Michael Serra

The Many Faces of Tic - Tac - Toe

The Many Faces of Tic-Tac-Toe is the fourth book in the series Developing Mathematical Thinking Through Game-based Learning. The connections between the games and puzzles in Tic-Tac-Toe and mathematical thinking are pointed out or left for you to discover in the process of solving puzzles. The Many Faces of Tic-Tac-Toe is intended for a variety of audiences from ages 5 to 90+. 

The first four chapters of The Many Faces of Tic-Tac-Toe introduce the basic game of Tic-Tac-Toe and then introduces simple changes to the rules and to the size of the playing board, giving us Reverse Tic-Tac-Toe, Three in a Row Tic-Tac-Toe on a 4x4 board, and Four in a Row Tic-Tac-Toe on larger boards.  These basic variations of the classic Tic-Tac-Toe game introduce the important idea of looking at what happens when a rule is changed in a system. This is what we often do in mathematical problem solving and what we often do in mathematical proof.

Chapter 8 introduces 3D Tic-Tac-Toe.  This is helpful in developing spatial reasoning and is a great introduction to graphing with three-dimensional coordinates.  Chapter 11, Tic-Tac-Square, is another version with an immediate application to school mathematics.

Sample Puzzles from Tic-Tac-Toe

The two games below are games of four-in-a-row tic-tac-toe on 5x5 boards.
7.                                                 8.   

                           

7.    Where should X play to guarantee a win for X?
8.    O just played at a3. Explain why that is a mistake. Where should O have played?

Tic-Tac-Square

The game of Tic-Tac-Square is another variation on Tic-Tac-Toe.  Instead of trying to get your X’s or O’s collinear, the object is to get your X’s or O’s arranged at the vertices of a square in a square grid.  The two games below are games of Tic-Tac -Square on 5x5 boards.

4.                                                  5.

       
Tic-Tac-Twice

Tic-Tac-Twice was invented by Andrzej Ehrenfeucht and Pat Baggett and is available as a commercial game distributed by Aristoplay.  It is played on two 4x4 boards.  Players take alternate turns by placing a colored token (a checker) on an empty spot on one board.  At the same move the player puts a token of his/her color on the same lettered square on a second board.  You win the game by having four of your colored tokens in a row (horizontally, vertically, or diagonally on either one or the other of the boards).  The two boards below show a game of Tic-Tac-Twice in progress.  It is the dark checker’s turn.  What letter square should the dark checker play next to guarantee a win on his/her next turn?

Sequential Reasoning

Sequential Reasoning is the third book in the series Developing Mathematical Thinking Through Game-based Learning.  Sequential Reasoning is designed to guide the active reader towards accomplishing a goal through a variety of sequential steps.  The connections between sequential reasoning and mathematical thinking are clearly displayed. 

Sequential reasoning is a broad and perhaps vague term given to the type of reasoning that is at the core of everyday problem solving and mathematical thinking.  Sequential reasoning is used in many projects, whether it is in party planning, remodeling a home, or solving complex issues in your work.  In mathematics instruction, sequential reasoning is used in problem solving and is fundamental to solving equations and creating proofs.  

Sequential Reasoning is intended for a variety of ages and audiences.  Sequential Reasoning puzzles can be played as part of your school math curriculum or as home entertainment for game players ages 9 to 90. 

Sample Sequential Reasoning Puzzles

Color Network Puzzle
Each circle is to be colored red (R), yellow (Y), or Blue (B).  However no two vertices of a triangle are permitted to be the same color. What must be the color of the circle marked with a question mark? 

                                     

Angle Chase Puzzles
Students rely on their recall of geometry properties, not their protractor, when solving Angle Chase puzzles.  The Angle Chase puzzle below uses properties of parallels, vertical angles, isosceles triangle properties, and polygon sum properties.  Find the measures of all the angles in the Angle Chase below.

Sudoku Puzzles
A Sudoku puzzle is a logical placement puzzle in the form of a square grid of squares called cells. Some cells contain given numerals.  The most common Sudoku puzzle, the puzzle that has been sweeping the planet, is a 9x9 Sudoku.  When a 9x9 Sudoku puzzle is completed every row, column, and non-overlapping 3x3 box contains every digit, 1 through 9.  The objective is to fill in all the empty cells. Logical sequential reasoning and perseverance is all that is required.
The most common Sudoku is the 9x9. A polyomino Sudoku puzzle is a Sudoku puzzle in which each box is a polyomino region rather than a square box.

Classic 9x9 Sudoku                    Polyomino Sudoku

               

NumbrixTM Puzzles

In the fall of 2008, Marilyn vos Savant began publishing a puzzle called NumbrixTM each week in her "Ask Marilyn" column.  The puzzle is played on a square array of cells, often a 9x9.  Some of the cells contain numbers. The objective in a 9x9 NumbrixTM puzzle is to fill the remaining empty cells with integers from 1 to 81 so that when completed there is a continuous path of numbers from 1 through 81.  The path moves horizontally or vertically from cell to cell but not diagonally.
Here are two puzzles I created.  Numbrix Puzzle 1 has the given numbers around the outer perimeter.  Numbrix Puzzle 2 has the given numbers along the two diagonals.  Add the missing numbers to complete the 7x7 puzzle below left and the missing numbers in the 9x9 puzzle below right to create a continuous path in each.

        Numbrix Puzzle 1.                  Numbrix Puzzle 2.

           

Magic Square

A Magic Square is a square array of distinct integers such that the numbers in any row, column, or main diagonal, have the same sum (called the magic sum).  If the n2 numbers in an n by n magic square are the positive integers 1 through n2 then the magic square is a normal magic square.  The magic square below is a normal 4x4 magic square.

               

Magic Square Puzzles.  A magic square puzzle is an incomplete magic square.  Complete the 5x5 magic square.  You will need to calculate the magic sum first.

                        

Lunar Lockout Puzzles

Lunar Lockout is a set of 40 puzzles created by Hiroshi Yamamoto with the assistance of Nob Yoshigahara for Binary Arts (now called ThinkFun.com).   Lunar Lockout is played on a 5x5 grid.  The objective is to maneuver the red piece onto the center red square. 
The Rules of Lunar Lockout.  In the Lunar Lockout puzzles to follow, the objective is to maneuver the piece (Spacepod) indicated by the letter Q, with the help of the other helper-bots indicated by the numbers 1-5 until the Q piece lands on the Emergency Entry Port (the center red square in the puzzles below).  All pieces (the Q as well as the helper bots 1-5) can only move according to the same two rules.

The Rules for moving pieces:   
• All pieces move only horizontally or vertically.
• A piece continues to move until its path is blocked by another piece.
Example:  How do you move the Q onto the red square?
The Q (red piece) cannot move straight down because there is nothing to stop it (so it would continue past the red square).  However, if the 1-bot moves to the right it is stopped by the 2-bot. The Q can then move down to the red square because it will be stopped by the 1-bot.

                                           

We symbolize the movements up, down, right, and left of the pieces by using arrows.  One solution is 1 to the right, then Q down. 

Symbolically it is written:                        

There is another solution. How would you write it?

Lunar Lockout Puzzles.  Solve these Lunar Lockout puzzles and write their solutions symbolically. 

             Lunar Lockout 1.                      Lunar Lockout 2.

                                

Summary

This article described six different sequential reasoning puzzles (and of course each puzzle has many variations).  These sequential reasoning puzzles, played as part of your school mathematics curriculum, will create an exciting change of pace.  They are novel ways to introduce the concept of sequential reasoning in problem solving and proof. Because these sequential reasoning puzzles are fun they are also motivational. Yet your students will develop strategies and reasoning skills faster and deeper with these games than with traditional methods.  Many parents are looking for opportunities to do mathematics with their children.  Send these puzzles home with your students to be played with by the whole family. Sequential reasoning puzzles provide a setting for students and parents to work on problem solving and mathematics together. Try them.  I think you’ll see the “reason” for including them in your lesson plans.

Buried Treasure

 Buried Treasure is the second book in the series Developing Mathematical Thinking Through Game-based Learning. Buried Treasure is not just another game book.  It is designed to guide the active reader towards discovery of strategies, to develop logical  reasoning, and learn mathematics. The connections between the games in Buried Treasure and mathematical thinking are pointed out. Buried Treasure introduces graphing on the Cartesian coordinate plane, on the Polar coordinate plane, with three dimensional coordinates, and using coordinates on the surface of a sphere. 

Buried Treasure is intended for a variety of audiences.  Buried Treasure is a resource for teachers of grade levels 5-12 and is also a fun tool for home schooling parents.  Buried Treasure can also complement the physical exercise program for active older adults who wish to exercise their brains as well.

Sample Games and Puzzles from Buried Treasure

In this set of Buried Treasure Puzzles with Tetrominoes, the goal is to find the two different tetrominoes hidden in a grid of 6x6 squares. The two tetrominoes will not occupy adjacent squares, horizontally, vertically, or diagonally. Occasionally, a portion of a tetromino is shown.  The numbers across the bottom and along the side indicate the number of squares from tetrominoes that occur in that row or column.

5.                                  6.                           7.

                                         

For Puzzles 15-19 below, sketch a pentomino given the coordinates of only some of its corners.  There are multiple solutions.  Different solutions may include the same pentomino in a different position or a completely different pentomino. Five of the ten corners include: (3,3), (4,4), (5,5), (6,6), (5,4)

15.                             16.                               17.                    

                     

 18.                             19.

                                  

Buried Treasure with the Cartesian Coordinate System

Now you are ready to play Buried Treasure Game Level 2A.  Locate the game sheet for the Level 2A game.  Select a partner and an opposing team.  Each team selects any one of the twelve pentominoes and hides it on the grid labeled your team’s buried treasure.                                                        

 

Rules:
Use the same rules (1-7) as Buried Treasure Level 1 with this addition:

•Teams still take three digs at a time but the response by the other team does not happen until after all three digs take place.

Basic Strategies:
1.    Create a system on the grid to determine if the “dig” is a hit or a miss just as suggested for Level 1 play (X for miss, C for corner, E for edge). 

2.    Do not place digs in a corner.  A corner dig covers only one square while a dig just  inside a corner covers the same corner square and it also covers three other squares.

3.    Do not place digs next to each other.  For example the two digs (1,5) and (2,5) cover six squares while the two digs (2,2) and (4,2) cover eight squares.

 

4.    The other team does not tell you which individual dig is a hit or miss until after all three of your digs are complete.  Therefore do not spread your digs too far apart.

                                       

 Not such a good idea                    A better idea

Using DG in the Classroom

In this section I’ve included a number of forms, outlines, and letters that I’ve used with my classes.  At the beginning of the year I pass out the description of how the class will be conducted, including the class rules, social skills, homework, and grading policies.  A letter is sent home to parents explaining all of this.  During the first few days of the school term I pass out the semester schedule of class activities, homework assignments, and last year’s final exam to the students.  The students know what material they are going to experience and when they are going to receive it.  With last year’s final exam in front of them, they see their year end objective.  Samples of these classroom handouts can be downloaded below.  They are titled:

•   Introductory Outline for our Geometry Course Download CMOutlineGeoCourse (PDF)

•   Geometry Class Policies and Expectations Download CMClassPolExpect (PDF)

•   Geometry Semester Calendar Download CMGeoSemesterCal (PDF)

•   Homework Sheet Spring Grading Period S.3 Download CMHomeworkSheet (PDF)

A very important part of my geometry curriculum is the student Notebook.  The student Notebook is a 3-ring binder with six sections: Notes/Homework, Chapter Summaries, Definitions, Investigations, Conjectures, Journal, and for my honors class, a seventh section, Postulates and Theorems.  After a large number of geometry vocabulary terms have been developed for van Hiele level 1 understanding, I give a vocabulary qualifying exam (VQE) [1].   When a student passes the VQE they have earned the right to use their notes for all tests for the rest of the year.  

Grading Notebooks is very time consuming and takes valuable energy away from the students.  Whenever you can turn a task of yours into a learning experience for your student, it helps everyone.  I create detailed rubrics for assessing the Notebooks and the students do most of the Notebook evaluations.  Samples of these classroom handouts can be downloaded below.  They are titled:

•   Vocabulary Qualifying Exam Download CMVocabQualExam (PDF)

•   Notebook Evaluation Sheet Download CMGeoNotebookEval (PDF)

 

I do not take homework home to grade.  I turn the task into a learning experience for the students by having them do the lion’s share of the homework evaluation. I created a homework log sheet so students become responsible for monitoring the amount of homework attempted by members of their group.   A sample of the Homework Log is available here. Download CMHomeworkLog (PDF)

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[1] The vocabulary qualifying exam is a 48-item exam.  On one side of the exam paper there are 24 terms.  The student is to sketch and correctly label the figure that matches each term.  On the other side of the exam paper there are 24 geometric figures.  The student is to give the term that correctly matches with the figure.

Introduction to Discovering Geometry

Discovering Geometry began in my classroom over 35 years ago. During my first ten years of teaching I did not use a textbook, but created my own daily lesson plans and classroom management system. I believe students learn with greater depth of understanding when they are actively engaged in the process of discovering concepts and we should delay the introduction of proof in geometry until students are ready. Until Discovering Geometry, no textbook followed that philosophy.

I was also involved in a Research In Industry grant where I repeatedly heard that the skills valued in all working environments were the ability to express ideas verbally and in writing, and the ability to work as part of a team. I wanted my students to be engaged daily in doing mathematics and exchanging ideas in small cooperative groups.

The fourth edition of Discovering Geometry includes new hands-on techniques, curriculum research, and technologies that enhance my vision of the ideal geometry class. I send my heartfelt appreciation to the many teachers who contributed their feedback during classroom use. Their students and future students will help continue the evolution of Discovering Geometry.

Institute Description

Institutes are one to five full-day sessions. This format is ideal for full immersion into Inquiry-based Learning.  At Michael’s institute, you will be engaged in exciting hands-on activities from start to finish.  Participants, working in a cooperative-learning environment, will perform investigations, discuss results with group members, make conjectures, test them, and use them to solve interesting problems.  In addition to the traditional geometry tools of compass, straightedge, ruler, and protractor, participants will have extensive experiences with Patty Paper and The Geometer’s Sketchpad. Some recent examples can be seen below.

Each institute is custom designed to meet your needs. Institutes can be set up at host school district sites or arrangements can be made for a full immersion geometry institute in San Francisco. Pricing is based on length of time, the number of visits, topics requiring special equipment and materials, and amount of travel involved. For more information please contact us at mserramath@gmail.com.

 

One Day Institute

One Day Institutes are ideal for teachers/school districts that want one day of PD covering a few content strands or a quick overview of the Discovering Geometry textbook and its approach. If the client wishes to focus on only one topic, such as Cooperative Learning, then a One Day Institute may be sufficient. A One Day Institute provides approximately five to six hours of instruction.

 

Two Day Institute

Two Day Institutes are ideal for teachers/school districts that want a more in-depth look at Inquiry-based Learning or a more detailed look at the Discovering Geometry textbook. In addition to a closer look at Discovering Geometry, topics may include Cooperative Learning, Discovering Learning, and Classroom Management. A Two Day Institute provides about 10 to 12 hours of instruction.

 

Three Day Institute

A Three Day Institute provides about 15 to 18 hours of instruction.

 

Four Day Institute

A Four Day Institute provides about 20 to 24 hours of instruction.

 

Five Day Institute

A Five Day Institute provides about 25 to 30 hours of instruction.