Purpose: Provide discovery, insight, and enrichment to a variety of mathematical concepts, many of which map directly to 4th Grade Standards.
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 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) 
View only  
Print KEY  View Key  
Finding Squares from Previous or Next Squares worksheet 

Print KEY  
Finding Products of NextDoor Neighbors worksheet  
Print KEY 
Reference(s)
Kelly, Gerard W. ShortCut Math. New York, NY: Dover Publications., 1984,
pp. 4548.
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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 142 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 coprime). 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
Reference(s)
Olivastro, Dominic. Ancient Puzzles. New York, NY: Banton Books., 1993, pp. 212219.
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Purpose
This activity, like Chinese Remainder Magic, focuses on basic division. It introduces students to a paper calculator used to
perform division with singledigit 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 multidigit 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).
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 singledigit 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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Purpose
This activity extends the Division Strips explored in "Division Strips  Basic" to multidigit dividends (beyond 2 digits).
The goal is twofold: learn the skill of short division for a single digit divisor and to motivate students to do division
with multidigit 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 multidigit 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 their entire creativity.
As a motivation, this activity begins with the magic of Division Strips (click slideshow at right). Two volunteers set up a division exercise consisting of a 4digit 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 singledigit 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. This section is incomplete.
Worksheets
Division Strips Advanced  KEY  
.  
Reference(s)
Colgan, Lynda. Mathemagic. Tonawonda, NY: Kids Can Press Ltd., 2011,
pp. 36  38.
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Purpose
This activity provides students with a set of multiplication problems for practicing multiplying a multidigit 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.
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. 910.
Woorksheets are original.
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Purpose
This activity provides students with a set of multiplication problems to practice multiplying a multidigit number (142857) by 2digit and 3digit 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 (M018) 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 2digit and 3digit 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 6digit 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.
Reference(s)
Devi, Shakuntala. Figuring  The Joy of Numbers. New Delhi, India: Oriental Paperbacks, 1993, p. 115118.
Worksheets are original.
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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 (29). 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 (M056).
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. 3031.
Worksheet concept and worksheet original.
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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" (O31) 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 (M037), 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.
Reference(s)
Original.
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Purpose
This miniactivity is of value in exploring number patterns.
Description
Fun patterns are explored when 1089 is multiplied by 19. An interesting followup activity of A Smart Trick (M006).
Reference(s)
Heath, Royal Vale. Math E Magic. New York, NY: Dover Publications, Inc, 1953, p. 36.
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 4Digit Magic and Beyond (M070) with the aid of worksheets.
Reference(s)
Gardner, Martin. Mathematics Magic and Mystery. New York, NY: Dover Publications, Inc, 1956, pp. 170172.
Worksheets are original.