Puzzling Pentominoes

Transcription

1 Puzzling Pentominoes Intended for Grade: Fourth Subject: Math Description: This activity employs a set of mathematical tools called "pentominoes" to explore patterns and investigate geometric concepts. Objective: The student will be able to solve a simple pentomino puzzle and systematically gather information to solve a problem. Mississippi Frameworks addressed: Math Framework 1a: Recognize, describe, and extend a given pattern. Math Framework 1b: Analyze a given pattern and generate a similar pattern. Math Framework 2a: Construct two and three-dimensional geometric figures with concrete materials. National Standards addressed: Math Standard: Algebra Math Standard: Communication Math Standard: Geometry Materials: Plastic Pentominoes (enough for each student to have a set of 12) 1 [5 x 5] grid (each cell 1 x 1 ) per student 1 [5 x 8] grid (each cell 1 x 1 ) per student Pentomino Code Worksheet 6 Manila Envelopes Mystery Game Worksheet Mystery Game Chart 5 clipboards NSF North Mississippi GK-8 1

2 At least 10 sets of paper pentominoes (cut out and laminated) Images of Famous Paintings Background: Pentominoes are a set of twelve shapes. Each shape is composed of 5 squares of equal size (sometimes called unit squares). The squares within each shape are lined up in such a way that no two sides are partially adjacent, that is any two sides are either perfectly aligned or do not touch at all. Each shape is named after a letter of the alphabet. In this project the W will occasionally be called by the name M. The pentomino shapes and their associated letters are shown below. Image from: A typical pentomino puzzle board is a rectangular grid containing 60 squares. Possible dimensions for each grid are 6 x 10, 5 x 12, 4 x 15, or 3 x 20 unit squares. A solution to a pentomino puzzle is any arrangement of the twelve pentominoes on the grid so that each unit square is covered and no pentomino pieces overlap. It is also possible to solve a smaller puzzle using some subset of pentominoes and a smaller grid. For example a 5 x 5 grid can be solved using 5 of the pentominoes. A smaller puzzle is typically simpler to complete than its larger counterparts. NSF North Mississippi GK-8 2

3 Pentominoes play a major part in the plot of the book Chasing Vermeer by Blue Balliett. The following activities were designed for a group of students who read this book. However, the activities can be completed by students who have not read the book. Procedure: Before the Activity: 1. Print enough 5x5 grids, 5x8 grids, and Pentomino Code worksheets so that each student will have one. 2. Print 6 copies of the Mystery Game worksheet and Mystery Game Chart. 3. Develop or buy enough sets of pentominoes so that each student may use one. [To develop your own pentominoes, visit the Chasing Vermeer website From this website, you can print the pentominoes. Once they have been printed, cut them out and laminate them if possible.] 4. Develop 10 additional sets of pentominoes (one for use in the Mystery Game and the rest for the winners to take home.) 5. Divide a set of pentominoes into two groups: L, I, V, P, Y, N and U, T, X, M (or W), Z, F. 6. Randomly choose one pentomino from each group and place both pentominoes in a manila envelope and put the envelope in a secret location. 7. Randomly choose one of the remaining pentominoes from each group and place both into each of the remaining 5 manila envelopes. 8. Print out the images of famous paintings included at the end of the activity and post them somewhere in your classroom. NSF North Mississippi GK-8 3

4 The Activity: 9. Distribute a set of 12 pentominoes and a 5 x 5 grid to each student. 10. Instruct the students to arrange the pentominoes in such a way that none of the squares on the grid are visible and none of the pentominoes overlap. 11. If a student finds multiple solutions to the 5 x 5 grid, give that student a 5 x 8 grid and the same instructions as in Step Distribute a Pentomino Code Worksheet to each student. 13. Instruct the students to decode the secret message using the given pentomino code. 14. When a student finishes Step 13, they may use the code to write a note to a friend. 15. Divide the class into 5 teams. 16. Label the teams: Team Degas, Team Monet, Team O Keeffe, Team Renoir, and Team Van Gogh. 17. Explain to the students that a painting has been stolen from a dear friend of yours and you need their help figuring out which painting was stolen and who took it. 18. Give each team a Mystery Game Worksheet and clipboard. 19. Tell the groups that your friend owned six extremely valuable paintings which are listed on the worksheet. 20. Explain to them that you have narrowed down the possible suspects to the six people also listed on the worksheet. 21. Explain that each painting and suspect corresponds to a pentomino (the underlined letter). 22. Tell the class that there is a secret envelope in a secret location which contains one pentomino identifying the stolen painting and one pentomino identifying the thief. Their job is to determine which pentominoes are in that envelope. 23. Give each team one of the remaining five envelopes. 24. Tell the teams that their envelopes also contain pentominoes for a painting and suspect which were not involved in the burglary. NSF North Mississippi GK-8 4

5 25. Instruct them to make sure that no other team sees what is in their envelope. 26. Give each team a minute to inspect the contents of their envelope. 27. Tell them that they will need to question the other groups to find out what is in that group s envelope so that they can determine the correct painting and suspect involved in the burglary by the process of elimination. 28. Explain that each team will have a chance to ask the other teams specific questions about what is in the other teams envelopes to try and further narrow their suspect list. 29. Have the teams take turns beginning with Team Degas and going in alphabetical order thereafter. 30. Team Degas will choose one other team to question. They will ask that team if their envelope contains the pentomino for a particular painting AND suspect. 31. If the questioned team has one or both pentominoes they must secretly show ONLY one pentomino to ONLY Team Degas. 32. If the questioned team has neither pentomino Team Degas asked about, they must tell everyone that they have nothing to show Team Degas. 33. Teams should keep track of all the pentominoes they are shown on their Mystery Game worksheet, as it is no longer possible for these to be involved in the burglary. They should also keep track of what the other teams do not have to prevent them from repeating a question. Have the teams use the Mystery Game Chart as a guide. 34. Take turns having each team execute steps as Team Degas did. 35. Continue in this fashion until the teams have gathered sufficient information to make an educated guess as to the identity of the painting and suspect involved in the burglary. This should take 5 or 6 rounds of play. NSF North Mississippi GK-8 5

6 36. Stop play and have each team fill out the Conclusion section of their Mystery Game worksheet. They should list ONE person and ONE painting that they think were involved in the burglary. 37. Collect all of the worksheets. 38. Have the teams tell their guess to the class one at a time. 39. Reveal the contents of the secret envelope to show the actual stolen painting and the identity of the thief. 40. Each member of the group with the correct answer wins a prize (a set of pentominoes). Evaluation: Students successfully solve the 5 x 5 pentomino puzzle. More advanced students should be able to solve the 5 x 8 puzzle as well. Students correctly decode the secret message. Students work together to make an educated guess regarding a solution to the mystery. Extended Activities: Visit the Scholastic Inc. website dedicated to Chasing Vermeer: Here you will find printable pentominoes as well as interactive pentomino puzzles generated for three levels of difficulty. Have the students develop their own secret pentomino code that only their class knows. Assign them a secret pen pal so that they may exchange notes. Have each individual research and write a report on the artist after whom his or her team was named or on one of the possible stolen paintings. NSF North Mississippi GK-8 6

7 Sources: Balliett, B Chasing Vermeer. New York, NY: Scholastic Inc. Pentomino, Wikipedia. Accessed March, < List of Artworks: Paintings, Wikipedia. Accessed March, < Purple Petunias. Accessed March, < Isabel Art Gallery Young Girl on Floor Reading. Accessed March, < Harley Hann Understanding Abstract Art. Accessed March, < WebMuseum, Paris. 19 Sept Rembrandt. Accessed March, < ArtCom Museum Tour Phillips Collection. Accessed March, < Oil Painting from Canvaz The Church at Varengeville. Accessed March, < Prepared by: Emlee Nicholson NSF NMGK-8 University of Mississippi March 2006 NSF North Mississippi GK-8 7

8 Pentomino Code F A M Y M B N Z U C O P D P T E Q N F R L G S I H T X I U Y J V Z K W V L X Secret message: (shhhhhhhhh!) X:1 N:1 F:3 U:2 X:2 N:2 P:2 T:1 M:2 I:2 U:2 F:2 X:1 M:2 U:2 U:1 U:2 P:1 T:1 X:1 L:2 P:1 X:1 N:1 N:1 T:1 N:2 T:1 M:2 I:2 N:1 N:2 U:2 F:2 U:1 F:1 V:1 P:1 T:1 N:2 L:2, Z:2 X:1 V:1 V:1 F:3 U:2 X:2 N:1 T:1 T:1 V:1 V:1 X:1 Z:1 T:1 F:1 M:1 V:1 X:2 T:1 U:2 M:2 T:1? NSF North Mississippi GK-8 8

9 Team Members: Mystery Game Circle your Team: Team Degas, Team Monet, Team O Keeffe, Team Renoir, Team Van Gogh Suspects: M - Mother F - Father U - Uncle X - ex-classmate Z - Zookeeper T - Teacher Names of possible stolen paintings: I Irises by van Gogh V The Church at Varengeville by Monet Y Young Girl Reading on the Floor by Degas L Luncheon of the Boating Party by Renoir P Purple Petunias by O Keeffe N Night Watch by Rembrandt Conclusion: Use the information you have gathered to determine which painting you think was stolen and who you think stole it. Painting: Suspect: NSF North Mississippi GK-8 9

10 Mystery Game Chart: **USE A PENCIL** The Church at Young Girl Reading on Luncheon of the Boating Purple Night Mother Father Uncle ex-classmate Zookeeper Teacher Irises Varengeville the Floor Party Petunias Watch Team Degas Team Monet Team O'Keeffe Team Renoir Team Van Gogh Based on the responses of the other teams to yours and your classmates' questions, use the following key to help you. If a team definitely has a certain pentomino, place a in that square. If a team definitely does not have a certain pentomino, place an x in that square. If you are not sure but want to question that team further about a certain pentomino, place a? in that square. NSF North Mississippi GK-8 10

11 Images of Famous Paintings Young Girl Reading on the Floor Degas The Church at Varengeville Monet NSF North Mississippi GK-8 11

12 Purple Petunias Georgia O Keeffe Night Watch Rembrandt NSF North Mississippi GK-8 12

13 Luncheon of the Boating Party Renoir Irises Vincent van Gogh NSF North Mississippi GK-8 13

14 5x5 GRID NSF North Mississippi GK-8 14

15 5x8 GRID NSF North Mississippi GK-8 15