Comenius "Maths is everywhere" - Iceland

Réttarholtsskóli Ásta Ólafsdóttir Project manager

Transformation Four transformations of the plane have special importance. They are the four rigid motions of the plane: translations, rotations, reflections and glide-reflections.

The definition for the Transformation of the Plane is the one-to-one correspondence of the set of points in the plane to itself. If point P corresponds to point P1, then P1 is called the images of P under the transformation. Point P is called the preimage of P1.

The definition for the rigid motion of the plane is the transformation of the plane if, and only if, the distance between any two points P and Q equals the distance between their image points P 1 and Q1.

That is, PQ=P1 Q1 for all points P and Q.

Reflection Reflection is also called a flip or a mirror reflection. Reflection is an exact copy of the original image.

When working with this type of transformation in the coordinate system the image is flipped over the reflection axes by means of coordinates. Reflection can be found everywhere, e.g. in nature, faces, bodies etc.

Here we can see the reflection of a mountain in the water and reflections in leaves.

Translation Translation, also known as a slide, is the rigid motion in which all points are moved the same distance in the same direction.

Here you can see it in fish skin and a wasp’s body.

Rotation A rotation, also called turn, is the movement where the image is rotated around a center-point through the same number of degrees - the turn angle or angle of rotation. It is turned counter clockwise.

Nature uses rotation to arrange petals in flowers and it can also be seen in tornados.

Pattern Pattern is usually made up of a certain pattern unit that is repeated over and over again, regularly or irregularly. Patterns can be both two dimensional and three-dimensional.

Dilation, also called size transformation, is the simplest transformation to change the size of a figure, but it preserves its shape and orientation. Patterns can be found all around you, including in animal fur, fabric and art.

Here we can see a dilation, or size transformation in the year rings of a tree and a puffin‘s nose

Polygons – Polyhedrons

Polygons are to be found all around us. They are only outlines of a subject such as triangles, squares and other quadrilaterals, pentagons, hedragons and circles. A plane of a polygon has an area and they are two dimensional. Polyhedrons are three dimensional such as all kinds of prisms, cylinders, cones and pyramids. They are made up from polygons and their surface area and volume can be measured.

Math and banks When banks advertise cakes are to be found on the internet we get interested in math  In Icelandic “kaka”– cake is a nickname of pie charts – “kökurit”. So how about baking the pie chart and work on math? Both cakes and pizza’s are good for learning about fractions and percentages.

An easy math cake Ingredients 25% of the recipe is liquid   

3 eggs 75 ml. water 25 ml. oil

75% is powder content 

Example flour and sugar

cream and food colors

And now it’s time to do fractions and percentages.

Are people thinking of getting a dog?

Taking on a dog is a very big decision. Dogs require a lot of love and care, discipline and time from the owner.

What does it cost to get a dog?

It is impossible to put a price on it. Pure-breed dogs can cost a fortune. It can be somewhere between 100.000kr. to 500.000kr. but mixed breeds are less expensive or between 30.000 50.000kr.

Below you can see the major expense of a dog:    

Food 2.000 – 5.000 kr/pr month.

A required dog license fee in Reykjavík is 17.500 kr. and they must be paid annually.

Vaccinations and physical examinations 3-4 times a year.

Brush, shampoo and conditioner need to be bought regularly.

The cost is probably not less than 100.000-120.000kr. per year.

And more the first year when you need to by:  Discipline course.  Chewing bones and toys are essential for puppies that are taking teeth.  Food and drinking bowl.  Neckband and leash.  Neck strap and leash  Dog cage.

Comparison of the distribution of employment in Iceland in the years 1900 and 2007

1900

2007

•

Agriculture

82,0 %

•

Agriculture

•

Services

11,0 %

•

Services

•

Fishing

6,0 %

•

Fishing

•

Industry

1,0 %

•

Industry

3,4 % 73,4 % 2,5 % 20,7 %

Measurements

Cup cakes with chocolate bits Fan oven heat 175°C 200 gr butter 3 dl sugar 3 eggs 6 ½ dl wheat 2 ¼ teaspoon baking powder 2 teaspoon vanilla 1 ¼ dl milk 1 ½ dl chocolate bits

Football The ball is 410-450 g Atli, one of our math teachers in Réttó

The radius of the midfield circle of a football field is 9.15 meters so the area is around 263.02 m² (9.15 ∙ 9.15 ∙ = 263.02 m²).

The midfield circle is then 3.5% of the whole field (263.02 / 7500 = 0.035 = 3.5%)

Basketball The Court

The ball is a sphere.

The diameter of a basketball hoop is 45cm.

The form of a basketball hoop is a ring.

The blackboard is rectangular.

The court lines form all kinds of shapes.

The height of the basketball hoop is 3.05 meters.

The court is divided into two equal parts.

The Game The game is played by 2 teams of 5 players each.

The court is 28m x 15m ( 420 m2) A basketball game is 60 minutes, with intervals.

The game consists of 4 x 10 minutes Half-time interval of play is 15 minutes (Âź of an hour).

The intervals between the 1st and 2nd is 120 sec and the same between 3rd and 4th.

Extension can occur if the teams have equal goals. Each extension is 5 min (1/12 of an hour).

The measurments of the court in feet

Championship

LA. Lakers Boston Celtics Miami Heat Chicago Bulls

American Football - Rugby •

Rugby is played on a rectangular field that measures 120 yards long and 53.3 yards wide. Lines marked along the ends and sides of the field are known respectively as the end lines and side lines, and goal lines are marked 10 yards outward from each end line.

•

The field is divided into 10 parts, so each part is 4.6 meters. The lines go across the field, and are labeled with a number that show how far it is to the next goal line.

•

The game is played by 2 teams of 11 players each and it consists of four 15-minute game parts with half-time after 2nd period and a short break between the other quarters.

The Golden Ratio The Golden Ratio is also called many other names such as the golden section, golden mean and mean ratio, medial section, divine proportion, golden proportion, and the golden number.

The Golden Ratio has been represented by the Greek letter (phi), after Phidias, who is said th to have employed it in his work. Phidias, who lived in the 5 century BC, was a Greek painter, sculptor and architect. He is commonly regarded as one of the greatest of all sculptors of Classical Greece .The Parthenon, was built in 447 to 438 BC, appears to use it in some aspects of its design to achieve beauty and balance its design.

The Golden Ratio is an irrational number that has special mathematical properties. The golden spiral is also said to hold aesthetic values.

The ancient Greek Euclid (365–300 BC) wrote of it in “Elements” as “dividing a line in the extreme and mean ratio.” Fibonacci (1170–1250) mentioned the numerical series now named after him. The ratio of sequential elements of the Fibonacci sequence approaches the golden ratio asymptotically. Johannes Kepler (1571–1630) described the Golden Ratio as one of the two great treasures of geometry, the other being the theorem of Pythagoras.

The Golden Ratio can be found in the design and beauty of nature as well as in being used to achieve beauty and balance in the design of art.

Phi and the Golden Ratio in Art Two quantities a and b are said to be in the golden ratio φ if:

The golden ratio, phi, is 1.6180339887…

Architecture Its use started perhaps as early as with the Egyptians in the design of the pyramids. When the basic phi relationships are used to create a right triangle, it forms the dimensions of the great pyramids of Egypt, with the geometry creating an angle of 51.83 degrees, the cosine of which is phi, or 0.618.

Paintings The Golden Ratio was used extensively by Leonardo Da Vinci. Here one can see his writing and drawing of the Vitruvian man. The proportions of the figure do not actually match it, and the text only mentions whole number ratios as in the Fibonacci sequence.

The Golden Section  Sometimes we speak of a certain proportion between the body parts  The ulna in the forearm of a person equals the length of the person’s feet.

 The distance between somebody’s neck and the tip of his nose equals the length of his foot.

 Golden Section in the forearm and hand

Dilation by a scale factor Scale is a ratio that describes a size of an object to the real object. For example if you find a dimension of an object is too large or too small you can use a model and scale it. A scale factor > 1 enlarges the shape. A scale factor < 1 reduces the shape.

Dilation is a size transformation for similar shape, it preserves the original shape and orientation.

Here in the picture is a dilation of the scale 1:4

The rhythms of life All aspects of musical form are „shaped“ by the relationship between time and the rythms of life. Rhythm is used to describe the time structures and tempo determines the speed of the beat. Like the rhythm in nature, the beating of the heart and the seasons‘ cycles, musical rhythm organizes notes into regularly occurring patterns. These patterns regulate the motion of the music.

Piano keyboard and the Fibonacci Series 1, 1, 2, 3, 5, 8, 13...

Diatonic scales are the basis of Western musical tradition, visually evident on a piano keyboard as a 7-note octave-repeating musical scale comprising 5 whole steps and 2 half steps. The piano keyboard is also structured using the Fibonacci series harmonic progression. The 13-note chromatic octave comprises 8 white keys (whole tones) and 5 black keys ( half-tones, sharpes and flats) arranged in groups of 3s and 2s to make a full octave (including the start of the next octave).

Pythagoras Pythagoras was fascinated by the study of music because it embodied numerical relationships. He discovered that tune strings on a musical instrument sounded pleasant, or in harmony, when their lengths were in unison; related to each as equals, but with pitches an octave apart. The strings‘ length are then divisible by 2, or in 1:2, 2:3, 3:4 (1, 2, 3, 4) proportions of length; note the key role of numbers 1, 2 and 3. These proportions also appear in the strongest overtones, known as „partials“ or „harmonics“. Harmonics are found blending in every single musical sound, like additional invisible strings being sounded at the same time. These additional harmonics turn mere noise into the art form „music

Note

American name

Value

Dotted Value

Double Dotted Value

whole note

4

6

7

half note

2

3

7/2 or 3 1/2

quarter note

1

3/2 or 1 1/2

7/4 or 1 3/4

eighth note

1/2

3/4

7/8

sixteenth note

1/4

3/8

7/16

thirtysecond note

1/8

3/16

7/32

At School

Í skólanum, í skólanum

er skemmtilegt að vera.

Við lærum þar að lesa strax

og leirinn hnoðum eins og vax.

Í skólanum, í skólanum

Er skemmtilegt að vera.

Math and dance Music and math play a big role in dance. Every step is a pattern of movements in different dimensions to the rhythm of music or sometimes to time in silence. Ballroom dancing is danced to music with tempo regulation such as waltz to the beat 28-30 bars per minute, 3/4 time.

Ballett  A pirouette is a turn. It is usually taken in whole circles but sometimes double, triple or more. Other turns can be a fraction of a whole circle.  A piqué and a grand battements, where the one leg is taken from the floor is placed in a certain angel or kicked in to a specific height.  Grand écart, also known as splits is when the dancer opens the legs in 180° front or sideways á la second.  Movement can be fast or slow and a tension can be made on a delay of the movement to the music.