Tiling the Plane

· Activity Name: Dragon Curves: Recursion & Fractals

· Objectives:

This activity focuses on the students’ ability to explain what is happening when a recursion takes place. It asks the students to do some writing and hypothesizing and then to rework their hypothesis if it is incorrect. The activity also touches upon the idea of Fractals in that the “dragon curve” that they make is a fractal

· EALR/Standards:

1.3 understand and apply concepts and procedures from geometric sense.

1.5 understand and apply concepts and procedures from algebraic sense.

2.1 investigate situations.

2.3 construct solutions.

3.1 analyze information.

3.3 draw conclusions and verify results.

4.1 gather information.

4.2 organize and interpret information.

· Materials:

Strips of paper 3 feet long (1 per student) – Adding machine tape is perfect

Colored markers – 4 or 5 different colors for each group

Worksheets – 1 set for each student

Graph paper – 1 piece for each student

· Teacher Notes

o Prerequisites for the learner:

None.

o Teacher hints for the activity:

This activity works on almost any schedule. Working in groups of four is best for this activity. Watch the students carefully to be sure that they are folding and unfolding correctly.

o Introductory questions:

What do you know about dragon curves or any other mathematical curve?

When you hear the word recursion, what do you think of?

o Wrap-up questions:

What will happen if you change the fold from the middle to the 1/3 mark?

How many times can you fold an object in half?

o Solutions:

Introductory questions: Many students will have heard about recursion through computers

programming.

Wrap-up questions: Have students actually fold at the 1/3 mark and then draw the results. The results will depend on how they decide to fold, and what they interpret the question to be.

If time permits or a few students are willing to work on it after class, have a

roll of toilet paper available for them to practice folding. A very long hall is

a must also. 11 times would be more than possible. This is due to the fact that the length is being cut in half, and the height (thickness) is doubling. Eventually, the thickness is more than the length.

o Assessment suggestions:

Collect the worksheets to verify participation and writing skills.

· The Activity:

Begin by handing out the worksheets and strips of paper. If the strips have been rolled, have the students hold the ends of the strips and then pull the strips tightly over the edge of the desks. Ask the students to fold their strips in half to the back (take the left edge and fold it under so that the fold crease will be on the left.). This should give them a valley fold if they open the paper by lifting the top flap. Explain that this is the first rule of the recursion and that they are to repeat this rule four or five times. Now, the students should be asked to open the paper up according to the following rules: 1) Each fold will open to a 90 degree angle. 2) Only open one “fold” each time. After each opening, the students should sketch a diagram of their papers. So that the diagrams will be the same, ask the students to always have the paper laying on the side that was last folded under (you may want them to glue this last side to a piece of card stock), and to then look at the “edge” of the paper that is closest to them. It is important that the fold is on the left so that everyone is looking at the same edge. With this information, the students should begin the worksheets. It will be important to check that all of the students are folding the left edge of the papers to the back each time, and that the folds are being unfolded correctly.

· Extensions:

The second set of worksheets can be used as an extension, or as a continuation in the class. Have the students do the first set of worksheets again, only this time alternate the fold to the back and the front. Other extensions include connecting the dragon curves to the Towers of Hanoi and ABACABA.

Recursions and Fractals

Dragon Curves (1)

Using a paper strip approximately 3 feet long, you will do a five-generation recursion using the following rule:

Fold the paper in half to the back – the left edge should be folded under so that the fold ends up on

the left side when done.

(Note: A recursion means to repeat the same rule, generations say how many times to repeat)

Begin by laying your paper flat on the table and labeling the right end as “A.” Fold according to the rule 5 times. “A” should always be on top.

Use your paper to answer the following:

1) Lay your paper on your desk with the “A” on the top, fold on the left. Hold the paper flat and look at the edge of the paper that is closest to you. Do the following:

Sketch:

Description:

_________________________

2) Label the side that is laying on the table “B.” Keep this “B” on the table for the rest of the questions. If you open fold number 5 to 90 degrees. What does your figure now look like? Sketch:

Description:

_________________________

3) If you were trying to tell someone how to draw this over the telephone, what “rule” would you give him or her to get from number 1 to number 2? _______________________________________________________________________

_______________________________________________________________________

4) With the fifth fold still open, open the fourth fold to 90 degrees. What does your new figure look like? “B” is still on the table.

Sketch:

Description:

_________________________

5) Does your “rule” in number three still work for the transition between questions two and four? Yes, or No If no, write a new description that will work for both transitions (1 to 2 and 2 to 4) _____________________________________________________________________________________

____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6) With the fifth and fourth folds still open, open the third fold to 90 degrees. What does your new figure look like? “B” is still on the table.

Sketch:

Description:

_________________________

7) Does your “rule” in number 5 still work? Yes or No If no, fix it again. If yes, do you think it will hold for the next transition? Why or why not? _____________________________________________________________________________________ _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

8) With the fifth, fourth and third folds still open, open the second fold to 90 degrees. What does your new figure look like? “B” is still on the table.

Sketch:

Description:

9) You are now talking to your friend on the telephone, write out your instructions on how they should draw an n-level dragon curve. _____________________________________________________________________________________ _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

_____________________________________________________________________________________ _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Using your graph paper, draw your level five dragon curve to scale.

Label each turn as a right or left starting at your “A” and turning left to get to “B.”

10) Using R for right and L for left, write your dragon curve’s RL pattern below.

11) Describe the pattern, if possible, tell what the next ten turns will be. (You may need to look at each level separately.)

Fold your curve back up, coloring each fold as it is folded. Color the top of the fold so that the color fades down enough for you to see the color on every layer. Unfold.

13) Using the first letter of each color for your key, write your dragon curve’s color pattern below.

_____________________________________________________________________________________

14) Describe the pattern, if possible, tell what the next ten colors would be if you added another fold.

***If you are finished before others, here is a bonus question/problem:

How many times can an object be folded in half? Feel free to use a strip of paper, notebook paper, Kleenex, toilet paper, etc.

Recursions and Fractals

Dragon Curves (2)

The rule for drawing this dragon curve is: Draw an isosceles right triangle on each straight line of the previous layer beginning with an “up” triangle and then alternating with a “down” triangle. The hypotenuse of the added triangle should be the existing edge.

Using your graph paper, draw a straight line in the middle of the page. Make the line approximately 4 inches long. (****Call this length 1)

Using a different colored pen or pencil for each layer and the drawing rule above, draw a five-layered curve.

1) Looking at the second layer, answer the following:

What is the RL pattern?__________________________________________