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Recreational Mathematics

Enrich Your Class... With Recreational Mathematics!

Purpose: Provide discovery, insight, and enrichment to a variety of mathematical concepts, many of which map directly to 4th Grade Standards.

Activities List

  Chinese Remainder Magic 2D Division Strips - Basic
   
  Magic of 142857 - Single Digit
   
  Magic of 142857 - Multi-Digit
   
  Lucky 7 Card Trick
   
  4-Digit Magic
   
  Exploring Handy Times
   

Chinese Remainder Magic 2D (M-033)

Standard
4.NBT.B.6

Purpose

This activity, Chinese Remainder Magic (CRM), promotes intrigue and curiosity by showing an imaginative result of the remainders of simple division problems. The produced grid of numbers allows students to explore patterns of numbers and remainders.

Description

Through a magic trick, students are introduced to an intriguing method of using only remainders to determine the number divided into (dividend). The numbers 1-42 are systematically written on a 6 by 7 grid by the presenter, and mimicked by the audience (click animation at the right and see worksheet "CRM Part I" below). While presenter's back is turned, a volunteer points to one of the 42 numbers for the audience to see. The volunteer and the audience divide the chosen number by both 6 and then 7. The presenter is only told the remainder portion in both results. The presenter turns around, looks at the grid and determines the chosen number.

The magic is a consequence of what is known as the Chinese Remainder Theorem that was first published in the 3rd to 5th centuries by the Chinese mathematician Sun Tzu. There is only one requirement for the trick to work: the two divisors (i.e., the dimensions of the grid) must not share any factors other than 1 (mathematicians say the numbers must be relatively prime or co-prime). In the example above, 6 is not prime, but the only common factors between 6 and 7 is 1. Each number from 1 to 42 has a unique remainder set when divided by 6 and 7. For example, 33 is the only number from 1 to 42 with a remainder set {5,3} when divided by 7 and 6 respectively. The filled in array provides the remainder set instantly for each number in the array.

For further details check out The Secret

Don't hesitate to try numbers that aren't relatively prime. A lot can be learned from experimentation. From above, we found that the numbers from 1 to 42 found unique spots in the 42 number grid. It just so happens that 42 is the LCM of 6 and 7. What happens when a 4 x 6 array is filled (click animation at the right and see worksheet "CRM Part II" below)?

The first 12 numbers fit quite nicely. Then, 13 came along and wanted 1's spot. But, 12 just happens to be the LCM of 4 and 6.

Worksheets

    Part I is used by the students to follow along with the opening magic.

    Part II is used to experiment with grid dimensions that have common factors other than 1 (i.e., not relatively prime).

    CRM Parts I and II (run off back to back)

    Part III (optional) can be used by students to pick their own grid dimensions. They can further experiment with any dimensions they choose. The grid is sized to accommodate up to 8 x 9. Unused rows and columns can be cut off or crossed out (from bottom or right side).

    CRM Part III


Reference(s)

Olivastro, Dominic. Ancient Puzzles. New York, NY: Banton Books., 1993, pp. 212-219.

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Division Strips - Basic
(Two Digit Numbers Divided by Single Digits)


Standard
4.NBT.B.6

Purpose

This activity, like Chinese Remainder Magic, focuses on basic division. It introduces students to a paper calculator used to perform division with single-digit divisors. The calculator can be used to explore a multitude of patterns as well as a means to check answers to exercises.

Description

Division Strips were developed in the late 1800s by two French mathematicians Henri Genaille and Edouard Lucas. They had the insight to realize that there are a finite number of calculations necessary to perform division by a single digit into a multi-digit number. The challenge was to organize these calculations on a set of strips, or dowels, so that division exercises can be constructed and the answers read off. Their success is captured in this basic activity as well as "Division Strips - Advanced" (see next activity).

As a motivator for both activities, this activity begins with the magic of Division Strips (click slideshow at right). Two volunteers set up a division exercise consisting of a 4-digit dividend. The divisor strip is set up to the left and the remainder strip is placed at the end. Then, a volunteer chooses one of the single-digit divisors on the divisor strip. The magician proceeds to read off the quotient and remainder upon examination of the strips.

The inner workings of Division Strips becomes clearer when one understands how to perform short division. Long division is the standard algorithm taught in schools consisting of the four major steps: divide, multiply, subtract and bring down. Short division correctly interprets the result of the subtraction step as the remainder of the previous division step. These remainders are carried along the dividend until a final remainder is obtained. This activity encourages, but does not mandate, students to use short division. The slideshow at the right contrasts long and short division. Short division can always be used when the multiples of the divisor are well know, which should be the case with any single-digit divisor.

For this activity, students will work in pairs. Each pair of students will get a set of division strips along with a display board that holds the strips neatly in place. The slideshow below illustrates the features of the strips and how they are used in conjunction with the display board. The worksheet below accompanies the division strips and display board.


TBD

Worksheets TBD


Reference(s)

Colgan, Lynda. Mathemagic. Tonawonda, NY: Kids Can Press Ltd., 2011, pp. 36 - 38.

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Exploring the Magic of 142857 - Part 1 (A-068) 
4.NBT.B5

Purpose

This activity provides students with a set of multiplication problems for practicing multiplying a multi-digit number by a single digit. They are also introduced to the strange behavior of the cyclic number 142857 and all its intriguing properties and patterns.

Description

This activity is introduced with the magic trick "Magic of 142857" (M62). The cyclic property of the number 142857 is all that is needed to provide an entertaining trick.

In this activity, the cyclic property that makes the trick work is explored. A worksheet guides students through a series of multiplication problems each multiplying 142857 by a single digit number. Once completed, students can examine the products and discover the cyclic nature of 142857. The bottom of the worksheet allows students to uncover a multitude of interesting patterns and properties of this special number.

worksheet   |   worksheet answers


Reference(s)

Heath, Royal Vale. Math E Magic. New York, NY: Dover Publications, Inc, 1953, p. 35.

Gardner, Martin. Mathematics Magic and Mystery. New York, NY: Dover Publications, Inc, 1956, pp. 9-10.

Woorksheets are original.

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Exploring the Magic of 142857 - Part 2 (A-069) 

Standard
Extending 4.NBT.B5

Purpose

This activity provides students with a set of multiplication problems to practice multiplying a multi-digit number (142857) by 2-digit and 3-digit numbers. They are also introduced to the strange behavior of the cyclic number 142857 as the products are rearranged to magically return back to 142857. This activity could be done using calculators in 4th Grade.

Description

This activity is an extension of the activity, Exploring the Magic of 142857 - Part 1, that introduces students to the cyclic number 142857. The magic trick Magic of 076923 (M-018) motivates this 4th grade activity. Unlike 142857, the number 076923 is a cyclic number involving two numbers: itself and 153846.

In this activity, the cyclic property of 142857 is explored by multiplying 142857 by 2-digit and 3-digit numbers. A worksheet is available to organize this activity. When each product is obtained, it is separated after 6 digits counting from the right.The remianing digits on the left are brought down (right justified) and added to the other 6-digit number. The results are quite surprising. See the worksheet answers for examples. Calculators can also be introduced to explore the behavior when larger numbers are used.

Additionally, students can discover the peculiar results produced when the cyclic number 076923 is multiplied by the numbers 2 through 12 and beyond. If time permits, students can break into groups and continue to play with this amazing property.

worksheet   |   worksheet answers


Reference(s)

Devi, Shakuntala. Figuring - The Joy of Numbers. New Delhi, India: Oriental Paperbacks, 1993, p. 115-118.

Worksheets are original.

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Exploring Lucky 7 Card Trick (A-070)

Standard
NS - Finding the LCM of two numbers

Purpose

Exploring Lucky 7 Card Trick provides a fun, enriching way for finding the LCM of two numbers. This activity allows students to explore the intriguing patterns that arise when a set of cards are manipulated in a magical way. Students discover wonderful geometric patterns that provide the necessary information to find the LCM, blending mathematics and art.

Description

Exploring Lucky 7 Card Trick is a unique blend of mathematics and art. Through card manipulations, this activity provides an exploratory approach to finding the LCM of two numbers (2-9). Beautiful geometrical art is produced as the relationship of two number's multiples is investigated. The LCM is determined by analyzing the characteristics of the art. This activity is introduced with Lucky 7 Card Trick (M-056).

Two numbers are chosen with the goal of finding their LCM. One number is used as the quantity of cards. A set of cards are moved one by one from the top of the deck to the bottom. The quantity of cards moved in a set is determined by the other number. A geometric pattern is drawn as students keep track of each number finishing a cycle (i.e., reaching its next multiple). The LCM becomes the total quantity of cards moved after the two cycles coincide for the first time. The geometric drawing provides all the necessary bookkeeping!

worksheet   |   worksheet answers

extended worksheet   |   extended worksheet answers


Reference(s)

Longe, Bob. The Magical Math Book. New York, NY: Sterling Publishing Co., 1997, pp. 30-31.

Worksheet concept and worksheet original.

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Exploring Handy Times - Multiplication Facts (A-008) 

Standard
NS - General - Place value and Other Bases

Purpose

Handy Times is a handy way to either review multiplication facts or enrich students with a unique finger math technique. Students are typically shown how to use their fingers to produce the multiplication facts of 9. This method can be extended to demonstrate all the basic multiplication facts by using a reduced set of fingers and patterns exhibited in other bases.

Description

The magic show opener Finger Math "6 thru 10" X "6 thru 10" (O-31) is used to demonstrate how the 10 fingers can be used to multiply any number from 6 thru 10 by another number 6 thru 10. Students are then reminded how the multiplication facts of 9 can be produced using their 10 fingers.

Motivated by the magic show event Handy Times - Multiplication Facts (M-037), this activity reveals how different sets of fingers can produce all the multiplication facts.

One finger from a wooden hand set can be removed to show how 9 fingers can be used to create the multiplication facts of 8. This strategy can be repeated until all multiplication facts are shown. A worksheet is available for each finger set to reinforce the place value and number base inherit in the finger set.

This activity can span multiple days, or a single day activity can select a few finger sets of interest (e.g., 9 fingers, 6 fingers, 3 fingers). Another option is to use the worksheets as extra credit to allow more demonstration time. An interesting pattern from finger set to finger set is also discovered and discussed.

worksheet   |   worksheet answers


Reference(s)

Original.

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1089 and All That (A-071) 

Standard
General Number Sense

Purpose

This mini-activity is of value in exploring number patterns.

Description

Fun patterns are explored when 1089 is multiplied by 1-9. An interesting follow-up activity of A Smart Trick (M-006).


Reference(s)

Heath, Royal Vale. Math E Magic. New York, NY: Dover Publications, Inc, 1953, p. 36.

 
Exploring 4-Digit Magic and Beyond (A-073) 

Standard
NS 3.1

Purpose

A great way to see how a solid understanding of place value and general number sense can produce instant sums.

Description

Exploring various ways to extend the magic trick 4-Digit Magic and Beyond (M-070) with the aid of worksheets.


Reference(s)

Gardner, Martin. Mathematics Magic and Mystery. New York, NY: Dover Publications, Inc, 1956, pp. 170-172.

Worksheets are original.