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Inquiry, Logic, and Puzzles Nicole Enzinger George Fox University, Nenzinger@Georgefox.Edu Digital Commons @ George Fox University Faculty Publications - School of Education School of Education 2013 Inquiry, Logic, and Puzzles Nicole Enzinger George Fox University, [email protected] Follow this and additional works at: http://digitalcommons.georgefox.edu/soe_faculty Part of the Science and Mathematics Education Commons Recommended Citation Published in CMC Communicator, 2013, 37(4), 28–30. This Article is brought to you for free and open access by the School of Education at Digital Commons @ George Fox University. It has been accepted for inclusion in Faculty Publications - School of Education by an authorized administrator of Digital Commons @ George Fox University. For more information, please contact [email protected]. Inquiry, Logic, and Puzzles by Nicole Wessman-Enzinger, Illinois State University [email protected] The greatest art doesn’t belong to the world. Do not be and the use of pentomino pieces became the intimidated by the experts. Trust your instincts. Do not be afraid to go against what you were taught, or what themes of many of the tasks we used in my you were told to see or believe. Every person, every set sixth grade class. of eyes, has the right to truth. —Blue Balliett in Chasing Vermeer Pentomino Pieces A pentomino is a polyomino constructed from he first time I read the book Chasing Ver- five congruent squares connected by edges. meer, I immediately fell in love with the The pentomino pieces are often given letter character of Ms. Hussey (Balliett 2005). names based on the similarity between the TMs. Hussey, an elementary teacher at the Uni- piece and the letter name. For example, the versity of Chicago Laboratory School, chal- “P” pentomino looks like the letter P. lenged her students and entertained whatever problems or questions her students brought to class. Ms. Hussey seemed beyond the epitome of an excellent teacher. One day in class, two of her students, Calder and Petra, informed the class about a recent local burglary of a spe- cial Vermeer painting. Through a series of events and suspense, Calder and Petra used Some students may be familiar with polyo- inquiry, logic, and puzzles to solve the mys- mino pieces from experiences in playing tery of the stolen painting. games. The game Tetris is composed of tetro- As a secondary mathematics teacher and minoes, which are polyominoes consisting of educator, I am always looking for a good book four congruent squares connected by edges. to implement in a middle school mathematics The game Blokus utilizes polyomino pieces classroom. Chasing Vermeer presented itself as consisting of one to five squares. Despite their a perfect book to implement in grades 5–8 presence in Chasing Vermeer and games like classrooms. Although Chasing Vermeer is not a Tetris and Blokus, many students are unfamil- book explicitly about mathematics, the mathe- iar with the pentomino pieces. matics and logic that emerged from the book spurred ideas that evolved into tasks for my Task 1: Creating the Pieces middle school students to solve (Wessman-En- Before my sixth-grade students started read- zinger, Wickstrom 2013). In this article I de- ing Chasing Vermeer, I introduced the book by scribe the book and a sequence of activities for showing them the first three polyominoes: the grades 5 –8. monomino, the domino, and the triomino. Af- ter discussing the rules for constructing a Mathematics Emerging from polyomino—a polyomino is constructed with Chasing Vermeer congruent squares that are connected to each other by at least one edge—I supplied my stu- The entire book can be thought of as a puzzle. dents with square-inch foam pieces and graph One of the main characters, Calder, used pen- paper, and I posed the first task: What would tomino pieces both as a fun puzzle and as a a tetromino look like, and how many tetromi- way to solve the mystery of the missing Ver- no pieces are there? meer painting. Not only did I love Ms. Students constructed as many tetrominoes Hussey, but as a lover of mathematics, I also as they could and then shared them with the loved the puzzles. The codes throughout the class. We noted that some of our pieces looked book, the hidden messages in the illustrations, the same and some looked different. From Page 28 CMC ComMuniCator Volume 37, Number 4 this, we then discussed the rigid transforma- matics, polyominoes present problems that tional properties of any polyomino and how promote the use of combinatorial mathemat- the pieces may look different but are still the ics, such as determining how many ways same, with transformations. After construct- these pieces can be arranged. Although math- ing and discussing the tetromino, we pro- ematicians have estimated and developed al- gressed to the pentomino, using the same gorithms for the estimates to determine how questions as before. many polyomino pieces there are, no formula Constructing the pentomino pieces seemed has as yet been found that relates the number more challenging for my class, perhaps be- of squares with the number pieces. cause there are significantly more pieces to This task promoted organizing data, spec- construct. After constructing the pentomino ulating about our data, and learning that pieces and discussing the shapes and number mathematics is a field of unanswered ques- of pieces, we displayed the solutions in class. I tions. I had given my students an unsolvable then gave each student a set of plastic pen- problem! Many students never get to experi- tominoes to use throughout the Chasing Ver- ence the wonders of unanswered but still re- meer unit. (Note: the Scholastic website, www. searchable questions in mathematics. By uti- scholastic.com/titles/chasingvermeer/pen- lizing pentominoes and extending Task 1, this tominoes.pdf, has a printable pentominoes became an outlet for these types of discus- sheet that students can cut out.) sions. I even shared the story about Andrew Wiles, who famously solved Fermat’s Last Task 2: Organizing and Looking for Theorem in 1994. He first fell in love with the Patterns problem in the 5th grade when his teacher Constructing all the polyomino pieces for a showed him an unsolvable problem. Follow- hexomino, a heptomino, or an octomino ing in Ms. Hussey’s footsteps, I wanted to would become an arduous task. Much like challenge my students to become the experts what Calder and Petra do in the story, I asked and promote autonomy in both creating and my class to organize what they already knew solving mathematical tasks. about the number of shapes for the previous polyominoes to predict how many shapes Task 3: Solving Puzzles there would be for hexominoes, heptominoes, Once or twice a week, as my students read the and octominoes without having to make book, I incorporated tasks that I felt were ap- them. My students eventually decided to or- propriate with discussion of the book. After ganize their data in a table that consisted of my students had created and discussed all the three columns (Name of Polyomino, Number pentomino pieces and encountered an unsolv- of Squares, and Number of Unique Pieces). able problem, I had the pentomino pieces The table also had a row for each kind of poly- available in the classroom for my students to omino—from monomino through octomino. use to complete puzzles. In addition to the Since the essence of the entire plot of Chas- physical puzzles pieces, Chasing Vermeer has a ing Vermeer is to look for patterns, I challenged website that provides varying levels of digital- the students to look for patterns once the data ly dynamic puzzles (www.scholastic.com/ was organized in a table. Middle school stu- blueballiett/games/index.htm). Before I pre- dents have used these types of tables mainly sented my class the two versions of the puz- for functions that have a linear pattern, and zles that I expected them to solve, I had them they started looking for linear patterns first play the less-challenging puzzles from the and then started looking for recursive relation- website. ships. This presented a real challenge for the Each pentomino piece consists of five students since the pattern was not so obvious. square units, and there are a total of 12 pieces. Some students resorted to modeling, which is The most challenging puzzles are those that feasible for the hexomino that has 35 unique use all 60 square units (12 pieces x 5 square pieces; however, modeling the heptomino, units). I gave my class the task of creating a with 108 unique pieces, is too time-consum- template for what they thought would be the ing. hardest puzzle and then to explain why. Some Like many puzzles in recreational mathe- June 2013 CMC ComMuniCator Page 29 puzzles were odd, complicated shapes; some shapes of different areas, related to Task 3. To puzzles used all of the pieces; and some tem- push my students, I wanted to know if they plates created by the students were not possi- could move beyond simply solving puzzles to ble because they did not use areas that were thinking about the connections between area multiples of five. and perimeter. For example, I told the stu- We discussed all of the templates that were dents to use four specific pentomino pieces, created and conjectured that the most chal- such as C, X, P, and V, and to find the shape lenging puzzle might be one that not only that maximized the perimeter. The students uses all of the pieces but is also a simple used the pentominoes and modeled the possi- shape, such as a rectangle. Some students ar- bilities to help them with the coordination be- gued this point since they thought there tween perimeter and area.
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