Summary
Students learn about twoaxis rotations, and specifically how to rotate objects both physically and mentally about two axes. A twoaxis rotation is a rotation of an object about a combination of x, y or zaxes, as opposed to a singleaxis rotation, which is about a single x, y or zaxis. Students practice drawing twoaxis rotations through an exercise using simple cube blocks to create shapes, and then drawing on triangledot paper the shapes from various x, y and zaxis rotation perspectives. They use the righthand rule to explore the rotations of objects. A worksheet is provided. This activity is part of a multiactivity series towards improving spatial visualization skills. At activity end, students retake the 12question quiz they took in the associated lesson (before conducting four associated activities) to measure how their spatial visualizations skills improved.Engineering Connection
Rotating objects is a spatial visualization technique that enables engineers to visualize complicated assemblies in mechanisms and other systems in a fields such as physics, chemistry, mathematics and engineering. Spatial visualization is an essential and learnable skill that engineers use to clearly communicate their ideas to other people so the ideas can ultimately be turned into realworld products, structures and systems.
Many engineering applications require the depiction of objects from multiple viewpoints, such as mechanical drawings used to manufacture components, architectural drawings, and depictions of chemical compounds. In order to produce these views, an object must be rotated across multiple axes, called twoaxis rotations. This skill is similar to lifting up an object and looking at it from all sides, which is not always possible since some objects/components may be too large, heavy or inaccessible for physical manipulation. Instead, engineers must be able to visualize what various viewpoints look like without physically manipulating objects. Typically, engineers use computeraided design (CAD) software to help visualize complicated rotations of objects. This activity helps students develop twoaxis rotation skills.
PreReq Knowledge
Before taking part in this spatial visualization activity, students should have taken the Spatial Visualization Practice Quiz and learned about spatial visualization in the associated lesson, Let’s Learn about Spatial Viz! They should know about isometric drawing, how to use triangledot paper and coded plans (as can be learned in the associated activity, Connect the Dots: Isometric Drawing and Coded Plans), orthographic views (as can be learned in the associated activity, Seeing All Sides: Orthographic Views), and have been introduced to rotations (as can be learned in the associated activity, Let’s Take a Spin: OneAxis Rotations).
Learning Objectives
After this activity, students should be able to:
 Rotate objects about two axes.
 Translate the rotation of one block to a second block.
 Use the righthand rule to explore rotations of objects.
More Curriculum Like This
Students learn about oneaxis rotations, and specifically how to rotate objects both physically and mentally to understand the concept. They practice drawing oneaxis rotations through a group exercise using cube blocks to create shapes and then drawing those shapes from various x, y and zaxis ro...
In this lesson, students are introduced to the concept of spatial visualization and measure their spatial visualization skills by taking the provided 12question quiz. Following the lesson, students complete the four associated spatial visualization activities and then retake the quiz to see how mu...
Students learn about isometric drawings and practice sketching on triangledot paper the shapes they make using multiple simple cubes. They also learn how to use coded plans to envision objects and draw them on triangledot paper.
Students learn how to create twodimensional representations of threedimensional objects by utilizing orthographic projection techniques. They build shapes using cube blocks and then draw orthographic and isometric views of those shapes—which are the side views, such as top, front, right—with no de...
Educational Standards
Each TeachEngineering lesson or activity is correlated to one or more K12 science,
technology, engineering or math (STEM) educational standards.
All 100,000+ K12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN),
a project of D2L (www.achievementstandards.org).
In the ASN, standards are hierarchically structured: first by source; e.g., by state; within source by type; e.g., science or mathematics;
within type by subtype, then by grade, etc.
Each TeachEngineering lesson or activity is correlated to one or more K12 science, technology, engineering or math (STEM) educational standards.
All 100,000+ K12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN), a project of D2L (www.achievementstandards.org).
In the ASN, standards are hierarchically structured: first by source; e.g., by state; within source by type; e.g., science or mathematics; within type by subtype, then by grade, etc.
NGSS: Next Generation Science Standards  Science

Evaluate competing design solutions using a systematic process to determine how well they meet the criteria and constraints of the problem.
(Grades 6  8)
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This standard focuses on the following Three Dimensional Learning aspects of NGSS:Science & Engineering Practices Disciplinary Core Ideas Crosscutting Concepts Evaluate competing design solutions based on jointly developed and agreedupon design criteria. There are systematic processes for evaluating solutions with respect to how well they meet the criteria and constraints of a problem.
Common Core State Standards  Math

Draw, construct, and describe geometrical figures and describe the relationships between them.
(Grade 7)
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International Technology and Engineering Educators Association  Technology

Students will develop abilities to apply the design process.
(Grades K  12)
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State Standards
Colorado  Math

Modeling geometric figures and relationships leads to informal spatial reasoning and proof.
(Grade
7)
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Objects in the real world can be modeled using geometric concepts.
(Grades
9 
12)
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Materials List
Each group needs:
 8 snap cubes (interlocking cubes); a set of 100 for $1013 at https://www.amazon.com/LearningResourcesLER4285MathlinkCubes100/dp/B000URL296 or https://www.amazon.com/LearningResourcesLER7584SnapCubes/dp/B000G3LR9Y
 pencil with eraser, for each student
 Blank TriangleDot Paper, two sheets per student
 Rotations Practice Worksheet, one per student
 Spatial Visualization Practice Quiz, one per student
To share with the entire class:
 (optional) computer and projector to show examples as provided in the Spatial Visualization Rotations Presentation, a PowerPoint® file; alternatively, draw the examples for students
Introduction/Motivation
(Have the slide presentation up and displayed to the class, starting with slide 26.) Today we are going to explore rotations about two axes. You experience twoaxis rotations throughout your everyday lives. How many times do you rotate a USB stick before getting it to correctly insert into a computer? How many different ways can you put on a tshirt? But first, let’s quickly review oneaxis rotations. Objects can be rotated about the three axes: x, y and z and any combination of the three.
(Show students slide 27; same as Figure 1.) Do you see that the first block on the top left is the same block as on the top right? We are simply looking at it from another angle. How would you describe this rotation? Take a look at this figure and determine what you think the answer is. Now explain to a partner what you think is the correct answer and why. What are some methods you used to determine how the block was rotated? How would you define the rotation in terms of a positive or negative rotation about the x, y and/or z axes? (Answer: Negative rotation about the yaxis, negative rotation about the zaxis.)
Now, let’s do a quick recap of the righthand rule (refer to slide 24 as a review from the previous associated activity). It works as follows: 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 axes shown in slide 23 (same as Figure 2) are the positive axes, and if you flip each axis 180°, you get the negative axes.
Vocabulary/Definitions
righthand rule : A useful memory tool in the rotation of objects that uses a person’s right hand and fingers to help in understanding orientation conventions for vectors in three dimensions. Often used in physics and math.
spatial visualization: The ability to mentally manipulate two and threedimensional objects. It is typically measured with cognitive tests and is a predictor of success in STEM fields. Also referred to as visualspatial ability.
triangledot paper: A grid of dots arranged equidistant from one another. Used in making isometric sketches. Also called isometric paper.
Procedure
Before the Activity
 Gather materials and make copies of the Blank TriangleDot Paper, Rotations Practice Worksheet and Spatial Visualization Practice Quiz.
 Prepare to project the Spatial Visualization Presentation, a PowerPoint® file, and use its content to aid in your instruction, as makes sense for your class. Slides 2629 support this activity, with review slides 2324 from the previous associated activity. The slides are animated so a mouse or keyboard click brings up the next image, text or slide.
With the Students
 Present to the class the Introduction/Motivation content. Also ask the preassessment question, as described in the Assessment section.
 Hand out to each student the cubes and triangledot paper.
 Have each student build a threeblocklong rectangular object.
 Have students draw the Cartesian coordinate system on triangledot paper.
 Direct students to draw the object before and after a positive x rotation followed by a positive y rotation. (Expect each student to finish with three consecutive drawings. As necessary, show students slides 2729 and their tips: Use the righthand rule; clockwise = negative rotation; counterclockwise = positive rotation; twoaxis rotation is not commutative [order matters!], and the “write a rule” approach to take note of their own logic and methods.)
 Share and compare students’ drawings with the class.
 Divide the class into student pairs.
 Have students complete the BlocknSwap Relay as follows:
 Have students sit with their partners.
 Have each student use the cubes to build an object and then proceed to define a rotation about two axes.
 Have each student draw the original isometric view of the object and then draw the object after the defined rotation.
 After a few minutes, have students pass each object and defined rotation to their partners.
 Then the partners draw the rotated views of the objects.
 Have partners compare the isometric drawings with each other.
 Discuss as a class, as described in the Assessment section.
 Assign students to complete the worksheet. Observe and assist as necessary.
 Readminister the Spatial Visualization Practice Quiz, as described in the Assessment section.
Worksheets and Attachments
Assessment
PreActivity Assessment
Question/Answer: Ask students: Why are twoaxis rotations are important to engineers? Why would a biomedical engineers designing a new heart valve need to see it from different views? Why is it important to see these different views? (Point to make: Our 3D world is difficult to represent on 2D screens and paper. The ability to rotate an object around in one’s mind helps complex, reallife challenges be understood more clearly. It is important for engineers to be able to visualize 3D objects in order to make design decisions that will work effectively in the 3D world in which our designs, products and inventions must operate.)
Activity Embedded Assessment
BlocknSwap: During the BlocknSwap Relay, observe students to make sure they are able to draw the rotated objects. If they are struggling, assist them as necessary.
Worksheet: After students complete the BlocknSwap Relay exercise, assign them to complete the Rotations Practice Worksheet. Observe whether students are able to draw the rotated objects or if they are struggling. Assist them as necessary. Review their answers to gauge their depth of understanding.
PostActivity Assessment
Discussion: Ask students to explain and describe their drawings with specific focus on twoaxis rotational views. What strategies did they use to draw their cube shapes? What were the limitations they experienced, if any? How did students solve any drawing challenges? Since everyone has worked through the same exercises, group sharing of their challenges and approaches informs the teacher of students’ depth of understanding and provides their peers with relevant ideas and tips.
Spatial Visualization PostQuiz: To conclude the series of four spatial visualization practice activities, readminister the Spatial Visualization Practice Quiz, which students took as a preassessment during the associated lesson. Now that they have practiced their spatial visualization skills throughout four activities, show them how much they have improved! A “passing score” is any score over eight points (out of 12 possible). However, it is most important to emphasize individual student improvement gains.
Activity Extensions
To further challenge students, direct them to draw the rotation views of various objects without looking at the blocks.
Activity Scaling
 For lower grades, provide students with a longer time on the BlocknSwap Relay. Also, spend more time introducing the concept by having the entire class make the same object, define a rotation and draw the rotation isometrically. Then, go over the correct answer with students. This provides lessadvanced students with more time to fully practice and grasp the topic before branching off in pairs.
 For higher grades, have students use more blocks to make more complicated objects, and during the BlocknSwap Relay, do not permit students to use the blocks to visually aid in their rotation drawings. Adjust the time allotment as needed.
Additional Multimedia Support
As an additional or alternative test, you may want to purchase for $25 a license to use a scanned digital version of the Purdue Spatial Visualization Test: Rotations from Educational Testing Service at http://store.digitalriver.com/store/ets/DisplayProductDetailsPage/productID.39353200.
Contributors
Emily C. Gill; Jacob Segil; Emily BreidtCopyright
© 2011 by Regents of the University of ColoradoSupporting Program
Engineering Plus Degree Program, University of Colorado BoulderAcknowledgements
This activity was developed by the Engineering Plus degree program in the College of Engineering and Applied Science at the University of Colorado Boulder.
This lesson plan and its associated activities were derived from a summer workshop taught by Jacob Segil for undergraduate engineers at the University of Colorado Boulder. The activities have been adapted to suit the skill level of middle school students, with suggestions on how to adapt activities to elementary or, in some instances, high school level.
Last modified: January 12, 2019
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