Title: A Visual Representation of the Fibonacci Sequence

Grade Level: 4^th

Time Length: one 50-minute period

Class Size: adaptable

Subject Areas: Math & Art

Objectives:

• Students will construct a visual replication of the Fibonacci sequence.
• Students will recognize the pattern found within this sequence both numerically and visually.

Materials Needed:

1.) Sheets of graph paper

2.) Sheets of computer paper

3.) Rulers

4.) Markers

5.) Pens/pencils

6.) Scissors

7.) Tape

Procedure:

1.) Teacher will begin by introducing the Fibonacci sequence to students (1,1,2,3,5,8,13…) This concept can be derived from the lesson introducing Pascal’s Triangle.

2.) Utilizing the inquiry method, the teacher will encourage students to pair or group up and discover the numerical pattern behind this sequence.

3.) After a period of “think time” has elapsed, the teacher will pull the class back together and record student discoveries on the black board. Allow time to discuss the patterns found.

4.) The teacher will pass out the graph paper, computer paper, and rulers to the students.

5.) Students will find the middle square on the graph paper. Students can find the middle square by folding the paper vertically and horizontally and then noting the intersection.

6.) Once the middle square is found, students will make a notch at this point.

7.) Student will then proceed to mark off the Fibonacci sequence on the horizontal line to the to right and left of the middle mark.

8 5 3 2 1 1 1 2 3 5 8

ß-----------------------------.-----------------------------à

8.) The graph paper normally accommodates up to “8” in either direction. It is crucial for students to leave a 1-box border after the “8 mark.”

9.) When the notches are complete, students will place the computer paper over the graph paper and draw horizontal lines that correspond with the notches. These lines should run side to side on the white paper. There should be a horizontal line running through each sequence notch including the extra one box border.

10.) Students will then turn the computer paper 90 degrees so that vertical lines can be drawn from the top to the bottom of the white paper. Once again, these lines should correspond with the notches of the Fibonacci sequence. Remember to include the one box border!

11.) Students will cut along the lines of the one box border to create a square sheet out of the computer paper.

12.) Once the square is cut, students will be asked to chose 1 or 2 colors of markers. Students will color every other rectangle that has been created out of the intersection Fibonacci lines. If the students choose one color, every other box will be white. If the students choose two colors, they will alternate the colors as they color the rectangles. Students should use a black marker to color the one box border

13.) Students need to be encourage to stay in the lines and color evenly and smoothly—This aids in the visual effect of the pattern.

14.) A space on the classroom wall needs to be cleared in order to display the patterns. When students are finished, they will hang their patterns next to each other on the wall with tape.

15.) When all the patterns are hung, the class will stand back and observe the visual effects of the Fibonacci sequence!

Post Discussion:

1.) The teacher will facilitate a discussion of the visual effects that they students experience while viewing their visual representation of the Fibonacci sequence.

2.) After gathering student feedback, follow up questionings should be used to discuss why the students see what they see. For example, why do the intersecting straight lines appear to be wavy or spiraling?

3.) Important—this discussion session should be centralized around student observation, opinion, and hypothesis. The teacher is the facilitator of this discussion, not the answer key!

Assessment:

• Informal—As the discussion facilitator, the teacher will keep a log of student responses and inquiry. It is the teacher’s role to make sure that all students are participating, contributing, and asking questions.
• Formal—The teacher will assess the finished visual replications of the Fibonacci sequence.

Fibonacci Pattern Rubric

Completion of Project	Followed correct procedure for the project; Project is completed and displayed; Student participated in assignment 3	Few mistakes made in terms of following directions; Project is mostly completed and displayed; Student participated in assignment 2	Did not follow directions; Project is mostly completed and displayed; Student did not fully participate in assignment/class community disruptions 1
Evidence of Fibonacci Sequence	Notches on graph paper correspond to Fibonacci sequence; Lines correspond to numerical sequence; 1-box border is present 3	Minor errors with notches on graph paper; Most lines correspond to numerical sequence; 1-box border is present 2	Notches on graph paper do not correspond to sequence; Lines correspond with incorrect notches; 1-box border is absent 1
Artistic Expression	Students stayed in the lines and colored smoothly/evenly; 1 or 2 colors are used; Colors alternate 3	Minor errors with coloring process (veer outside lines, blotchy coloring, etc.); 1 or 2 colors used; Difficulty alternating different colors 2	Errors with coloring process (veer outside lines, blotchy coloring, etc); 1 or 2 colors used; Problems with color alternation 1