TITLE:Puzzling with Pentominoes

CONTENT AREAS:  Mathematics, Art

GRADE LEVEL:  Can be adjusted for all grades K – 12, our lesson will focus on 4-5^th grade

MATERIALS NEEDED:  Scissors for each pair, graph paper as attached, coloring utensils (pencils, markers, crayons), glue

KEY CONCEPTS:  Geometry, Congruence, Line Symmetry, Area, Perimeter

EALR’s:

·         Math 1.2 - Understand concepts of perimeter, area and volume. 

·         Math 1.3 – Understand and apply concepts from geometric sense.

·         Math 1.4 – Understand and apply concepts from probability and statistics.

·         Math 1.5 – Understand and apply concepts from algebraic sense.

·         Math 2.1 - Investigate situations.

·         Math 2.3 – Construct solutions.

·         Math 3.3 – Draw conclusions and verify results.

·         Math 4.1 -   Gather information—read, listen, and observe to access and extract mathematical information.

·         Math 4.2 - Organize and interpret information.

·         Math 4.3 -   Represent and share information—express and explain mathematical ideas using language and notation in ways appropriate for audience and purposes.

·         Math 5.3 -   Relate mathematical concepts and procedures to real-life situations and understand the connections between mathematics and problem-solving skills used everyday at work and at home.

·         Communication 3.1 - Use language to interact effectively and responsibly with others.

·         Communication 3.2 -Work cooperatively as a member of a group.

·         Communication 3.3 -Seek agreement and solutions through discussion.

GOALS:  For the student to manipulate pentominoes, find a relationship between 2-D and 3-D, develop and discover spatial perception, develop problem solving, give them a better understanding of area and perimeter, group cooperation, and develop note taking skills.

OBJECTIVES: 

• The students will construct pentominoes with their own understanding of putting shapes and puzzles together. 
• The students will learn the basic history behind pentominoes
• The students will discover shapes, patterns and then construct a 3-D box.
• The students will understand that pentominoes have the same area.  All 12 pentominoes put together in different shapes will also have the same area.
• The students will be able to share their creation with their peers and communicate their findings.
• The students will take notes on key terms of geometry.

PROCEDURES:

• Arrange on tables sets of 5 squares at each chair, with chairs arranged in pairs.  Instruct students to sit at one of the “stations”.  The stations also include 2 pieces of graph paper, a puzzle grid, a 1-inch square grid, vocabulary sheet, glue, scissors, and markers.   Instruct them to play with the squares and find as many shapes as they can with these.
• Ask students to find shapes with the square sides touching each other. 
• After they have played around for a time, tell them to trace each shape they find onto the graph paper.  Color your shapes.  Then cut them out.  Discuss congruence.
• “What could these shapes be called?”  (Let students answer)  A man named Solomon W. Golomb “invented” pentominoes in 1953.  He came up with the name, but the puzzle was published in 1907 in the Canterbury Puzzles.
• “How many shapes did you come up with?”  “Do there seem to be any patterns?”   Have groups share the shapes that they found.  When one shape is found, display the corresponding “big” pentominoe on the board.  Do this until all 12 shapes are found. “Are there any more shapes?  Why aren’t there anymore?  How do you know?”
• “Which shapes can you make a box with? (3-D)” Let students explore this. And report out their answers.
• “With these 12 pentominoes, what would the area of each of them be?”  “What is the perimeter of the pentominoes?” They are do not have the same perimeter, but have the same area.
• “Can you make another shape with combining all 12 of your pentominoes?”
• You have in front of you, another graph sheet of paper.  This is a puzzle grid.  Try and put all your pentominoes into this shape. 
• What area would it have?  What would the perimeter be?
• Then have students share their shape with the class.
• If the students desire, glue their puzzle together.  Or if they want to make another shape, allow them to create another shape with their pentominoes.
• Analyze why certain people got the shapes they did. How about the area?  Is the area of all of these shapes the same?  Why would all the areas of these shapes be the same? 
• Discuss the beautiful collage of color that each shape made.
• How could this information be used in their lives today?

PENTOMINOE GAME:

"Besides its intrigue as a puzzle, the placement of pentominoes on a checkerboard also makes it an exciting competitive game of skill. Played by two or three players, the object of the game is to be the last player to place a pentominoe piece on the checkerboard. Players take turns choosing a piece and placing it on the board. The pieces must not overlap or extend beyond the boundary of the board, but they do not have to be adjacent. The game will last at least five, and at the most twelve moves. "