Purpose: Provide discovery, insight, and enrichment to a variety of mathematical concepts, many of which map directly to 5th Grade Standards.
Purpose
The primary purpose of this activity is to provide students with a basic understanding of how mathematics plays a major role in music  both in note pitch and
note duration. The 2500 year history of music dates back to the Greek philosophers known as the
Pythagoreans  back when music was a discipline within mathematics. Students will understand and experience what that early Greek society knew:
"Harmonious sounds are given off by strings whose lengths are to each other as the ratios of whole numbers [i.e., fractions]". The relationship of one note to
another can be demonstrated through fraction multiplication (e.g., the octave above any note is 1/2 its length and twice the frequency).
Description
The first lesson begins with an explanation that fractions are used in both note duration and note pitch. The notes and their durations are presented as fractional parts
of a whole note. Twinkle, Twinkle Little Star is examined to introduce the rudimentary components of a sheet of music: time signature, one measure.
The students are then challenged to fill in the missing note of several measures of Oh Susanna by adding up the durations of the other notes in each measure.
To visualize the "harmonious sounds given off by strings whose lengths are to each other as the ratios of whole numbers" students shade in the appropriate
fractional portion of a whole representing middle C.
The lesson concludes with students breaking up into two groups to master the classic Row, Row, Row Your Boat. Each note has an associated tube. The goal is to
assemble the notes in the correct order and successfully play the tune.
The second lesson focuses on how the notes are related to each other in pitch (as ratios of whole numbers). The fractional part of each note is multipled
by the length of the middle C tube, to produce the length of each successive tube to play a C Major Scale.
The remainder of the session is dedicated to playing Camptown Races. The song is broken up into 5 parts. Partial instruments are used to play each
of the five parts. It becomes a battle of the bands as up to three groups strive for victory.
Handouts/Worksheets
Music and Fractions
(4 pages  print back to back)
Musical Pitch
KEY
Camptown Races Sheet Music
(two per page)
Reference(s)
www.philtulga.com  interactive musical website
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Purpose
The primary purpose of this activity is to set the stage for the activity Using Magic Squares (A016). This activity introduces many clever techniques for generating Magic Squares of different sizes. Magic Squares have intrigued mathematicians and nonmathematicians for centuries. Students share in the excitement as they realize they can generate nontrivial Magic Squares with the tools presented here.
Description
For centuries, the construction and examination of Magic Squares have occupied people's time of all levels of mathematical talent. Ben Franklin and the German painter Albrecht Dürer (1500s) spent countless hours developing their own techniques of construction.
This activity serves as a motivation for many follow up activities for 5th grade and beyond. In particular, Magic Squares can be constructed using fractions, decimals, and negative numbers which are the focus of activity Using Magic Squares (A016).
Using an interactive approach, students are provided a Magic Squares packet that guides the activity through a variety of strategies for constructing Magic Squares. Additional patterns and properties are highlighted for each Magic Square following construction.
Reference(s)
Heath, Royal Vale. Math E Magic. New York, NY: Dover Publications, Inc, 1953, pp. 8789.
Simon, William. Mathematical Magic. New York, NY: Charles Scribnerâ€™s Sons, 1964, pp. 110, 127129.
Pickover, Clifford. The Zen of Magic Squares, Circles, and Stars. Princeton, NJ: Princeton University Press, 2002, pp. 4849.
Kenda, Margaret. Math Wizardry for Kids. New York, NY: Scholastic Inc., 1995, pp. 289290.
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Purpose
The focus of this activity is to provide practice in adding and subtracting fractions, decimals, and negative numbers. This activity can be tailored to meet the needs of any 5th grade class. Separate worksheets are available for varying levels of each topic. For example, an entire session can be devoted to working strictly with decimals. On the other hand, the activity can be extended to work with multiplication of fractions or decimals.
Description
Magic Squares can be used to motivate students to practice addition and subtraction of fractions, decimals, and negative numbers. Engaging the students into the mysteries of Magic Squares is best accomplished through the recreational activity Interactive Magic Squares (A075), which provides the necessary tools for constructing Magic Square needed for this activity.
Using Magic Squares is a very versatile activity. It can focus on a single topic (e.g., addition of fractions), or it can encompass multiple topics. It is even possible for this activity to span multiple days. There are a few rules that must be followed to successfully generate a Magic Square that sums to the same value for each of its rows, columns, and diagonals. First, a starting number is chosen (it need not be "1"). Second, an increment needs to be decided upon (i.e., the difference between successive values). For example, one could generate values for a Magic Square by choosing 0.1 as a starting value with an increment of 1.1. This would generate the 9 values: 0.1, 1.2, 2.3, 3.4, 4.5, 5.6, 6.7, 7.8, and 8.9. This set of numbers can be used to generate a Magic Square with sum = 13.5.
For more advanced lessons, a formula can be introduced to predict the sum before the magic square is generated, providing an intriguing setting for 5th grade Algebra and Functions standards. It is also possible to construct Magic Squares with all rows, columns, and diagonals multiplying to a common value.
Fractions 
worksheet

worksheet answers
Decimals 
worksheet

worksheet answers
Negative #s 
pending

pending
Reference(s)
Andrews, W. S. Magic Squares and Cubes. New York, NY: Dover Publications, Inc, 1960, pp. 5462.
Pickover, Clifford. The Zen of Magic Squares, Circles, and Stars. Princeton, NJ: Princeton University Press, 2002, pp. 111113.
Worksheets are original.
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Purpose
This activity provides practice in computing the perimeter and area of squares and rectangles. Equally important, it provides insight in how area and perimeter are interrelated. In particular, how is area affected when a rectangle is transformed into another rectangle maintaining the same perimeter? What happens when the roles are reversed? Most people's intuition is challenged as these questions are posed. The relationships can even be graphed to organize the results. This session is a great example of why we need mathematics.
Description
Perimeter vs. Area is a recreational look at how area and perimeter are intertwined. What happens to the area when a rectangle is transformed into other rectangles, maintaining the same perimeter each time? What does your intuition say? What does mathematics say? What happens when the roles of perimeter and area are reversed? These are the questions that are explored and answered in this activity.
After the students' predictions have been tallied, the diagrams are examined using the definitions of perimeter and area along with a mathematical interpretation of the transformations. The intuition of students is aided by a manipulative rectangle that demonstrates simple transformations without the need to perform any computations. A combination of worksheets and graph paper allow students to experiment with various transformations. Students can even organize their findings by graphing each transformation as the perimeter or area is held constant. Which rectangle provides the greatest area for a fixed perimeter?
The honeycomb illustration suggests that bees know something about area and perimeter. Do bees really use a minimum amount of wax in creating the storage for honey, pollen, and their young?
Reference(s)
Bolt, Brian. Mathematical Activities. New York, NY: Cambridge University Press, 1982, p. 28 (#38).
Burns, Marilyn. About Teaching Mathematics. Sausalito, CA: Math Solutions Publications, 2000, pp. 54, 57, 256, 260.
Worksheets are original.
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Purpose
This activity provides a challenge to one's intuition. Students learn that two figures with the same surface area don't necessarily have the same volume.
In the process, they get lots of practice computing surface area and volume of rectangular prisms.
Description
The opener for this activity challenges the students to decide whether or not the volume changes when two identical sheets are rolled:
one lenghtwise and the other widthwise (see the minislideshow above). Students will not do any calculations with the cylinders, but they
will see what happens when one of them is filled with split peas and poured into the other one.
The above demonstration serves as a motivation for the students to experiment with rectangular prisms. Students will fold cardstock lengthwise and widthwise to form
two different regular prisms with identical surface area. The surface area can be computed two ways: compute the surface area of the four sides of the rectangular prism or
finding the area of the cardstock sheet. The volumes of both rectangular prisms are then computed. Are they the same?
For more indepth analysis, students can ponder these questions:
Worksheet
Surface Area vs. Volume worksheet

KEY
Reference(s) Opening activity: http://figurethis.nctm.org/challenges/c03/challenge.htm
Worksheets are original.
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Purpose
The primary purpose of this activity is to reenforce the concepts of Base 10 by providing students with an opportunity to experience another
number system: Base 2 or Binary  the language of computers. One of the best ways to understand Base 10 is to see the correlations with other bases.
Description
This activity begins with two magic tricks:
Base 10 is reviewed using the worksheet "Defining Base 10" (see below). The primary components are emphasized: digits, place value, and basic operations.
Requiring fast action responses, students are challenged to write the successor of a set of called out numbers as well as the predecessor of another set. There are a few
that might catch them off guard.
Next, students are introduced to Napier's Chessboard Calculator (click animation at the right). It is a device for introducing Base 2, or Binary, to students.
The device can be used to show
how to convert Base 10 numbers to Base 2, count by adding 1 to existing numbers and perform the basic operations of addition, subtraction, and multiplication.
Twenty such devises are available for student use, allowing students to discover Base 2 in pairs. A second worksheet, "Defining Base 2", will facilitate
the instruction.
The animation at the right helps to illustrates further the major concepts encapsulated in the Base 2 number system: place values double as you move to the left and halve as you move to the right, any whole number can be represented as the sum of a subset of the Base 2 Place Values (e.g., 13 = 8 + 4 + 1) and that representation is unique.
With a basic understanding of Base 2, the final step provides students with laminated devices that they manipulate to determine the number chosen by the magician. There are a few key points to highlight in closing:
Handouts/Worksheets
Defining Base 10:
worksheet

completed worksheet
Defining Base 2:
worksheet

completed worksheet
Reference(s)
Napier's Chessboard Calculator
 interactive website for learning Base 2 on a Chessboard Calculator
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Purpose
This activity focuses on different strategies for adding and subtracting decimals: complements, rounding and adjusting, a subtraction method
called Equal Additions. Of course, the key to adding and subtracting decimals is to line up the decimal points. So let's put the points down
first!
Description
TBD
Worksheet
Add & Sub Decimals  KEY  Print (back to back). 
Reference(s)
Round & Excess Method:Handley,Bill. Speed Mathematics. John Wiley and Sons Inc, 2003, pp. 6263.
Equal Additions: Maxfield, Clive. How Computers Do Math. Hoboken, NJ: Wiley & Sons Inc. Publications, 2005, pp. 8285.
Worksheet is original.
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Purpose
This activity provides a roller coaster analogy for visualizing the rounding of numbers. The notes section contains a number sense approach to understanding
how to find the midpoints used in rounding, instead of just a mechanical method of performing the problems.
Description
Students are guided through a series of notes in order to understand the pivotal midpoints used in rounding. They are then introduced to a Roller Coaster analogy
for determining where a decimal number is rounded given a place value.
Worksheets/Notes
This section contains a template from which students can take notes. A worksheet follows. They can be printed back to back.
Notes  KEY  
Worksheet  KEY  Notes and Worksheet may be printed back to back 
Reference(s)
Reys, Robert E. Helping Children Learn Mathematics. Hoboken, NJ.
John Wiley and Sons Inc., 2007, pp. 36  38.
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Purpose
This activity extends the Division Strips explored in the 4th Grade activities.
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 4th Grade activities, 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/Notes
The first worksheet contains a set of division problems that students complete using long or short division. Then, they use Division Strips to check
their answers.
The second printout is a combination of notes (students will fill out during lesson) and worksheets. The first page is notes
and should be printed a a separate page. The remaining two pages are the worksheets, and should be printed back to back.
The bonus worksheet
contains four empty Division Strip boxes that need quotients, remainders, and all the lines!
Worksheet #1  KEY  
Notes & Worksheet #2  KEY  Print Page 1 (notes) separate. Pages 2 & 3 (worksheets) back to back 
Bonus Worksheet  KEY  
Reference(s)
Colgan, Lynda. Mathemagic. Tonawonda, NY: Kids Can Press Ltd., 2011,
pp. 36  38.
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Purpose
This activity provides an introduction to division with multidigit divisors: all divisors divide evenly into the dividends.
Additionally, students are taught to look for ways of factoring the divisors and
extending their use of short division. It also provides number sense as it incorporates the skills learned in the activities
Place Value Slider and
Division Strips.
Description
Division FactorEase is a secret branch (or division) of Mathematics Magic. It is so secret that the rest of the world refers to it as
"Division Factories" (you kinda have to play along). To the outside world, we are an organization that specializes in performing
long division with multidigit divisors. That's especially true with our main customer "So Long Division". However, internally we do more than just long
division. To check our work, we factor the divisors, and perform a "stack" of short divisions. We then play with our favorite tools to further verify
our work. Our employees are highly trained in the use of Place Value Sliders
(both Base 10 and Base 2) and Division Strips.
Here at Division FactorEase we enjoy our work. It doesn't matter that we go far beyond what the rest of the world expects. The organization
was founded on the same principles that make teachers go into teaching: the love of sharing knowledge and helping employees (students) grow.
First students will complete the external verification forms consisting of the long division exercises requested by our customer "So Long Division". Then,
they will play with the divisors to come up with a factorization that will creatively use the tools mentioned above, keeping in mind that there are no
remainders in the original problems. In the process they will complete internal verification forms by computing a series, or stack, of short divisions, and then a flow
diagram illustrating the input and output of each tool used in the verification process.
Worksheets (i.e., Verification Forms)
External Verification Form  
Print KEY  
Internal Verification Form Examples (for Notes) 
Print (back to back)  View only 
Print KEY (two pages)  View Key  
Internal Verification Forms  View only  
Print KEY  View Key  
Reference(s)
Stoddard, Edward. Speed Mathematics Siplified. New York, NY:
Dover Publications, Inc., 1994, pp. 198  203.
Incorporating the Place Value slider and Division Strips is original.
The FactorEase story is also original and should not be shared with the outside world, especially So Long Division.
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Purpose
Strategy games are a way to challenge students to think logically or practice a skill such as divisibility (see Divide and Conquer)
Description
Strategy games are a great way to reward students and
provide them with an environment that promotes insight as well as success.
Strategy Game  Reference  
Divide and Conquer Game  Zeitz, Paul, "The Art and Craft of Mathematical Problem Solving" The Great Courses DVD 2010.  
Print View Only  
Reference(s) listed alongside each game
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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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