Fractals Using the Koch Snowflake

Transcription

1 Fractals Using the Koch Snowflae Grade level: 8 th Time needed: One class period *May be incorporated with a geometry unit. Objectives Recognize and construct similar figures Find perimeter and area of polygons Materials/Equipment White paper (at least 1 X 1 ) Pencil Ruler Protractor Colored pencils Description of the Activity Students will construct the Koch snowflae. They will also examine the impact of the construction on the area and perimeter of the completed figure. Students will begin by constructing an equilateral triangle with side lengths of 9 inches on their paper, using protractors (angles 60 degrees) and rulers. *Students should draw this first triangle in the middle of the paper. They will be adding smaller triangles on every edge. Discuss the perimeter and the area of the triangle. Instruct the students to record these measurements on their Observations page. Iterations are made by first dividing each existing side into 3 equal segments, then drawing 3 more equilateral triangles on the exterior of the first triangle using the middle segments on each of the 3 existing sides as the bases of the new triangles. See Figure 1. Figure 1

2 Allow each student enough time to mae these iterations and color them all the same color (student s choice)(i am confused by the picture. Which three rianges above have the same color?). Then discuss the area of these new triangles and have the students add their areas to the original triangle and record these measurements on their Observations page. Students then need to erase the bases of the 3 new triangles. If constructed correctly, the figure will loo lie a sixpointed star. Students then need to measure or calculate the perimeter of the new figure and record the measurement. They should then be instructed to continue with more iterations, dividing each segment in the perimeter of the figure into 3 equal parts, creating new triangles on the outside of the figure, as before, coloring each additional iteration with a different color, and then erasing the bases. See Figure. Figure Unless I am not seeing (figure ) in the same way as intended, the coloring instructions are confusing.) Be sure to have students complete the table on the Observations page with the appropriate figures as they add new triangles. This lesson could be adapted for younger students by having colored paper pre-printed with similar triangles (for instance, yellow paper for 1 large triangle with side lengths of 9 inches, blue paper for 3 triangles with side length 3 inches, green paper for 1 triangles with side lengths of 1 inch, and maybe one more iteration pin paper for 8 triangles with side lengths of one-third inch.) Students could cut out the triangles and glue them in the right place on construction paper to form the Koch Snowflae, and could still examine patterns they see. Discussion Questions Students may only need to complete row number 3 in order to recognize a pattern in the table, regarding the area and the perimeter. Mae sure you allow ample time for all students to discover the pattern before discussing it with the class as a whole.

3 1. What would the area and perimeter be if another iteration was completed? (See Teacher s Guide on Page 8). From this pattern, let s see if we can come up with a formula for finding the area and perimeter for any number of iterations. Area of th iteration = 3! 1 3 ' s $ 3 3 % = s & 3 & Perimeter of th iteration =! # 3 s $ % 3 " *s=the length of the side of the original equilateral triangle 3. What do the angles of the snowflae have in common? (All are 60 degrees). What can we say about the perimeter? (There is no limit to the perimeter if more and more triangles are added, perimeter approaches infinity.) 5. What can we say about the area? (Area is finite. The amount of the area added gets smaller with each iteration, so eventually the difference is so small that it becomes insignificant.) Assessment The completed tables will provide insight into students understanding of perimeter and area of unusual figures that are made up of smaller polygons. A formal grade may be given for the accuracy of their measurements and calculations as well as the accuracy of their construction of the Koch Snowflae. Reference and Resources Mandelbrot, Benoit B. (198). The Fractal Geometry of Nature. New Yor: W.H. Freeman & Company. Lesmoir-Gordon, Nigel. (001). Introducing Fractal Geometry. Kallista, Australia: Totem Boos. " # 9! 1

4 National Science Standards Life Science: CONTENT STANDARD C Science and Technology: CONTENT STANDARD E Process Standards Communication: COM.1, COM., COM. Reasoning and Proof: RP.1, RP. NCTM Standards Content Standards Algebra: A.1, A.3 Data Analysis and Probability: DAP.3 Geometry: G.1 Measurement: M.1, M. Observations Date: Name:! s! s Sides Sides Perimeter Perimeter Area Area Orig Teacher s Guide

5 ! s! s Sides Sides Perimeter Perimeter Area Area Orig Math Review Area of a triangle = 1 X base X height. We then need to find the height of an equilateral triangle. Remember that for a right triangle, Pythagorean s Theorem tells us: a b c + =. We can then construct the height of an equilateral triangle by drawing a broen line from one vertex to the midpoint of the opposite side. Label this broen line, a (height). One-half the side of the equilateral will be the b part of Pythagorean s Theorem. The c part of the theorem will be the length of one of the sides of the equilateral triangle. In the Koch Snowflae activity, the equilateral triangle has sides with length 9, so we see that a b c + =! & 9 # 81 a + $! = 9! + = 81 % " a! a = 81! 81 ()(81) (1)(81) a =!! (3)(81) a =! 9 3 a =.

6 In general, we see that: Height of an equilateral triangle = 3 X length of one side. Since any side of an equilateral triangle can serve as the base, it follows that the area of an equilateral triangle is: Area of a equilateral triangle = 1 X base X height! 1 3 Area of an equilateral triangle = X base X X base! Area of an equilateral triangle = 3 X base. Remember that the perimeter of any figure is equal to the sum of all its sides. The derivation of the formula for the area of th iteration: Let s=the length of the side of the original equilateral triangle. Area of the original triangle = 3 s. Area of 1 st iteration = 3 s 3 & s # + 3! " $ % 3 3 & 3 # = s $ 1 +!. % 9 " Area of nd 3 # iteration = & 3 3 & s # s $ 1 +! + 3 $! = 3 & 3 3 # s $ 1+ +!. % 9 " % 9 " % 9 9 " Area of 3 rd iteration = 3 & 3 3 # 3 & s # s $ 1+ +! + 3 $ 3! % 9 9 " % 3 ".

7 ! 1! 1 3 ' s $ 3 3 Area of the th iteration = 3 % = s ". & 3 # 9 The formula for the perimeter of the th iteration can be similarly derived and is: & # 3 s $!. % 3 " Contour Lines Grade level: 7 th Time needed: one class period Objectives Solve problems involving scale factors using ratios and proportions Apply geometric ideas and relationships in areas outside the mathematics classroom Materials/Equipment Contour line maps (may be found at the online resources below and laminated for long term use) Playdoh (various colors) Wax paper Ruler Description of Activity Before beginning the activity, as the discussion questions below while showing the students different inds of contour maps. Some examples of contour maps are: 1. Contour maps of the oceans are made to show the different water temperatures to aid fishermen and scientists.. Contour maps of the oceans are also made to show the different depths, i.e. underground mountains and valleys. 3. Contour maps of the land are made to show the different elevations.. Not a clear example. Is it needed? Allow the students to wor in groups of 3-. Each group will receive a copy (today s fols don t now what a ditto is) of a contour map. Have

8 them place a sheet of wax paper over their contour map. The students will then use Playdoh to build a 3-D figure of their contour map, on top of the wax paper. Encourage them to use a ruler to construct layers of the same depth. Outlining the elevations with a small rope of Playdoh and then filling in the middle of the elevations seems to mae the tas easier. If time allows, have the students roll out thin layers of Playdoh to drape over their Playdoh formations or simply blend the edges of the layers to mae their formations more lifelie. An extension of this project is to have each group mae a topographic map of a local area with which they are all familiar. Discussion Questions 1. What do close contour lines on a map of elevation tell us about the terrain? (A steep slope, cliff, or cavern can be located there.). What do we now about the terrain if the contour lines on a map of elevation are far apart? (The land gently slopes.) 3. Why do contour lines never intersect on a topological map? (Each contour line represents one unique elevation; therefore, since a point of intersection cannot represent different values, contour lines cannot intersect.) Assessment The level of the students understanding will be reflected in the construction of their 3-D maps. A suggested critical thining question might be: 1. How would the topographic map show the difference between a mountain and a canyon? The lowest and highest elevations should be labeled numerically. References and Resources urspacing.html National Science

9 Standards Science as Inquiry: CONTENT STANDARD A Earth and Space Science: CONTENT STANDARD D Process Standards Communications: COM.1, COM., COM.3, COM. Connections: CON.1, CON., CON.3 Representation: R.1, R.3 NCTM Standards Content Standards Geometry: G.1, G. Measurement: M.1, M. Number and Operations: NO.1 Ice and Gravel Glaciers Grade level: 8 th Time needed: three class periods Objectives Use ratios and proportions to represent real-world situations Interpret graphs which represent rates of change Apply geometric ideas and relationships in areas outside the mathematics classroom Solve real-world problems involving rate/time/distance (d=rt) Solve problems involving scale factors using ratios and proportions Determine the mean of a given set of real-world data Materials/Equipment Balance or scales for mass measurement 1 X 1 pieces of plywood ( per group) 35 mm film containers ( per group) Masing tape Popsicle stics Glue Small bathroom Dixie cups ( per group)a vague definition - 16oz cup is a small at a convenience store Aquarium gravel Sand Description of the Activity This activity is intended to explore several different qualities of glaciers, glacier movement, and glacier melting. The activity will tae three class periods to complete over a three day period as the glaciers will need to be made and frozen over night. Students will wor in groups of 3 or.

10 DAY 1: *Glacier Creation Provide each group with Dixie cups. Have them label the bottoms of all cups. Each group will create ice glaciers by filling cups ¾ full of water. They will then create glaciers of ice and gravel by filling the remaining cups ½ full of gravel and then filling up to ¾ full of water. The cups from all of the groups will need to be placed in a freezer overnight. *Mountain Creation Students will create ramps of different surfaces, sand and gravel. Each group will receive pieces of plywood, film containers, Popsicle stics, masing tape, sand and gravel. (1)Sand Surface Incline: Students will begin by taping 1 film container to the bottom of each piece of plywood (close to one edge), forming an incline when placed on the des or floor. The students will then pac a half inch layer of damp sand over the entire surface of the plywood. When the sand is in place, the Popsicle stics should be inserted into the sand to divide the incline into two separate but equal inclines. ()Gravel Surface Incline: This incline will be constructed in much the same manner as the sand surface incline but students will glue a layer of gravel to the surface of the plywood. After the glue has set, apply another layer of gravel on top of the glued layer (is this correct?) (Yes) but this time without gluing. As with the sand surface incline, Popsicle stics should be inserted into the gravel to divide the incline into two separate but equal inclines. DAY : Each group will measure and record the mass of each of their glaciers. Have students place a glacier of each ind at the top of their sand incline and a glacier of each type at the top of their gravel incline. Time of placement should be recorded. Students will measure the distance each glacier travels at 5 minute intervals until each glacier has moved to the bottom of each incline. All measurements of distance should be made from the top of the incline. Rate of change should be calculated at the end of each 5 minute interval. Once the glaciers reach the bottom of their respective inclines, students should measure and record their masses. Proportions of remaining mass compared to initial mass will be calculated and recorded. Average rates will also be calculated for each glacier on each surface incline.

11 DAY 3: Groups will then create a line graph representing the rate of change per 5 minute intervals for each glacier on each surface. All lines should be displayed on the same graph so comparisons of the glacial movements can be more easily made. Each line can be drawn using a different color marer. Discussion Questions Allow each group to present their line graphs and observations. Lead the groups into answering the following questions in their presentations:if all of the answers to these questions are will vary. What is the message that the students should draw? Is there any soecic nowledge to be learned or is this an area where little is nown. The activity is fantastic, but these questions and the outcomes are unsettling. Why would a teacher do this if he/she might end up looing lie they were just having fun and passing the time? (See explanation below) 1. Which glacier on which surface moved at the highest rate? (Answers will vary.). Which glacier on which surface moved at the lowest rate? (Answers will vary.) 3. Did you see any movement of sand or gravel from water? (Answers will vary.). Did you see any accumulation of water at the bottom of your incline? (Answers will vary.) 5. Which glacier appeared to be melting quicer? (Answers will vary.) 6. Which glacier appeared to be melting slower? (Answers will vary.) Lead the class in a discussion on how their findings were similar and how they were different. * Glaciers, in nature, often melt and refreeze along their journey, so although an experiment in the classroom is informative, it is limited. Because conditions are liely to vary from classroom to classroom, we thin it best to refrain from giving definite outcomes. The activity,

12 however, is excellent for collecting and recording data and drawing line graphs of results. Assessment Students and groups will be assessed on the accuracy of their calculations regarding proportions, rates and averages. Students will also be assessed on their line graphs and presentations of their findings. Resources and References ml/changes.htm&direct=yes osion.htm National Science Standards Science as Inquiry: CONTENT STANDARD A Physical Science: CONTENT STANDARD B Science in Personal and Social Perspectives: CONENT STANDARD F NCTM Standards Process Standards Content Standards Communication: COM.1, Data Analysis and COM., COM.3, COM. Probability: DAP. Connections: CON.1, Measurement: M.1, M. CON., CON.3 Representation: R.1, R.3 Observations Date: Names: Glacier Initial Mass Final Mass Remaining Proportion

13 Ice on Sand Ice/gravel on Sand Ice on Gravel Ice/gravel on Gravel Which glacier melted slower than the others? Which glacier melted faster than the others? Glacier Ice on Sand Ice/gravel on Sand Ice on Gravel Ice/gravel on Gravel Distance Traveled Glacier Ice on Sand Ice/gravel on Sand Ice on Gravel Ice/gravel on Gravel Rates

14 Glacier Ice on Sand Ice/gravel on Sand Ice on Gravel Ice/gravel on Gravel Average Rate Glacier Ice on Sand Ice/gravel on Sand Ice on Gravel Ice/gravel on Gravel Average Rate per Minute Math Review Proportion of remaining glacier = remaining mass initial mass X 100% Rate = dis tan ce traveled time Average rate = sum of all rates number of rates added Average rate = number of average rate per min ute min utes in deno min ator of average rate