Puzzling Pentominoes - University of Mississippi

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
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• 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: www.wikipedia.org/wiki/Pentominoes.
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.
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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
http://www.scholastic.com/titles/chasingvermeer/. 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.
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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 10.
12. 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.
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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 30-33 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.
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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:
http://www.scholastic.com/titles/chasingvermeer/. 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.
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Sources:
Balliett, B. 2004. Chasing Vermeer. New York, NY: Scholastic Inc.
Pentomino, Wikipedia. Accessed March, 2006.
List of Artworks: Paintings, Wikipedia. Accessed March, 2006.
Purple Petunias. Accessed March, 2006.
Isabel Art Gallery. 1995-2006. Young Girl on Floor
Reading. Accessed March, 2006.
Harley Hann. 2006. Understanding Abstract Art. Accessed March, 2006.
WebMuseum, Paris. 19 Sept 2002. Rembrandt. Accessed March, 2006.
ArtCom Museum Tour. 1995-2005. Phillips Collection. Accessed
March, 2006.
Oil Painting from Canvaz. 1999-2003. The Church at
Varengeville. Accessed March, 2006.
Prepared by:
Emlee Nicholson
NSF NMGK-8
University of Mississippi
March 2006
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Pentomino Code
1 2 3
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
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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:______________________________________
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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.
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Images of Famous Paintings
Young Girl Reading on the Floor
Degas
The Church
at
Varengeville
Monet
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Purple Petunias
Georgia O’Keeffe
Night Watch
Rembrandt
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Luncheon of
the Boating
Party
Renoir
Irises
Vincent van
Gogh
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5x5 GRID
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5x8 GRID
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