1. Spatial Visualization Workshops
Spatial visualization is an essential and learnable skill that engineers use to clearly communicate their ideas to other people so the ideas can evolve and become real-world products, structures and systems.
5. Inclined Planes and Curved Surfaces
6. Reflections and Symmetry
7. Write a Rule
Orthographic Views
Isometric Drawings and Coded Plans
One-Axis Rotations
Two-Axis Rotations
Spatial Visualization Workshops (a series of 7 workshops/lessons)
by Zoe A. Welz and Jacob L. Segil, College of Engineering and Applied Science, University of Colorado Boulder 2017
Note: The slides are animated so a mouse or keyboard click brings up the next image, text or slide.

2. Orthographic Views Workshop 1
Spatial Visualization lesson 1/workshop 1
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3. Orthographic Drawings
Also called “multi-view” drawings
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Image source: (orthographic drawings of 3 sides of multi-cube object)
2012 Cdang, Wikimedia Commons

4. Orthographic Drawings
Side View (from left)
Top View
Front View
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Image source:
(3 view drawing) 2009 Biezl, Wikimedia Commons
(3-view details of house shape) 2009 Biezl, Wikimedia Commons

5. Orthographic Drawings
Always draw the three views in this arrangement
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Image source: 2016 Jacob L. Segil, College of Engineering and Applied Science, University of Colorado Boulder

6. Orthographic Drawings
Always draw the three views in this arrangement
-so they align with
each other
Top
Top
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Image source: 2016 Jacob L. Segil, College of Engineering and Applied Science, University of Colorado Boulder
Front
Side
Front
Side

7. Orthographic Drawings
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Image sources:
2007 Zink, Wikimedia Commons
2007 Zink, Wikimedia Commons

8. Orthographic Drawings
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Image sources:
(house shape inside clear cube worthographic projections marked) 2004 MdeVicente, Wikimedia Commons
(technical drawing exercise) 2011 Andrevruas, Wikimedia Commons

9. Orthographic Drawings
Engineering examples
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Image sources:
(wooden bench photo & top/front/side view hand drawings): © 2009 Richard M. Weidner. Used with permission.
(front and side elevation of a German church in Tilsit): 1980 Siegfried Harbrucker, Wikimedia Commons

10. Orthographic Drawings
Tips:
Draw views in order: top → front → side
Draw lines where edges exist (changes in plane)
Show hidden edges using dotted lines
Solid lines trump dotted lines

11. Station 4: Computer-Aided Visualization
Workshop 1 Stations
Station 3: No-Look Pass
In pairs, one student wears a blindfold while the other student builds an object with 5 blocks
The blindfolded student holds the object and describes the three orthographic views out loud
The partner draws ONLY what is described by the blindfolded student
When finished, correct the drawing and switch roles
Repeat and practice, moving to larger objects
Station 4: Computer-Aided Visualization
Read Module 4 – Orthographic Views
Complete the orthographic views exercises
Station 1: Block Relay
Build an object with 8 blocks
Draw the three orthographic views
Then pass the object to a neighbor and repeat
Check drawings for accuracy with neighbor
Station 2: Workbook Drill
Individually, answer questions on ODD pages ortho1 to ortho24 on scrap paper
Check your answers with the answer key
Leave this slide up so students know what to accomplish at each station

12. Isometric Drawings & Coded Plans
Workshop 2
Isometric Drawings & Coded Plans
Spatial Visualization lesson 2/workshop 2
Note: The slides are animated so a mouse or keyboard click brings up the next image, text or slide.

13. Depicting a 3D Cube Non-isometric view of a cube ↘
Corner angles are not equal
Sides have different areas
↙ Isometric view of a cube
Sides connect in a corner
All corner angles are equal (120°)
Sides are the same size
Shown on triangle-dot paper
Image source: (2 drawings of a cube)
2015 Jacob L. Segil, College of Engineering and Applied Science, University of Colorado Boulder

14. Isometric Drawing Example
Isometric means “equal measure”.
A house depicted isometrically using AutoCAD ➔
Useful for blueprints and design plans
Think of the cube:
Equal side faces
Equal corner angles (120°)
Triangle-dot paper: dots are 120° from each other
Image source: (drawing of house)
2015 Jacob L. Segil, College of Engineering and Applied Science, University of Colorado Boulder

15. Click mouse/keyboard to reveal
Isometric Views
Click to reveal
Click to reveal
Click mouse/keyboard to reveal
Use this slide to support activity 1 on the topic of isometric views and coded plans.
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Image source: (drawings of various cube shapes)
2015 Jacob L. Segil, College of Engineering and Applied Science, University of Colorado Boulder

16. Isometric Views Four views of the same multi-cube object
Image sources: (cube and letter shapes)
2015 Jacob L. Segil, College of Engineering and Applied Science, University of Colorado Boulder
Four views of the same multi-cube object
Two capital letters drawn isometrically

17. Isometric Views Tips: Align paper in landscape orientation
Define your axes on object and isometric paper
Only draw lines where edges exist

18. A coded plan of the same image ↓
Isometric Views
A coded plan of the same image ↓
Image source: (isometric and coded plan drawings of same object)
2015 Jacob L. Segil, College of Engineering and Applied Science, University of Colorado Boulder
Corners are labeled by letters
# in squares = # cubes stacked up

19. A 1 3 2 Coded Plans to Isometric Views Tips:
Image source: (coded plan)
2016 Jacob L. Segil, College of Engineering and Applied Science, University of Colorado Boulder
Tips:
Define your axes on a coded plan and isometric paper
Start drawing from perspective

20. Station 4: Computer-Aided Visualization
Workshop 2 Stations
Station 1: Coded Plan Relay
Draw a coded plan
Then, build the object
Draw the isometric view
Pass the coded plan to a neighbor and repeat
Check drawings for accuracy with neighbor
Station 2: Workbook Drill
Individually, answer workbook questions on pages iso1 to iso12 on scrap paper
Check your answers with the answer key
Station 3: Peer Teach
In pairs, choose 2 viewpoints of the provided coded plan
Draw the isometric views from the 2 viewpoints
Then, redraw the isometric view while describing out loud how you did it
Station 4: Computer-Aided Visualization
Read Module 3: Isometric Drawings and Coded Plans
Complete the isometric drawings and coded plans exercises
Leave this slide up so students know what to accomplish at each station

21. Click mouse/keyboard to reveal the possible solutions
Coded Plans to Isometric Views (Station 3 answers)
Click mouse/keyboard to reveal the possible solutions
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Image source: (cube shapes drawn from a coded plan on triangle-dot paper)
2015 Jacob L. Segil, College of Engineering and Applied Science, University of Colorado Boulder

22. One-Axis Rotations Workshop 3
Spatial Visualization lesson 3/workshop 3
Note: The slides are animated so a mouse or keyboard click brings up the next image, text or slide.

23. One-Axis Rotations
is rotated to ➔
Can you find the rotation of the gray object that is analogous to the rotation of the white object?
is rotated to:
Answer = B:
The middle gray object in the last row is rotated 90 degrees about the z-axis—just like the white object in the top row.
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Image source: 2015 Jacob L. Segil, College of Engineering and Applied Science, University of Colorado Boulder

24. One-Axis Rotations Three positive axes, x, y and z:
X = horizontal axis
Y = vertical axis
Z = axis coming towards us
vertical
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Image source: (x, y, z axes)
2016 Jacob L. Segil, College of Engineering and Applied Science, University of Colorado Boulder
horizontal axis
depth

25. One-Axis Rotations How to do the right-hand rule:
Point your thumb parallel to the axis you are rotating about and curve your fingers naturally towards the palm of your hand
Your fingers will move in the same way that the object will move
The right-hand rule is a mnemonic—which is a learning tool that helps us remember something.
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Image sources:
(x, y, z axes) 2016 Jacob L. Segil, College of Engineering and Applied Science, University of Colorado Boulder
(drawing with hand) 2009 Schorschi2 and 2011 Das steinerne Herz, Wikimedia Commons {PD}

26. One-Axis Rotations Tips: Right-hand rule!
Clockwise = negative rotation; counter-clockwise = positive rotation
90°, 180°, 270° rotations only
“Flag around a flagpole”

27. Workshop 3 Stations Station 1: Block Relay Station 2: Workbook Drill
Build an object with 6 blocks
Define the axes and a one-axis rotation
Draw the isometric views before and after rotation
Then pass the object to a neighbor and repeat
Check drawings for accuracy with neighbor
Station 2: Workbook Drill
Individually, answer questions on ODD pages rot1-5 to rot1-17 on scrap paper
Check your answers with the answer key
Station 3: Frozen
In pairs, build an object of any size
Define the axes and a one-axis rotation
Switch objects with partner and draw the isometric view before and after rotation without moving the object – it’s frozen!
Compare drawings with partner and repeat
Station 4: Computer-Aided Visualization
Read Module 7 – Rotations about a Single Axis
Complete the rotations about a single axis exercises
Leave this slide up so students know what to accomplish at each station

28. Two-Axis Rotations Workshop 4
Spatial Visualization lesson 4/workshop 4
Note: The slides are animated so a mouse or keyboard click brings up the next image, text or slide.

29. Two-Axis Rotations
Can you find the rotation of the gray object that is analogous to the rotation of the white object?
Answer = A:
The far left gray object in the last row is rotated 90 degrees about the negative y-axis and 90 degrees about the negative z-axis—just like the white object in the top row.
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Image source: 2015 Jacob L. Segil, College of Engineering and Applied Science, University of Colorado Boulder

30. Two-Axis Rotations Tips: Right-hand rule!
Clockwise = negative rotation; counter-clockwise = positive rotation
Two-axis rotation is NOT commutative (order matters!)

31. Workshop 4 Stations Station 1: Block Relay Station 2: Workbook Drill
Build an object with 6 blocks
Define the axes and a two-axis rotation
Draw the isometric views before and after rotations
Then pass the object to a neighbor and repeat
Check drawings for accuracy with neighbor
Station 2: Workbook Drill
Individually, answer questions on ODD pages rot2-5 to rot2-17 on scrap paper
Check your answers with the answer key
Station 3: Frozen
In pairs, build an object of any size
Define the axes and a two-axis rotation
Switch objects with partner and draw the isometric view before and after rotation without moving the object – it’s frozen!
Compare drawings with partner and repeat
Station 4: Computer-Aided Visualization
Read Module 8 – Rotations about Two Axes
Complete the rotations about two axes exercises
Leave this slide up so students know what to accomplish at each station

32. Inclined Planes & Curved Surfaces
Workshop 5
Inclined Planes & Curved Surfaces
Spatial Visualization lesson 5/workshop 5
Note: The slides are animated so a mouse or keyboard click brings up the next image, text or slide.

33. Inclined Planes & Curved Surfaces
For this 3D object with an L-shaped inclined plane, follow the L-shape:
Appears as a line in the front view (bottom left corner)
Appears as a plane in the top and side views

34. Workshop 5 Stations Station 3: Orthographic Telephone
Build an object with an inclined plane or curved surface
Pass object to neighbor; the neighbor draws orthographic views of object and disassembles it
Second student passes orthographic drawings to different neighbor who rebuilds the object
Continue in circle until object arrives at originating student
Did final version match original object?
Station 4: Computer-Aided Visualization
Read Module 5 - Inclined Planes and Curved Surfaces
Complete the inclined planes and curved surfaces exercises
Station 1: Block Relay
Use cubes to build a shape with an inclined plane or curved shape
Use Play-Doh to add inclined and curved shapes to the blocks
Draw the isometric view
Pass object to a neighbor and repeat (draw the isometric view of your neighbor’s object)
Check both drawings for accuracy with neighbor
Station 2: Workbook Drill
Individually, answer questions on ODD pages inc1 to inc15
Check your answers with the answer key
Leave this slide up so students know what to accomplish at each station.

35. Reflections & Symmetry
Workshop 6
Reflections & Symmetry
Spatial Visualization lesson 6/workshop 6
Note: The slides are animated so a mouse or keyboard click brings up the next image, text or slide.

36. Object Reflections Thoughts:
The plane acts as a mirror to create new orientation of object (a reflection)
Use corresponding points to determine where edges/surfaces are reflected

37. Drawing a Reflection Tips:
First, project the object onto the reflection plane
Draw the edges from the projection
Edges and surfaces that were close are now the farthest away

38. Symmetry
Thoughts:
Symmetrical objects have one or more planes that create identical shapes
A 180-degree rotation about the plane of symmetry creates the same shape

39. Workshop 6 Stations Station 1: Block Relay Station 2: Workbook Drill
Build an object with 6 blocks and define a reflection plane
Draw isometric views of the object before and after reflection
Pass object and reflection plane to neighbor and repeat
Check drawings for accuracy with neighbor
Station 2: Workbook Drill
Individually, answer questions on EVEN pages reflx/sym04 to reflx/sym14
Check your answers as you go
Station 3: Meet with the Expert
Bring your questions to the expert(ish)
Station 4: Computer-Aided Visualization
Read Module 9 – Object Reflections on Symmetry
Complete the reflections and symmetry exercises
Leave this slide up so students know what to accomplish at each station

40. Write a Rule Workshop 7 Spatial Visualization lesson 7/workshop 7
Note: The slides are animated so a mouse or keyboard click brings up the next image, text or slide.

41. Purdue Spatial Visualization Test: Rotations (PSVT:R)
Example question:
You have taken the Purdue Spatial Visualization Test: Rotations, the hardest test. You were given 30 minutes to answer 30 questions that look like this.
The format is similar to SAT questions, but with shapes instead of words.
What do you think is the answer? (Answer is D)
Today we are going to explore a test-taking technique called “write a rule” that is helpful for answering these sorts of spatial visualization problems, on tests and in the real world.
A B C D E

42. Write a Rule Steps Look at the example and pick a side
Find the same side after its rotation
Write a rule!
Locate the same side on the new object
Follow your rule
Compare to the provided answers and find the correct one

43. Workshop 7 Stations Station 1: Write a Rule Practice
Individually, complete the first 8 problems on the practice test
For each problem, write a rule and then select the answer, writing it on scrap paper
Check your answers with the answer key
Station 2: Workbook Drill
Individually, answer questions on EVEN pages rot2-1 to rot2 -17 on scrap paper
Check your answers with the answer key
Leave this slide up so students know what to accomplish at each station