Students use mathematical processes & knowledge to solve problems. Students apply problem-solving & decision-making techniques, & communicate mathematical ideas.
Benchmark B: Apply mathematical knowledge & skills routinely in other content areas and practical situations.
4. Patterns, Functions & Algebra: Students use patterns, relations & functions to model, represent & analyze problem situations that involve variable quantities.
5. Data Analysis and Probability:
Students pose questions & collect, organize, represent, interpret & analyze data to answer those questions. Students develop & evaluate inferences, predictions & arguments that are based on data.
K. Make predictions based on theoretical probabilities & experimental results.
5 Scientific Inquiry.
Students develop scientific habits of mind as they use the processes of scientific inquiry to ask valid questions and to gather and analyze information.
Benchmark A. Participate in and apply the processes of scientific investigation to create models and to design, conduct, evaluate and communicate the results of these investigations.
6. Draw logical conclusions based on scientific knowledge and evidence from investigations.
English/Language Arts: 9. Research Standard: Students define & investigate self-selected or assigned issues, topics & problems.
Benchmark: B. Evaluate the usefulness & credibility of data & sources.
C. Organize information from various resources & select appropriate sources to support central ideas, concepts & themes.
Technology: 5. Technology & Information Literacy: Students engage in information literacy strategies, use the Internet, technology tools & resources, & apply information-management skills to answer questions & expand knowledge.
Benchmark B: Apply a research process model to conduct research & meet information needs.
Library 5: Effective school library media programs provide information literacy skills instruction.
Benchmark C: Explores information through independent discovery, peer collaboration & inquiry learning.
Benchmark D: Utilizes a research model to locate, use, & evaluate information
Students will play two different games that invoke the students’ ability to think ahead two or three steps. They will make predictions on the strategies of the games. Students will research the games and then write a short essay explaining what data predicted, what research told them and how the two could be resolved.
8-54 minute class periods.
Commentary: This lesson will help enhance a student’s ability to think 2 or 3 steps ahead on problems through playing a number of thinking games and then studying the strategies behind them.
Pre-assessment can be done by looking at how a student solves a two or three step equation. This lesson is designed to help those students, typically in lower levels of mathematics, who have a hard time looking beyond the next step. Teachers must also play both Hex and Lines and Dots with each of the students to get a “read” on their thought processes and strategies or to determine which students haven’t yet formulated a strategy.
A teacher should make notes on each of the students as well as keep a sample of how they solve the algebraic equations. This will offer a point of comparison at the end of the lesson.
Teachers should keep in close touch with each student during this process. As a game is played more and more, and competitors see how the others play, students should begin to formulate their own strategies. If not, the teacher may want to intervene and offer some help in building a strategy. The teacher did a great deal of intervention during the writing stage with those students on IEPs because they had writing disabilities. Many students struggled with the explanation offered in the research and needed extra help through oral explanation.
Students’ papers need to be graded on the following criteria:
1. Did the student address his/her hypothesis for Hex and Lines and Dots from playing each game at least 30 times?
2. Did the student come to the correct conclusion that it is advantageous to go first in Hex and Lines and Dots? Did the student discuss the winning strategy of each?
3 points for making a correct hypothesis based on the data from playing the games.1.5 points for making a hypothesis that does not follow their own personal data.
3 points for correctly interpreting the theoretical argument surrounding the game and explaining it in their own words. 2 points for explaining the argument in their own words and not being completely accurate. 1 point for more or less copying the argument from the research and no points if research was not done.
6 pts=A, 5pts=B, 3-4 pts.=C, 2pts=D and 0 or 1 pt=F
Different reading and writing levels must be addressed and more support must be given to those students needing it. Teacher and librarian must interact with students on all levels to make sure proper progress is occurring.
Students who excel can be asked to study the game of Chess. Chess is much more difficult, though there are some simple beginning strategies that may be considered. Students created their own Hex boards and game boards of their own creation after studying hex and completing their research. Their project was designed to promote understanding of probability and to help them to anticipate the moves in a game. They were asked to transfer their knowledge from simple games such as tic-tac-toe and zero sum games.
Students were assigned to play each of the two games 10 times with a parent, sibling or friend outside of class to enlarge their sample space for analysis. Second readings of research were assigned. The final drafts of their essays were to be completed outside of class.
Problem solving is needed in many other classes and this activity was created to help students to make predictions and to evaluate the cause and effect of these actions. The lesson also provides practice in creating and evaluating hypotheses, an important standard in their science classes.
A list of websites for research on Hex and Lines and Dots. Copies of the playing boards for both games.
MazeWorks—Hex 7 http://www.mazeworks.com/hex7/index.htm
Offers the chance to play an online version of the game HEX. This site also provides an “(I)ntroduction to the game, its inventors, the basic strategy and three challenging hex problems.”
Use Britannica Online for information on “game theory,” and “Two-person nonzero-sum games.”
Students will also utilize the Hex website above for practice sessions with the online version of the game.
Hypotheses, game theory, winning strategy,
Students will conduct research utilizing the computers in the library or the mobile cart. The teacher and the librarian will assist students to ensure that they are on task and are successful in locating materials both relevant to the project and at an appropriate reading level.
For background information, the encyclopedias available through the INFOhio site are appropriate. Students can find simple articles and can select the difficulty of the encyclopedia that they wish to use. Articles found through databases are often too detailed for beginning students or are intended for an audience of graduate level students and professors. Use Google or another reputable search engine to locate web sites dealing with games.
Record observations and suggested modifications to facilitate instruction and student learning in the lesson. Do not ask students to do research on the open topic of game theory without guiding them to a database or search engine. Many of the articles and sites that we located dealt with philosophical discussions between experts in the field; the finer points of economic game theory are often confusing and do not help students to understand the concepts.