Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 71370
Slice of a Cone
Students are asked to sketch, describe, and compare three horizontal cross sections of a cone.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, three-dimensional, two-dimensional, cross section, cone, slice, parallel
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_SliceofaCone_Worksheet.docx
FORMATIVE ASSESSMENT TASK
Instructions for Implementing the Task
This task can be implemented individually, with small groups, or with the whole class.
1. The teacher asks the student to complete the problems on the Slice of a Cone worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
The student is not able to identify the shape of a horizontal cross section of a cone.
Examples of Student Work at this Level
The student:
Describes the three-dimensional parts of the cone that result from slicing it horizontally rather than two-dimensional cross sections of the cone.
page 1 of 4 Identifies the three-dimensional pieces in the first question and then describes flat shapes in the second question, but does not demonstrate an understanding of the
definition of a cross section.
Describes the cross section as a triangle and then writes a vague statement about the shape being bigger at different levels.
Questions Eliciting Thinking
What is a cross section?
Can you show me how you would cut this cone horizontally?
Are the three-dimensional pieces you described cross sections?
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 meanings of terms used in this task: cross section, horizontal, and height. Model for the student how to draw and describe a horizontal cross section of a cone.
Relate the cross section to the shape of the base. Make it clear that a horizontal cross section of a cone is a circle, and relate the radius of the cross section to the radius of
the base of the cone. Guide the student to observe and describe the relationship between the radius of the circular cross section and the height at which it occurs.
Provide additional experience with identifying and drawing cross sections of three-dimensional figures and describing their dimensions.
Consider using a virtual manipulative such as Cross Section Flyer – Shodor (CPALMS Resource ID#: 25314) to help the student visualize cross sections of various three
dimensional solids.
Moving Forward
Misconception/Error
The student incorrectly or ambiguously describes the relationship between the radius of the circular cross section and the height at which it occurs.
Examples of Student Work at this Level
The student understands that the cross sections are circles but:
States the shorter the height, the smaller the circle.
Says the circles keep “getting bigger and bigger” but does not relate this to the height at which the cross section occurs.
page 2 of 4 Questions Eliciting Thinking
Suppose I slice the cone half way up its height. Where could you slice the cone to get a circle with a smaller radius? A larger radius?
Instructional Implications
Provide the student with a diagram in which the dimensions of the cone are labeled. Instruct the student to sketch three horizontal cross sections at three different given
heights. Use the height of the cone, the radius of the base of the cone, and the heights at which the cross sections occur to calculate the radius of each cross section.
Assist the student in making a table comparing the radius of the cross section to the height at which it occurred. Discuss with the student the relationship between the
radius of the cross section and the height at which it occurs.
Consider implementing CPALMS Lesson Plan 2D Representations of 3D Objects (ID 32549). This lesson will help students visualize two dimensional cross sections at different
levels of a three dimensional object.
Almost There
Misconception/Error
The student uses imprecise language to describe the cross-sections or the relationship between the sizes of the cross sections and the height at which they occur.
Examples of Student Work at this Level
The student:
Does not explicitly describe the cross sections as circles and describes the cross sections as getting “wider” the closer they are to the base.
Uses terms like “larger” and “smaller” rather than describing the size of the circular cross sections in terms of their radii.
Does not precisely describe the change in size.
Questions Eliciting Thinking
What word describes each cross section? What kind of geometric figure is the cross section?
What measure can you use to describe the size of a circle? Can you explain the changes in the circular cross sections in terms of their radii?
Instructional Implications
Provide feedback to the student on the use of terminology and allow the student to revise his or her response. Encourage the student to learn and use appropriate
mathematical terminology.
Consider implementing CPALMS Lesson Plan 2D Representations of 3D Objects (ID 32549). This lesson will help students visualize two dimensional cross sections at different
levels of a three dimensional object.
Consider implementing MFAS tasks Slice It (G-GMD.2.4) and Inside the Box (G-GMD.2.4).
Got It
Misconception/Error
The student provides complete and correct responses to all components of the task.
Examples of Student Work at this Level
page 3 of 4 The student sketches three horizontal circles that vary in size and explicitly describes the cross sections as circles. The student states that the closer the cross sections are
to the base, the greater their radii (or diameters or areas).
Questions Eliciting Thinking
Could you slice this cone to find a triangular cross section? How? Where?
Could you imagine a cross section of this cone that is not a circle or a triangle? What other possible shapes of cross sections can you find? Can you sketch them?
Instructional Implications
Challenge the student to mathematically describe the radius, r, of a cross section in terms of the height, H, of the cone, the radius, R, of its base, and the height, h, at
which the cross section occurs. Define the height of the cross section as a distance along the cone’s height either from the vertex or from the base of the cone.
Consider implementing MFAS tasks Slice It (G-GMD.2.4) and Inside the Box (G-GMD.2.4).
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Slice of a Cone worksheet
SOURCE AND ACCESS INFORMATION
Contributed by: MFAS FCRSTEM
Name of Author/Source: MFAS FCRSTEM
District/Organization of Contributor(s): Okaloosa
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.G-GMD.2.4:
Description
Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional
objects generated by rotations of two-dimensional objects.
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