Smart Moves: Developing Mathematical Reasoning with Games and Puzzles

Michael Serra’s Discovering Geometry, my favorite Geometry text, includes mathematical puzzles in each chapter. Most students love to work on these just for fun, often not realizing how well the puzzles are helping them develop mathematical and logical thinking skills. Serra uses puzzles like those in Discovering Geometry and many others as a regular part of his teaching, whether with fifth graders or high school students.

In Smart Moves, he has pulled together an assortment of sequential reasoning puzzles that might be used with students as young as fifth grade but that will also work through high school and college, and even for adults who enjoy the mental challenge. This book is the first in a planned series of books that will each focus on a particular type of mathematical reasoning.These puzzles include some that are great for a person to try on his or her own, some that require two players, and some that will accommodate more players. You might try some as a family.

Puzzles are divided into groups of various types of sequential reasoning puzzles: warm up puzzles (see sample to the left), racetrack games, movement puzzles, tour puzzles, magic square puzzles, sequential reasoning and algebra puzzles, and sequential reasoning and geometry puzzles. Within each of these groups except for geometry there are two or more types of puzzles.

Most of these puzzles are difficult to describe, but one of the simplest forms is the magic square puzzle in which there “is a square array of distinct integers such that the numbers in any row, column or main diagonal have the same sum.” You might already be familiar with the most basic form with three squares in each direction where the digits add up to 15 horizontally, vertically, and diagonally. Magic squares progress from this simple version all the way up to a 12 by 12 square with two- and three-digit numbers. Then there are algebraic magic squares!

Sliding block puzzles are similar to the small plastic four-by-four squares where you slide the numbers around using the one empty space to try to put them in order. Racetrack puzzles allow movement along a grid presented in the form of a racetrack, but movement is only done according to the prescribed number of moves along the x and y axes. Some of the tour puzzles allow movement only in the way a knight or a rook or a king might move—only one of these types of moves per puzzle. For those of us who enjoyed those geometry problems where you are presented with a number of lines crossing each other at different angles and asked to determine the size of the unidentified angles, there are a number of “angle chase” puzzles.

Puzzle solving tips, strategies, and even some hints are provided, and there is an answer key for all puzzles. There are reproducible racetrack and “puzzle board” pages in an appendix.

The publisher offers a set of 178 free color, downloadable PDF worksheets for people who purchase directly from their website: www.michaelserra.net. These are not the same as the pages in the book. They take puzzles presented in miniature in the book, and blow them up to a workable size. These will save you a lot of work in enlarging, photocopying, and other work to create immediately usable puzzles. To get them, you must purchase through the publisher's website rather than elsewhere.

Puzzles are challenging. Some students will gravitate towards these types of puzzles more than others, but many will enjoy doing them as a joint effort rather than a competition. Do not reserve them for only those students who are naturally interested. Within the 210 pages of this book, there are bound to be puzzles that appeal to everyone.

The goal throughout this book is to strengthen sequential reasoning abilities since such skills are critical in life in many areas besides mathematics. These types of puzzles are one of the best ways to accomplish this goal.

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List Price \$32.95

Smart Moves: Developing Mathematical Reasoning with Games and Puzzles

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