Fractal Wallhanging.

Fractal Wallhanging

Goals Participants will: Analyze and decode fractal images.
Identify the properties of self-similar figures. Identify the eight symmetry transformations. Determine the effect of composition of transformations. Solve equations involving composition of transformations. Construct fractal images.

Common Core State Standards

Standards for Mathematical Practice
MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP7: Look for and make use of structure. MP8: Look for and express regularity in repeated reasoning

Review of Transformations
What is a transformation? When all points on one geometric figure are mapped to corresponding points on another geometric figure. Translation Rotation Reflection Dilation

Symmetry A rigid motion of a geometric figure that determines a one-to-one mapping onto itself. Symmetry is a type of invariance: the property that something does not change under a set of transformations. An object (a set of points in the plane) has symmetry if you can find some transformation such that the set of points before the transformation is the same set as the set of points after the transformation.

Symmetry Transformations
Translational Symmetry The figure can be divided into equal parts The parts can be placed on top of each other by a sliding motion

Reflectional Symmetry The figure can be divided into equal parts The parts can be placed on top of each other by a reflecting motion

Rotational Symmetry The figure can be divided into equal parts The parts can be placed on top of each other by a rotating motion

What are Fractals? A fractal is a graphical representation of an iterative process with properties of self-similarity and non-integral complexity. Fractal patterns are extremely familiar, since nature is full of fractals. For instance: trees, rivers, coastlines, mountains, clouds, seashells, hurricanes, etc. Break down the definition of fractal with participants. A fractal is a graphical representation (a picture) of an iterative process (ongoing feedback loop) with properties of self-similarity (Self-similarity is when each of the small parts look like the whole figure, only smaller in size) and non-integral complexity.

Generate the Sierpinski-Triangle
Reduce Replicate and Rebuild

Reduction-Copying-Machine Similarity-Machine
1/2

1/2 Iteration

… Iteration

RRR: Multiple-Reduction-Copying Machine Three Lens System
Each generates a 50% copy of the Original and places the copy at a particular position on the output paper The copy machine takes an image as input. It has three independent lenses each of which reduces the image and places it somewhere on the output image.

Iteration An output image becomes the input image for the next step
Then the output image is placed on the copy machine, the three independent lenses reduce the image and places it in the same manner on the output image. An output image becomes the input image for the next step

Iteration of RRR An output image becomes the input image for the next step 1. 2. 3. 4. 5. This process is repeated over and over. Eventually, …

Similarity Machine …

Similarity Machine … Self-Similarity

How would the figure change if we apply one or even several transformations during the construction process?

RRR: Multiple-Reduction-Copying Machine
How would the image change if one of our three lenses rotated the image 90 degrees during the RRR process?

Iteration What will happen in the next step?

Iteration of RRR What happens to the image if we were to continue the process? 2. …

Iteration of RRR Have participants take note of how different the images are with only one transformation.

Before showing the grid, ask the participants, “What do you see
Before showing the grid, ask the participants, “What do you see?” Answers will vary. Ask participants to describe, in geometric terms, any outstanding features of the figure (triangles, diagonal lines, arrows, etc.) Distribute the Solve the Puzzle 1 Handout Ask students to cut out the three reduced images at the bottom. They should notice that these images are exactly the same as the large image at the top of the sheet, only reduced in size. Students are to place the three images in the empty grid so that when completed, the image created is identical to the large image. Students should write their description of how they solved the puzzle before they glue the pieces in place. Just like a jigsaw puzzle, the pieces may need to be turned (or rotated) in order to make the pieces “fit” to complete the puzzle.

As students write their explanations, have them ask themselves, “What did I need to do to each piece to solve the puzzle?” Formal mathematical language is not needed at this time. It will be developed in later activities. Make sure that students understand that the grid is not part of the image. The grid provides a framework to order the image. As the students are solving the puzzles, a discussion about the three reduced copies needs to take place. Possibly some students have discovered on their own that the three small images are identical to the large image at the top of the activity sheet, only smaller. If not, ask them, “What do you notice about the smaller pieces?” Re-iterate the concept of self-similarity. Have participants share out within groups or with the whole class.

Have participants make observations about this fractal image.
Distribute “Solve the Puzzle 2” Handout. The handout has three parts. At the top left is a large image. Next to it is a grid where the puzzle pieces will be placed (just like Solve the Puzzle 1). (2) At the bottom of the activity sheet, there are three smaller copies of the larger image, called the tool. The tool has a “front” and a “back” side. This design allows more options when solving the puzzle. The letters “I” and “V” are used to differentiate between the front and back. The letter I (which stands for Identity position) was chosen because the smaller copy in this position looks exactly like the given image at the top of the activity sheet when the I tab is pointed down. The letter V (which stands for Vertical reflection) was chosen because the small copy looks like the reflection of the large image on itsvertical axis at the top of the activity sheet when the V tab is pointed down. (3) In the middle of the activity sheet, students provide a written description of how they manipulated each of the three small copies to replicate the image just as they did in Solve the Puzzle 1. Students will also translate their sentences into some kind of mathematical shorthand. Allow students to use symbols and terms of their own choosing. In the next activity, formal mathematical notation will be introduced.

Students should carefully cut out the three tools along the solid lines. Have students fold each tool along the dotted line so there is an image on the front and back. Glue or tape the sides together so that the three tools will stay closed during the manipulation process. Once the three tools are assembled, students will discover (as in Solve the Puzzle 1) what to do with each tool to create an image identical to that at the top of the page. In the left hand column, students are to record, or describe, how they manipulated each tool to get proper placement in each cell for a successful copy. In the right hand column, students should write a more “ mathematical” description to describe the steps they followed. These descriptions provide a form of mathematical shorthand. Examples might be: • “I used the “I” side and turned the tool left one time” translates in mathematical shorthand to I, L, 1. • “I flipped the tool to the V side and turned left twice” translates in mathematical shorthand to V, L, 2. Have participants share out within groups or with the whole class.

Transformation Tool The tool just has an I and a V on it.
I: Identity - There is no rotation or reflection. The image on the tool is in the identical position as the given image. The I tab is facing front and pointing down. This is known as the Identity position. V: Vertical Reflection - Begin with the tool in the Identity position. Reflect or flip the tool over the vertical axis. (Hold the tool by the top and bottom tabs and flip the tool.) After this is done, the V tab will be facing front and pointed down. As participants manipulate the tool and perform the transformations, they will fill in the names of the remaining transformations on each tab (R90, R180, R270, H, D+, D-). Distribute the Spidey Tool Handout, Participants should prepare the Blank Tool the same way they did in Solve the Puzzle 2: carefully cut out the three tools along the solid lines; fold the tool along the dotted line so the I and V appear on the front and back; glue or tape the backs together so that the tool will stay closed during the manipulation process.

What Happened?

What Happened? 2 moves? 3 moves?

Rotations Identity or starting
R90: Rotation 90 degrees counterclockwise- Begin with the tool in the Identity position. Rotate the tool 90º counterclockwise. The tab pointed down will be blank. Label it R90.

Rotations

Rotations 90 degree (R90)

Rotations

Rotations 180 degree (R180) R180: Rotation 180 degrees counterclockwise - Begin with the tool in the Identity position. Rotate the tool 180º counterclockwise. The tab pointed down will be blank. Label it R180.

Rotations

Rotations 270 degree (R270) R270: Rotation 270 degrees counterclockwise - Begin with the tool in the Identity position. Rotate the tool 270º counterclockwise. The tab pointed down will be blank. Label it R270.

Rotations

Rotations 360 degree (identity)

Reflections Vertical flip (V) Identity Horizontal flip (H)
H: Horizontal Reflection – Begin with the tool in the Identity position. Reflect or flip the tool over the horizontal axis. (Hold the tool by the left and right tabs and flip the tool.) The tab pointed down will be blank. Label it H.

Reflection Identity

Reflection Identity D+ Flip
D+: Positive Diagonal Reflection - Begin with the tool in the Identity position. Reflect or flip the tool over the positive slope diagonal axis. (Hold the tool by the lower left and upper right corners and flip the tool.) The tab pointed down will be blank. Label it D+.

Reflection Identity

Reflection/Composition
D- Flip Identity D-: Negative Diagonal Reflection – Begin with the tool in the Identity position. Reflect or flip the tool over the negative slope diagonal axis. (Hold the tool by the upper left and lower right corners and flip the tool.) The tab pointed down will be blank. Label it D- .

Looking at a Limit Figure
Have participants look at this image and try to find the code. Click to place the grid over the image to make it easier for participants to find the code. Have them discuss within small groups what they think the code is.

How were the parts Manipulated
Diagonal reflection + Diagonal Reflection - Horizontal Reflection Vertical Reflection Rotation 180 Rotation 270 Rotation 90 Identity Click through the animations to show each transformation and see which ones are used to create this image. Which transformation do you apply to the reduced figure to match Each piece?

Can you figure out the code for this figure?
Have participants look at this image and try to find the code. Click to place the grid over the image to make it easier for participants to find the code. Have them discuss within small groups what they think the code is. Link to game

Discuss observations about the fractal image.
In Solve the Puzzle 2, participants described how they manipulated the tools in terms of I and V to make some kind of turn or flip. With the development of the six other transformations, participants can now begin using mathematical notation to describe the moves needed to solve the puzzles. Using mathematical notation also allows for more precise and consistent language. Distribute the Find the Code Handout. Students need to carefully cut out the tool along the solid lines. Have them fold the tool along the dotted lines so there is an image on the front and back. Glue or tape the sides together so that the tool will stay closed during the manipulation process.

Remind participants to always begin their moves and their statements with the Identity position.
The handout is designed with three columns headed 3 moves, 2 moves, 1 move. As participants find the code, they write down multiple solutions for each cell. A discussion with the students about why so many solutions are possible is important. Participants may not understand that the puzzle can always be solved with just one move per cell. The reason for this is that any composition of two transformations from I, R90, R180, R270, V, H, D+, D- yields one of these transformations again. Therefore, if the puzzle is solved with several moves, students can work the solution down to just one move step by step.

Review possible answers with participants.
Have participants share out within groups or with the whole class.

Symmetry Computer The idea that the composition of two transformations is equivalent to a single transformation leads to the development of the Look-Up Table. Students should work with a partner to complete the Look-Up Table. Using the Symmetry Computer assists students in completing this task. Give participants the Symmetry Computer Participant Handout. Have the students cut out the symmetry computer and board. The cutout corner of the board becomes the viewing window. Transformations can be applied to the board (either one or several) and the resulting transformation appears in the window.

Symmetry Computer Identity
NOTE: When using the symmetry computer, always start at the Identity position.

Symmetry Computer Rotation 90 degrees
Place the board on top of the symmetry computer so that the I is showing in the viewing window. Rotate the board 90 degrees counterclockwise and you should see R90 in the viewing window.

Symmetry Computer Identity

Symmetry Computer Vertical Flip
Place the board on the symmetry computer so that the I is showing in the viewing window. Flip the board over the vertical axis of symmetry and the V will show in the viewing window.

Symmetry Computer Horizontal Flip
Place the board on the symmetry computer so that the I is showing in the viewing window. Flip the board over the horizontal axis of symmetry and the H will show in the viewing window.

Positive Diagonal Flip
Symmetry Computer Positive Diagonal Flip Place the board on the symmetry computer so that the I is showing in the viewing window. Flip the board over the positive slope diagonal axis of symmetry and the D+ will show in the viewing window.

Negative Diagonal Flip
Symmetry Computer Negative Diagonal Flip Place the board on the symmetry computer so that the I is showing in the viewing window. Flip the board over the negative slope diagonal axis of symmetry and the D- will show in the viewing window.

Fill in each circle by finding the correct composition.
Distribute the Look-Up Participant Handout. The Look-Up Table is a concise format for displaying the results when two transformations are performed one after another. The transformation applied first is given in the left-hand column of the Look-Up Table. The second transformation is given in the top row of the Look-Up Table. Using the symmetry computer, determine the results when an R90 transformation is followed by a V transformation. When R90 is followed by V, D+ should be seen in the viewing window. To enter this result in the Look-Up Table: 1) Move down the left column of the table until Row R90 is located. 2) Move across the top row of the table until Column V is located. 3) At the intersection of Row R90 and Column V, write in the result, D+. Using the symmetry computer, determine the results when an R270 transformation is followed by an R90 transformation. When R270 is followed by R90, I should be seen in the viewing window. Using the symmetry computer, determine the results when a D+ transformation is followed by an R270 transformation. When D+ is followed by R270, H should be seen in the viewing window. Have participants work with a partner to complete the table. Each student could complete four rows and then exchange his results with his partner. H

After the table has been completed and answers verified, ask participants to examine the Look-Up Table and identify the patterns they see. Discuss the patterns the students find. What patterns do you notice? Are the composition of transformations commutative? What happens when you do a rotation followed by a rotation? What happens when you do a reflection followed by a reflection?

What’s missing? 2nd ? V 1st D- result =
Have participants use the Lookup Table to complete the equation.

R90 V = ___ ___ H = R270 D+ ___ = H Practice Quiz
At this point, students have practiced performing multiple symmetry transformations to obtain a result. For example, they have learned that from the I position they can apply an R90 and then a V on the same image. A formal mathematical way of representing this combination is necessary. Composition notation can be used (similar to that used on functions in algebra) as the way of describing mathematically what is taking place. The example given above would be written in the following way: V o R In composition notation, expressions are read from left to right. This expression would be read as: ”Apply V, then apply an R90” That is, starting in Identity position, a V is applied to the image first, and an R90 is applied second. Remember that before any transformation is applied, the image must begin in the Identity position. More than two transformations can be applied. Composition notation can be used to represent this idea. For example, “First apply an H, then a V, then a D+” would be written as: D + o V o H D+ ___ = H

Now that the participants are familiar with composition notation, the Look-Up Table, and the Symmetry Computer, they will be able to solve transformation equations. As in algebra, the variable X is used to represent the unknown transformation. For example, V o X = R90 is read, “What transformation X, when composed with a V, will result in an R90?” Participants will learn how to solve such equations in this activity. Give the participants the Solving Equations Participant Handout. Participants may use the Look-Up Table, the Symmetry Computer, or both to solve the equations. Allow participants to work with a partner, if necessary, to solve the equations. Allow groups to check their answers with one another.

Building a Fractal from One Filled Square
Stage 0 Reduce – by 1/2 Replicate – make three copies Rebuild In this activity, students will construct a fractal image from one filled square. The power of this simple process—iterating the transformations, or code, over and over—is made clear to the students. Distribute the Fractal From One Filled Square Participant Handout. Provide the students with the code of the fractal they will be building: Cell A: R90 Cell B: R180 Cell C: I The important points to remember in sketching each successive stage are: • Reduce the previous stage by 1/2. This is accomplished by using the smaller squares on the graph paper on the activity sheet. • Replicate the reduced copy three times. • Rebuild the next stage by applying the appropriate transformation (code) in the appropriate cell. Allow the students to work in groups so they may assist each other in sketching the stages. When they reach Stage 5, the image begins to become more detailed. R90 R180 I

Stage 0 Reduce Replicate Rebuild Have participants sketch successive stages of the image. Sketching the first two stages will be easy for most students, but Stages 3 and 4 will be more difficult and may be frustrating for some students. Allow the students to work in groups so they may assist each other in sketching the stages. When they reach Stage 5, the image begins to become more detailed. R90 R180 I Stage 1

Stage 1 Reduce Replicate Rebuild R90 R180 I Stage 2

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 Stage 5

Can you build a Wallhanging?
R90 I In the Fractal from One Filled Square activity, participants began with one filled square and sketched successive stages of a fractal according to its code. In this activity, participants will create a Fractal Wallhanging from these Stage 5 images. Distribute the Wallhanging Stage 5 Images Participant Handout and the Wallhanging from One Filled Square Grid Sheet Participant Handout. The steps in building a Fractal Wallhanging are described below. Step 1--Determine the Code Begin with the stage 5 fractal image. Display the Image Without Grid on the overhead projector. Determine the transformation, or code, needed in each cell to recreate the image. If necessary, sketch a grid on the image to assist students as they complete this task. Step 2--Construct Stage 1 Begin constructing the wallhanging. Three copies of the image and one Grid Sheet square are needed. Glue or tape the pieces onto the Grid Sheet. At this point, a Stage 1 image has been completed. Step 3--Construct Stage 2 Repeat the process with three Stage 1 images. Make three copies of Stage 1 and put them together according to the code for each cell. Step 4--Continue Construction Process Stages 3 and 4 can be completed in the same manner. The process can be repeated until the physical limitations of the classroom prevent the students from going further. Typically, a fractal wallhanging can be constructed to Stage 5, 6, or higher. The size of the final stage will depend on the size of Stage 0, but usually will be feet square. R180

Goals Participants will: Analyze and decode fractal images.
Identify the properties of self-similar figures. Identify the eight symmetry transformations. Determine the effect of composition of transformations. Solve equations involving composition of transformations. Construct fractal images.

Fractal Wallhanging.

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