I. Curricular Framework

Concept:

Patterns

Essential Question:

How do patterns help us solve mathematical problems?

Bridge:

Pattern Comparison

Content:

Fibonacci Sequence and the Golden Rule

Outcomes:

II. Standards Aligned

III. Instruction and Assessment

1. Connect: Connecting to the Concept Experientially

Objective: Students will enhance observation skills as they search for patterns in nature.

Activity: Students need a grocery sack, paper and a pencil. The teacher takes the group on a nature walk. Students look for items in nature that contain patterns. When students observe an item with a pattern that can be taken with them (example: a leaf), it should be placed in their sack. If the item cannot be taken (example: a tree), they are to sketch the item and make notes of the patterns observed.

Assessment: Quality of student involvement.

2. Attend: Attending to the Connection

Objective: Students' curiosity about patterns will be aroused as they reflect upon the patterns observed in nature.

Activity: Students share the items found on the nature walk and show the patterns exhibited. Magnifying glasses may be needed to examine patterns in the smaller specimens. Students discuss why they think nature uses patterns.

Assessment: Quality and depth of student sharing and discussion.

Assessment, Phase One, Level of Engagement, Fascination:

3. Image: Creating a Mental Picture

Objective: Students will make a connection between mathematics and the patterns found in nature.

Activity: Students view "Donald Duck in Mathemagicland" and look for patterns in nature shown in the film. Students discuss the patterns observed and responses are written on the chalkboard.

Assessment: Attentiveness of students and quality of discussion.

Assessment, Phase Two, Seeing the Big Picture:

4. Inform: Receiving Facts & Knowledge

Objective: Students will learn the concepts and vocabulary of the Fibonacci Sequence and The Golden Rule. Note: From this point, the unit is divided into two main topics (Fibonacci Sequence and The Golden Rule). There are four Fibonacci activities and six Golden Rule activities. You may wish to teach the material (step 4), and have the students reinforce the learning (step 5) for one activity at a time. All of the activities under a topic should be completed before beginning step 6 for that topic. Students may need to review the three following concepts in decimal use if they are to experience success in the third Fibonacci activity:

Meaning of the decimal part of a number, i.e., that it is part of a whole and that the places represent tenths, hundredths, and thousandths of a whole.

Conversion of any fraction to a decimal by dividing with the calculator.

Measurement of items with a centimeter ruler to the nearest millimeter and naming the result as a decimal number of centimeters. (This review can be found in any good math text.)

Activity: Teach the following Fibonacci Sequence* concepts and vocabulary:

Helices, helix, sequence, Fibonacci Sequence, infinite, finite (correlates with "The Pinecone Numbers" worksheets).

Clockwise and counterclockwise curves (correlates with "Coloring Sunflower Spirals" worksheets).

"Babies," "youngsters," and "adults" in rabbit life cycles; "ideal" versus "real" answers in science (correlates with "Leonardo's Rabbits" worksheets).

Hexagon, parthenogenesis (correlates with "The Number Secret of the Bees" worksheets). Teach the following Golden Rule* concepts and vocabulary:

Ratio, relative and absolute comparisons (correlates with "Introducing Ratios" worksheets).

The Golden Ratio, "Classic" Greek, profile of characteristics, trend, sample, random (correlates with "The Greeks and Their Golden Ratio" worksheets).

"Perfect" human face (correlates with "Golden Faces" worksheets).

Golden Rectangle, arcs, spiral (correlates with "The Golden Rectangle" worksheets).

Calculus, limit (correlates with "The Fibonacci Numbers Strike Again" worksheets).

Commutative (correlates with "The Fibonaccis, The Golden Ratio, and Some Calculator Tips" worksheets). * See resources listed in bibliography for teaching the above concepts.

Assessment: Subjective measure of student understanding.

Assessment, Phase Three, Success with Acquiring Knowledge:

5. Practice: Developing Skills

Objective: Students will enhance higher level thinking skills while reinforcing the concepts.

Activity: Students complete the following worksheets** on the Fibonacci Sequence: The Pinecone Numbers, Coloring Sunflower Spirals, Leonardo's Rabbits, Number Secrets of the Bees. (These correlate with the concepts and vocabulary taught in step four.) Students complete the following worksheets** on The Golden Rule: Introducing Ratios, The Greeks and Their Golden Ratio, Golden Faces, The Golden Rectangle, A Golden Rectangle Puzzle (optional), The Fibonacci Numbers Strike Again, The Fibonaccis, The Golden Rule and Some Calculator Tips. (These correlate with the concepts and vocabulary taught in step four.) ** See resource listed in the bibliography for student worksheets.

Assessment: Quality of student activities and worksheets.

Assessment, Phase Four, Success with Acquiring Skills:

6. Extend: Extending Learning to the Outside World

Objective: Students will personalize their learning through a choice of learning activities.

Activity: Students select one of the following projects and design a plan for its completion (correlate with the Fibonacci Sequence):

Research and report on the life of Fibonacci.

Make and demonstrate an abacus.

Research how the Moslem's invented their number system. Include a diagram of the number system.

Complete a detailed, colored drawing or design using spirals or helices.

Research the spiral and its meaning in ancient art, philosophy, and architecture.

Research and diagram or build the DNA double helix molecule. ï Research and report on infinity. ï Research and report on Einstein and his theory of relativity. ï Research and construct a model of the Leaning Tower of Pisa. ï Research and report on the town of Pisa, Italy. ï Create a detailed, colored drawing of fantasy flowers that shows Fibonacci numbers operating on them in different ways. ï Make a display of climates and soils that encourage growth of different species of sunflowers and daisies. Use a flower field guide as a resource. ï Count the clockwise and counterclockwise spirals and petals of different species of daisies. Record and diagram your findings. ï Find at least three different pine cone species. Paint the scales of each helix a different color. Try to paint four cones of the same species to show the four Fibonacci Numbers for that sequence. Create a display of your work. ï Check the hexagons on a pineapple for number sequences. Chart or diagram your results for a display. ï Draw and color at least five different species of Fibonacci flowers. Take notes on their Fibonacci Sequences and numbers. Use the notes and drawings to create a display. ï Report why ancient Greeks and Egyptians considered "7" a number of "completion." ï Report on the dawning of the modern scientific age with the discovery of Uranus in 1781, by William Herschel. ï Visit a beekeeper and secure a honeycomb to be used in connection with a report on the life of bees. ï Research and report on the "Sieve of Eratosthenes." Students will select one of the following projects and design a plan for its completion (correlates with the Golden Ratio): ï Measure and diagram the ratios of your family's bodies. ï Find at least ten pictures of animals for a booklet. Measure each for the Golden Ratio. Report your findings. ï Measure and diagram at least five real statues for the Golden Ratio. ï Design and build a "Golden Ratio Robot." ï Draw and color five strange faces for a booklet full of Golden Ratios. ï Draw, color and redesign a Greed face and statue so they have no Golden Ratios. ï Create a new ratio, name it, and write about how you discovered it. Draw a diagram showing how it would be used. ï Find faces of at least five different nationalities in magazines. Measure to see which have more or fewer Golden Ratios. Diagram your results. ï Diagram animal species that have spiral or helical horns, tusks, teeth, etc. Include animal parts that would grow in the spiral curve it continued: rhino horn, wolf fang, beaver tooth, fingernail. ï Compare and contrast (include diagrams) spirals of five different shells. ï Make a bulletin board exhibit of spirals in nature. Use pictures or tracings that are colored. Make the exhibit in spiral form. ï Using a ruler and calculator, check at least ten common rectangular objects, both big (door, playground, school wall) and small (light switch, page) to determine if they are Golden Rectangles. Document findings, both negative and positive). ï Report on the United Nations building, its architect, when it was built, how it was built, its floor plan, etc. Was it designed to be a Golden Rectangle? ï Research and report on "calculus" a kind of closer-and-closer study approach to a number. What is a "limit"?

Assessment: Quality of student plans.

7. Refine: Refining the Extension

Objective: Students will develop their creativity and higher level thinking skills as they refine and complete their projects.

Activity: Students meet with teacher periodically to discuss and evaluate various aspects of the project such as resources, objectives, final project form, sharing method, and completion date. Necessary changes and refinements are made.

Assessment: Ability to carry out plan and make necessary changes.

8. Perform: Creative Manifestation of Material Learned

Objective: Students will develop their creativity and higher level thinking skills as they refine and complete their projects.

Activity: Students meet with teacher periodically to discuss and evaluate various aspects of the project such as resources, objectives, final project form, sharing method, and completion date. Necessary changes and refinements are made.

Assessment: Ability to carry out plan and make necessary changes.

Assessment, Phase Five,Performance, Creative Use of Material Learned: