Getting Started 
Misconception/Error The student does not understand the relationships between the bases of the pyramids, the volumes of the pyramids, and how they relate to the volume of the prism. 
Examples of Student Work at this Level The student is unable to explain why the areas of the triangles are congruent and why the volumes of the pyramids are equal. Additionally, the student cannot explain that the volume of each pyramid resulting from the decomposition of the prism is the volume of the prism.
The student:
 Describes the spatial relationship between the two triangles rather than the relationship between their areas.
 Recognizes the areas of the two triangles are equal but is unable to explain why.
 States that the triangles are congruent but does not address the relationship between their areas.

Questions Eliciting Thinking What is the formula for the area of a triangle? How can you use the diagram to explain the relationship between the bases of the triangles and the heights of the triangles?
Every pyramid has a base and a vertex. Can you identify a base and a vertex for pyramid ABDC and pyramid EDBC? What is the relationship between the vertices of the two pyramids? How does this relate to the heights of the pyramids?
In question one, what did you determine about and ? What can you say about the bases for pyramid ABDC and pyramid EDBC?
What is required by Cavalieri’s Principle to justify that the pyramids in questions two and four have equal volumes? Can this requirement be satisfied with these pyramids? Why or why not?
What does question five suggest about the three different pyramids that combine to make the prism? What property of equality is applied?
If three equal pyramids combine to make one prism, then each pyramid is what fraction of the prism? 
Instructional Implications If necessary, explain that if two triangles have the same base and height, their areas will be equal. Remind the student that the lateral faces of all prisms are parallelograms but in this case, they are rectangles. The most direct way to explain that the areas of the triangles are the same combines a parallelogram property (opposite sides are congruent) with the definition of a rectangle (e.g., a polygon with four angles). Since a rectangle has four angles, the sides are perpendicular and the triangles have equal heights. Triangles with equal bases and heights have equal areas.
Help the student recognize that if the pyramids have congruent bases and the same height, then the pyramids have congruent cross sections at any height. This allows Cavalieri’s Principle to be applied and justifies why the pyramids have the same volume. Remind the student that the congruent bases for the pyramids have already been verified in the first question. However question two requires the relationship between the heights and the volumes to be explained. To explain the congruent heights of the pyramids, remind the student that the bases are coplanar and they share the same vertex. The distance from a point to a plane is measured along the perpendicular from that point to the plane. In this case, the base of each pyramid represents the plane and the vertex of each pyramid (C) represents the point. Since these pyramids (ABDC and EDBC) have coplanar bases and their vertices are the same point, the pyramids have congruent heights.
After congruent bases and equal heights are established for the two pyramids, ask the student to reread the first paragraph on the worksheet to determine the next step (e.g., cross sections). Assist the student, as needed, to understand that pyramids with congruent bases and the same height have congruent cross sections at any height. Since Cavalieri’s Principle requires solids to have bases of the same area and cross sections of the same area, it can now be applied and used to verify that the pyramids have equal volumes.
Next, help the student recognize that question five applies the Transitive Property of Equality to the answers for questions two and four. As such, the three unique pyramids combine without gaps or overlaps to create the prism and these three pyramids have equal volumes. Once the congruency of the three pyramids and their volumes is established, the relationship between one pyramid and the prism can be addressed. Since three pyramids of equal volume comprise the prism, then each pyramid is the volume of the prism. 
Moving Forward 
Misconception/Error The student does not understand the relationship between the volume of each pyramid and the prism. 
Examples of Student Work at this Level The student indicates an understanding of the relationships between the areas of the bases, the heights, and the volumes of the pyramids, but does not understand the relationship between the volume of each pyramid and the volume of the prism. The student writes a correct answer for question five, but does not seem to understand what questions six and seven are asking or is unable to make the connections with the concepts previously verified.
The student:
 Suggests that the volume of the pyramids with base and height AD would have unequal volumes.
 Writes that the volume of a pyramid with base and height AD is a fraction of the volume of the prism other than .

Questions Eliciting Thinking Can you reread question number five? What property of equality is applied? How many pyramids does question number five indicate completely make up prism ABCDEF without any overlaps or gaps? In questions one through five, what is indicated about the relationship between the volumes of each of these pyramids?
If three equal pyramids combine to make one prism, then each pyramid is what fraction of the prism? 
Instructional Implications Help the student recognize that three pyramids of equal volume can be combined (without overlap or gaps) like puzzle pieces, to create one complete prism, so that each pyramid is the volume of the prism. Use the Transitive Property of Equality to illustrate, as needed.
If necessary, after the student is able to correctly answer questions six and seven, ask the student to examine the response written for question one to make sure an explanation includes a statement about congruent bases and heights. Provide feedback as necessary.
If necessary, after the student is able to correctly answer questions one, six, and seven, ask the student to examine the response written for question two. The pyramids have the same base and the same height (and consequently, equal cross sections at each height), so by Cavalieri’s Principle, their volumes are the same. Provide feedback as necessary.
Consider implementing other MFAS tasks for GGMD.1.1. 
Almost There 
Misconception/Error The student indicates an overall understanding about how the volume formula for a pyramid is derived from the volume of a prism, but some written communication needs finetuning. 
Examples of Student Work at this Level The student indicates a conceptual understanding of the derivation of the pyramid volume formula, but:
 Omits some details when explaining the relationship between the areas of the triangles and/or how the volumes of the pyramids are congruent.
 Does not sufficiently explain why the areas of the bases are congruent, although he or she correctly explains the relationships between the volumes of the pyramids.
 Does not write that pyramids with congruent bases and heights also have congruent cross sections so that Cavalieri’s Principle can be applied.

Questions Eliciting Thinking You seem to have a good understanding of how the formula for the volume of a pyramid can be derived from the volume of a prism, but some of your explanations are missing important details. Can you tell me more information about why or how the areas of and are the same?
Can you give me more information about why or how the volumes of the pyramids in question two are the same?
Can you identify the base for pyramid ABDC? For pyramid EDBC? In question one, what did you determine about and ? What can you say about the bases for these two pyramids?
What specifically allows the heights of the pyramids in question two to be the same? Are the bases for the pyramids coplanar? Why or why not?
What is the vertex of each pyramid? What is the relationship between the vertices of the two pyramids?
If two pyramids have bases that are coplanar (same lateral face) and they have the same vertex, will the pyramids have the same height? Why or why not?
What does the first paragraph on the worksheet say about the cross sections of two pyramids when the pyramids have congruent bases and the same height?
How can Cavalieri’s Principle justify two pyramids as having equal volumes? Have you addressed all of the requirements for using Cavalieri’s Principle? What is the relationship between the cross sections of each pair of pyramids? 
Instructional Implications If necessary, ask the student to use the formula for the area of a triangle to provide details explaining why the triangles in question one have the same areas. Provide feedback as necessary.
If necessary, ask the student to write more specific details about why or how the volumes of the pyramids in question two are the same, including why the heights are the same. If necessary, remind the student that the bases of the pyramids are coplanar because they are part of the same lateral face of the prism and they have the same vertex. Next, remind the student that the distance from a point to a plane is measured along the perpendicular from that point to the plane. Since the triangular bases of the pyramids are coplanar and they both have the same vertex, then the distance (height) of both pyramids will be the same.
Once it is established that the two pyramids have bases with congruent areas and equal heights, ask the student to reread the first paragraph on the worksheet to determine the next step (e.g., cross sections). From the worksheet, the sentence after Cavalieri’s Principle can be used to verify the cross sections as congruent because the areas of the bases and the heights are the same. As a result, Cavalieri’s Principle, which requires bases of the same area and solids with equal cross sections at any given height, can be applied to relate the volumes of the pyramids.
Consider implementing other MFAS tasks for GGMD.1.1. 
Got It 
Misconception/Error The student provides complete and correct response to all components of the task. 
Examples of Student Work at this Level The student explains:
 The areas of and are the same because they have equal bases and heights. The bases are equal because the lateral faces of the prism are rectangles (parallelograms) so the opposite sides are congruent. Since a rectangle has four angles, the sides are perpendicular and the triangles have equal heights. Triangles with equal bases and heights have equal areas.
 Pyramids ABDC and EDBC, have equal bases (areas of and are the same) and equal heights (the bases of the pyramids are coplanar and they have the same vertex), so their cross sections at any height are the same and by Cavalieri’s Principle, they have equal volumes.
 The areas of and are the same because they have equal bases and heights using the same reasoning as in question one.
 Pyramids EBCD and CFED, have equal bases, heights, and volumes (see question two).
 The volumes of pyramids ABDC, EDBC, and CFED are the same by the transitive property of equality.
 The volume of any one of these pyramids is the volume of prism ABCDEF.
 The volume of any pyramid, with base and height AD, is the volume of prism ABCDEF.

Questions Eliciting Thinking Does the relationship between the volume of these three pyramids and the volume of this prism also apply to other triangular pyramids and prisms? Can you write a conjecture?
What about pyramids and prisms with hexagonal or pentagonal bases? What parts must be congruent? 
Instructional Implications Challenge the student to explore this formula (and these relationships) with prisms and pyramids having polygon bases containing more than three sides. Ask the student to write a conjecture relating the volumes of pyramids and prisms with any polygon base (ngons).
Consider implementing other MFAS tasks for GGMD.1.1. 