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
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. www.youcandothecube.com 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
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. 2 www.youcandothecube.com 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 through research (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.)
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. 3 www.youcandothecube.com 4) Groups take turns sharing their presentations with the class in order by topic (man, origin, impact, then fun facts). Students will take notes on the 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-minute clip from a Time interview with Ernö Rubik found on YouTube at: https://goo.gl/jhe9BV or https://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.
4 ©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. www.youcandothecube.com 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.
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. 6 www.youcandothecube.com 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?
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. www.youcandothecube.com 7 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?
Presentation Plans: Who is going to say what?
What is going to be displayed? And how?
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. 8 www.youcandothecube.com 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?
Presentation Plans: Who is going to say what?
What is going to be displayed? And how?
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. www.youcandothecube.com 9 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?
Presentation Plans: Who is going to say what?
What is going to be displayed? And how?
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. 10 www.youcandothecube.com 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 on the Classifying Polygons worksheet.
2) Students share and compare their definitions since they may find alternative 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 not considered regular? Show or explain your reasoning. *Can you construct a pentagon with 5 congruent sides but is not considered 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.
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. www.youcandothecube.com 11 Classifying Polygons
Find the following definitions.
Two Dimensional – Having two dimensions, length and width
Line Segment – A line bound by two endpoints
Polygon – A two dimensional shape made up of three or more line segments
Congruent – Having the same shape and size
Regular polygon – A polygon where all sides are equal and all angles are equal
Irregular polygon – Not regular; at least one side or angle is of different measure
Convex polygon – A polygon with no reflex angles (all interior angles are less than 180)
Concave polygon – A polygon with at least one reflex angle
Give the prefixes for the following numbers. Ex: Decade means 10 years, and century means 100 years.
Tri Quadri Penta Hexa
Hepta Octa Nona
Hendeca Dodeca Icosa
What does the suffix –gon mean?
A shape having a specific number of angles
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. 12 www.youcandothecube.com 3 5 1 7 8 4 10 6 2 9
Fill in the blanks using the shapes above. shape # concave/convex regular/irregular # of sides name
ex) convex irregular 3 triangle
1) Convex Irregular 4 Quadrilateral
2) Convex Regular 3 Triangle
3) Concave Irregular 12 Dodecagon
4) Convex Regular 5 Pentagon
5) Concave Irregular 4 Quadrilateral
6) Convex Irregular 7 Heptagon
7) Concave Irregular 9 Nonagon
8) Convex Regular 11 Hendecagon
9) Convex Regular 8 Octagon
10) Concave Irregular 6 Hexagon
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. www.youcandothecube.com 13 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 of faces, 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 online for definitions.
2) While students are working, set up 8 stations around the classroom by placing one numbered notecard and one shape at each location (the Rubik’s Cube counts as one of the 8 shapes).
3) When done with the worksheet, students share and compare the definitions they found, and discuss the similarities and differences of their findings.
4) As students get out their Classifying Polyhedrons worksheet, explain to them that they will be visiting each station in an attempt to name each shape (using two words) and determine how many faces, edges, and vertices it is comprised of.
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. 14 www.youcandothecube.com 5) Spread the students out at the different stations and let them start filling out their worksheet, writing down their answers for each station in the corresponding numbered row.
6) Students compare their answers and compile what they think the answer key is; and then compare to the actual key.
7) Students work on their Writing Rules worksheet. They will focus on specific groups of polyhedrons when counting faces, edges, and vertices to discover shortcuts in the counting process. Then they will translate those 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.
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. www.youcandothecube.com 15 The Third Dimension
Define the following words.
Space figure – A 3-dimensional shape that has depth in addition to
length and width
Polyhedron – A 3-dimensional shape made up of polygons (no curves)
Parts of a Polyhedron: Face – A flat surface of a 3-dimensional shape
Edge – A line segment that connects two faces
Vertex – A point where the edges meet
Special Polyhedrons
Prism - Platonic Solid - Pyramid - Others:
A 3D shape with A 3D shape where all A 3D shape with a Concave, Truncated, two congruent faces are congruent, polygon base etc... parallel bases all faces are regular, connected to triangles connected by and all vertices have that all extend to the rectangles. the same number of same point
edges
Name the shape that is both Name the shape that is both a prism and a platonic solid: a pyramid and a platonic solid:
Cube Regular Tetrahedron (Regular Hexahedron) (Triangular Pyramid) (Rectangular Prism)
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. 16 www.youcandothecube.com Leave the last column blank. It will be used later in the lesson. document the number of faces, edges, and vertices of each shape. correctly name each space figure Visit each of the shape stations. At each station, attempt to Classifying Polyhedrons
(using two words). Then
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. www.youcandothecube.com 17 Writing Rules
Prisms 1) Count the faces, edges, and vertices of the following prisms.
Name n faces vertices edges triangular prism 3 5 6 9 rectangular prism 4 6 8 12 pentagonal prism 5 7 10 15 *n represents the number of sides of one of the bases
Take n and: +2 x2 x3
2) Find the patterns in the table above and use it to make a prediction.
Name n faces vertices edges hexagonal prism 6 8 12 18
3) Draw a hexagonal prism and check your prediction.
4) Write function rules that describe the shortcuts in finding the number of faces, edges, and vertices 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 4 4 6 rectangular pyramid 4 5 5 8 pentagonal pyramid 5 6 6 10 *n represents the number of sides of the base
Take n and: +1 +1 x2
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. 18 www.youcandothecube.com 2) Find the patterns in the previous table and use it to make a prediction.
Name n faces vertices edges hexagonal pyramid 6 7 7 12
3) Draw a hexagonal pyramid and check your prediction.
4) Write function rules that describe the shortcuts in finding the number of faces, edges, and vertices 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.
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.
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.
F + V = E + 2 this formula works for every polyhedron
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. www.youcandothecube.com 19 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 the Platonic 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 3D shapes. Two examples that you could show them are the nets of a cylinder, and a square-based pyramid.
4) Students can decorate their nets by drawing pictures or designs on them. (Platonic nets are in the student workbooks.)
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. 20 www.youcandothecube.com 5) Students cut out the nets, fold them, and tape them shut.
6) Optional: Students can use the string to connect their Platonic Solids to a 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
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. www.youcandothecube.com 21 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 6 squares 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 you demonstrate how the two example problems are done.
3) Students cut out the 6 squares from their workbooks, then explore their own 6-square formations, testing them and compiling their discoveries.
4) Save the last 5-10 minutes to let students share their lists of unique nets 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.
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. 22 www.youcandothecube.com Here are the 11 possible unique net variations of the cube:
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. www.youcandothecube.com 23 Mix & Map & Match
Common 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 to 3-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 templated hexomino/cubic nets. Each student will need 2 nets.
2) As students are getting out their Rubik’s Cube and Mix & Map & Match worksheet, hand each student one of the four templated nets (disperse them 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 all to see. Then students will randomly trade their Rubik’s Cubes. Make sure to express the importance of not twisting or turning any of the cubes.
4) Students should continue along with the steps outlined in their instructions. As students are working on step 5, hand each of them another templated net (different from the one they currently have).
5) For step 7, you could either have a class discussion, or have the students write down their responses.
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. 24 www.youcandothecube.com 6) After step 9, take the note of any student unable to match their Rubik’s Cube back to their two nets. You will need to group them differently 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 find and write down all of the pairs of nets.
8) At stage 13, any student that was unable to rematch after step 9 will wait until all others are paired up, and will then join a group of two. One of the other two partners (with their matching nets) will cut up both of their nets in step 15 to trade.
9) Students can use the same rolling method done in the Testing Net Variations 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.
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. www.youcandothecube.com 25 26 27 28 29 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’s Cube 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 in small groups. (Meeting the Cube & Labeling the Cube)
2) Have students start working through the Solution Guide with their Rubik’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 other students. *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 the patterns 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]
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. 30 www.youcandothecube.com 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 teaching others.
2) Students will gain experience using technology by creating and editing a 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 stage of the solution: 1. Meeting the Cube 2. Reading Algorithms 3. Layer 1 – Making a Cross (Plus Sign) 4. Layer 1 – Permuting the Cross 5. Layer 1 – Solving the Corners 6. Solving Layer 2 7. Layer 3 – Making a Cross 8. Layer 3 – Permuting the Cross 9. Layer 3 – Permuting the Corners 10. 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 complete their assigned stage. Then they should develop a script using their How To Videos worksheets.
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. www.youcandothecube.com 31 3) Have groups check in after they have completed their scripts, and after they 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 a viewing 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.
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. 32 www.youcandothecube.com 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 idea of inverse operations.
3) Students will learn to rewrite algorithms by applying the Geometric idea 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 to Speed Solving worksheet.
2) Read through part 1 with the class. Have students test examples 1-2 to verify those algorithms’ cycles. *Sadly, if a student makes a mistake during 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 more algorithms.
4) In part 3, students will make up their own random algorithm and test for its cycle value. *Some cycles values can be rather high, so make sure students stick with a 3-5 turn algorithm.
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. www.youcandothecube.com 33 5) Have students get out their Writing Inverse Algorithms worksheet and read through the intro together and walk through the example together.
6) Students will then write three inverse algorithms, test them, and describe their benefits when solving the Rubik’s Cube.
7) When you get to the Writing Mirrored Algorithms worksheet, you can either have all the students pair up and do the mirror simulation, or you can take two volunteers and have them do the simulation in front of the class while the rest do the observations.
8) After going through the intro with the class, have the students write the three algorithms. This is where the optional mirrors could come in. I have had students perform the algorithms for #1-3 while observing the moves of their reflection. It is challenging, but is a really good way for students to experience reflections.
9) Students should then test their newly written algorithms and describe their 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 out their Rubik’s Cube, solve layers 1 & 2, and then explore and attempt using 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.
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. 34 www.youcandothecube.com 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 ______.6
2) Layer 3 – Orienting the Cross (F R U R’ U’ F’) has a cycle of ______.6
3) Layer 3 – Permute the Corners (U R U’ L’ U R’ U’ L) has a cycle of ______.3
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.
©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. www.youcandothecube.com 35 Writing Inverse Algorithms
Inverse operations can be seen in writing “return” directions off of a map, or solving an algebraic equation (as seen below).