Problem Solving with Pentominoes

Overview: Students create the 12 different pentomino shapes (letters) on the computer using the Logo programming language. Students build each letter based on a 30 by 30 square. Students use super procedures to create each letter. After creating the 12 pentominoes letters, students use their letters to create new "flip" letters. Students then combine letters to create five different puzzles: two different rectangles, two different sets of stairs, and a hat shape.

Goals and objectives:

The student will be able to use the Logo programming language to create defined shapes and reorient those shapes. The student will develop spatial visualization skills necessary to solve the five puzzle problems.

 

Math TEKS

Problem solving; translations; reflections; rotations; Geometry (Spatial Sense, Conceptualization, Connections, Reasoning, Problem solving)

Technology Application TEKS

Use of foundation and enrichment curricula in the creation of products; Use of GroupWare, collaborative software, and productivity tools to create products; Resolve information conflicts and validate information through research and comparison of data.

 

Prerequisite skills/knowledge:

Vocabulary: A pentomino is a shape made of five squares. All of the squares must have at least one common side with at least one other square.

Skills: Students must recognize that when a pentomino is rotated or flipped, it is still the same pentomino.

Levels of understanding: Students must have a basic understanding of the Logo programming language.

Materials/equipment:

Hardware

Macintosh Lab with a computer for each student

Software

Terrapin Logo

Printed Materials

Packet of Pentominoes
Classroom Warm Up Activities Packets
Summary of Logo commands [PDF file1 ] [PDF file 2 ]
Logo Pentominoes Unit grade sheet
Letters [PDF file ]
Flip letters [PDF file ]
Pictures [PDF file ]
There are copyrighted handouts available through Creative Publications 1-800-624-0822.

Other resources:

One set of pentominoes for each student.

Picciotto, Henri. Pentomino Activities. Lessons, and Puzzles. Mountain View, California:
	Creative Publications, 1984.
Fitch, Dorothy. 101 Ideas for Logo. Portland, Maine: Terrapin Software, Inc., 1993.

 

Instructional activities/strategies:

Classroom Management

(1) For 10 days, during the warm up time in the mathematics classroom, students use pentominoes to fill in puzzles. This hands-on experience is essential for the success of the project. It allows time for students to work concretely on the very difficult tasks of translations, rotations, and reflections so that later, on the computer screen, they can visualize the necessary moves to create their products.

(2) The students spend 5 days in the computer lab writing procedures for the various pentomino shapes. The students rotate, flip, and reflect them, as well as combine them to create new puzzle shapes.

Sequence of Lesson

Part I - The mathematics classroom: For 10 days, the students work independently at their own pace on filling in puzzles with pentomino pieces. The difficulty of the puzzles increases with each puzzle. The students use a checklist so that they can keep track of where they left off the day before. There are 83 puzzles for the students to work through. During this time, basic Logo programming commands such as fd 30, right 90 and repeat 4 [fd 30 right 90] are reviewed.

 

Part II - The computer lab: During the first day in the lab, students are assisted in writing a procedure to create a 30 by 30 square. Using the procedure for the square, a procedure for the I pentomino shape is then written, named and saved. The students are given a LOGO Pentominoes Unit worksheet to monitor their progress. As they work, they check off the shapes they have written procedures for and saved. The students continue creating the 12 pentomino shapes. At the beginning of the second day, how to color a shape black is demonstrated to the students. Because Logo-created shapes lose shared edges when they are combined, it is necessary to shade certain pieces to maintain their identity when they are put together later. The students are shown how to flip letters by rotating the turtle prior to typing in the command to create the letter. The students then continue to work individually creating the pentomino shapes. They use problem solving skills to combine letters to solve the five puzzle problems. At the end of each day, the students save their work. After they finish the Logo Pentominoes Unit worksheet, there are other shapes for them to create. To create these shapes, they use the pentomino sets at their desks, draw the results, and then write Logo procedures and create the shape on their computer.

Communication:

Students communicate their results by typing in the names of their procedures and showing the teacher the results.

 Assessment options:

Formative:

(1) From experience working with 6th and 7th grade students using the Logo programming language, teachers realize the difficulty students have with using the same figure more than once in the same work area. They think that if it is oriented differently on the screen, it must be approached as a different shape. Students do not see that the shape is the same.

(2) During the classroom part of the pentominoes unit, the teacher observes the students solving puzzle problems. This observation provides the teacher with valuable information about the students' spatial sense and problem-solving ability.

Summative:

The teacher grades the students' work by having all of the students type in the names of their procedures one by one. The teacher can call out the name of a particular procedure and walk around the room quickly and check that everyone's shape is identical. Some preliminary grading can occur before the last day of the project if desired. On the last day of the project, students stop creating 20 minutes before the end of the period. The teacher calls out the names of each procedure and walks around the room as the students type in the name called out. The teacher can check 30 students' work visually very quickly.

How will students demonstrate they have learned the objectives?

Students will demonstrate they have learned the objectives by creating the 12 pentomino shapes and using them to solve the five puzzle problems.

 

Teaching Suggestions/Extensions:

Other puzzles to create include stamp, fish, bat, escalator, pyramid, and building. To write Logo procedures for these puzzles, the students must first determine which pentominoes to use. The students use a set of pentominoes to analyze the problem. After making their decision, they write a Logo procedure for each of the puzzles. These puzzles require more pentominoes than the simpler ones.

 

Contributors:

Mindy Hensen
Cedar Valley Middle School
Round Rock I.S.D.