Math crafts have been invented since ancient times for various purposes and practices. A piece of rope with 12 knots of equal distance was used to form a right triangle for measuring and making sure that two construction components were perpendicular to each other. Abacus was invented for daily accounting in oriental countries until being replaced by calculator. Napier's bones are a multiplication tool taught by some of the middle-school textbooks today. Of course, calculator is the most powerful math tool for all purposes. However, the math crafts invented in history are studied as part of culture. And yet, gaming with math crafts can be creative and entertaining.
Magic Squares follow the idea of modeling arithmetic by geometry. Based on the concept of area, revealing never before relations between numbers and squares, calculating operations and manual manipulations, prime numbers and rectangle screens, the method is straightforward. Numbers are viewed from a new perspective. Instead of following rules step by step, a student is able to visualize what are product, quotient, square root, etc. And playing games artistically with patterns of squares will certainly make learning more interesting.
Old way of rote learning (simply memorizing multiplication table, for example) is not enough. In agreement with the Common Core standards, Magic Squares are a method for learning math in a sense of construction, symbolism, and creativity. Four operations (addition, subtraction, multiplication, and division) can be carried out on a baseboard by manipulating square chips—counting the number of chips, building the rectangles, reading the numbers from the sidelines, etc. Associative, commutative, and distributive properties can be tested easily by the same token. Additive identity (zero is nothing), multiplicative identity (unit-side square), and impossibility of dividing by zero, are intrinsic to the craft. The square of a number can be found by building a square, and the irrational numbers like √2 can be found by rebuilding a square on and on. And yet, some of the basic principles of prime numbers can be shown by simple geometry—building rectangles on the baseboard. A rectangle screen of prime numbers can be made for middle-school students to understand.
The arithmetic craft of Magic Squares is an assembly of: a square baseboard with a size of 10×10 unit squares; unit square chips; hundredth square chips; and lumped hundredth square chips, in one embodiment.
In accordance with one embodiment, a math craft comprises unit square chips, hundredth square chips, lumped hundredth square chips, and a square baseboard on which all the arithmetic games, including screening prime numbers, can be played.
Accordingly several advantages of one or more aspects are as follows: to provide a math craft that aids arithmetic teaching and learning by novel geometric modeling—all the four operations can be performed on numbers with decimal point; arithmetic properties including the ones intrinsic to the craft can be easily tested; squaring a number is same as building a square; the digits of an irrational number resulted from square root can be obtained in the steps of square rebuilding; prime numbers can be studied by geometry—showing some of the fundamental principles and making the rectangle screen of prime numbers; factoring a large number is made easy by steps of making up a rectangle; and after all, a relation between the numbers and square geometry in the arithmetic framework is unraveled by such a simple mechanical set-up. Other advantages of one or more aspects will be apparent from a consideration of the drawings and ensuing description.
In the drawings, closely related figures have the same number but different alphabetic suffixes.
In the first embodiment, there are 100 unit square chips and 10000 hundredth square chips lumped in various ways for quick fit-in. The operations of addition, subtraction, multiplication, division, square, and square root can be carried out on numbers with decimal point in the range between 0 and 100 on a square baseboard. The limitation on digits comes solely from the difficulty in subdivision of hundredth square chip into ten-thousandth. On the other hand, large numbers like 1000, 10000, etc. can be represented by unit square chip as well.
In the second embodiment, there are 25 larger unit square chips with a smaller baseboard for playing easier games of all the operations. In this set-up, the hundredth square chips are larger for playing the games of screening prime numbers more easily. As a hundredth square chip can be chosen to represent the unit of natural numbers, the rectangle screen is capable of sieving out prime numbers in the range between 1 and 2500.
Fitting in lumped prime testers, the games can be played quickly. Lumped 7-testers, for example, are a column of 7 square chips, two columns of 14 square chips, etc. The prime testers like 7-testers, 11-testers, 13-testers, and so forth, form rectangle screens of prime numbers in the corresponding ranges of natural numbers.
Numbers with digits after decimal point between 0 and 100 are represented by unit chips and hundredth chips (in the first embodiment).
Multiplication (
Division (
Square (
Square-root (
Rectangle screen of prime numbers (
Screening begins with building a square as close as possible by the number (remember any number falls in between two neighboring square numbers). If it is a square, then the number is composite; if it is not, then build a rectangle by the side of the square (skip if it is not prime); if it fails, then the next rectangle to be built must have a side of a prime number next to and smaller than the number of the side of the square; if a rectangle is successfully built by the number, then the number is composite. The process continues until all the smaller prime numbers are tested. All fails screen out the number as a prime number.
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Based on the principles of rectangle screen, factoring a large number can be made easy and interesting. For factoring 713, a unit square chip in the second embodiment is chosen to represent 100. In
A restriction on the number of digits, say, from √5 is due to the difficulty of subdividing a hundredth square chip into ten-thousandth square chips in the first embodiment. An alternative embodiment is to make the chips bigger for having ten-thousandth square chips which can be lumped in various ways. Games can be played on flat floor without any baseboard.
Accordingly, the reader will see that this craft of various embodiments can be used to perform virtually all the arithmetic operations: addition, subtraction, multiplication, division, square, square root, factoring, screening prime numbers, and finding LCM. Kids knowing how to count should be able to learn basic math by such a vivid method that let them see math of real things which can be played with fun. Square, the most fundamental shape of geometry, should be able to impress the kids with its beauty and power in those games. They should be able to learn more, and more quickly. The Magic Squares certainly support the idea of modeling in secondary or elementary math education.
With foldable or aired cubes, the arithmetic games can be played in 3-dimension on flat floor. An assembly is shown in
Although the description above contains several specificities, these should not be construed as limiting the scope of the embodiments but as merely providing illustrations of some of the embodiments. The scope of the embodiments should be determined by the appended claims and their legal equivalents, rather than by the examples given.