Rubik’sCubeUni tStudy:Teacher’sGuidelghttp.60951.nexcesscdn.net/80487FB/Downloads/resources/...Ernö Rubik was born on July 13, 1944 in Budapest, Hungary to his parents, Erno Rubik,

©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. www.youcandothecube.com

Rubik’s Cube Unit Study: Teacher’s Guide Table of Contents

Page # Days Introduction 2 Lessons The Man, The Cube, Its Impact 3-‐4 2-‐3

Background article (Ernö Rubik) 5-‐6 Guiding Questions handouts 7-‐10

Classifying Polygons 11 1 Classifying Polygons answer key 12-‐13

Classifying 3-‐dimensional Shapes 14-‐15 2-‐3 The Third Dimension answer key 16 Classifying Polyhedrons answer key 17 Writing Rules answer key 18-‐19

Platonic Solids 20-‐21 1-‐2

Testing Net Variations 22 1 Testing Net Variations answer key 23

Mix & Map & Match 24-‐25 1-‐2 Templated Nets handouts 26-‐29

Solving the Rubik’s® Cube 30 5-‐10

How To Videos 31-‐32 2-‐3

Writing Algorithms-‐ Intro to Speed Solving 33-‐34 2-‐3 Intro to Speed Solving answer key 35 Writing Inverse Algorithms answer key 36 Writing Mirrored Algorithms answer key 37 Exploring Adv. Speed Solving Algorithms answer key 38

Creating Rubik’s Art 39-‐40 6-‐10

Making a 2x2 Rubik’s Cube 41 1 Making a 2x2 Rubik’s Cube answer key 42 Making a 2x2 Solution Guide answer key 43


Introduction

This unit study is designed for students in grades 5-8. The lessons have been tested in classroom settings as well as during out of school time programming. You do not have to do all of the lessons in the unit, or in the order they are presented. Feel free to choose the activities that are right for your class and rearrange the order based on your students’ preferences.

Some benefits of using Rubik’s® Cubes in the classroom: • Helps build mathematics skills in disciplines such as STEM, Geometry,

Algebra, and General Math concepts• Enables students to develop a more positive attitude towards math• Promotes 21st Century Skills such as problem solving, critical thinking,

perseverance and logical thinking• Supports STEM content and teaching using an authentic learning

experience• Builds student confidence

Teachers and youth leaders can borrow sets of 12, 24, or 36 Rubik’s Cubes FREE through the You CAN Do the Rubik’s Cube Lending Library. For six weeks, your students can enjoy the fun of learning STEM and 21st Century skills at no cost other than return shipping. Materials are also available to purchase at a discounted cost for educational use.

Request to borrow a class set of Rubik's Cubes www.youcandothecube.com/lending-library

2

©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. www.youcandothecube.com 3

The Man, The Cube, Its Impact Common Core: Determine central ideas or themes of a text and analyze their

development; summarize the key supporting details and ideas. (CCRA.R.2)

Integrate and evaluate content presented in diverse media and formats, including visually and quantitatively, as well as in words. (CCRA.SL.2)

Present information, findings, and supportive evidence such that listeners can follow the line of reasoning and the organization, development, and style are appropriate to task, purpose, and audience. (CCRA.SL.4)

Objectives: 1) Students will learn about the history of the Rubik’s® Cube throughresearch (and presentations).

2) Students will practice collecting and organizing information.

3) Students will prepare and share a presentation for the class.

Materials: Guiding Questions worksheets (found in Teacher’s Edition) Presentations Notes worksheet (found in Student Workbook) Computers/devices with internet access (for research) Art supplies* (poster paper, markers, tape, etc.) Projector* Speakers* *depending on presentation expectations

Procedure: 1) Explain your presentation expectations (duration, number of facts,type of display, etc.).

2) Break the class up into groups of 2-‐3, and assign each group a topic:Who is Ernö Rubik? How did the Rubik’s® Cube come to be? What impact has the Rubik’s Cube had over the years? What else? (fun & random facts about the Rubik’s Cube)

Hand each group the appropriate Guiding Questions worksheet.

3) Groups work on collecting, organizing, and displaying information.(This could take one or more class periods.)


4) Groups take turns sharing their presentations with the class in order bytopic (man, origin, impact, then fun facts). Students will take notes onthe Presentation Notes worksheet during classmates’ presentations.

After all of the presentations, give the students some time to answer the questions at the bottom of the worksheet.

5) If not already shown in a presentation, show the class a five-‐minuteclip from a Time interview with Ernö Rubik found on YouTube at:https://goo.gl/jhe9BV orhttps://www.youtube.com/watch?v=0poQ8q8RzSg

Notes to Teacher: If you have more than four groups, topics may be assigned to multiple groups.

Also, check your school’s internet filter. I have to get mine temporarily altered so that sites aren’t blocked when students are researching.

An article about Ernö Rubik is included to give you some background information.

Ernö Rubik:

Inventor of the

Rubik’s® Cube

By Diana Gettman Flores

www.youcandothecube.com

Ernö Rubik was born on July 13, 1944 in Budapest, Hungary to his parents,

Erno Rubik, Sr., an aeronautical engineer who designed gliders and light aircraft,

and Magdolna Szántó, a poet. Rubik studied sculpture at the Technical

University in Budapest and then architecture at the Academy of Applied Arts and

Design, also in Budapest. Rubik was a professor at the Academy when he

invented the Rubik’s Cube in 1974.

Rubik’s first design was made of 27 wooden blocks. It took Rubik about six

weeks to design a mechanism that would allow the rows to rotate, thus

rearranging the smaller cubes, but reforming a large cube. Once he had a

working cube, Rubik spent no less than a month figuring out a solution method

never writing anything down, just working through the solution in his head.

As a teacher, Rubik was always looking for new ways to present

information to his students. He used his cube invention to explain spatial

relationships as well as algebraic group theory. Rubik also considered his

invention to be a work of art, a mobile sculpture that may look very simple at

first, but is, in fact, rather complex. Rubik marveled at the fact that throughout

its many transformations in colors and patterns, the cube remains a single unit.

The first cubes were made and sold in Hungary as “Magic Cubes.” When

Ideal Toy Company began the sale of the cube in the United States in 1980, the

name was changed to the Rubik’s® Cube, which is thought to be one of the first

toys named after its inventor.

©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved.

www.youcandothecube.com 5

The popularity of the Rubik’s Cube quickly spread and the first

international speedcubing competition was held in 1982. Today hundreds of

speedcubing competitions are held regularly around the world. As of June 2016,

Lucas Etter, a 14yearold from Kentucky, holds the world record for the fastest

single solve of a 3x3 Rubik’s Cube. Etter solved the puzzle in 4.904 seconds in

November of 2015.

The Rubik’s Cube holds the record as the world’s bestselling puzzle/ toy

with over 350 million units sold. It is estimated that 1 in 5 people worldwide has

held a Rubik’s Cube. There have been over 50 books published describing how to

solve the puzzle, as well as humorous books poking fun at cube solvers and the

previous cube solving books.

The Rubik’s Cube is back on track to being as popular today as it was in

the early 1980s when it was first available in the United States. Part of the

resurgence in popularity is credited to today’s Internet culture and the

availability of solution guides and techniques for solving that can be learned

from streaming videos online, as well as teachers that are bringing Rubik’s Cubes

into their classrooms to empower today’s youth to learn to solve the Cube, and

incorporating the powerful tool into math and science lessons.

References:

"Erno Rubik". Encyclopædia Britannica. Encyclopædia Britannica Online. Encyclopædia Britannica Inc., 2016.

Web. 17 Jun. 2016

"Erno Rubik Biography." The Famous People website. The Famous People, n.d. Web. 17 June 2016.

Fisher, Dave. "Erno Rubik and the Invention of the Rubik's Cube." About.com Home. N.p., 3 Nov. 2015. Web. 6

June 2016.

Lynch, Kevin. "Confirmed: Teenager Lucas Etter Sets New Fastest Time to Solve a Rubik's Cube World Record."

Guinness World Records. Guinness World Records, 24 Nov. 2015. Web. 6 June 2016.

Slocum, Jerry. The Cube: The Ultimate Guide to the World's Bestselling Puzzle: Secrets, Stories, Solutions. New

York: Black Dog & Leventhal, 2009. Print.

"The History of the Rubik’s Cube." Rubiks. Rubiks.com, n.d. Web. 17 June 2016.


www.youcandothecube.com 6


Guiding Questions

Topic: Who is Ernö Rubik?

Partners:

Questions: Where was Ernö born?

Where did he grow up?

What was he like as a child?

What did he like as a child?

Where was he educated?

What kind of work did he do?

What was his family like?

What is he doing now?

Presentation Plans: Who is going to say what?

What is going to be displayed? And how?


Guiding Questions

Topic: How did the Rubik’s® Cube come to be?

Partners:

Questions: When was the Rubik’s Cube made?

Where was it made?

How was it made?

Why was it made?

Where did Ernö get the idea?

How long did it take to make?

What did it look like?




Guiding Questions

Topic: What impact has the Rubik’s® Cube had over the years?

Partners:

Questions: How, and when, did the Rubik’s Cube make it around the world?

What Rubik’s brand toys came after?

How has the Rubik’s Cube inspired the world of art?

What other twisty puzzles have been made?

How, and when, did speed solving become a sport?

What is the World Cubing Association?




Guiding Questions

Topic: What else? (fun facts about the Rubik’s® Cube)

Partners:

Questions: How many Rubik’s Cubes have been sold?

Where is the Rubik’s museum, and what is in it?

What is the most valuable Rubik’s Cube?

How many ways can a Rubik’s Cube be scrambled? How long would it take to see each?

What is the world record for solving the Rubik’s Cube?

What size is the smallest Rubik’s Cube? The largest?

What percent of the population owns a Rubik’s Cube? What percent can solve one?

What are some other talented things that have been done with a Rubik’s Cube?




Classifying Polygons Common Core: Understand that attributes belonging to a category of two-‐dimensional

figures also belong to all subcategories of that category. (5.G.B.3)

Classify two-‐dimensional figures in a hierarchy based on properties. (5.G.B.4)

Objectives: 1) Students will learn vocabulary related to polygons.

2) Students will use that vocabulary to classify polygons.

Materials: Classifying Polygons worksheet (found in Student Workbook) Internet access (for looking up definitions)

Procedure: 1) Students search the Internet for the definitions and record them onthe Classifying Polygons worksheet.

2) Students share and compare their definitions since they may findalternative definitions.

3) Introduce or review prefixes and suffixes.

4) Students fill in the Prefixes section, and share answers.

5) Students classify the polygons found on page 2 of the worksheet.

6) With time remaining, have students explore some extension questions:*Can a polygon be regular and concave? Show or explain your reasoning.*Can a triangle be concave? Show or explain your reasoning.*Could we simplify the definition of regular to just…

All sides congruent? or All angles congruent? Show or explain your reasoning.

*Can you construct a pentagon with 5 congruent angles but is notconsidered regular? Show or explain your reasoning.*Can you construct a pentagon with 5 congruent sides but is notconsidered regular? Show or explain your reasoning.

Notes to Teacher: I have my students search for these answers and definitions online, however I am sure that some math textbook glossaries may be a good alternative resource.


Classifying Polygons

Find the following definitions.

Two Dimensional –

Line Segment –

Polygon –

Congruent –

Regular polygon –

Irregular polygon –

Convex polygon –

Concave polygon –

Give the prefixes for the following numbers. Ex: Decade means 10 years, and century means 100 years.

What does the suffix –gon mean?

Having two dimensions, length and width

A line bound by two endpoints

A two dimensional shape made up of three or more line segments

Having the same shape and size

A polygon where all sides are equal and all angles are equal

Not regular; at least one side or angle is of different measure

A polygon with no reflex angles (all interior angles are less than 180)

A polygon with at least one reflex angle

Tri Quadri Penta Hexa

Hepta Octa Nona

Hendeca Dodeca Icosa

A shape having a specific number of angles


Fill in the blanks using the shapes above. shape # concave/convex regular/irregular # of sides name

ex) convex irregular 3 triangle

1)

2)

3)

4)

5)

6)

7)

8)

9)

10)

5 1

4

3

2

8 7 10

9 6

Convex Irregular 4 Quadrilateral

Convex Regular 3 Triangle

Concave Irregular 12 Dodecagon

Convex Regular 5 Pentagon

Concave Irregular 4 Quadrilateral

Convex Irregular 7 Heptagon

Concave Irregular 9 Nonagon

Convex Regular 11 Hendecagon

Convex Regular 8 Octagon

Concave Irregular 6 Hexagon


Classifying 3-Dimensional Shapes Common Core: Write, read, and evaluate expressions in which letters stand for numbers.

(6.EE.A.2)

Use variables to represent numbers and write expressions when solving a real-‐world or mathematical problem. (6.EE.B.6)

Objectives: 1) Students will learn vocabulary related to space figures.

2) Students will be able to name 3-‐dimensional objects.

3) Students will be able to identify parts of 3-‐dimensional objects.

4) Students will write algebraic expressions that help find the amount offaces, edges, or vertices of a particular polyhedron.

Materials: The Third Dimension worksheet (found in Student Workbook) Internet access (for looking up definitions) Classifying Polyhedrons worksheet (found in Student Workbook) Notecards labeled 1-‐8 (for numbering 8 different stations) 1 Rubik’s® Cube 7 Shape blocks (preferably: a regular tetrahedron, rectangular pyramid,

triangular prism, hexagonal prism, regular octahedron, regular dodecahedron, and regular icosahedron)

Writing Rules worksheet (found in Student Workbook)

Procedure: 1) Students fill out The Third Dimension worksheet by searching onlinefor definitions.

2) While students are working, set up 8 stations around the classroom byplacing one numbered notecard and one shape at each location (theRubik’s Cube counts as one of the 8 shapes).

3) When done with the worksheet, students share and compare thedefinitions they found, and discuss the similarities and differences oftheir findings.

4) As students get out their Classifying Polyhedrons worksheet, explain tothem that they will be visiting each station in an attempt to name eachshape (using two words) and determine how many faces, edges, andvertices it is comprised of.


5) Spread the students out at the different stations and let them startfilling out their worksheet, writing down their answers for each station inthe corresponding numbered row.

6) Students compare their answers and compile what they think theanswer key is; and then compare to the actual key.

7) Students work on their Writing Rules worksheet. They will focus onspecific groups of polyhedrons when counting faces, edges, and verticesto discover shortcuts in the counting process. Then they will translatethose shortcuts into algebraic expressions.

Notes to Teacher: Depending on the length of your class, good breaks would be after procedure 5, or after procedures 3 and 6.

My students tend to struggle with naming 3-‐dimensional shapes while visiting the stations. I encourage them to make educated guesses based on the definitions that we have been studying.

You will need to create your own answer key for the Classifying Polyhedrons worksheet, due to the fact that you may use different shapes, more or less shapes, or put shapes at different stations.

During the Writing Rules worksheet, give students access to as many of the shape blocks as you can. It gives them a hands-‐on resource while they are trying to develop hypotheses and test theories.


The Third Dimension

Define the following words.

Space figure –

Polyhedron –

Parts of a Polyhedron: Face –

Edge –

Vertex –

Special Polyhedrons

Prism -‐ Platonic Solid -‐ Pyramid -‐ Others:

Concave, Truncated, etc...

Name the shape that is both Name the shape that is both a prism and a platonic solid: a pyramid and a platonic solid:

A 3-dimensional shape that has depth in addition to

length and width

A 3-dimensional shape made up of polygons (no curves)

A flat surface of a 3-dimensional shape

A line segment that connects two faces

A point where the edges meet

A 3D shape with two congruent parallel bases connected by rectangles.

A 3D shape where all faces are congruent, all faces are regular, and all vertices have the same number of edges

A 3D shape with a polygon base connected to triangles that all extend to the same point

Cube (Regular Hexahedron) (Rectangular Prism)

Regular Tetrahedron (Triangular Pyramid)


Classifying Polyhedrons Visit each of the shape stations. At each station, attem

pt to correctly nam

e each space figure (using two w

ords). Then docum

ent the number of faces, edges, and vertices of each shape.

Leave the last column blank. It w

ill be used later in the lesson.


Writing Rules

Prisms 1) Count the faces, edges, and vertices of the following prisms.

Name n faces vertices edges triangular prism 3 rectangular prism 4 pentagonal prism 5

*n represents the number of sides ofone of the bases

2) Find the patterns in the table above and use it to make a prediction.

Name n faces vertices edges hexagonal prism 6

3) Draw a hexagonal prism and check your prediction.

4) Write function rules that describe the shortcuts in finding the number of faces, edges, andvertices of a prism. (vertices is already completed as an example)

Faces: F = n + 2

Edges: E = 3n or E = 3·∙n

Vertices: V = 2n or V = 2·∙n

Pyramids 1) Count the faces, edges, and vertices of the following pyramids.

Name n faces vertices edges triangular pyramid 3 rectangular pyramid 4 pentagonal pyramid 5

*n represents the number of sides ofthe base

5 6 9 6 8 12 7 10 15

Take n and: +2 x2 x3

8 12 18

4 4 6 5 5 8 6 6 10

Take n and: +1 +1 x2


2) Find the patterns in the previous table and use it to make a prediction.

Name n faces vertices edges hexagonal pyramid 6

3) Draw a hexagonal pyramid and check your prediction.

4) Write function rules that describe the shortcuts in finding the number of faces, edges, andvertices of a pyramid.

Faces: F = n + 1

Edges: E = 2n

Vertices: V = n + 1

Platonic Solids

What shortcuts could be used to count the faces, edges, and vertices of these regular polyhedrons? Write a description and try it.

Euler’s Formula

Go back to the Classifying Polyhedrons worksheet. Label the blank column “F + V”, and in each row add the number of faces to the number of vertices and record that in the new column. When you get done with that, compare column “F + V” with column “E”, and write down a function rule describing the pattern.

7 7 12

For the dodecahedron on the right, there are 12 faces. Each face is a pentagon, which has 5 edges (sides). 5x12=60; however an edge connects two faces, so we are counting each edge twice. 60/2=30; so there are 30 edges. Each face also has 5 vertices (angles). 5x12=60; however a vertex connects three edges of three different faces. 60/3=20; so there are 20 vertices.

F + V = E + 2 this formula works for every polyhedron


Platonic Solids Common Core: The Platonic Solids are not specifically found in the Common Core,

however this activity is a beneficial precursor to some of the standards.

Describe the two-‐dimensional figures that result from slicing three-‐dimensional figures. (7.G.A.3)

Represent three-‐dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-‐world and mathematical problems. (6.G.A.4)

Objectives: 1) Students will learn more about Platonic Solids.

2) Students will use nets to create the Platonic Solids.

Materials: Computer, internet, and projector (for watching a YouTube video) Platonic nets (found in Student Workbooks) Scissors Tape Crayons/markers/colored pencils String and hangers (optional)

Procedure: 1) Have the students watch parts 1 (length 8:33) & 2 (length 9:15) of thePlatonic Solids video.

Part 1: https://www.youtube.com/watch?v=voUVDAgFtho Part 2: https://www.youtube.com/watch?v=BsaOP5NMcCM

2) Introduce Nets as flat (2D) shapes that can be folded to create a 3Dshapes. Two examples that you could show them are the nets of acylinder, and a square-‐based pyramid.

4) Students can decorate their nets by drawing pictures or designs onthem. (Platonic nets are in the student workbooks.)


5) Students cut out the nets, fold them, and tape them shut.

6) Optional: Students can use the string to connect their Platonic Solids toa hanger, creating a mobile.

Notes to Teacher: The platonic nets in the student workbook can also be found in Wikimedia Commons:

Tetrahedron: https://commons.wikimedia.org/wiki/File%3AFoldable_tetrahedron_(blank).jpg Hexahedron: https://commons.wikimedia.org/wiki/File%3AFoldable_hexahedron_(blank).jpg Octahedron: https://commons.wikimedia.org/wiki/File%3AFoldable_octahedron_(blank).jpg Dodecahedron: https://commons.wikimedia.org/wiki/File%3AFoldable_dodecahedron_(blank).jpg Icosahedron: https://commons.wikimedia.org/wiki/File%3AFoldable_icosahedron_(blank).jpg


Testing Net Variations Common Core: Represent three-‐dimensional figures using nets made up of

rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-‐world and mathematical problems. (6.G.A.4)

Understand that a two-‐dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations. (8.G.A.2)

Objectives: 1) Students will learn that a 3D shape may have multiple unique nets.

2) Students will use a Rubik’s® Cube to test if a potential formation of 6squares is a cubic net, also known as a hexomino net.

3) Students will compile a list of all the unique cubic nets.

Materials: Solved Rubik’s Cube (1 per student) Testing Net Variations worksheet (found in Student Workbook) Square cut-‐outs sheet (found in Student Workbook) Scissors Tape (optional) Paper for notes (optional)

Procedure: 1) Prepare 6 squares (cut out) prior to class for demonstration purposes,and tape or magnets for sticking them on the front board.

2) Student follow along on their Testing Net Variations worksheet as youdemonstrate how the two example problems are done.

3) Students cut out the 6 squares from their workbooks, then exploretheir own 6-‐square formations, testing them and compiling theirdiscoveries.

4) Save the last 5-‐10 minutes to let students share their lists of uniquenets and formations that didn’t work.

Notes to Teacher: Students can roll pieces of tape into circles and place them on the backs of their squares. That way, when they make a formation on their desktops, the squares will not slide around while rolling the Rubik’s Cube over them. As for taking notes, I have my student compile their notes on the backs of their Testing Net Variations worksheets.


Here are the 11 possible unique net variations of the cube:


Mix & Map & MatchCommon Core: Understand that a two-‐dimensional figure is congruent to another if the

second can be obtained from the first by a sequence of rotations, reflections, and translations. (8.G.A.2)

Represent three-‐dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-‐world and mathematical problems. (6.G.A.4)

Objectives: 1) Students will map 3-‐dimensional shapes onto 2-‐dimensional nets.

2) Students will be able to read and rebuild nets to match to3-‐dimensional shapes.

Materials: Templated Nets; multiple copies of each (found in Teacher’s Edition) Mix & Map & Match worksheet (found in Student Workbook) Crayons/markers/colored pencils Scissors Rubik’s® Cubes (1 per student) Tape/magnets/clips (for steps 3 & 10)

Procedure: 1) Prior to the lesson, make multiple copies of the templatedhexomino/cubic nets. Each student will need 2 nets.

2) As students are getting out their Rubik’s Cube and Mix & Map & Matchworksheet, hand each student one of the four templated nets (dispersethem randomly).

3) Help the class work through the steps on their worksheet. At step 3,you will collect up all of the nets and tape, magnet, or clip them up for allto see. Then students will randomly trade their Rubik’s Cubes. Makesure to express the importance of not twisting or turning any of thecubes.

4) Students should continue along with the steps outlined in theirinstructions. As students are working on step 5, hand each of themanother templated net (different from the one they currently have).

5) For step 7, you could either have a class discussion, or have thestudents write down their responses.


6) After step 9, take the note of any student unable to match theirRubik’s Cube back to their two nets. You will need to group themdifferently for steps 13-‐17.

7) At step 10, you will need to collect all of the nets, shuffle (randomize)them, label them 1,2,3,…, and then display them around the room.Students can set aside their Rubik’s Cube until step 13. Students will findand write down all of the pairs of nets.

8) At stage 13, any student that was unable to rematch after step 9 willwait until all others are paired up, and will then join a group of two. Oneof the other two partners (with their matching nets) will cut up both oftheir nets in step 15 to trade.

9) Students can use the same rolling method done in the Testing NetVariations activity when checking their answers in step 17.

Notes to Teacher: For larger class sizes, you could divide the room into halves, and for steps 3 and 10 display the two groups’ nets separately.

When displaying the nets at step 10, you will want to spread the nets out enough for all students to be able to view, but avoid spreading them out too much since students are trying to identify matches.

26

27

28

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Solving the Rubik’s® Cube Common Core: Mathematical Practice Standards:

1. Make sense of problems and persevere in solving them.2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the reasoning of others.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated reasoning.

Objectives: 1) Students will reference previous knowledge to understand the Rubik’sCube better.2) Students will learn how to solve the Rubik’s® Cube.

Materials: Introductory Information sheets (found in Student Workbook: Meeting the Cube, Labeling the Cube) Solution Guide sheets (found in Student Workbook) “What’s Next?” worksheet (found in Student Workbook) Rubik’s Cubes (1 per student) Stopwatches (optional) Internet access (optional)

Procedure: 1) Read through the Introductory Information sheets as a class, or insmall groups. (Meeting the Cube & Labeling the Cube)

2) Have students start working through the Solution Guide with theirRubik’s Cube. Feel free to read through some of these pages as a class.That way if questions come up, students may be able to learn from otherstudents. *Working through the Solution Guide will take multiple days,and some students will progress much faster than others.

3) When students succeed in solving the Rubik’s Cube, have them try thepatterns on the “What’s Next?” worksheet.

Notes to Teacher: There are many solutions to solving the Rubik’s Cube. A popular ‘beginner’s method’ can also be found in the You CAN Do the Rubik’s Cube Solution Guide and online at http://www.youcandothecube.com/secret-‐unlocked/ You can also request printed Solution Guides for your class by writing to [email protected]

30


How To Videos Common Core: Present information, findings, and supporting evidence such that

listeners can follow the line of reasoning and the organization, development, and style are appropriate to task, purpose, and audience. (CCRA.SL.4)

Make strategic use of digital media and visual displays of data to express information and enhance understanding of presentations. (CCRA.SL.5)

Objectives: 1) Students will solidify their skills of solving a Rubik’s® Cube by teachingothers.

2) Students will gain experience using technology by creating and editinga how to video.

Materials: Rubik’s Cubes (possibly 1 per student) Recording devices (iPads, smartphones, computers, etc.) How To Videos worksheet (found in Student Workbook)

Procedure: 1) Organize the class up into groups of two and assign each group a stageof the solution:

1. Meeting the Cube2. Reading Algorithms3. Layer 1 – Making a Cross (Plus Sign)4. Layer 1 – Permuting the Cross5. Layer 1 – Solving the Corners6. Solving Layer 27. Layer 3 – Making a Cross8. Layer 3 – Permuting the Cross9. Layer 3 – Permuting the Corners10. Layer 3 – Orienting the Corners

If you don’t have enough students to make 10 groups you could have some work individually, you could assign some groups two stages (pair up 1 & 2, pair up 3 & 4), or you could omit stages 1 & 2. If you have more than 10 groups, assign some stages twice.

2) Groups should review the steps and algorithms needed to completetheir assigned stage. Then they should develop a script using their HowTo Videos worksheets.

31


3) Have groups check in after they have completed their scripts, and afterthey record take 1 of their video, so that you may give feedback.

4) If there is time after all the videos are complete, you could have aviewing party and let the class watch them all in order.

Notes to Teacher: Constructive feedback about their progress is very beneficial. If the videos turn out well enough, I save them and use them as a resource for my next group of students.

My students used iMovie. One great feature was that students were able insert text on top of the recording, which allowed students to have the algorithms displayed on the screen during their video.

The video lengths of each individual stage seem to range from 1.5-‐3 minutes.


Writing Algorithms- Intro to Speed Solving Common Core: Solve word problems leading to equations of the

form px + q= r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. (7.EE.B.4.A)

Describe the effect of dilations, translations, rotations, and reflections on two-‐dimensional figures using coordinates. (8.G.A.3)

Objectives: 1) Students will learn about the cycles of algorithms.

2) Students will learn to rewrite algorithms by applying the Algebraic ideaof inverse operations.

3) Students will learn to rewrite algorithms by applying the Geometricidea of mirror/reflective operations.

Materials: Rubik’s® Cubes (1 per student) Access to mirrors (optional) Intro to Speed Solving worksheet (found in Student Workbook) Writing Inverse Algorithms worksheet (found in Student Workbook) Writing Mirrored Algorithms worksheet (found in Student Workbook) Exploring Advanced Speed Solving Algorithms worksheet (found in

Student Workbook) Internet access (for students)

Procedure: 1) Students should start with a solved Rubik’s Cube and their Intro toSpeed Solving worksheet.

2) Read through part 1 with the class. Have students test examples 1-‐2 toverify those algorithms’ cycles. *Sadly, if a student makes a mistakeduring a cycle, they will have to resolve their cube and start over.

3) In part 2, students are going to determine the cycle values of 3 morealgorithms.

4) In part 3, students will make up their own random algorithm and testfor its cycle value. *Some cycles values can be rather high, so make surestudents stick with a 3-‐5 turn algorithm.


5) Have students get out their Writing Inverse Algorithms worksheet andread through the intro together and walk through the example together.

6) Students will then write three inverse algorithms, test them, anddescribe their benefits when solving the Rubik’s Cube.

7) When you get to the Writing Mirrored Algorithms worksheet, you caneither have all the students pair up and do the mirror simulation, or youcan take two volunteers and have them do the simulation in front of theclass while the rest do the observations.

8) After going through the intro with the class, have the students writethe three algorithms. This is where the optional mirrors could come in. Ihave had students perform the algorithms for #1-‐3 while observing themoves of their reflection. It is challenging, but is a really good way forstudents to experience reflections.

9) Students should then test their newly written algorithms and describetheir benefits when solving the Rubik’s Cube.

10) With the Exploring Advanced Speed Solving Algorithms worksheet,have students start by looking up the answers to #1-‐2 online.

11) Read through the new notations together. Have students get outtheir Rubik’s Cube, solve layers 1 & 2, and then explore and attemptusing OLL and PLL.

Notes to Teacher: You will want to verify that your school’s internet does not block the websites needed for this activity.

Examples of mirrored algorithms can be clearly seen in the steps used to solve the middle layer of the Rubik’s Cube. The directions for moving a piece to the right edge mirror those that move a piece to the left edge.


Intro to Speed Solving

Part 1: The algorithms that we use have cycles. This means that if you do the same algorithm over and over again, the puzzle will eventually go back to the state it started in prior to the repeated steps.

Examples: Layer 3 – Permute the Cross (U R U R’ U R U2 R’ U) has a cycle of 3.

Layer 1 – Orient the Cross (R U’ R’ U R) has a cycle of 12.

Layer 2 algorithm U R U’ R’ U’ F’ U F has a cycle of 15.

Starting with a solved Rubik’s Cube, test examples 1 & 2 by repeating each given algorithm the specified number of times. When you are done , the puzzle should be returned to the solved state.

Part 2: Determine the cycle values of the following algorithms:

1) Layer 1 – Solving the Corners (R’ D’ R D) has a cycle of ______.

2) Layer 3 – Orienting the Cross (F R U R’ U’ F’) has a cycle of ______.

3) Layer 3 – Permute the Corners (U R U’ L’ U R’ U’ L) has a cycle of ______.

Part 3: Make up your own algorithm consisting of 3-‐5 turns. The algorithm can be completely random; it does not need to contribute to solving the cube. Once you have made your algorithm, test it to determine its cycle value.

6

6

3


www.youcandothecube.com

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Writing Inverse Algorithms

Inverse operations can be seen in writing “return” directions off of a map, or solving an algebraic equation (as seen below).

Inverse operations have us “undo” everything that has been done. In other words, inverse operations make us do the opposite of each step AND in the reverse order.

Here is an example (this is not one of the learned algorithms): Original algorithm: L, U’, R’, U

Inverse algorithm: U’, R, U, L’

1) Layer 1/3 – Solving the Corners: R’ D’ R DWrite the inverse algorithm:

2) Layer 3 – Permute the Cross: U R U R’ U R U2 R’ UWrite the inverse algorithm:

3) Layer 3 – Permute the Corners: U R U’ L’ U R’ U’ LWrite the inverse algorithm:

4) When done, compare answers with a neighbor. Then test out your new algorithms.5) When will these algorithms be beneficial?

Looking at the equation 5 = #$%&'

Following the order of operations Following inverse operations Start at x End at what x equals 1) multiply by 3 3) divide by 32) add 2 2) subtract 23) divide by 4 1) multiply by 4Get an answer of 5 Start at the answer 5

D’, R’, D, R

U’, R, U2, R’, U’, R, U’, R’, U’

L’, U, R, U’, L, U, R’, U’

These algorithms reverse the order of a cycle. For example, if I wastrying to solve the corner piece shown, I would R’ D’ R D once.However, if that corner piece was in the same place, but white tilewas on the front face, that same algorithm would need to be used 5times (however its inverse would only be needed once).


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Writing Mirrored Algorithms

Mirrored (or reflective) operations can be seen by observing movements in a mirror. Let’s compile some observations by simulating mirrors. Find a partner and stand facing each other. Indicate who will be the “model” and who will be the mirror. The model will perform a couple of movements, and the partner will act as if they were the reflection in the mirror.

Movements by model Movements by mirror Wave with left hand What hand is Mirror waving with? Pat your head w/right hand Is Mirror patting head? With what hand? Put hands behind your back Where are Mirror’s hands? Slowly rotate right arm clockwise Which direction is Mirror’s arm rotating? Turn body right, rotating 90° clockwise Which way did Mirror rotate? Lower your chin, then raise it When did Mirror’s chin go down? Up?

What we hopefully noticed, was that with mirrored operations, up is still up, down is still down, front is still front, and back is still back. We may have also noticed that right becomes left, left becomes right, clockwise becomes counterclockwise, and counterclockwise becomes clockwise.

Here is an example: Original algorithm: L, U’, R’, U

Mirrored algorithm: R’, U, L, U’

1) Layer 1/3 – Solving the Corners: R’ D’ R DWrite the mirrored algorithm:

2) Layer 3 – Permute the Cross: U R U R’ U R U2 R’ UWrite the mirrored algorithm:

3) Layer 3 – Permute the Corners: U R U’ L’ U R’ U’ LWrite the mirrored algorithm:

4) When done, compare answers with a neighbor. Then test out your new algorithms.5) When will these algorithms be beneficial?

L, D, L’, D’

U’, L’, U’, L, U’, L’, U2, L, U’

U’, L’, U, R, U’, L, U, R’

Again, these algorithms reverse the order of the cycles, but with a coupleof difference: First, pieces that you would line up on the right prior to solving will now need to be lined up on the left (and vice versa); and second, algorithms that were originally right-hand dependent will now be left-hand dependent (and vice versa).


Exploring Advanced Speed Solving Algorithms

1) What does acronym OLL stand for? What does OLL mean? (search the internet)

2) What does acronym PLL stand for? What does PLL mean? (search the internet)

In order to use them, we will need more abbreviated notations. Here are the notations we have used and some of the new letters/sets that may come up in more complex algorithms:

• F (front) – the side facing toward, as viewed by the solver• B (back) – the side that is opposite the front, as viewed by the solver• L (left) – the side to the left of the front, as viewed by the solver• R (right) – the side to the right of the front, as viewed by the solver• U (up) -‐ the side on top, as viewed by the solver• D (down) – the side on bottom, as viewed by the solver

*The six letters above assume a 90° clockwise rotation.• 2 (two) – turn the given face twice• ‘ (apostrophe) – turn counterclockwise• f (front two faces)• b (back two faces)• l (left two faces)• r (right two faces)• u (upper two faces)• d (downward two faces)• x (rotate entire cube) – D will become F• y (rotate entire cube) – R will become F• z (rotate entire cube) – U will become R

• Solve layers 1 and 2 of a scrambled Rubik’s Cube, but not the last layer. Then go to oneof the following sites (top preferred):

• http://www.cubezone.be/oll.html• https://ruwix.com/the-‐rubiks-‐cube/rubiks-‐cube-‐solution-‐with-‐advanced-‐

friedrich-‐method-‐tutorial/orient-‐the-‐last-‐layer-‐oll/• https://www.speedsolving.com/wiki/index.php/OLL

Match up your mixed-‐up third layer to the same scenario from the site’s list. Then follow the provided algorithm to orient the last layer.

3) When successful in part 3, go to one of the following sites (top preferred):• http://www.cubezone.be/pll.html• https://ruwix.com/the-‐rubiks-‐cube/rubiks-‐cube-‐solution-‐with-‐advanced-‐

friedrich-‐method-‐tutorial/permutate-‐the-‐last-‐layer-‐pll/• https://www.speedsolving.com/wiki/index.php/PLL

Match your puzzle to the same scenario on the site and follow the corresponding algorithm.

x

y

z

Orient Last Layer: to get all remaining pieces turned the right way

Permute Last Layer: to get all remaining pieces moved to correct locations


Creating Rubik’s Art Common Core: Without matching this to specific standards, this block of activities

involves creativity, collaboration, cooperation, computer skills, photo editing, blueprinting, and pattern recognition.

Objectives: 1) Students will be able to replicate color configurations on one face of aRubik’s® Cube.

2) Students will design their own 81-‐pixel picture/pattern, and thenreplicate it using 9 Rubik’s Cubes.

3) Students will gain exposure to photo editing, and will use twoparticular programs to create larger-‐scaled Rubik’s mosaics.

Materials: Crayons/markers/colored pencils Scissors Rubik’s Cubes (more cubes means larger mosaics and more detail)

*can be borrowed through YouCanDoTheCube.comPractice & Pixelate worksheet (found in Student Workbook) Building a Mini-‐mosaic worksheet (found in Student Workbook) Designing a Rubik’s Mosaic using Gimp worksheet (found in Student

Workbook) Designing a Rubik’s Mosaic using Twist the Web worksheet (found in

Student Workbook) Computers, with Gimp 2 software installed (free software) Computers, with Internet access (specifically Google Chrome) Printer (optional)

Procedure: 1) With the Practice & Pixelate worksheet and a Rubik’s Cube, havestudents complete tasks #1-‐2.

2) As a class, discuss responses to #2 and share strategies.

3) Have students get some coloring utensils (only yellow, blue, orange,red, and green) and draw a picture or pattern for #3. I recommendhaving extra copies of this worksheet ready in case any students makemistakes, want to start over, or want to draw a second picture/pattern.

4) Students transfer their picture/pattern onto the Building a Mini-‐Mosaic worksheet. Then they cut out the 9 3x3 squares.


5) When a number of students are ready, group them together, give thegroup 9 Rubik’s Cubes, and have them help each other build their mini-‐mosaics. *Your group sizes depend on access to Rubik’s Cubes. Take thenumber of Rubik’s Cubes you have and divide that by 9. That is howmany groups you can have.

6) Students will log onto a computer that contains Gimp 2. They will thenfollow the instructions on the Designing a Rubik’s Mosaic using Gimpworksheet. *If computers are limited, pair students up and have them doit together. *Depending on time and classroom management, you maywant to set expectations on step #1, “getting a picture”.

7) Now that students have experienced the photo editing needed tocreate a Rubik’s mosaic, they will use a web-‐based program that does alittle more of the work for them. They will need a computer with internetaccess through Google Chrome and the Designing a Rubik’s Mosaic usingTwist the Web worksheet. Students will follow the instructions on theworksheet to create a Rubik’s mosaic. *For #9, you may want to instructstudents to save their work instead of printing, because printing uses alot of paper.

8) Show the class some of the art made using Twist the Web. Havestudents vote on one, print the blueprint for the mosaic, and have theclass work together on building it. *Art designed on Gimp 2 can be used,but it won’t print in a nicely blueprinted format like Twist the Web.

Notes to Teacher: Some of my students tend to spend too much time looking for (or taking) pictures to use when making mosaics with Gimp and Twist the Web. Again, setting expectations about how long they have to take or find a photo (and even what the photos can contain) will help keep students on task.

Students will find out that some pictures transform into Rubik’s mosaics better than others. Pictures with a lot of colors, lots of details, and small main objects tend to transfer poorly. I will, sometimes, steer students to cartoon or clip-‐art pictures.

Gimp 2 software can be downloaded at: https://www.gimp.org/downloads/


Making a 2x2 Rubik’s Cube Common Core: Solve real-‐world and mathematical problems involving area, volume and

surface area of two-‐ and three-‐dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. (7.G.B.6)

Objectives: 1) Students will learn how a 3x3 Rubik’s Cube can be modified to make a2x2 Rubik’s Cube.

2) Students will create a solution guide for solving a 2x2 Rubik’s Cube bymodifying their solution guide for a 3x3 Rubik’s Cube.

Materials: Crayons/marker/colored pencils Scissors Rubik’s Cubes (1 per student) Tape Making a 2x2 Rubik’s Cube worksheet (found in Student Workbook) Cut-‐outs for a 2x2 Rubik’s Cube worksheet (found in Student Workbook) Making a 2x2 Solution Guide worksheet (found in Student Workbook)

Procedure: 1) Have students gather needed materials: Making a 2x2 Rubik’s Cubeworksheet, a Rubik’s Cube, coloring utensils, scissors, tape, and the cut-‐outs for a 2x2 Rubik’s Cube worksheet.

2) Students will then follow the directions on the Making a 2x2 Rubik’sCube worksheet.

3) When students finish, have them share their responses to #5 (ordiscuss as a class).

4) Have students complete the Making a 2x2 Solution Guide worksheet,and use it to solve their 2x2.

Notes to Teacher: Relating this way back to “The Man, The Cube, Its Impact”, I speculate to my students that this is why the 3x3 Rubik’s Cube was constructed/completed first, because the easiest way to make a 2x2, is by modifying a 3x3.


Making a 2x2 Rubik’s® Cube You are going to make a 2x2 Rubik’s Cube.

1) Cut out 24 squares (found on the Cut-‐Outs page). Color them: 4 yellow, 4 blue, 4 orange, 4 red, and 4 green (leaving 4 white). The squares should measure about 1” x 1”.

2) Tape the squares onto the corner pieces of the corresponding sides.

*Do not do any taping on the edge pieces.

3) You now have a 2x2 Rubik’s Cube. Turn it slowly to begin, as some pieces of paper may catch. If so, curl those edges upward.

4) Mix it up and then solve it. Have some tape nearby just in case some of your squares come

loose.

5) What are the similarities and differences between solving the 3x3 and the 2x2? Similarities:

Differences:

The two main similarities are: Solving the first layer corners Solving the last layer corners

Some of the big differences are: There are no center pieces to indicate the color the side should be There is no layer one cross There is no layer 2 There is no layer 3 cross

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Making a 2x2 Solution Guide

Let’s make a solution guide for the 2x2 Rubik’s Cube. U

se our 3x3 guide, and cross out any step that applies to an edge piece. W

e now have a guide for

solving the 2x2 Rubik’s Cube. Try it out.

Documents

Rubik’sCubeUni tStudy:Teacher’sGuidelghttp.60951.nexcesscdn.net/80487FB/Downloads/resources/...Ernö Rubik was born on July 13, 1944 in Budapest, Hungary to his parents, Erno Rubik,