Purpose: Provide discovery, insight, and enrichment to a variety of mathematical concepts, many of which map directly to 5th Grade Standards.
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 octive above any note is 1/2 its length and twice the frequency).
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.
www.philtulga.com - interactive mulsical website
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.
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.
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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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.
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.
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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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.
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?
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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