulas and see if they
are close.  
 
Other
Evidence
 
Stage
3 – Learning Plan 
Learning
Activities Consider the W.H.E.R.E.T.O. elements.

1. 5 minutes: Explain the social context in which the pyramids were built. Show pictures. Explain how ancient Egypt was one of the first advanced civilizations, the importance of the Nile River and its annual flooding, and the general idea of how and why the pyramids were built. 2. 5 minutes: Explain the two formulas discovered by the Egyptians regarding the volume of a square pyramid ([1/3]a squared h]) and the volume of a frustum big pyramid minus small pyramid. 3. 3040 minutes: Challenge the students to test the veracity of these formulas. Pass out worksheet which explains the procedure and contains blanks for students to fill in their values. Explain the procedure while drawing diagrams on the board. Use the equilateral triangles to draw 4 adjacent triangles on poster board. Mark the midpoint of each triangle and connect them with dark lines. Measure side of each triangle (side of the base of the pyramid) and side of each segment that connects midpoint (side of bas of "halfpyramid"). Cut, fold, and tape these triangles together to form a square pyramid. Invert and measure height of pyramid. (Assume height of "halfpyramid" is half this measurement, which can be roughly confirmed with a ruler). Measure how much sand fits in the "halfpyramid" (up to the segments connecting the midpoints). Measure how much sand fits in the whole pyramid. Subtract values to find the volume of frustum. Compute theoretical values using the formulas and see if they are close. 4. Students then get into the groups and do the same experiment just demonstrated. 5. Collect worksheets at end of class. 6. Ask the class the questions in "Other Evidence." 