Category: Left Right Brain Learning

Math and The Chromatic Scale: Loving Music, Loving Math


Gold Harp wiki Nov 7Harmony
occurs in music when two pitches vibrate at frequencies in small integer ratios.  Long ago, Greek people realized the concept of harmony occurred when sounds and frequencies are in rational proportion. i.e., One Octave is equal to when the frequency is doubled, and a tripling of frequency brings the key One Octave higher, and is called a perfect fifth. Though not knowing this in relation to “frequency”, ancient Greeks knew this in relation to lengths of vibrating strings; http://www.math.uwaterloo.ca/~mrubinst/tuning/12.html  (Why 12 Notes to The Octave?)

 

1/1  unison              C

2/1  octave              C

3/2  perfect fifth       G

4/3  fourth              F

5/4  major third         E

8/5  minor 6th           Ab

6/5  minor 3rd           Eb

5/3  major 6th           A

9/8  major 2nd           D

16/9 minor 7th           Bb

15/8  major 7th          B

16/15 minor 2nd          C#

Piano Player Creating and LearningThe most common conception of the chromatic scale before the 13th century was the Pythagorean chromatic scale. Due to a different tuning technique, the twelve semitones in this scale have two slightly different sizes. (En. Wikipedia. org/wiki/Chromatic_scale)  Thus, the scale is not perfectly symmetric.  http://strathmaths.wordpress.com/2012/02/22/tipping-the-scales-some-of-the-mathematics-behind-music/.  Pythagoras, 13thC Greek mathematician, was famous in geometry for the Pythagorean theorem (en.  Wikipedia. org / wiki/Pythagoras).  The theorem states that in a right-angled triangle, the area of the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares of the other two sides—that is, a^2 + b^2 = c^2.  Pythagoras experimented with a monochord, noticing that subdividing a vibrating string into rational proportions produced resonant sounds. When the frequency of the string is inversely proportional to its length, its other frequencies are simply whole number multiples of the fundamental.  (En. Wikipedia. org/wiki/Chromatic_scale)

The term chromatic derives from the Greek word chroma, meaning color, where the total chromatic / aggregate is the set of all twelve pitch classes; an array being a succession of aggregates.  Shí-èr-lǜ (Chinese: 十二律 (twelve-pitch scale) is a standardized gamut of twelve notes. The Chinese scale uses the same intervals as the Pythagorean scale, based on 2/3 ratios (2:3, 8:9, 16:27, 64:81, etc.). The gamut or its subsets were used for tuning and are preserved in bells and pipes.  In China, the first reference to “the standardization of bells and pitch,” dates back to around 600 BCE.  According to ancient scroll/script literature, Pythagoras taught that music was not intended for entertainment, though for calming the mind and bringing about order from chaos of life and the universe using spiritual instruments.  Music of the Spheres is one of the phrases used to describe Ancient Greek Pythagorean Music.  Here is a sample of what this music sounds like: http://www.youtube.com/watch?v=Bm2Pn_8Oxww  This clip is a short educational video on a Pythagorean Tone Generator: Pythagorean Tone Generator: http://www.youtube.com/watch?v=BhqgOH0gDIc

James Hopkins, a student and practitioner of Pythagorean Monochords visually shows us his handmade Monochord Stringed instruments: http://www.youtube.com/watch?v=tbCZO6rPcY8

For more cool Learning Math Games, feel free to visit us here:

http://www.math-lessons.ca/fraction-games-activities/

http://butterflybooks.ca/math-activities/

 

Golden Mean Ratio: Egyptian Sculpture

Math in Ancient Sculpture is one of the most interesting and intriguing applications of mathematics, both formula equation and geometric structure.  This article examines the Golden Mean Ratio and Fibonacci FibonacciChamomile wikip Oct 7 2013  Nefertiti Neues Museum Wikip Oct 7 2015Sequence found in the Bust of Nefertiti in Egypt in the early 1900’s. Nefertiti literally, means, “the beautiful one has come”, and is the 14th-century BC Great Royal Wife (chief co-regent) of the Egyptian Pharaoh Akhenaten of the Eighteenth dynasty of Egypt, 1352 BC to 1336 BC.  The iconic Bust of Nefertiti is in Neues Museum of Berlin; though originally in Egypt (En.   Wikipedia.   Org / wiki/Nefertiti_Bust   (230px-Nofretete_Neues_Museum 2 .jpg)  During excavation of Nefertit’s Bust, one of the items found noted Thutmose as being the court sculptor of Egyptian Pharaoh Akhenaten, Nefertiti’s Royal Husband.  In Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science, Stakho speaks of the Berlusian philosopher Edward Soroko’s attempts to determine what mathematical ideas were used in the creation of Her Bust, “…Harmony was the perogative of the Divine order that dominated the universe, and geometry was the main tool of its expression. 

Nefertiti, playing role of Goddess, thus Her image personifying the Wisdom of the world, must have been formed with geometrical perfection and irreproachable harmony, beauty and clarity.  As a matter of fact, the main idea of ancient Egyptian aesthetic philosophy was to glorify the eternal, the measured, and the perfect in a constantly changing universe…In his analysis, he found a harmonious system of regular geometric figures such as triangles, squares, and rhombi.”  (Alekseĭ Petrovich Stakho; Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science; Scott Anthony Olsen. Book. English. 2009; pp 57-59)  Soroko believed that the Bust of Nefertiti was created based upon the Geometries of the Golden Mean Ratio and the Fibonacci Sequence.  A graphical image of Her Bust, as was drawn by Soroko is shown in this link:  http://www.pinterest.com/pin/529454499915732124/  Highly recommended to click on this link to have a finer and more detailed idea of the geometries that Thutmose is thought to have in mind when sculpting Her Bust.  One can appreciate, in particular, how the front top vertex of Her headdress is in perfect alignment with Her Heart and Breast Bone.

Golden Mean Squares Wiki Oct 7 2014Depicted here, the Golden Mean Ratio, and in the geometries thought to be applied by Thutmose, the Egyptian Court Sculptor, in the creation of Nefertiti’s Bust:

n is to m, as m is to n − m,

or, algebraically,

 

n            =         m

_____            ______

m                     n – m

A more clear example of the pattern he believed Her Bust was created upon, the Fibonacci Sequence, can more easily be seen in examining nature’s pattern in pine cones, and seashells.  This ratio is also the most efficient mathematical equation for trees absorbing the most amount of sustenance from the sun’s rays, hence growing in a spiral.  In the same way, today, we can apply solar panels onto rooftops mimicking this pattern to most efficiently absorb the sun’s rays.  For further discussion on Fibonacci Sequence, visit here:  http://www.math-lessons.ca/fibonacci-sequence.html/

See in the photo at the top, the chamomile spiral, as well as in this link on “fibonacci-sequence” showing the pattern.  Adding two consecutive numbers from the sequence to equal the next one following, the basic mathematical sequence looks like this:

0,         1,            1,            2,            3,            5,            8,            13,            21,            34,            55,            89,            144,            233,            377, etc., etc.

That is:

0+1=1

1+1=2

2+1 (the “number” before)=3

3+2 (the number before)=5

5+3=8

8+5=13

13+8=21

34+21=55

55+34=89

89+34=144

144+233=377

etc., etc., etc.

Whereas the fibonacci sequence, cannot technically be exacted in real time, (similarly, there is no real exact end found as of yet to pi’s “3.14159265359…..”), mathematicians drew swirling squares around the spiral in the attempt to exact the formula – basically giving the left side of the human brain a way of understanding the right-brained infinite spiral, being that the left side of the brain requires finiteness / exactness to feel satisfied, shall we say.  

What kind of intriguing application can your class find to prove that math is interesting and fun?

We wonder if the geometries of Nefertiti’s Bust were also the same geometries of Nefertiti’s real-life human head.  The skulls of Ancient Egyptians, as well as those of Ancient Greece and other cultures, have definitely been shown to have geometrically perfect structures.  How can that be?

For samples of our Fun Learning Math Games, feel free to visit here:

http://butterflybooks.ca/geometry-games/ 

http://butterflybooks.ca/math-activities/

 

 

 

 

Developing Number Theory and Fraction Concepts

Many students can begin to feel challenged in math in middle school. Students who have been good at, and have even enjoyed, math suddenly look to their teachers, friends or parents for assistance. Why does this happen? If you look at the concepts that are significant in middle school grades (fractions, decimals and integers), you find that these concepts appear to break all the rules their teachers have told them up to this point.

Typically, students are taught that when you multiply two numbers, the product is always larger. When you divide two numbers, the quotient is always smaller. However, these rules apply to whole numbers, not fractions. When you multiply two fractions, the resulting product may be smaller! When you divide two fractions, the quotient may be larger!  Many students become frustrated, confused and give up on math. As teachers, we need to make sure our students understand concepts, not just memorize rules about them. Students need time to explore and discuss real life examples of the concepts we are teaching. Below are some high-level tasks that allow students to explore number theory and fraction concepts. As with all tasks, students should represent their work in numbers, pictures and/or words. They should have time to communicate their thoughts and findings with others.

Topic: Factors and Multiples

Task: Max is making table favors for a party. The candles come in boxes of 15 and the candleholders come in boxes of 9. Max does not want any leftover candles or holders. What is the fewest number of candles and candleholders he needs without any leftover? How many boxes of each should he buy? Task: At a day camp, there are 12 girls and 18 boys. The camp counselors would like to split the campers into teams. However, they must follow these rules: 1) All campers must be on a team; nobody can be left out, 2) all teams must have the same number of campers, and 3) each team can only have all boys or all girls; no boys and girls can be on the same team. What is the greatest number of camperseach team could have?

Topic: Understanding Fractions

Task: In Penny’s Pet Shop,  of the pets were dogs,  of the pets were cats,  of the pets were birds and the rest were gerbils. There were 48 pets in all. How many of each type of pet were there? Task: Ms. Kinny has  tank of gas in her Volkswagen Beetle. Miss Jamison has  tank of gas in her Ford Mustang. Dr. Beck has  tank of gas in her Honda Accord. Mrs. Hughey has  tank of gas in her Toyota Prius. Without finding common denominators, list the women in order from the person who has the least amount of gas in her car to the person who has the greatest amount of gas in her car.

And for more of our Fun Learning Math Games, you can visit here:

http://www.math-lessons.ca/activities/index.html

http://www.math-lessons.ca/activities/FractionsBoard5.html

http://www.math-lessons.ca/activities/Geometry.html

Developing Deductive Reasoning with Hula Hoop

Here is a fun game to help students in your elementary math classroom – develop their observation skills while at the same time practice their deductive reasoning. My students have named this game “Soup,” and we pretend that we are cooking up a delicious soup. Feel free to adapt it to your own students’ interests.

Materials: Attribute blocks(these are our ingredients) and a hula hoop (this is our pot in which to cook).

 

How to Play the Game:

1)    Have your students sit around the outside of the hula hoop so that they can all see and reach it. The teacher begins the game by creating a rule for the “soup” (e.g. square soup). Without telling the students the rule, the teacher places one attribute block into the center of the hula hoop, saying “This piece belongs in my soup today.”

2)    The first student in the circle chooses any other piece, places in in the “pot” and asks, “Does this belong in your soup today?” If the piece matches the rule, the teacher says, “Yes it does,” and the student gets another turn. If it does not, the student removes that piece from the center, and her turn is over.

3)    Students continue to take turns going around the circle. A student may guess the rule only during her turn. (e.g. “I think you are making blue soup.”) If the student is wrong, her turn is over. If she is correct, she wins the game.

4)    You can continue to play the game by creating a new rule or allowing the winning student to create a new rule for her classmates to figure out.

Notes:

1)    Attribute blocks are excellent tools for this game because they contain four different attributes (color, shape, size and thickness). When I play with very young students, I choose only one attribute (e.g. red soup or triangle soup). However, when I play with older students, I use several attributes (e.g. thick yellow soup or small red triangle soup).

2)    When allowing students to create the rule and start the game, it is a good idea to have them whisper their “soup recipe” in your ear. They tend to forget their rule and provide false information at times!

3)    I find that continuing around the circle after a game is won keeps students from arguing about whose turn it is and gives everyone a chance to play. For example, if the sixth child in the circle correctly guessed the soup recipe, the next game starts with the seventh child in the circle.

For more fun and interesting Learning Math Games, you can visit us here:
http://www.math-lessons.ca/activities/FractionsBoard5.html
http://www.math-lessons.ca/timestables/times-tables.html
http://www.math-lessons.ca/activities/FractionsBoard4.html
http://www.math-lessons.ca/index.html

Primary Geometry All Around

Here’s a fun way to integrate math, technology and language in one project for your youngest learners. Mathematically, students learn about solid figures and their properties. Technologically, they use digital cameras and work with word processing programs to insert pictures, word process and format documents. Students practice writing original thoughts and grammatically correct sentences as they describe the objects they have discovered.

Start the project by having students take digital pictures of solid figures in their everyday lives. They can either do this at home or (if they do not have a digital camera) at school. The following chart lists the common solids primary students learn about and some everyday items children would be familiar with. Hopefully, your students will find numerous examples of each.

Cones

Spheres

Cubes

Rectangular Prisms

Cylinders

-ice cream cone

-construction cone

-cone used for sporting events

-party hat

-ball

-globe

-scoop of ice cream

-marble

-play block

-sugar cube

-dice

-tissue box

-cereal box

-stick of butter

-pack of gum

-juice box

-soup can

-marker

-dowel rod

-stove pipe

-rolling pin

 

Once students have taken their pictures, visit the computer lab so students can create their “Book of Geometric Solids.” Students can organize their books in a variety of ways. However, each picture should have one to three sentences to serve as a caption. Captions must identify the type of solid the object is and must tell something about that object. For example, the student inserts a picture of a soccer ball. He then writes, “A soccer ball is an example of a sphere. I play soccer every Saturday morning. It is my favorite sport.”

This activity not only allows students to discover math in their everyday lives, but it also helps develop multiple academic skills.

For more fun and interesting Learning Math Games, you can visit us here:
http://www.math-lessons.ca/activities/FractionsBoard5.html
http://www.math-lessons.ca/timestables/times-tables.html
http://www.math-lessons.ca/activities/FractionsBoard4.html
http://www.math-lessons.ca/index.html

Teaching Math: A Breeze when Incorporating Favorite Interests

Teaching math in the Elementary classroom can be a challenge, as all elementary math teachers know, but when favorite activities of students are incorporated into the curriculum, it can be a breeze. When students have a required learning skill to attain, and they incorporate their favorite interests, then there is inspiration – and the student becomes “self-motivated”.    This is a great First Item to address with the rounds of new students returning for the fall season.  Find out Learners’ Favorites, and keep the list in a special file.

Then, as the year progresses, if a student is having a challenge learning a particular new math lesson, Teachers can refer to the student’s personal file of “Favorites Activities List”.  At this time, then, introducing the association and how that interest relates to the new lesson.

Example No. 1:  Young Matthew enjoys playing or watching the game of baseball. That is included in their list of favorite activities.  November rolls around, and the lesson of drawing shapes in geometry arises, but Matthew is not grasping the concepts.  If looking down onto a Baseball Diamond from an aerial perspective, the shape of Square is easily seen in the formation of the 4 bases on the ball field.  As well, the shape of the bases individually, is a square.

Then show in sequence what happens when the player runs to first base, second, third and fourth, demonstrating the making of a Straight Line 4 times, and in consecutive order. Within each corner, while the player stands on the base, the player looks down at both straight lines that connect, and the player can then see a perfect Right Angle of 90 degrees.  Drawing a line across from the base to the left to the base t the right demonstrates a perfect Right Triangle.

Suddenly a light is switched on in the child’s brain, and Matthew is on the way to understanding the concept of geometry.  Not only do they understand it on paper in 2-D form, but now in 3-D form, in the context of a baseball game, in a real life scenario.


Example No. 2:
  Ashley likes archery.  Archery is included in her favorite activities list.  While imaging and practicing her archery skills, she sees concentric circles – one inside the other.  When a line is drawn from her bow to the target, she demonstrates a perfect straight line.  Hence, she has a different yet equally effective association of a favorite interest to relate to the concepts in geometry –  as Mathew’s love of baseball.  Imagine now that Susie, not only is attaining the required skills the in geometry lesson, but is also having fun while doing it, and has developed self-motivation and interest in learning math.

In these examples, both sides of the Brain are exercised, (Left Right Brain Learning and Thinking) new neural connectors and dendrites grow, and you have encouraged the growth of a healthy developing young brain.

http://www.funderstanding.com/brain/right-brain-vs-left-brain/.

http://www.oecd.org/edu/ceri/neuromyth6.htm.

Start the school year off right, and find out what your students’ favorite activities are. Keep the lists on file, and refer to them from time to time during the school year.  You may be surprised at the effectiveness of this subtle teaching tool.

For one of Our Fun Learning Math Games, feel free to visit here:

http://www.math-lessons.ca/activities/index.html

http://www.math-lessons.ca/activities/Geometry.html

http://www.math-lessons.ca/activities/chocolate.html