SPIRALS LESSON

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Click on this poster to download a large copy for display.

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• Brochure text
• Spiffy Spirals and Palpable Patterns
• From nature to numbers to nations, become a mathemARTician by creating three dimensional spirals which mimic those formed by living things. Travel across the globe, from Africa to Asia to Europe and then North America to uncover the mathematics within the artwork of other cultures.
• Materials
• Boxes for student pairs
• three sets of polyhedron wooden blocks:
• 6 with 4 side lengths colored purple
• 10 with two side lengths red and two side lengths purple
• 12 with one side length white, two green and one yellow
• three strings
• These paper shapes can be cut out to resemble the blocks. The arrows indicate how the blocks should join. the colors are coded as Cuisenaire rods, showing some of the side lengths. Click on an image for a larger image template.
• 2 uncolored copies of the Celtic spiral
• 4 colored markers or crayons, 2 overhead projector markers
• Van Gogh, Miro, and Calder pictures
• 1 or more pine cones, shells, and (if possible) grapevine tendrils
• Cuisenaire rods (regular size)
• 10 white
• 4 red
• 5 light green
• 2 ìBarneyî- purple
• 5 yellow
• 2 dark green
• 3 black
• 9 brown
• 2 copies of Adinkra cloth symbols
 
• Teacher props
• laminated pictures on display, including:
• jaguar deity and snakes from Peru - from a drawing of a stone relief in the ruins of Chavin de Huantar, Peru, A.D. 200-400
• natural objects
• nautilus, hurricane, galaxy, ramís horn, spider web...
• Adinkra cloth-
• Celtic spiral -
• Mohawk Indian bead work
• Mankara
• other art works (see background from countries and descriptions)
• poster map of the world
• oversized Cuisenaire rods with magnetic strips attached, or clear plastic representations on an overhead projector
• 4 white, 4 red, 5 light green, 2 ìBarneyî- purple, 7 yellow, 2 dark green, 3 black, 10 brown
• Three different wooden bead sets or paper beads
• Magnetic board with ìSpiralsî , your name, and ìFibonacci" written or posted on it with room for writing the Fibonacci sequence later, and for displaying Cuisenaire rods  - or - and overhead projector
• template for drawing Fibonacci points on the board ahead of time
• acetate overlay with the same spiral drawn on it
• hooks for overlay
• Not on display, but for reference later - the colored Celtic spiral poster. 
• glued models of wooden bead put together in spirals
• red cabbage (optional) and cleaver with chopping block or other cutting surface

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• Concepts and objectives
• Spirals vocabulary
• spiral- a ìcurly shapeî in which a line travels in one direction around a fixed point (or, in the case of some such as the Fibonacci spiral, a fixed area)
• continent- one of the seven largest land masses on the earth
• Fibonacci sequence- a pattern of numbers in which a number is the sum of its preceding two numbers. 0,1,1,2,3,5,8,13,21,34,55,89,144...
• Fibonacci spiral (also known as the ìgolden spiralî) which is one kind of growth spiral) a spiral formed in which the areas between the lines have the successive ratios of the Fibonacci sequence
• Archimedian (Donít use unless asked) a sort of spiral that does not have a growing area between the lines. The distance between the lines is always the same.
• Objectives
• Students will use rods to show the growth patterns of two kinds of spirals.
• Students will string wooden blocks to demonstrate that some growth patterns can create spiraling shapes and some cannot
• Activities
• Introduction
• Art and science are ways humans have related to nature, explored nature, represented nature, and explained nature. Mathematics is the language of science and can also be found in art. One shape that appears in nature, in art and in mathematics is the spiral. Imagine that today youíre going to be an archeologist. An archeologist studies manmade things that are very old. Youíll making an imaginary voyage around the world. Youíll be seeing things that are not manmade, and youíll be seeing works of art that are made by humans. Youíll be using your mathematics skills. Today youíre going to be looking for spiffy spirals. Youíll need your imaginary lucky hat so put it on!
• Natural Objects
• Invite students to look in their boxes and see if they can find some natural objects, [shells and pine cones] Ask how they know they are natural [because they are not man-made]. Point out the spiral shapes on the shells.
• Point out that the pine cones are painted with dots to show the spiraling patterns they have.

cone with painted dots	spirals clockwise and counter clock-wise

• Pictures - Ask students to point out pictures of natural objects in the room. Point out that some natural spirals are really big, like the hurricane or the galaxy. Others are tiny, like the insides of some flowers.
• posters for the classroom

radar image of Hurricane Hugo	spiral galaxies	spider web	seed head
sliced nautilus shell	bighorn sheep

• Countries on the map
• Invite students to point to areas on the map what will be mentioned later.
• India
• Mohawk Nation
• Ireland
• Peru
• Ghana
• Locate spirals
• Have students look at pictures in their box (done by famous 20th century painters Van Gogh, Miro, and Calder (Their works can be found at the Baltimore Museum of Art)
• Have them trace spirals with their fingers as you define spiral.
• Have students mark 2 spirals on each of the laminated pictures found around the room.
• You may wish to do this as a cooperative game. Have the class estimate how long it would take for Everyone to find and mark one spiral and then try to beat the estimated time. (Class is done when each student is standing by their mark with their arms folded)
• Spirals as symbols
• Point out the silkscreened imitation of the Africa Adinkra cloth and tell them the story of its origin. Adinkra also means ìgoodbyeî to the Ashanti people from Ghana. The cloth was used for burials and as gifts for those leaving. The symbols on this cloth are the same used on an Adinkra cloth given to an Ashanti king a little over 100 years ago.
• Be sure a student points out the location on the map.
• Ask students what qualities would make a good king.
• Have students refer to the shapes key and tell the meaning of the shapes with spirals.
• Count the spiral lines
• Have students take out the copies of the Celtic spirals from Ireland. This design was cut onto a stone 3000 years ago and was found in the innermost chamber of a megalithic (great stone) tomb.
• Be sure a student points out the location on a map
• Ask them to predict how many lines they will find in this design.
• Have them trace each separate line with a different color marker.
• Show the colored solution that shows three lines.
• Building spirals
• Point out that the Archimedian spiral of the Mohawk design has the same amount of space between the rows of beads (just about NO space in this case). Be sure a student points out the location of the Mohawk Nation on the map. (Northern NY along the St. Lawrence river)
• Show how the Indian beam and the nautilus shell has a growing growth pattern. Be sure a student points out India on a map.
• Discuss the dimensions of the Cuisenaire rods. Point out that the rods are in standard sizes starting with a one centimeter cube. While all are one centimeter wide, they have different lengths of 1,2,3,4,5,6,7, and 8 centimeters.
• Discuss the Fibonacci sequence. Have students mentally add to help you write a sequence on the chalkboard.

1,1,2,3,5,8,13,21,34,55,89,144,...

• Draw the growth spiral on the chalkboard (following the dots)
• Have the students make squares out of rods with the first numbers in the sequence. Point out that these squares can be put together to form a rectangle on which a Fibonacci spiral can be traced. Model this on the chalkboard. Then use the acetate overlay to show how the spiral shape fits over the squares.
• Refer to the Fibonacci exhibit components, especially the artwork by Amalie Rothschild. (Pronounced AH-muh-lee)
• Stringing a growth spiral.
• Have students take one of each kind of wooden block and see if they can tell if and how each is different. [one has all lengths the same from end to end. Another has two different lengths from end to end. The third has three different lengths from end to end.] Point out how the lengths are color coded by length [like the Cuisenaire rods]
• Tell the students that as some things grow in a certain direction, the sides grow at different rates. Think of one block as one day of growth. One side grows longer than the other. Point out that the side of the bead with the black mark on the top is the side that points to the direction of growth. Have students predict what each string of beads will look like if several ìdaysî go by and the beads are string with the lines of the same color touching.
• Direct students to string the beads with the sides of the same length on the same side of the string. have students put their thread into the plain hole so that it comes out of the black hole as they string. (Let students revise their predictions as you demonstrate with your set of beads). With the bead for helix spirals, have them focus on having only the white edges actually touch.
• Point out that the tendrils of peas, the horns on a male sheep ( a ram), and grape vines have this growth pattern.
• Background info: sometimes the growth of one side is inhibited, sometimes the one side is accelerated - both have the same results. In the case of squash or grape vines , when the straight growing tendril rubs on something, (which may potentially support the upward growth of the plant) hormones make the opposite side grow faster
• Summary
• Humans describe nature with art and with mathematics.
• One shape found in nature is the spiral.
• Different kinds of spirals show different kinds of mathematical patterns.
• One pattern of a growth spiral is the Fibonacci sequence of numbers.
• Spirals have been found in art ranging in date from thousands of years ago of today.
• Let me leave you with a question to discuss with your teacher: Which came first: art or mathematics?î
• Bibliography

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