Triangle Town

Hi, I’m Elke. Pleased to be your guide through Triangle Town. Don’t ever worry about getting lost. I’ll stay with you the whole time. Now click on the arrow below to get going. Click on the forward button.

Each person will have a chance to control the mouse. The first person, numerically, will have the first turn. I will tell you when it is the next person’s turn. Click on the forward button.

Sierpinski Triangle Click on the forward button.

The first person will control the mouse. It is important not to move ahead until everyone in your group is ready. Let’s practice! Click on the forward button.

This design is called a Sierpinski Triangle. Do you think it is a fractal ? Why? Click on the forward button.

If you said, “YES!”, you are correct. The triangle is made of smaller, similar parts that are part of the whole, so they are self-similar. The fractal is formed by repeating stages. Click on the forward button.

Directions Everyone in your group needs an activity sheet labeled “Sierpinski Triangle” Look in your traveler’s journal. Each person should begin with a red crayon. Click on the forward button.

Directions On your dotted grid, each of you will create a Sierpinski Triangle, zooming down to the third stage while following these instructions. Click on the forward button.

Directions As you create each stage, you will fill in the information in the chart below the triangle. Click on the forward button.

Sides are different lengths Stage 0 The triangle that is drawn in dots on your grid paper is an equilateral triangle. What is true about an equilateral triangle? Select your answer and click when a hand appears. Sides are all same length

Click on the return button. Try Again

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Switch to the second person for Stage 1 The second person will now control the mouse. Click on the forward button.

Stage 1 On the bottom side of the triangle on your paper, use red to color the point in the middle of the side. This point is called the midpoint of a side. Click on the forward button. midpoint

Stage 1 Find the midpoints of the other two sides of the triangle, and color them red. Click on the forward button.

Stage 1 Connect the midpoints with line segments, using a straightedge if you would like. Click on the forward button.

Stage 1 Click on the forward button. How many small triangles do you have? –On your chart look at the first box in Stage 1 to see if you have the right answer.

Stage 1 Do the new triangles all look to be the same size and shape? Triangles that are the same size and shape are called congruent triangles. Click on the forward button.

Stage 1 With a red crayon shade in the center triangle on your paper. Click on the forward button.

Stage 1 How many uncolored triangles do you now have? Put the answer in the 2 nd box in Stage 1 on your chart. Click on the forward button.

What did you write? (select your answer and click when a hand appears) Stage 1

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Click on the forward button. Yes, there are 4 triangles of this size, but only 3 are uncolored.

What fraction of the original triangle is now still uncolored? Place your answer in the 3rd box in Stage 1 on your chart. Stage 1 Click on the forward button.

Stage 1 Are the new small triangles the same shape as the original big triangle? Select your answer and click when a hand appears. YesNo

Click on return button. Try Again Hint: Are all of the angles the same?

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Are all the triangles in your picture the same size? Select your answer and click when a hand appears. YesNo Stage 1

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Triangles that have the same shape but not necessarily the same size are called similar triangles. Stage 1 triangles are similar to the Stage 0 triangle, but not the same size. Click on the forward button.

Now it’s time to switch back to the first person to control the mouse for Stage 2 Click on the forward button.

Each person in the group should put away their red pencil and grab a purple pencil. Click on the forward button.

Stage 2 For the uncolored triangle at the top, find the midpoint of each side and then connect the points with line segments. Do the same for the other two uncolored triangles. Click on the forward button.

Stage 2 How many new uncolored triangles do you have altogether? Put your answer in the first box in Stage 2 on your chart. Click on the forward button.

Stage 2 Are the new uncolored triangles congruent? Click on the correct answer where a hand appears. YesNo

Click on the return button. Try Again

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Stage 2 Shade in the middle triangle in each group. Click on the forward button.

Stage 2 How many new triangles are still uncolored? Place answer in the 2 nd box in Stage 2. Click on the forward button.

What fraction of the new triangles is still uncolored? Put your answer in the 3 rd box in Stage 2 in your chart. Challenge: Try to reduce your answer to lowest terms. Stage 2 Click on the forward button.

Each person in the group should put away their purple crayon and grab a green crayon. Click on the forward button.

It’s time for person number two to control the mouse once again. Click on the forward button.

Stage 3 Find the midpoint of each of the sides of the remaining uncolored triangles and draw them with your green crayon. Connect the midpoints with line segments. Click on the forward button.

Stage 3 Count the number of new uncolored triangles Compare to the answer in your table for stage 3. Click on the forward button.

Stage 3 Use your green crayon to fill in all center triangles. Count the number of remaining uncolored triangles and compare to the answer in the Stage 3 box. Click on the forward button.

Stage 3 Compute and record in your table: The fraction of uncolored triangles to total new triangles. Click on the forward button.

Sierpinski Triangle Stage 3 Click on the forward button. Does your design look like this?

BONUS: Stage 4 Study the growing patterns in Stages 1, 2, and 3. Talk about it in your group. Then, try to fill in all of the boxes for Stage 4. When complete, raise your hand to receive the secret code from the Fractal Keys tour guide if you are correct.

Zooming Sierpinski ! There is more! Click on the forward button.

Four applets! Any three points See stages again Here’s a puzzle Musical gasket Try them all before you move on Click on the forward button.

There is more! Click on the forward button.

There is more! Click on the forward button.

There is more! Click on the forward button.

This is the last slide.