exponential growth/decay, geometric sequence, fractals, iteration, self-similarity, ratio, tessellation, golden ratio, golden rectangle,

Fractals - figures that are formed from Iterations that follow self similarity

Iteration – formed by repeating steps (this is referred to as Iteration)

Self – similarity-  each successive stage of the object becomes a reduced copy (that is proportional) of the entire object from the previous stage.

ratio - the comparison of two numbers/values

tessellation - patterns (for our purposes figures in the coordinate plane)

golden ratio - the ratio of a rectangle that is divided into a square and a rectangle where each successive rectangle is similar to the original rectangle

golden ratio - the similarity ratio of the similar rectangles formed in the golden rectangle.  This ratio is 1:1.618

geometric sequence - a sequence where each number is multiplied or divided by the same number (pattern)

exponential growth/decay - is a constant growth of rate (or constant rate of decay)

In this lesson students will look at fractals and their properties using an online interactive fractal tool.  The rest of the lesson will have students explore fractals, compile what they find on fractals following a rubric, and presenting this information.  In this lesson students will be able to:

• demonstrate that they understand what fractals are;
• describe where fractals are found;
• show comprehension of how fractals are used in real world situations including tessellations;
• determine how each successive stage of a fractal can be found mathematically.

80 to 120 minutes

plus includes time student's own time: homework time

geometry book (chapter on similar triangles that includes fractals),

PowerPoint presentation (Fractals),

worksheet on fractals (fractals worksheet),

laptop/computers with links to essential websites.

Fractal Tool : From Thinkfinity; interactive tool for exploring fractals. 

Additional Links they will use to complete assignment

Patterns in Nature : From pbs.org Beauty Patterns in Nature through Thinkfinity; an interactive site for the mathematical study of plant pattern formation. 

Visual Math : From pbs.org Beauty Patterns in Nature through Thinkfinity; an exploration of the Golden Ratio.

The Anthill : From pbs.org Beauty Patterns in Nature through Thinkfinity; an exploration of biological and chemical reactions using mathematics through the Ant Hill java applet

Java Man : From pbs.org Beauty Patterns in Nature through Thinkfinity; an exploration of fractals through color assignment.

Gold Fractal II : From pbs.org Beauty Patterns in Nature through Thinkfinity; an exploration of fractals through art.

Interactive Tessellate : From pbs.org Beauty Patterns in Nature through Thinkfinity; interactive tool for exploring fractals.

W:  Students will copy Fractal notes from PowerPoint (Fractals) introducing the concepts of fractals after they have completed a lesson on similar figures (what similar figures are, ratio, proportion, etc.).  Once they have copied the notes, students will look at the activity shown in the book on fractals (This should be a lesson in the book on fractals, usually found with similar triangles).  They then will be introduced to an online sight from Thinkfinity (Fractal Tool) called the "Fractal tool" and will be given a supplement worksheet which will be completed using the fractal tool (fractals worksheet). The worksheet is used by the entire class as a guided instruction with the teacher, being prompted to solve problems and answer using the fractal tool.  After completion of the worksheet, students will use computers and the identified sites on fractals (Patterns in Nature, Visual Math, The Anthill, Java Man, Gold Fractal II, Interactive Tessellate). The rubric on fractals will be explained as well as the expectations of the project.  Students will be given time to explore the websites on fractals and also time to begin planning what to put into their presentations.  Students will be given the next class period to collect materials to be included in their presentations using the computers. The remainder of the project assembly will be given as a homework assignment.  Students must complete the poster/presentation using one of the approved websites (www.prezi.com, www.glogster.com). These tools work best as a once and done assembly after all tools are gathered.

H:  The interactive tools and links are really the key to success in attracting the students' interest and enthusiasm.  Also, the homework assignment is not a traditional rote and practice drill, but a presentation/poster that students can put together using the online resource tool of "Prezi" or "Glogster" or on a traditional poster. 

E:  Modeling, scaffolding, and guided instruction will help provide students with the understanding they will need to succeed in this lesson.  Students will be able to see how fractals are formed as well as the math behind fractals using the fractal tool.  Students will then get the opportunity to explore fractals themselves using the recommended sites.  After students explore, they will synthesize their findings into a presentation meeting the elements on the rubric on fractals. 

R:  Students will need to reflect on knowledge of fractals and the relationship that fractals have to the real world as well as examples they have discovered that represent fractals.  Students will need to revisit the information provided in completing their presentations.  They will revise the information they discover so that it is original and in their own words.  Students will need to rethink the way information was presented by the teacher and how they can present it to the class.

E:  Students will be able to express their comprehension through the guided instruction process as well as the presentations created by the students using the websites provided.

T:  Possibilities for Differentiated Instruction may include:

• Differentiation for higher level learners may be able to go deeper into how they answer the essential questions in the rubric as well delving deeper into the math concepts behind fractals during guided instruction.  This may include more work with exponential growth (pre-calculus concepts) and geometric sequence. 
• Differentiation for both lower or for higher level learners could be where the teacher may allow students to come up with their own fractal/iteration and then decide which math concept (exponential growth or geometric sequence) their own creations follow.
• Differentiation could occur for the lower level learners by making the project a group project and assigning roles.  Students can be grouped where each student is assigned to a specific part(s) of the rubric thus allowing for a decrease in the amount of work for each student but still allowing the content to remain the same. 
• Less developed learners may choose to work with Glogster, which is a little less flashy than Prezi, as well as use less of the resources when answering the essential questions.  Glogster will allow students to still follow the criteria of the rubric, but allow students to work with a simplified media technology.

O: 

1. Students will enter the room with their books and notebooks ready to take notes (as they do in the established routine)
2. Students will take notes from PowerPoint on the concepts of fractals Fractals
3. Students will receive the fractal worksheet (fractals worksheet) as the teacher uses the interactive white board to display the website that contains the fractal tool (Fractal Tool). 
4. Students will engage in a guided instruction which reviews the mathematical concepts dealing with fractals as well as the visual examples that reinforce the math they will be working on.
   • Students will review with the teacher the first part of the worksheet which gives the formula for geometric sequence and the parts of the formula.  Explain the parts of the formula and answer any questions the students may have about the formula
   • Students will participate as the teacher will demonstrate the fractal tool using the first example on the worksheet.  The students will then complete the other problems using the formula which has been demonstrated.   
   • The students will then be introduced to the second example on the worksheet as well as the fractal tool.  This example shows exponential growth.  This example shows exponential growth and the formula that should be utilized.  Students will then complete the other problems on the worksheet.
   • The students will then be shown examples 3 and 4 on the fractal tool.  We can ask students to determine if the problem is geometric sequence (like example 1 was) or exponential growth (like example 2 was).  Example 3 is asking for the area of the blue square.  The area of the blue square decreases each time by 1/4 and then is continuously added onto the previous area.  This is an example of exponential decay because each square's area is successfully 1/4 less than the previous although the area still increases until it reaches an eventual asymptote.  The fourth example shows both geometric sequence and exponential decay.  This is because the number of green triangles becomes three new green triangles in each successive stage.  The area of each green triangles becomes less by 1/4 in each successive stage this is subtracted from the previous area. 
5. After students complete the worksheet, they will be assigned computers that already have the bookmarked links on fractals in the favorite tools or handed a sheet with the URLs.  Students will receive the rubric (rubric on fractals) before they are allowed to open their computer screens.  The rubric/project will be discussed with students, as well as expectation of the remainder of the period, the agenda of the next class session, procedures, grading, use with computers, and any other issues or questions the students have about the project.
6. Students will be given time to explore the websites for fractals (websites).  They will gather the ideas, pictures, etc., to be included in their presentations.
7. Students will continue to work on this the following class session.  The same routine will be followed where students will be assigned their laptops/computers and students will review again the expectations of the project.  Students will be given time to work on the assignment.
8. The remainder of the assignment will be given as homework.  After the teacher grades the assignments, they may allow students to link projects to class website to view the completed assignments and/or through presentation of projects in a later class session. 

Students will use the rubric and the recommended websites to create a presentation or poster that shows what they have learned from the lessons on fractals.

• Rubric for presentation:  rubric on fractals;
• www.prezi.com :  a website that allows students to put together presentations in an interactive format;
• www.glogster.com: a website that allows students to put together poster presentations online.

Fractal Tool : From Thinkfinity; interactive tool for exploring fractals. 

Patterns in Nature : From pbs.org Beauty Patterns in Nature through Thinkfinity; an interactive site for the mathematical study of plant pattern formation. 

Visual Math : From pbs.org Beauty Patterns in Nature through Thinkfinity; an exploration of the Golden Ratio.

The Anthill : From pbs.org Beauty Patterns in Nature through Thinkfinity; an exploration of biological and chemical reactions using mathematics through the Ant Hill java applet

Java Man : From pbs.org Beauty Patterns in Nature through Thinkfinity; an exploration of fractals through color assignment.

Gold Fractal II : From pbs.org Beauty Patterns in Nature through Thinkfinity; an exploration of fractals through art.

Interactive Tessellate : From pbs.org Beauty Patterns in Nature through Thinkfinity; interactive tool for exploring fractals.