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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 5th Grade Standards.

Activities List

  Music and Mathematics Introduction to Base 2
   
  Interactive Magic Squares Add & Sub Decimals
   
  Using Magic Squares Roller Coaster Rounding
   
  Perimeter vs. Area Advanced Division Strips
   
  Surface Area vs. Volume Division Factor-Ease
   
  Strategy Games Magic of 142857 - Multi-Digit
   

Music and Mathematics
The Magic of Fractions
(ratios of whole numbers)



Standard
5.NF


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 FractionsPrint  (4 pages - back to back) View Only

Musical Pitch   KEY  

Camptown Races Sheet Music   (two per page)


Reference(s)

www.philtulga.com - interactive musical website

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Interactive Magic Squares (A-075)

Standard
General Number Sense

Purpose

The primary purpose of this activity is to set the stage for the activity Using Magic Squares (A-016). This activity introduces many clever techniques for generating Magic Squares of different sizes. Magic Squares have intrigued mathematicians and non-mathematicians for centuries. Students share in the excitement as they realize they can generate non-trivial 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 (A-016).

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.

worksheet   |   worksheet answers


Reference(s)

Heath, Royal Vale. Math E Magic. New York, NY: Dover Publications, Inc, 1953, pp. 87-89.

Simon, William. Mathematical Magic. New York, NY: Charles Scribner’s Sons, 1964, pp. 110, 127-129.

Pickover, Clifford. The Zen of Magic Squares, Circles, and Stars. Princeton, NJ: Princeton University Press, 2002, pp. 48-49.

Kenda, Margaret. Math Wizardry for Kids. New York, NY: Scholastic Inc., 1995, pp. 289-290.

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Using Magic Squares (A-016) 

Standard
NS 2.0, 2.1 Fractions, Decimals, Negative Numbers. AF 1.2

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 (A-075), 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. 54-62.

Pickover, Clifford. The Zen of Magic Squares, Circles, and Stars. Princeton, NJ: Princeton University Press, 2002, pp. 111-113.

Worksheets are original.

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Perimeter vs. Area (A-012) 

Standard
MG 1.4, DA 1.2 Perimeter and Area

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?

worksheet   |   worksheet answers


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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Surface Area vs. Volume 

Standard
5.MD.5 - Voume of Rectangular Prisms

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 mini-slideshow 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 in-depth analysis, students can ponder these questions:

  • What about the ratio of the volumes vs. the ratio of the sides of the original cardstock?
  • What is the only side ratio that will produce identical volumes?


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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Introduction to Base 2 (Binary)
The Language of Computers

Standard
5.NBT

Purpose

The primary purpose of this activity is to re-enforce 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:

  • Crystal Cube Revisited  Click here to see details
    A motivational opener that elicits mystery and intrigue. A volunteer chooses a number from 1 to 60, and the magician determines the number through a series of questions. The key to unraveling the mystery lies in an understanding of the Base 2 Number System, which just happens to be the focus of the entire lesson.

  • Crystal Planets Revisited (optional)   Click here to see details
    Here the tables are turned: the magician chooses a number from 1 to 99, and the students determine the number with the use of a card. Crystal Planets is the Base 10 version of Crystal Cube. Reasoning and a little guidance exposes how the trick works, providing some insight into how a magic trick can be based on place value.

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:

  • Each side of the cube represents a predetermined place value within Base 2 (e.g., the yellow side represents the ones place). Therefore, it does not matter what order the sides are examined.
  • The numbers on a given side of the cube are the set of all numbers from 1 to 63 that have a "1" in that particular place value, and the absense of a number on a given side means that number has a "0" in that place value.
  • In the end, the series of "yes" and "no" responses for a chosen number translate to the series of 1s and 0s that represent that chosen number's binary string.


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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Adding and Subtracting Decimals 

Standard
5.NBT.B7

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. 62-63.

Equal Additions: Maxfield, Clive. How Computers Do Math. Hoboken, NJ: Wiley & Sons Inc. Publications, 2005, pp. 82-85.

Worksheet is original.

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Roller Coaster Rounding
A Visual Approach


Standard
5.NBT.A.4

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


Standard
5.NBT.B.6

Purpose

This activity extends the Division Strips explored in the 4th Grade activities. 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 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 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 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 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. 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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Division Factor-Ease
(Factoring Multi-Digit Divisors)


Standard
5.NBT.B.6

Purpose

This activity provides an introduction to division with multi-digit 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 Factor-Ease 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 multi-digit 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 Factor-Ease 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  
  Print KEY  
 
Internal Verification Form Examples
(for Notes)
Print (back to back) View only
  Print KEY (two pages) View Key
 
Internal Verification Forms Print 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 Factor-Ease story is also original and should not be shared with the outside world, especially So Long Division.

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Prime Number Activities (A-049) 

Coming soon!

 
Prime Factorization Activities (A-076) 

Coming soon!

 
Strategy Games

Standard
General Logic and Reasoning

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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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 multi-digit numbers by 2-digit and 3-digit numbers. They are also introduced to the strange behavior of the cyclic numbers 142857 and 076923, as the products are rearranged to magically return back to cyclic patterns.

Description

This activity is an extension of the activity, Exploring the Magic of 142857 - Part 1 (4th Grade), that introduces students to the cyclic number 142857. The magic trick Magic of 0769 23 (M-018) motivates this 5th 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 and 076923 are explored by multiplying them 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 remaining digits on the left are brought down (right justified) and added to the right-most 6-digits. 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.

worksheet   |   worksheet answers


Reference(s)

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

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

Worksheets are original.

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