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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
   
  Using Magic Squares
   
  Perimeter vs. Area
   
  Surface Area vs. Volume
   

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 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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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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Prime Number Activities (A-049) 

Coming soon!

 
Prime Factorization Activities (A-076) 

Coming soon!