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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

  Multiplication Table Diagonals Chinese Remainder Magic 2D
   
  Magic of 142857 - Single Digit Basic Division Strips
   
  Lucky 7 Card Trick Intermediate Division Strips
   
  4-Digit Magic Sum Magic
  
  Map Coloring Number Magic
   
  Exploring Handy Times Games, Puzzles and Challenges
   

Multiplcation Strategies - The Diagonals

Standard
4.NBT.B.5

Purpose

To explore patterns in the diagonals of the multiplication table and discover multiplication strategies that extend beyond the basic facts.

Description

By exploring the diagonals of the multiplication table, one can discover many patterns that extend beyond just the basic fact. Much of the exploration centers around the numbers on the main diagonal, commonly know as the Square Numbers. Students discover useful patterns within the Square numbers, and learn how to derive new square numbers from previously learned ones. Careful examination of the diagonals off the main diagonal reveals that they group the factors into logical sets: factors that are one apart (e.g., 3 x 4), factors that are two apart (e.g., 6 x 8), factors that are three apart (e.g., 4 x 7), and factors that are 4 apart (e.g., (4 x 8). The square numbers can even be thought of as being 0 apart. It is also possible to investigate factors that are separated by more than 4, and discuss the benefits of the observations.

This activity is divided into multiple activities, because it is not recommended to bombard the students with all of it at once. One or two of the patterns should be introduced at a time for a total of 5 or 6 sessions. Hence,portions of this activity can be coupled with other small activities to comprise full sessions.

Worksheets / Notes

Finding 10s and 100s to Explain Multiplication worksheet Print  
  Print KEY  
 
The Squares Arithmetic and Geometric
(for Notes)
Print (back to back) View only
  Print KEY (two pages) View Key
 
Finding Squares with a Little Help
(for Notes)
Print View only
  Print KEY View Key
 
Finding Squares from Previous or
Next Squares worksheet
Print  
  Print KEY  
 
Finding Products of Next-Door Neighbors worksheet Print  
  Print KEY  
 
Finding Products of Numbers 2 Apart worksheet Print  
  Print KEY  
 
Finding Products of Numbers 3 Apart worksheet Print  
  Print KEY  
 
Finding Products of Numbers 4 Apart worksheet Print  
  Print KEY  


Reference(s)

Kelly, Gerard W. Short-Cut Math. New York, NY: Dover Publications., 1984,
pp. 45-48.

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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 - Intermediate" (see next activity).

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 at the right 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.


Worksheets / Notes

The first handout below, A Basic Understanding of Division, provides an understanding of division by contrasting it with multiplication using an acronym (GET) that ties together the three main component in any multiplication or division problem: the number of groups (G), the number in each group (E) and the total number (T). When completed, through the interactive lesson, it becomes a set of notes for the student.

The second and third handouts are worksheets. The second one can be combined with the first handout. It provides practice with 1-digit divisors, 2-digit dividends, and 1-digit quotients with remainders. The third handout extends from the second by including two digit quotients.

Basic Understanding of Division    (Notes) Completed notes  
Basic Division Strips Part 1
   (worksheet)
KEY  
 
Basic Division Algorithm Explained    (Notes) Completed notes  
Basic Division Strips Part 2
   (worksheet)
KEY  
 


Reference(s)

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

Woorksheets are original.

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Division Strips - Intermediate
(Multi-Digit Numbers Divided by Single Digits)


Standard
4.NBT.B.6

Purpose

This activity extends the Division Strips explored in "Division Strips - Basic" to multi-digit dividends (beyond 2 digits). The goal is two-fold: learn the skill of short division for a single digit divisor and to motivate students to do division with multi-digit dividends and single digit divisors. In the later, Division strips is used to check their answers to worksheet exercises.

Description

As stated in the earlier activity, 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 activity that illustrates the brilliance of their creativity.

The inner workings of Division Strips becomes clearer when one understands how to perform short division. This section is incomplete.

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. But, wait, there is another cyclic number, 076923 (I know, what's up with that leading "0"?), and it brought along a friend.

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 or 0768923 by a single digit number. Once completed, students can examine the products and discover the cyclic nature of these numbers. The bottom of page 1 of the worksheet allows students to uncover a multitude of interesting patterns and properties of 142857.

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.

Lobosco, Michael. Mental Math Challenges. New York, NY: Tamos Books Inc., 2000, p. 115-118.

Woorksheets 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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Sum Magic

Standard
4.NBT.A.4 & 4.NF.B.3b

Purpose

To provide A challenging and competitive setting to work with addition and subtraction of whole numbers, money and fractions.

Description

Sum Magic is an original game that was devised through the study of magic squares. The goal of the game is to come up with triplets of entities that total to a given Magic Sum. The entities are from one of three categories (much like the categories in the game show Jeopardy):

  • Whole numbers
  • Money
  • Fractions
Each category has challenges ranging from 200 points to 1000 points. For each challenge, students are presented with 9 entities and a Magic Sum. Against the clock, they need to find as many of the possible 8 triplets that total the Magic Sum.


Handouts

   Preliminary Fraction Examples Print View only
  Print Key View KEY
 
   Sample Group Challenges TBD TBD
 


Reference(s)

     The Sum Magic Game/Contest is original. It was developed through studying Magic Squares.

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Number Magic

Standard
General number sense

Purpose

To provide a setting for intrigue, investigation, and verification.

Description

Students are presented with various number relationships that result in number exploration and verification.


Handouts

Arithmetic Pyramid of Equations Print View only
   Explanation of Equations Print View only
 


Reference(s)

Arithmetic Pyramid of Equations:
     - Julius, Edward H. Arithmetrics. Jossey Bass, 1995, p.131.
     - Benjamin, Arthur. The Magic of Math. Basic Books, 2015.
       pp. 9-10.

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What's the Minimum Number of Colors
Required to Color a Map?


Standard
General logic & Strategy Building

Purpose

To explore one of the most historically interesting theorems and engage in strategies for using the fewest number of colors.

Description

Students are presented with a set of maps and asked to "color" them using the fewest number of colors. In order to focus on the strategies and logic, students will be using numbers to keep track of the number of colors needed. Regions that share a border must be different colors, although regions that only share a point of intersection may be the same color. Next, students will be presented with a map that used five colors, and their task is to re-color it using only four. The final task allows students to attempt to create a map that requires more than four colors.

Ever since the mathematician Mobius posed the problem in a lecture in 1840, mathematicians have tried to prove that 4 colors are sufficient to color any map. According to Wikipedia, "The four color theorem was proven in 1976 by Kenneth Appel and Wolfgang Haken. It was the first major theorem to be proved using a computer."

Handouts

Map Coloring Activity Print Print KEY
 

Reference(s)

     - Exercises 1-6 are original
     - Exercise 7: Bolt, Brian. Mathematical Activities.
       Cambridge University Press, 1982. p. 56

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Games, Puzzles and Challenges

Standard
General - visualization skills,
logic, number sense

Purpose

These activities, consisting of games, puzzles and other challenges, can be used as an introduction for larger sessions. They are designed as either a warmup for another activity or a way of getting students focused on mathematics after a recess or other lesson.

Description

Most sessions with Mathematics Magic have an introductory activity or opener. The activities in this section contains some of these openers.

The Games, Puzzles, and Challenges

The Locker Puzzle SOLUTION (ref: Zeitz, Paul. The Art and Craft of Mathematical Problem Solving. The Great Courses (DVD) Lecture 7 - Parity)


Reference(s) listed alongside each puzzle

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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.