Getting Started |
| Misconception/Error The student does not understand that two-dimensional figures can result from slicing three-dimensional figures. |
| Examples of Student Work at this Level The student does not describe the two-dimensional cross section of the pyramid, but instead:
- Draws the three-dimensional pieces that result from slicing the pyramid.
- Draws the net of the three-dimensional shape resulting from the slice.
- Draws a two-dimensional view showing the face and where the slice will be made.
- Draws the results of a slice other than the one given.
The student may also confuse some or all of the terms: horizontal, vertical, parallel, perpendicular, vertex, and base.
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| Questions Eliciting Thinking What is the difference between a two-dimensional figure and a three-dimensional figure? Can you give me an example of each?
Do you know what cross section means? Can you imagine the cross section of the pyramid that is revealed by the slicing? How would this cross section be different than a net?
Which way is horizontal (vertical)? What does parallel (perpendicular) mean?
Where should the slice go that is perpendicular to the base but not through the vertex? |
| Instructional Implications Review the difference between two-dimensional and three-dimensional figures. Provide the student with examples of figures to be classified as either two-dimensional or three-dimensional. Ask the student to classify the figures and identify the dimensions of each. Clarify the difference between a “net” and a “slice” of the figure, explaining that a section of a net represents a face of the three-dimensional figure. Therefore, a net is always congruent to the corresponding faces; however a slice may or may not be congruent to a face.
Consider implementing the CPALMS Lesson Plan Can You Cut It? Slicing Three-Dimensional Figures (ID 47309). This lesson guides the student to sketch and describe a two-dimensional figure resulting from the horizontal or vertical slicing of a three-dimensional figure. Be sure the student understands the difference between horizontal and vertical, parallel, and perpendicular. Model horizontal and vertical slices. Define parallel and perpendicular, and then model parallel and perpendicular slices in relation to the base. If needed, review the dimensions of a pyramid (e.g., the length and width of the base, height, lateral edge, and slant height). Show the student that the dimensions of the slices can be described in terms of the dimensions of the original pyramid. Provide additional experience with identifying and drawing two-dimensional slices of three-dimensional figures and describing their dimensions. Consider implementing this task again to assess if the student can sketch and describe the two-dimensional cross section resulting from each slice. |
Making Progress |
| Misconception/Error The student does not adequately describe the dimensions of the cross section in terms of the dimensions of the original figure. |
| Examples of Student Work at this Level The student can draw the shape of the correct two-dimensional cross section resulting from each slice, but:
- Does not clearly describe how the dimensions compare to the original figure.
- Describes the dimensions in terms of W, B, and/or H but does not specify what these variables represent.
- Is not specific in describing the dimensions of the cross sections and only indicates that they “are not the same.”
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| Questions Eliciting Thinking What do you mean by B (or W) and H? Where are these lengths on the original figure?
To what part of the pyramid can you compare the square? How does the size of the slice compare to the size of the base of the pyramid?
What are the two dimensions of the triangle resulting from the slice? To what part of the pyramid can you compare the base and height of the triangle?
What type of quadrilateral resulted from the horizontal (or vertical, non-vertex) slice? How can you describe the lengths of the sides? |
| Instructional Implications Guide the student to relate the dimensions of the two-dimensional figure to the dimensions of the original three-dimensional figure. Model a concise comparison (e.g., the height of the triangle is equal to the height of the pyramid and the base of the triangle is equal to the length of the base edge of the pyramid). Provide additional opportunities to precisely describe cross sections of three-dimensional figures.
Consider implementing the MFAS tasks Cylinder Slices, Cone Slices, and Rectangular Prism Slices for additional practice. |
Got It |
| Misconception/Error The student provides complete and correct responses to all components of the task. |
| Examples of Student Work at this Level The student correctly identifies and draws the plane figures resulting from each slice, and describes each using specific dimensions. For example, the student says:
- The cross section is a triangle with a height equal to the height of the pyramid and a base equal to the length of the base edge of the pyramid (or if the student sliced vertically through the diagonal of the base, the length of the base of the triangle would equal the length of the diagonal of the base of the pyramid).
- The cross section is a trapezoid with the length of the bottom base equal to the length of the base edge of the pyramid, the length of the top base is less than the length of the base edge of the pyramid, and the height of the trapezoid is less than the height of the pyramid.
- The cross section is a square with the length of each side less than the length of the base edge of the pyramid.
Note: The student may label the dimensions of the pyramid and use the associated dimension labels on the two-dimensional shapes, rather than use a word description of the resulting dimensions. |
| Questions Eliciting Thinking Does the slice have to be in the middle (halfway) in order to be horizontal? Can the horizontal slice be close to the bottom (or top) base of the pyramid? How will the place where it is sliced change its shape or dimensions?
How will the shape of the vertical slice (not through the vertex) be different if your slice is not parallel to a side of the base? |
| Instructional Implications Challenge the student to:
- Find as many different two-dimensional shapes as possible while describing the slice needed to make each one, including slices that are neither parallel nor perpendicular to the base.
- Describe slices from more complex figures such as a double cone or a pentagonal prism.
Consider implementing the MFAS tasks Cylinder Slices, Cone Slices, and Rectangular Prism Slices for additional practice.
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