StackUps
Spatial Reasoning Using Cubes
and Isometric Drawings
Teacher's Guidebook
Copyright © 2007
Louisville, Kentucky. All Rights Reserved
Product Designer/Author:
Karen J. Poppe, Tactile Graphics Project Leader
Tactile/Visual Designer:
Tom Poppe, Pattern/Model Maker
Project Assistants:
Matt Smith
Monica Vaught
Guidebook/Cover Layout:
Bisig Impact Group
StackUps: Spatial Reasoning Using Cubes and Isometric Drawings
APH Catalog Number: 1-08960-00
Large Print Teacher's Guidebook
APH Catalog Number: 7-08960-00
Braille Teacher's Guidebook
APH Catalog Number: 5-08960-00
American Printing House for the Blind, Inc.
1839 Frankfort Avenue
P.O. Box 6085
Louisville, Kentucky 40206-0085, USA
Phone: 502-895-2405
Toll Free: 800-223-1839
Fax: 502-899-2274
Email: info@aph.org
Web site: www.aph.org
Acknowledgements
The following teachers of the visually impaired contributed their time, knowledge, and creativity to the evaluation of StackUps: Spatial Understanding Using Cubes and Isometric Drawings:
Diane Clay, Teacher of the Visually Impaired, Catholic Charities Maine, Lewiston, Maine
Donna Fridgant, Teacher of the Visually Impaired, Tennessee School for the Blind, Nashville, Tennessee
Sister Elaine George, Materials Assistant, St. Lucy Day School for Children with Visual Impairments, Philadelphia, Pennsylvania
Michelle Harbison, Teacher of the Visually Impaired, Anne Arundel County Public Schools, Linthicum, Maryland
Lisa Hodge, Teacher of the Visually Impaired, Washington School for the Blind, Vancouver, Washington
Jennifer F. Johnston, Teacher of the Visually Impaired, Gardendale High School/Jefferson County Public School System, Birmingham, Alabama
Brenda Ireland, Teacher of the Visually Impaired/Certified O&M Specialist, Payette School District, Payette, Idaho
Yvonne Martinez, Teacher of the Visually Impaired, Mesquite Independent School District, Mesquite, Texas
Jill R. Pariso, Teacher of the Visually Impaired, Orleans-Niagara BOCES, Medina, New York
Shari Scott, Teacher of the Visually Impaired, Tennessee School for the Blind, Nashville, Tennessee
Christy J. Shepard, Teacher of the Visually Impaired, Cypress-Fairbanks Independent School District, Houston, Texas
Janet M. Ulwick-Sacca, Teacher of the Visually Impaired, Carroll Center for the Blind, Newton, Massachusetts
Danelle Volk-Heimbach, Teacher of the Visually Impaired, Anne Arundel County Public Schools, Linthicum, Maryland
Sally Walton, Teacher of the Visually Impaired, Frederick County Public Schools, Walkersville, Maryland
JoAnn Weatherall, Teacher of the Visually Impaired, Tennessee School for the Blind, Nashville, Tennessee
David Wixom, Teacher of the Visually Impaired, Industrial Arts and Technology, Missouri School for the Blind, St. Louis, Missouri
Special thanks to the 56 students who were involved in the field test of this product.
Introduction
Purpose of StackUps
StackUps: Spatial Reasoning Using Cubes and Isometric Drawings is a set of materials designed to assist students with visual impairments and blindness in the interpretation of raised-line graphics depicting 3-dimensional figures, specifically stacked cube arrangements. Using the materials provided, students can practice the following:
- building 3-dimensional models using Velcro® cubes in combination with tactile displays,
- using Mat Plans to construct and create stacked cube arrangements,
- interpreting Front-Right-Top views,
- determining volume and surface area, and
- interpreting cube nets
StackUps addresses the following Geometry Standards from the NCTM Principles and Standards for School Mathematics:
- Analyze characteristics and properties of 2- and 3-dimensional geometric shapes and develop mathematical arguments about geometric relationships
- Use visualization, spatial reasoning, and geometric modeling to solve problems
StackUps encourages students to 1) recognize, name, build, draw, compare and sort 2- and 3-dimensional shapes; 2) describe attributes and parts of 2- and 3-dimensional shapes; and 3) investigate and predict the results of putting together and taking apart 2- and 3-dimensional shapes.
Getting Started with StackUps
The StackUps kit includes 20 cubes, each covered with hook Velcro (white and rough texture) and loop Velcro (blue and soft texture). These contrasting colors and textures correspond with the accompanying tactile and visual Stacked Cube Arrangement Cards [32 total]. The sides of the cubes covered with soft, blue Velcro are intended to represent the visible sides of a cube when shown 2-dimensionally in a tactile display. Therefore, in the accompanying Stacked Cube Arrangement Cards, the sides of each cube always have a smooth texture. Conversely, the side of a cube covered with rough, white Velcro is intended to always represent the visible top of a cube when shown 2-dimensionally in a tactile display. On the Stacked Cube Arrangement Cards, the top of each cube is indicated by a rough, pebbled texture.
The Stacked Cube Arrangement Cards depict arrangements that vary in difficulty and are progressive with regard to the number of cubes depicted. For example, one of the Stacked Cube Arrangement Cards shows a single cube. Use Stacked Cube Arrangement #1 [see Appendix A] to first acquaint a student with how the textures on the tactile card correspond to the actual StackUps cubes. Have the student position the StackUps cube on a flat surface that mimics the presentation of the single cube shown on the tactile card. The rough, white Velcro side should be positioned upward; the two facing sides will have soft, blue Velcro.
Important: As StackUps cubes are joined together to build various stacked cube arrangements, it is important that all the cubes be oriented in the same direction to each other. Therefore, white, rough surfaces of the cubes should always attach to blue, soft surfaces of the cubes. This consistent orientation of the cubes becomes essential as cubes are added behind and above the first row of cubes.
Orient cubes so that rough, white Velcro is on top, and the two facing sides are soft, blue Velcro. Keep all cubes oriented in the same direction as they are stacked.
Use the provided 5 x 5 Raised-Line Grid to assist students in keeping the cubes in straight rows and columns as they build stacked cube arrangements.
Before performing the activities described in the next section of this guidebook, allow the student to simply practice stacking the StackUps cubes together until she becomes familiar with how the rough, white sides fasten to the soft, blue sides. Document the student's mastering of this skill in the StackUps Skills Checklist found in Appendix D.
Suggested Activities
Determining Possible Stacked Cube Arrangements
Learning Objective:
- Determine possible cube arrangements using the same number of cubes.
Provide the student with three cubes and ask him to build as many stacked cube arrangements as possible using only three cubes. Continue to do this activity using only four cubes, only five cubes, and so forth. How many different configurations are possible with each number of cubes provided? Some of the student's constructed cube arrangements will match those shown throughout the Stacked Cube Arrangement Cards.
Example: Possible arrangements using only three cubes:
Example: Possible arrangements using only four cubes:
Constructing 3-D Models and Interpreting Tactile Displays of Stacked Cube Arrangements
Learning Objectives:
- Build a 3-dimensional model that matches a 2-dimensional tactile display.
- Identify a 2-dimensional tactile display that matches a 3-dimensional model.
Build a 3-dimensional model that matches a 2-dimensional tactile display.
Provide the student with only two StackUps cubes and have him join the two cubes together. Then ask him to position the two linked cubes on a flat surface in a variety of positions -- with only one cube against the table's surface, with only two cubes against the table's surface, and so forth.
Notice that Stacked Cube Arrangement Cards #2, #3, and #4 show two cubes linked together, but from different perspectives. These three tactile cards can be used to illustrate how 2-dimensional stacked cube arrangements can be shown from a variety of angles. Have the student build each perspective by constructing a StackUps model and positioning it on a flat surface in the same perspective shown on the tactile card. Continue to practice building the models so that the sides of the cubes "seen" or facing the student have a soft texture and the tops of the cubes have a rough texture.
Draw the student's attention to the tactile difference between the raised lines used to depict the outer edges of each cube in the Stacked Cube Arrangement Cards. A solid line indicates edges shared by two adjacent cubes OR the outer boundaries of the 2-dimensional drawing. A broken or dashed line is always vertical and indicates the joining line of the two faces of a single cube.
Once the student gains understanding and experience building simpler 3-dimensional models, then progress to more complex stacked cube arrangements. Ask the student to recreate each figure presented 2-dimensionally in the Stacked Cube Arrangement cards. Begin with cards involving only three or four cubes, progressing in the order shown in Appendix A. Stacked Cube Arrangement Card #32 involves all 20 cubes, the maximum number of cubes provided with the StackUps kit. [Additional StackUps cubes are available separately in packages of 20 cubes: APH Catalog #: 1-08960-01.] Note that all the Stacked Cube Arrangement Cards have a diagonal cut in the upper-right corner; this cut serves as a cue for correct orientation.
2-D to 3-D
Identify a 2-dimensional tactile display that matches a 3-dimensional model.
Provide the student with a pre-assembled 3-dimensional model, made with StackUps cubes, that matches one of the provided Stacked Cube Arrangement Cards. Ask the student to locate the tactile card that matches the 3-dimensional model. Narrow the selection of cards by providing only three or four possibilities for the student to choose from.
3-D to 2-D
Stacked Cube Arrangement Cards #23 and #24 represent the same solid. These two cards are added to illustrate mirrored images of the same 3-dimensional shape.
Instructional Tip:
Use a Raised-Line Grid from the StackUps kit to provide a stable working surface as cubes are being stacked. The blue, soft Velcro sides of the cubes will stick to the rough Velcro circles on the grid. The grid will also assist the student in aligning the cubes in straight rows and columns.
Using Mat Plans
Learning Objectives:
- Build a 3-dimensional solid when given a Mat Plan.
- Create a Mat Plan when given a 3-dimensional solid.
Build a 3-dimensional solid when given a Mat Plan.
A Mat Plan is a top view of a 3-dimensional cube arrangement, with the number of cubes appearing in each vertical column recorded in the corresponding box.
The number within each box indicates how many cubes are stacked vertically in that specific spot. It is essentially a blueprint or building plan for constructing the 3-dimensional model of a stacked cube arrangement. A tactile/print Mat Plan is provided for each of the Stacked Cube Arrangement Cards. Appendix B pairs each Mat Plan with its corresponding Stacked Cube Arrangement Card.
Present the Mat Plans one at a time to the student and ask her to build a corresponding 3-dimensional solid. Demonstrate that sometimes a Mat Plan can represent more than one stacked cube arrangement. For example, the Mat Plan for Stacked Cube Arrangement #13 also represents Stacked Cube Arrangement #14, according to NCTM.
However, a Mat Plan will always be unique to a specific cube arrangement if instructions are given to always build each vertical column of cubes from the base upward, with the bottom cube of each column against a flat surface. With this instruction, Mat Plans can provide a foolproof construction plan for stacked cube arrangements. This directive will ensure, for example, that the Mat Plan shown above will only represent Stacked Cube Arrangement #13, and not the suspended cube presentation as encountered in Stacked Cube Arrangement #14 or any other arrangement.
Note that the Mat Plan Cards are intentionally unnumbered so that students cannot memorize which Mat Plan corresponds to which Stacked Cube Arrangement Card. The Mat Plan Cards also have a curved orientation cut in the upper-right corner.
Create a Mat Plan when given a 3-dimensional solid.
The StackUps kit includes a package of 25 consumable Mat Plan Worksheets. [Additional Mat Plan Worksheets are available separately in packages of 25 sheets: APH Catalog #: 1-08960-02.] Each Mat Plan Worksheet has an embossed raised-line grid in a 5 x 5 configuration. These embossed grid sheets can be used by the student to "draw" Mat Plans of stacked cube arrangements. By using a braillewriter, slate and stylus, or by applying adhesive braille number stickers, the student can independently build a Mat Plan by plotting the number of cubes in each vertical column within the tactile squares. Numbers can be handwritten inside the grid as well. Note: A print version of the Mat Plan Worksheet is found in Appendix C. Permission is given to photocopy this page as needed for low vision students.
Have the student create a Mat Plan to match one of the provided Stacked Cube Arrangement Cards. Start with simple cube arrangements and progress to more challenging ones. Then ask the student to design Mat Plans that build cube arrangements not provided within the StackUps kit. For each newly designed Mat Plan, the student can build a 3-dimensional solid that corresponds to it. The teacher can also use Mat Plan Worksheets to generate additional Mat Plans from which the student can practice building matching 3-D solids with the StackUps cubes. After numbers have been brailled and/or printed on the Mat Plan Worksheet for a particular stacked cube arrangement, the unused squares can be cut away to present the final Mat Plan.
To make the Mat Plan Worksheets reusable, use them in conjunction with adhesive braille and/or print number stickers that are affixed to magnetic sheets. Simply place the Mat Plan Worksheet on top of a metal surface (e.g., a cookie sheet) and place magnetic stickers within the raised outline squares.
Interpreting Front-Right-Top Views
Learning Objectives:
- Create a Front-Right-Top view when given a 3-dimensional stacked cube arrangement.
- Create a 3-dimensional stacked cube arrangement when given a Front-Right-Top view.
A Front-Right-Top view provides three separate perspectives of a cube arrangement. For example, the three views below represent the construction of the Stacked Cube Arrangement #29.
Top View
Isometric View
Front View
Right Side View
Below is another example of a Front-Right-Top view using Stacked Cube Arrangement #21.
Top View
Isometric View
Front View
Right Side View
The Front-Right-Top view for each of the Stacked Cube Arrangements included in StackUps is found in Appendix A.
Create a Front-Right-Top view when given a 3-dimensional stacked cube arrangement.
Provide the student with a constructed 3-dimensional stacked cube arrangement, as well as all three Raised-Line Grids. Have the student "draw" each perspective on a separate grid by inserting Velcro-backed blue squares into the grid compartments.
Top View
Stacked Cube Arrangement #21
Front View
Right Side View
Create a 3-dimensional stacked cube arrangement when given a Front-Right-Top view.
Creating a 3-dimensional model of a stacked cube arrangement given a Front-Right-Top view can be more challenging. Begin with simple cube arrangements like Stacked Cube Arrangement #6 [shown below] and then progress to more complicated ones.
Top View
Stacked Cube Arrangement #6
Front View
Right Side View
Determining the Volume of Stacked Cube Arrangements
Learning Objective:
- Find the volume of stacked cube arrangements.
Have the student find the volume of the figure depicted on a selected Stacked Cube Arrangement Card in combination with a constructed StackUps 3-dimensional model that matches the card. The volume of each figure can be found by counting the number of cubes it contains. Have the student indicate the volume in "cubic units," not in inches. The volume of each Stacked Cube Arrangement Card is given in Appendix A.
Instruction Tips for the Tactile Learner:
Always begin with the 3-dimensional model of a stacked cube arrangement when determining volume. Determining volume from the tactile, 2-dimensional display will be difficult even for the most proficient tactile reader. Complement the 3-dimensional model by showing the 2-dimensional display in a side-by-side manner.
When attempting to count the number of cubes in a tactile 2-dimensional display, encourage the tactile reader to start by counting tops of cubes that can be touched, that is, the number of individual rough areas surrounded by solid raised lines. Using Cube Arrangement #16 as an example, the student would first count four rough areas and determine that there are at least four cubes in the arrangement.
He would then count the number of cubes in each vertical column, going left to right. Still using card #16 as an example, the number of cubes per column (left to right) would be "2, 1, 1, 2," with a total volume of 6 cubic units.
Determining the Surface Area of Stacked Cube Arrangements
Learning Objective
- Find the surface area of stacked cube arrangements.
Using a constructed 3-dimensional StackUps model of a selected Stacked Cube Arrangement Card, have the student determine the surface area of the figure, that is, the number of square units that are exposed on the outside. The student should report the surface area in square units instead of inches. The surface area of each Stacked Cube Arrangement Card is provided in Appendix A.
When finding the surface area, remind the student to count the surface squares that cannot be touched when the figure is sitting flat on a table. For example, only five sides of a single cube can be touched when it sits flat on a table, but when the cube is held in one's hand, all six sides can be detected. Therefore, the surface area of a single cube is 6 square units.
Instruction Tips for the Tactile Learner:
When asking the student to determine the surface area of a stacked cube arrangement, it is advisable to use a 3-dimensional model whenever possible, either alone or in combination with the tactile 2-dimensional drawing. Providing a model will allow the student to more readily count the exposed or outside square surfaces. Using Stacked Cube Arrangement #7 as an example, first provide a 3-dimensional model of the tactile display for the student's exploration.
When handling the model, the student might recount the same surfaces more than once as he examines the model. To prevent this, the student can mark off the sides of the cubes as they are counted (e.g., by attaching temporary tactile stickers). The student can then count each square surface, one by one, as stickers are added, allowing the student to count without retracing his steps. He should arrive at the answer "16 square units" for this cube arrangement.
Determining surface area based upon a tactile display alone is extremely challenging, even for the proficient tactile reader. In the 2-dimensional display, the tactile reader will not be able to feel the bottom of each cube, nor the hidden sides of each cube. A side-by-side instruction, using both the 3-dimensional model and the tactile display would be advantageous to the student's understanding of surface area.
Identifying Cube Nets
Learning Objective
- Identify cube nets that build 3-dimensional cubes.
A net is a 2-dimensional figure that can be folded into a 3-dimensional object. The following nets will fold to build a cube:
Create foldable cube nets by cutting them out of embossed Mat Plan Worksheets. Then have the student fold each to determine if a 3-dimensional cube results. Mix in cube nets that will not produce 3-dimensional cubes such as the following:
Appendix A: Stacked Cube Arrangements
StackUps: Stacked Cube Arrangement 1
Volume: 1 cubic unit
Surface Area: 6 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 2
Volume: 2 cubic units
Surface Area: 10 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 3
Volume: 2 cubic units
Surface Area: 10 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 4
Volume: 2 cubic units
Surface Area: 10 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 5
Volume: 3 cubic units
Surface Area: 14 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 6
Volume: 3 cubic units
Surface Area: 14 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 7
Volume: 4 cubic units
Surface Area: 16 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 8
Volume: 4 cubic units
Surface Area: 18 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 9
Volume: 4 cubic units
Surface Area: 18 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 10
Volume: 4 cubic units
Surface Area: 18 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 11
Volume: 4 cubic units
Surface Area: 16 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 12
Volume: 4 cubic units
Surface Area: 18 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 13
Volume: 5 cubic units
Surface Area: 22 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 14
Volume: 5 cubic units
Surface Area: 22 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 15
Volume: 6 cubic units
Surface Area: 22 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 16
Volume: 6 cubic units
Surface Area: 24 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 17
Volume: 6 cubic units
Surface Area: 26 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 18
Volume: 6 cubic units
Surface Area: 22 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 19
Volume: 7 cubic units
Surface Area: 24 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 20
Volume: 7 cubic units
Surface Area: 26 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 21
Volume: 7 cubic units
Surface Area: 26 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 22
Volume: 7 cubic units
Surface Area: 30 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 23
Volume: 8 cubic units
Surface Area: 28 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 24
Volume: 8 cubic units
Surface Area: 28 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 25
Volume: 9 cubic units
Surface Area: 30 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 26
Volume: 10 cubic units
Surface Area: 36 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 27
Volume: 12 cubic units
Surface Area: 32 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 28
Volume: 12 cubic units
Surface Area: 38 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 29
Volume: 12 cubic units
Surface Area: 36 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 30
Volume: 14 cubic units
Surface Area: 44 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 31
Volume: 15 cubic units
Surface Area: 42 square units
Front View
Right Side
Top View
StackUps: Stacked Cube Arrangement 32
Volume: 20 cubic units
Surface Area: 48 square units
Front View
Right Side
Top View
Appendix B: Mat Plans
This Mat Plan corresponds to Stacked Cube Arrangement 1.
This Mat Plan corresponds to Stacked Cube Arrangement 2.
This Mat Plan corresponds to Stacked Cube Arrangement 3.
This Mat Plan corresponds to Stacked Cube Arrangement 4.
This Mat Plan corresponds to Stacked Cube Arrangement 5.
This Mat Plan corresponds to Stacked Cube Arrangement 6.
This Mat Plan corresponds to Stacked Cube Arrangement 7.
This Mat Plan corresponds to Stacked Cube Arrangement 8.
This Mat Plan corresponds to Stacked Cube Arrangement 9.
This Mat Plan corresponds to Stacked Cube Arrangement 10.
This Mat Plan corresponds to Stacked Cube Arrangement 11.
This Mat Plan corresponds to Stacked Cube Arrangement 12.
This Mat Plan corresponds to Stacked Cube Arrangement 13 and 14.
This Mat Plan corresponds to Stacked Cube Arrangement 15.
This Mat Plan corresponds to Stacked Cube Arrangement 16.
This Mat Plan corresponds to Stacked Cube Arrangement 17.
This Mat Plan corresponds to Stacked Cube Arrangement 18.
This Mat Plan corresponds to Stacked Cube Arrangement 19.
This Mat Plan corresponds to Stacked Cube Arrangement 20.
This Mat Plan corresponds to Stacked Cube Arrangement 21.
This Mat Plan corresponds to Stacked Cube Arrangement 22.
This Mat Plan corresponds to Stacked Cube Arrangement 23.
This Mat Plan corresponds to Stacked Cube Arrangement 24.
This Mat Plan corresponds to Stacked Cube Arrangement 25.
This Mat Plan corresponds to Stacked Cube Arrangement 26.
This Mat Plan corresponds to Stacked Cube Arrangement 27.
This Mat Plan corresponds to Stacked Cube Arrangement 28.
This Mat Plan corresponds to Stacked Cube Arrangement 29.
This Mat Plan corresponds to Stacked Cube Arrangement 30.
This Mat Plan corresponds to Stacked Cube Arrangement 31.
This Mat Plan corresponds to Stacked Cube Arrangement 32.
Appendix C: Mat Plan Worksheet
Permission is given to photocopy this page as needed for low vision students.
Appendix D: StackUps Skills Checklist
Student's Name: _______________________________________________________
Instructor's Name: ______________________________________________________
Directions: Use the following rating scale to indicate the student's current level of understanding of each skill/concept:
1 = Beginning level of performance
2 = Developing level of performance
3 = Mastered/Accomplished level of performance
StackUps Skills Checklist
Objective/Skill
Stack StackUps cubes independently. |
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Determine possible cube arrangements from a given number of cubes. |
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Build a 3-D solid that matches a 2-D tactile display. |
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Identify a 2-D tactile display that matches a 3-D solid. |
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Build a 3-D solid when given a Mat Plan. |
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Create a Mat Plan when given a 3-D solid. |
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Create a Front-Right-Top view when given a 3-D solid. |
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Build a 3-D solid when given a Front-Right-Top view. |
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Determine the volume of a stacked cube arrangement. |
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Determine the surface area of a stacked cube arrangement. |
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Identify cube nets that build 3-D cubes. |
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Permission is given to make copies of this checklist as needed. You may also complete this form electronically using the StackUps Skills Checklist provided as an MSWord file on the accompanying CD.
Bibliography
National Council of Teachers of Mathematics. (2005). Illuminations: i-Math Investigation/6-8: Spatial Reasoning and Using Cubes and Isometric Drawings. Retrieved July 15, 2005 from http://illumtest.nctm.org/imath/6-8/isometric/index.html
National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: The National Council of Teachers of Mathematics, Inc.