Concept of the Lesson:
Using the provided step-by-step directions, the students will create an iterative fractal tree in Geometer’s Sketchpad. An example of this tree is to the right.
After the tree has been generated, the students will find two attributes of the tree to compare. The first is the number of branches the tree has at each iteration; this they can count by sight. The second is the length of the stem at each iteration; this length will need to be measured using the Calculate -> Measure feature for one path of the tree.
There are two accompanying sketchpad files, a student file with the sliders set up, and a teacher file with a sample ‘answer’. /

Exponential Curve Fitting:

Once the students have found the two data sets they are ready to plot their points. First have them plot the number of branches vs. number of iterations (iterations, branches). Have them make conjectures about what type of graph this looks like, add slides for the ‘a’ and ‘b’ value and then graph the function y = abx and adjust the ‘a’ and ‘b’ sliders until the curve fits the data points as accurately as possible. They will notice that the ‘b’ value is equal to the number of branches were told to make when they created their tree – it demonstrates the exponential expansion of the tree. If each stem branched into two new stems, then their ‘b’ value would = 2.

Secondly have them plot the points for the (iteration, stem length) and have them notice this is now exponential decay. They will notice that the amount of dilation (greater than 1) used is equal to the ‘b’ value. This shows that for every iteration the stem is getting smaller and smaller by a set amount, or decaying.

Variations of the lesson include having the students change the process that the tree was created to include more branches at each iteration or to change the ratio of dilation. Another variation would be to assign students different ratio/branch combinations to try and then have them compile their results to draw conclusions at the end. You may want to consider giving different instruction sheets on how to create a different type of tree to eliminate confusion or error for those students that are less proficient with Sketchpad.

Part I: Creating the Fractal

1. Open Sketchpad and maximize both screens. Select the segment tool and draw a vertical segment about and label it AB as shown. (Note: Hold Shift key to get a straight line.) Make sure the line width is ‘thin’ and that all other lines drawn are ‘thin’. You can change its thickness by right clicking or going to the Display menu. /
2. ‘Mark’ A as the center by double clicking on it. You will see black rings vibrate around it for a second to show it was marked. Then unselect A and click on B.
Go to Transform -> Dilate and dilate point Athe distance (shown lower left)
3. Label this new point C. Now ‘Mark’ B as the center, unselect it, and click on C.
Go to Transform -> Rotate and rotate point C 30° and -30° about point B. (shown center)
4. Hide point C. Label the new rotate points C’. Draw two segments to connect BC’. (shown lower right)
5. Now it is time to start the iterative process. We need to tell Sketchpad to do this process over an over (so we don’t have to  ). To do this follow the following steps. As you have the iterative screen open, notice you can see a ‘pre-image’ of where the tree will be each step of the way. Make sure this image looks like the tree we have started, and that it is a new branch off of each branch.
6. Click on A, then B and go to Transform -> Iterate.
A window will pop up and you will notice A is highlighted. Click on B and then C’ on the left branch. Your Iterate pop-up window should look like the screen on the right.
7. Then click on the Structure drop down button and Choose ‘Add new Map’. Then repeat the process but click on B and then C’ on the right branch. Your Iterate pop-window should look like the second screen.
8. Click on the Structure drop down button and select “All Object Images”. /
9. Click on the Display drop down button and select “Full Orbit”
10. Click the Iterate button and watch your tree grow! /

Reflection

Describe what is happening as sketchpad is creating the tree. What patterns or trends do you notice?

Part II: Collecting the Data

Now that we have our fractal tree we need to collect two sets of data to analyze. Be sure to record your data in the appropriate tables below so you can refer to them later.

  1. Count the number of branches the tree has at each iteration. Iteration 0 is considered segment AB only, therefore there is only 1 branch. At iteration 1, the first split occurs, and there are now two branches. Fill in the first column using this procedure below.
  1. Trace the main stem AB. Change the thickness from thin to thick. Measure this length by going to Measure -> Length. Record this length for iteration 0 because this was the length before any iteration took place.
  2. Continue tracing the right-most stems (so they are thick) and measuring them. Record these lengths in the table. When you are done your tree should look like the one to the right below.

# Iterations / Number of Branches / Length of Stem
0
1
2
3
4
5
/
  1. What types of observations can you make about the relationship between the number of iterations and the number of branches?
  2. How about the number of iterations and the length of the stem?
  3. How about the number of branches and the length of the stem?

Part III: Curve-Fitting and Drawing Conclusions

  1. Open the file called ExploreFractals-Students.gsp
  2. You will notice directions at the top on how to plot data points and how to plot a new function. You may ‘hide’ these directions by clicking on the hide/show button in the top left of the screen.
  3. Notice the tabs on the bottom of the screen, these tell you which window to do each comparison exploration in so your work will be separated.
  4. On each tab, do the following:
  5. Plot the data points using the number of iteration and the specified data
  6. What type of function does this look like?
  7. Plot a general function using ‘a’ and ‘b’ and the sliders as described in the on screen directions
  8. Adjust the blue endpoint on the sliders until the curve appears to fit through a many points as possible.
  9. Repeat the process on the other tab for the other data set.
  1. What equations did you have for the Iterations and Stem length data set?
  2. What meaning do you think the ‘a’ and ‘b’ values have in this equation? Be sure to talk in terms of the tree when phrasing your answer.
  1. What equations did you have for the Iterations and Number of Branches data set?
  2. What meaning do you think the ‘a’ and ‘b’ values have in this equation? Be sure to talk in terms of the tree when phrasing your answer.

Shannon Denna - FremdHigh School 10/8/2018 Page 1