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Fractions, Fractals, and Pascal

Subject: Mathematics

Materials: This lesson asks students to have the following each day: a centimeter ruler, pencil, and a "Mathematics Today!" magazine packet. Students are also given time to explore fractals and Pascal's Triangle (later in the lesson) on computers. They are sent to these sites: http://shodor.org/interactivate/activities/SierpinskiTriangle/ and http://shodor.org/interactivate/activities/ColoringMultiples/ This is done through wireless Internet and a laptop cart. A projector is used to have students come up to present patterns of the multiples in Pascal's Triangle. The goal is to have students find that the multiples of 2 strongly resemble Sierpinski's Triangle.

About: Students create Sierpinski's Triangle using the centimeter ruler and a pre-designed equilateral triangle with sides of 16cm. On the projector, I show the classroom examples of other fractals. I ask the students to look for similarities between all fractals and see if they can create a definition. We discuss the definition of a fractal and how this is the process that creates one. We talk about what an iteration is and find the relationship between the area of each smaller triangle and the largest triangle. We then ask about the areas after the second iteration. The students color their fractal in after 3 iterations and then find the area of each color. Students use fractions and the powers of fractions to find the areas. They also learn about Pascal's Triangle and how to create it. Using the ideas of multiples, they see the relationship between Pascal's Triangle and Sierpinski's Triangle.

The final project culminates in a "Mathematics Today" magazine. The magazine includes a cover containing Sierpinski's Triangle, directions on creating a fractal, examples of fractals, definition of a fractal, Pascal's Triangle, written observations of Pascal's Triangle and its relevance to the project, and the final sheet that includes answers to the areas of each color on the cover.

This project covers many of the topics that need to be learned in mathematics. In the New York State curriculum, students are required to use exponents with fractions, find the area of a triangle, identify patterns, make comparisons, create definitions from observations, be creative with colors, and learn to use the computer to aid in instruction. I feel that the computer/Internet adds vital practice to the project. Students are allowed to "play". This strengthens instruction and aids in conversations about what Pascal's Triangle has to with Sierpinski's Triangle. This connection is a key part of comparisons between Pascal's Triangle and Sierpinski's Trangle.

One of the great things about this project is that it can be run in any mathematics classroom. Teachers should arrange time in the computer lab if portable laptops are not available. Markers or crayons should also be available. Students tend to rush through the measurements for midpoints. These poorly measured points affect the cover sheet for the project. Students may not see a connection between Pascal's Triangle and Sierpinski's Triangle. A few of the students struggle with the idea of 1/4 raised to a power. The nice thing about this project is that once that proper measuring is stressed, teachers can work with these students to make sure areas work out correctly.

http://eev3.liu.edu/oc0507/bm/fracfracs.htm

 Objectives
 For this lesson I would like the students to understand the idea of a fractal. They may use the Internet to look at various fractals and see how multiple versions are all similar to each other. All students have a chance to propose a definition during class. Students will understand how to create a fractal called Sierpinski's Triangle. They use the midpoints of each side and realize that connecting them is called an iteration. Each iteration will create triangles that are one-fourth of the triangle that they are drawn from. Students will understand how Pascal's Triangle is formed. They fill in missing numbers and see that there are some interesting patterns that appear within the triangle. Students continue with Pascal's Triangle on the Internet and look at the multiples of numbers within the pattern. In their attempt to discover the relationship between Pascal and Sierpinski, students look at what happens when the multiples of 2 are filled in on Pascal's Triangle. We first look at this on the Internet. Then we look at it on the overhead. When the two are overlapped, they match up (roughly). Students continue to explore patterns of multiples in Pascal's Triangle. Students color in Sierpinski's Triangle and find the area's of each color. They use the area formula once and, like the problem states on the magazine cover, find the area of each color. This is done by multiplying the total areas by powers of 1/4. Students begin to find patterns in their answers and use those to get answers for other parts of the problem. They are able to check their answers by adding all of the area together to get the original area. The project will finish with the comparison, observation, and conclusion about Pascal's Triangle, Sierpinski's Triangle, and the relationship that the two figures have. Students will have to write coherent sentences that outline this information.

 Websites
 This link provides the inspiration to the lesson. It is a great way for students to understand how to create a fractal called Sierpinski's Triangle. The students can look at the steps involved and understand what is included in an iteration. http://shodor.org/interactivate/activities/SierpinskiTriangle/ This link shows students that Sierpinski's Triangle does not have to be an equilateral triangle. Sierpinski's Triangle can actually be created on any type of triangle. Students can "play" to see this. Again, this play will increase comprehension. http://serendip.brynmawr.edu/complexity/sierpinski.html This site enables students to see other types of fractals call the Mandelbrot Set and the Julia Set. This will drive students towards a definition of the word "fractal." http://facstaff.unca.edu/mcmcclur/java/Julia/ This is another site that lets students explore different types of fractals. Students should not only make a definition for the word fractal but try to think of a reason for creating a fractal. http://nlvm.usu.edu/en/nav/topic_t_3.html This is another site that allows students to see fractals being created. These other fractals are called the "Jurassic Park" fractal, Koch's Snowflake, and the Anti-Snowflake. These sites offer even more opportunities for students to come up with a definition and decide the purpose of a fractal. http://math.rice.edu/~lanius/frac/index.html This site lets students explore Pascal's Triangle. The students can see how the triangle is created and what happens when specific multiples are colored in. This enables students to see the similarity between Pascal's Triangle and Sierpinski's Triangle. http://shodor.org/interactivate/activities/ColoringMultiples/ Students see other patterns that exist in Pascal's Triangle. The creation of the triangle is a small part of the entire figure. This identifies items called natural numbers, triangular numbers, tetrahedral numbers, and Fibonacci numbers. There are other patterns that students may research on their own. http://mathforum.org/workshops/usi/pascal/mo.pascal.html There are a host of other links that are important to the project. I have compiled them into one link that encourages the students to conduct further investigation and study. On the link, sites that provide students a chance to further explore fractals and Pascal's Triangles are listed under the explorations link on the left menu. http://eev3.liu.edu/oc0507/bm/fracfracs.htm

 Standards
 Students understand numbers, ways of representing numbers, relationships among numbers, and number systems. They work with fractions, exponential expressions, and multiples, and understand what an exponent is and how to find the answer to an exponential expression that contains fractions. Students must also develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use. NCTM 6-8 Mathematics In this project, students practice creating and critiquing inductive and deductive arguments concerning geometric ideas and relationships such as congruence and similarity. They analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships and recognize and apply geometric ideas and relationships in areas outside the mathematics classroom such as art, science, and everyday life. NCTM 6-8 Mathematics Students apply appropriate techniques, tools, and formulas to determine measurements. They select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision. They learn how to check their answers by adding results to obtain the original area. They also check their observations through sharing them with the class. NCTM 6-8 Mathematics Besides the mathematics that is involved in the Sierpinski vs Pascal project, there are many parts of the New York State curriculum that are really discussed and developed within the student. There are few times where students are asked to defend their ideas, present new ideas, make comparisons, conclusions, use physical objects as representations, apply strategies, etc. These are a big part of the middle school mathematics curriculum. New York State Math Curriculum (6-8) Mathematics Students understand and use the elements and principles of art (line, color, texture, shape) in order to communicate their ideas. They use their colors in Sierpinski's Triangle. This brings mathematics into the Creative Arts classroom and gives evidence that they have developed an emerging personal style. The Arts ( K-12 ) Visual Arts How were such ideas created so long ago? Blaise Pascal lived in the emergence of the first global age. Waclaw Sierpinski lived in the age of revolutions. Students in upper grades may benefit from some discussion about whose time was harder to live through. Who had the harder life? Social Studies (6-9) Social Studies Students read, write, listen, and speak for critical analysis and evaluation. As listeners and readers, they analyze experiences, ideas, information, and issues presented by others using a variety of established criteria. As speakers and writers, they present, in oral and written language and from a variety of perspectives, their opinions and judgments on experiences, ideas, information, and issues. English Language Arts (7-10) English Language Arts Students are asked to look at pictures and make written observations about what they see. This asks them to organize their thoughts and put them into clear sentences. Students are also asked to make comparisons between two different pictures. In their written observations, they will see that Pascal's Triangle and Sierpinski's Triangle have a relationship.

 Day 1: Creating a fractal
 Objectives
 Students are shown how to create a fractal on the Internet. They follow the process and look at the other types of fractals. Students create their own definition for a fractal and then discuss it with the class. We end with one final definition for the word fractal. The class discusses the use of fractals and what the point is of creating such things. Students should discuss the artistic form of a fractal and how they create shapes that are pleasing to look at. Students finish the day with coloring the fractal with at least four different colors. If the coloring is not completed, they must complete it for homework.
 Materials
 Students need all of the items required for all math classes. They need laptop computers, rulers, calculators, pencils, pens, erasers, computers, graph paper, lined paper, "Fractions, Fractals, and Pascal" packet, and a positive attitude. Students need a laptop computer and links to the Shodor.Net site. They will use this to see the steps on how to create a fractal. Students continue to use the laptops to see the different types of fractals. They use these to create a definition for the word fractal. They need the second sheet of the packet as well. The class also requires markers to creatively color the fractal that they created with instructions given at the beginning of class and modeled on the Internet. Students will be able to play with the fractal on sites provided to increase comprehension.
 Procedures
 Students are given a "Fractions, Fractals, and Pascal" packet. We start with the definition of the word "midpoint." The example given on the Internet is a great way for students to see the idea of the midpoint. Students are asked how to find the midpoint using the supplies that they have with them. We draw the midpoint on each side of the triangle and connect them. This is called an iteration. I ask the students to complete a second iteration on their own. A second iteration includes three more triangles that point down, not just one. We then look at other types of fractals. Students are asked to look at all of the fractals and create a definition in groups of three or four. We discuss the word and then give the definition. We start a discussion about the reason for fractals. Fractals are important in art and in determining reason in the randomness of life. Many think that there are events that can be predicted in the picture of the Mandelbrot set. It is similar to the digits in Pi. Students begin to create their covers. Once they are done with creating the fractal called Seirpinski's Triangle to four iterations, they can begin coloring. They must use at least four different colors.
 Homework
 Students complete coloring in their Sierpinski's Triangle and construct a theory on how to solve the "Mathematics Today" question. Answers will be discussed at the beginning of the next class.
 Assessment
 As students are working, I walk around to see what they are writing on their papers and inspecting their use of the Internet. The goal of today's assessment is to correct any misunderstandings so that more students can reach the large goal at the end.

 Day 2: How do we find the area?
 Objectives
 Students need to find the area of each color. In the evening prior, students were to find ways to get their answer. Students will all have a way to get the answer in the first 10 minutes of class. We will see how to express that way in terms of exponents and compute it using a calculator. Students continue to color, explore, find the areas of each color. Those who are finished will come up with a method to check their area answers. They present their answers in the last five minutes of class. This gives the class time to catch up to each other so that we can move together as a group.
 Materials
 Students need markers, laptop computers, rulers, pencils, "Fractions, Fractals, and Pascal" packet, pencils, pens, erasers, calculators, and a positive attitude.
 Procedures
 Students are asked to take out a sheet of paper and write "Method of Area Calculation" at the top. They present their methods to the rest of the class so that every student will see that each iteration breeds triangles that are 1/4 the size of the bigger triangle. Students look at the Explorations sites from the web page and make observations about types of fractals, Pascal's Triangle, or the relationship between the two. Students present methods that will prove area answers are correct. They should see the method where you add all of the colors areas together and end up with the original area.
 Homework
 Students take home the Pascal Triangle sheet and look for ways to fill in the missing numbers. They should also create one pattern that exists within the triangle.
 Assessment
 Students are assessed on how they find the areas of triangles in their Sierpinski's Triangle. Make sure that they only use the formula for area one time. All of the colors must be calculated using the powers of 1/4.

 Day 3: What is Pascal's Triangle?
 Objectives
 Students share how they filled in the missing numbers. They should see that each number is the sum of the two numbers directly above it. Students see if they can find any other patterns in Pascal's Triangle. We will present them to the class. Students again travel to Shodor.net and look at Pascal's Triangle. We are looking for similarities between Pascal and Sierpinski. The students see that Pascal's triangle is made up of patterns of multiples. Students explore this and eventually see that when the multiples of 2 (even numbers) are colored in, Pascal's Triangle turns into Sierpinski's Triangle. Students make observations about multiples in Pascal's Triangle. They also make conclusions about the relationship between Pascal and Sierpinski.
 Materials
 On a sheet of lined paper, tudents write down different patterns that they see in Pascal's Triangle. Of course, the students need lined paper, pencils, "Fractions, Fractals, and Pascal" packet, calculator, laptop computer, etc.
 Procedures
 We spend time discussing missing spaces on Pascal's Triangle. Students look at the triangle and see that each number is the sum of the two numbers above it. We see other patterns like the second diagonal is the list of Natural Numbers. Other students will know what triangular numbers are, or even tetrahedral numbers. Students may even find the sum of each row to be the powers of two. We see some great patterns when the students color the multiples of specific numbers. Before they start, they must extend the triangle in the picture to its largest size. We end with interesting patterns the students found. Students share their findings with the rest of the class at the end of the period.
 Homework
 Students must make sure that all parts of the "Fraction, Fractals, and Pascal" packet are filled in.
 Assessment
 Each student choses a multiple and colors their Pascal Triangle on the computer. It is great practice with factors and multiples. Discussions are a bit harder to assess. However, all students benefit from the class discussions. If not seen during classtime, students will know that their grade will be lower because of the attached rubric.

 Day 4: Final Work Day
 Objectives
 Students are asked to complete work on Sierpinski's Triangle. All students should have completed and colored their Sierpinski Triangle. Students that are completed will sit with students who are not and aid in their completion. Students are asked to complete the area for each color in Sierpinski's Triangle. Their work page will include the use of "Powers of 1/4". Students who have completed should help. Students are asked to have Pascal's Triangle completed. All numbers should be filled in and all patterns recognized in class should be written down. Students who have completed should help. Students should have observations about the similarities between Pascal and Sierpinski written down. They should have observations about what happens when they color in multiples of any number--specifically, what happens when they color in the multiples of 2. Students who have completed should help. If all students are finished, they should work on comparisons between the two triangles and set up the magazine according to instructions. They may use the rubric to grade their own project before it is turned in.
 Materials
 Students should bring in all materials from the previous three days of work. `
 Procedures
 Students are divided into groups. The teacher works with students who have not yet completed Sierpinski's Triangle. Those who have finished will work with students who have yet to complete the areas of each color. Once all students are finished creating the triangle, I will begin working with my students in that group. Once they are done, I will work with the last group that has to create observations about the project.
 Homework
 Students are asked to complete the project for homework. The next day is spent on discussion and fine-tuning before the project is turned in. All parts of the magazine must be completed.
 Assessment
 While helping students, they are assessed on whether they are behind for lack of effort or from being absent.

 Day 5: Discussion - Fractions, Fractals, and Pascal
 Objectives
 We discuss the reason for fractals one last time. We then talk about the fractal called Sierpinski's Triangle and its relationship to Pascal's Triangle. We also discuss Pascal's Triangle and the groups of multiples within the triangle. We discuss why this happens. We talk about other parts of the project that the students liked or did not like.
 Materials
 Students need to bring all of the supplies necessary to complete the "Mathematics Today" magazine. They also need to bring the ideas and supplies to share with other students about the project.
 Procedures
 Students are asked questions from the packet. When the answer is given, students make sure their answer is correct.
 Homework
 Relax.
 Assessment
 The project is collected and graded according to the rubric that was provided.

Brian McCarty

bmccarty@hhh.k12.ny.us

West Hollow Middle School
250 Old East Neck Road
Melville, NY 11747

I have been teaching for many years and have noticed that students are more attentive when they can play. Oddly enough, they learn more, too. I have made it my job to find places on the Internet where students can play and learn. I try to incorporate these "games" in my lessons as often as I can. The students appreciate it and even offer examples of interactive thinking games that they find on the Internet. I feel that this attitude has helped me in my career. I have won Redwood Middle School's "Viking of the Year" and "Teacher of the Year" awards. I look forward to continuing my career in the same fashion.

Important documents for this lesson plan.

 fractallesson.doc relationfrac.jpg pascalwrk.jpg

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