Puzzling Pentominoes - University of Mississippi
Puzzling Pentominoes - University of Mississippi
Puzzling Pentominoes - University of Mississippi
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<strong>Puzzling</strong> <strong>Pentominoes</strong><br />
Intended for Grade: Fourth<br />
Subject: Math<br />
Description: This activity employs a set <strong>of</strong> mathematical tools called<br />
"pentominoes" to explore patterns and investigate geometric concepts.<br />
Objective: The student will be able to solve a simple pentomino puzzle<br />
and systematically gather information to solve a problem.<br />
<strong>Mississippi</strong> Frameworks addressed:<br />
• Math Framework 1a: Recognize, describe, and extend a given pattern.<br />
• Math Framework 1b: Analyze a given pattern and generate a similar<br />
pattern.<br />
• Math Framework 2a: Construct two and three-dimensional geometric<br />
figures with concrete materials.<br />
National Standards addressed:<br />
• Math Standard: Algebra<br />
• Math Standard: Communication<br />
• Math Standard: Geometry<br />
Materials:<br />
• Plastic <strong>Pentominoes</strong> (enough for each student to have a set <strong>of</strong> 12)<br />
• 1 [5 x 5] grid (each cell 1” x 1”) per student<br />
• 1 [5 x 8] grid (each cell 1” x 1”) per student<br />
• Pentomino Code Worksheet<br />
• 6 Manila Envelopes<br />
• Mystery Game Worksheet<br />
• Mystery Game Chart<br />
• 5 clipboards<br />
NSF North <strong>Mississippi</strong> GK-8 1
• At least 10 sets <strong>of</strong> paper pentominoes (cut out and laminated)<br />
• Images <strong>of</strong> Famous Paintings<br />
Background:<br />
<strong>Pentominoes</strong> are a set <strong>of</strong> twelve shapes. Each shape is<br />
composed <strong>of</strong> 5 squares <strong>of</strong> equal size (sometimes called “unit” squares).<br />
The squares within each shape are lined up in such a way that no two<br />
sides are partially adjacent, that is any two sides are either perfectly<br />
aligned or do not touch at all. Each shape is named after a letter <strong>of</strong><br />
the alphabet. In this project the “W” will occasionally be called by<br />
the name “M”. The pentomino shapes and their associated letters are<br />
shown below.<br />
Image from: www.wikipedia.org/wiki/<strong>Pentominoes</strong>.<br />
A typical pentomino puzzle board is a rectangular grid<br />
containing 60 squares. Possible dimensions for each grid are 6<br />
x 10, 5 x 12, 4 x 15, or 3 x 20 unit squares. A solution to a<br />
pentomino puzzle is any arrangement <strong>of</strong> the twelve pentominoes<br />
on the grid so that each unit square is covered and no<br />
pentomino pieces overlap. It is also possible to “solve” a smaller<br />
puzzle using some subset <strong>of</strong> pentominoes and a smaller grid.<br />
For example a 5 x 5 grid can be solved using 5 <strong>of</strong> the<br />
pentominoes. A smaller puzzle is typically simpler to complete<br />
than its larger counterparts.<br />
NSF North <strong>Mississippi</strong> GK-8 2
<strong>Pentominoes</strong> play a major part in the plot <strong>of</strong> the book<br />
Chasing Vermeer by Blue Balliett. The following activities were<br />
designed for a group <strong>of</strong> students who read this book. However,<br />
the activities can be completed by students who have not read<br />
the book.<br />
Procedure:<br />
Before the Activity:<br />
1. Print enough 5x5 grids, 5x8 grids, and Pentomino Code<br />
worksheets so that each student will have one.<br />
2. Print 6 copies <strong>of</strong> the Mystery Game worksheet and Mystery<br />
Game Chart.<br />
3. Develop or buy enough sets <strong>of</strong> pentominoes so that each student<br />
may use one. [To develop your own pentominoes, visit the Chasing<br />
Vermeer website<br />
http://www.scholastic.com/titles/chasingvermeer/. From this<br />
website, you can print the pentominoes. Once they have been<br />
printed, cut them out and laminate them if possible.]<br />
4. Develop 10 additional sets <strong>of</strong> pentominoes (one for use in the<br />
Mystery Game and the rest for the winners to take home.)<br />
5. Divide a set <strong>of</strong> pentominoes into two groups: L, I, V, P, Y, N and<br />
U, T, X, M (or W), Z, F.<br />
6. Randomly choose one pentomino from each group and place both<br />
pentominoes in a manila envelope and put the envelope in a secret<br />
location.<br />
7. Randomly choose one <strong>of</strong> the remaining pentominoes from each<br />
group and place both into each <strong>of</strong> the remaining 5 manila<br />
envelopes.<br />
8. Print out the images <strong>of</strong> famous paintings included at the end <strong>of</strong><br />
the activity and post them somewhere in your classroom.<br />
NSF North <strong>Mississippi</strong> GK-8 3
The Activity:<br />
9. Distribute a set <strong>of</strong> 12 pentominoes and a 5 x 5 grid to each<br />
student.<br />
10. Instruct the students to arrange the pentominoes in such a way<br />
that none <strong>of</strong> the squares on the grid are visible and none <strong>of</strong> the<br />
pentominoes overlap.<br />
11. If a student finds multiple solutions to the 5 x 5 grid, give that<br />
student a 5 x 8 grid and the same instructions as in Step 10.<br />
12. Distribute a “Pentomino Code” Worksheet to each student.<br />
13. Instruct the students to decode the secret message using the<br />
given pentomino code.<br />
14. When a student finishes Step 13, they may use the code to write<br />
a note to a friend.<br />
15. Divide the class into 5 teams.<br />
16. Label the teams: Team Degas, Team Monet, Team O’Keeffe,<br />
Team Renoir, and Team Van Gogh.<br />
17. Explain to the students that a painting has been stolen from a<br />
dear friend <strong>of</strong> yours and you need their help figuring out which<br />
painting was stolen and who took it.<br />
18. Give each team a “Mystery Game” Worksheet and clipboard.<br />
19. Tell the groups that your friend owned six extremely valuable<br />
paintings which are listed on the worksheet.<br />
20. Explain to them that you have narrowed down the possible<br />
suspects to the six people also listed on the worksheet.<br />
21. Explain that each painting and suspect corresponds to a<br />
pentomino (the underlined letter).<br />
22. Tell the class that there is a secret envelope in a secret location<br />
which contains one pentomino identifying the stolen painting and<br />
one pentomino identifying the thief. Their job is to determine<br />
which pentominoes are in that envelope.<br />
23. Give each team one <strong>of</strong> the remaining five envelopes.<br />
24. Tell the teams that their envelopes also contain pentominoes for<br />
a painting and suspect which were not involved in the burglary.<br />
NSF North <strong>Mississippi</strong> GK-8 4
25. Instruct them to make sure that no other team sees what is in<br />
their envelope.<br />
26. Give each team a minute to inspect the contents <strong>of</strong> their<br />
envelope.<br />
27. Tell them that they will need to question the other groups to find<br />
out what is in that group’s envelope so that they can determine<br />
the correct painting and suspect involved in the burglary by the<br />
process <strong>of</strong> elimination.<br />
28. Explain that each team will have a chance to ask the other teams<br />
specific questions about what is in the other teams’ envelopes to<br />
try and further narrow their suspect list.<br />
29. Have the teams take turns beginning with Team Degas and going<br />
in alphabetical order thereafter.<br />
30. Team Degas will choose one other team to question. They will ask<br />
that team if their envelope contains the pentomino for a<br />
particular painting AND suspect.<br />
31. If the questioned team has one or both pentominoes they must<br />
secretly show ONLY one pentomino to ONLY Team Degas.<br />
32. If the questioned team has neither pentomino Team Degas asked<br />
about, they must tell everyone that they have nothing to show<br />
Team Degas.<br />
33. Teams should keep track <strong>of</strong> all the pentominoes they are shown<br />
on their Mystery Game worksheet, as it is no longer possible for<br />
these to be involved in the burglary. They should also keep track<br />
<strong>of</strong> what the other teams do not have to prevent them from<br />
repeating a question. Have the teams use the Mystery Game<br />
Chart as a guide.<br />
34. Take turns having each team execute steps 30-33 as Team Degas<br />
did.<br />
35. Continue in this fashion until the teams have gathered sufficient<br />
information to make an educated guess as to the identity <strong>of</strong> the<br />
painting and suspect involved in the burglary. This should take 5<br />
or 6 rounds <strong>of</strong> play.<br />
NSF North <strong>Mississippi</strong> GK-8 5
36. Stop play and have each team fill out the “Conclusion” section <strong>of</strong><br />
their Mystery Game worksheet. They should list ONE person<br />
and ONE painting that they think were involved in the burglary.<br />
37. Collect all <strong>of</strong> the worksheets.<br />
38. Have the teams tell their guess to the class one at a time.<br />
39. Reveal the contents <strong>of</strong> the secret envelope to show the actual<br />
stolen painting and the identity <strong>of</strong> the thief.<br />
40. Each member <strong>of</strong> the group with the correct answer wins a prize<br />
(a set <strong>of</strong> pentominoes).<br />
Evaluation:<br />
Students successfully solve the 5 x 5 pentomino puzzle. More advanced<br />
students should be able to solve the 5 x 8 puzzle as well.<br />
Students correctly decode the secret message.<br />
Students work together to make an educated guess regarding a solution to<br />
the mystery.<br />
Extended Activities:<br />
Visit the Scholastic Inc. website dedicated to Chasing Vermeer:<br />
http://www.scholastic.com/titles/chasingvermeer/. Here you will find<br />
“printable pentominoes” as well as interactive pentomino puzzles<br />
generated for three levels <strong>of</strong> difficulty.<br />
Have the students develop their own secret pentomino code that only<br />
their class knows. Assign them a secret pen pal so that they may<br />
exchange notes.<br />
Have each individual research and write a report on the artist after<br />
whom his or her team was named or on one <strong>of</strong> the possible stolen<br />
paintings.<br />
NSF North <strong>Mississippi</strong> GK-8 6
Sources:<br />
Balliett, B. 2004. Chasing Vermeer. New York, NY: Scholastic Inc.<br />
Pentomino, Wikipedia. Accessed March, 2006.<br />
<br />
List <strong>of</strong> Artworks: Paintings, Wikipedia. Accessed March, 2006.<br />
<br />
Purple Petunias. Accessed March, 2006.<br />
<br />
Isabel Art Gallery. 1995-2006. Young Girl on Floor<br />
Reading. Accessed March, 2006.<br />
<br />
Harley Hann. 2006. Understanding Abstract Art. Accessed March, 2006.<br />
<br />
WebMuseum, Paris. 19 Sept 2002. Rembrandt. Accessed March, 2006.<br />
<br />
ArtCom Museum Tour. 1995-2005. Phillips Collection. Accessed<br />
March, 2006.<br />
<br />
Oil Painting from Canvaz. 1999-2003. The Church at<br />
Varengeville. Accessed March, 2006.<br />
<br />
Prepared by:<br />
Emlee Nicholson<br />
NSF NMGK-8<br />
<strong>University</strong> <strong>of</strong> <strong>Mississippi</strong><br />
March 2006<br />
NSF North <strong>Mississippi</strong> GK-8 7
Pentomino Code<br />
1 2 3<br />
F A M Y<br />
M B N Z<br />
U C O<br />
P D P<br />
T E Q<br />
N F R<br />
L G S<br />
I H T<br />
X I U<br />
Y J V<br />
Z K W<br />
V L X<br />
Secret message: (shhhhhhhhh!)<br />
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<br />
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<br />
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<br />
F:3 U:2 X:2 N:1 T:1 T:1 V:1 V:1 X:1 Z:1 T:1 F:1<br />
M:1 V:1 X:2 T:1 U:2 M:2 T:1<br />
NSF North <strong>Mississippi</strong> GK-8 8
Team Members:_________________________________<br />
Mystery Game<br />
Circle your Team:<br />
Team Degas, Team Monet, Team O’Keeffe, Team Renoir, Team Van Gogh<br />
Suspects:<br />
M - Mother<br />
F - Father<br />
U - Uncle<br />
X - eX-classmate<br />
Z - Zookeeper<br />
T - Teacher<br />
Names <strong>of</strong> possible stolen paintings:<br />
I – Irises by van Gogh<br />
V – The Church at Varengeville by Monet<br />
Y – Young Girl Reading on the Floor by Degas<br />
L – Luncheon <strong>of</strong> the Boating Party by Renoir<br />
P – Purple Petunias by O’Keeffe<br />
N – Night Watch by Rembrandt<br />
Conclusion: Use the information you have gathered to determine<br />
which painting you think was stolen and who you think stole it.<br />
Painting:______________________________________<br />
Suspect:______________________________________<br />
NSF North <strong>Mississippi</strong> GK-8 9
Mystery Game Chart: **USE A PENCIL**<br />
The Church<br />
at<br />
Young Girl<br />
Reading on<br />
Luncheon <strong>of</strong><br />
the Boating<br />
Purple<br />
Night<br />
Mother Father Uncle eX-classmate Zookeeper Teacher<br />
Irises<br />
Varengeville<br />
the Floor<br />
Party<br />
Petunias<br />
Watch<br />
Team Degas<br />
Team Monet<br />
Team O'Keeffe<br />
Team Renoir<br />
Team Van Gogh<br />
Based on the responses <strong>of</strong> the other teams to yours and your classmates' questions, use the following key to help you.<br />
If a team definitely has a certain pentomino, place a in that square.<br />
If a team definitely does not have a certain pentomino, place an x in that square.<br />
If you are not sure but want to question that team further about a certain pentomino, place a in that square.<br />
NSF North <strong>Mississippi</strong> GK-8 10
Images <strong>of</strong> Famous Paintings<br />
Young Girl Reading on the Floor<br />
Degas<br />
The Church<br />
at<br />
Varengeville<br />
Monet<br />
NSF North <strong>Mississippi</strong> GK-8 11
Purple Petunias<br />
Georgia O’Keeffe<br />
Night Watch<br />
Rembrandt<br />
NSF North <strong>Mississippi</strong> GK-8 12
Luncheon <strong>of</strong><br />
the Boating<br />
Party<br />
Renoir<br />
Irises<br />
Vincent van<br />
Gogh<br />
NSF North <strong>Mississippi</strong> GK-8 13
5x5 GRID<br />
NSF North <strong>Mississippi</strong> GK-8 14
5x8 GRID<br />
NSF North <strong>Mississippi</strong> GK-8 15