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.

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• 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.

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Sierpinski Triangle

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• The first person will control the mouse. It is important not to move ahead until everyone in your group is ready.

Let’s practice!

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This design is called a Sierpinski

Triangle.

Do you think it is a fractal ? Why?

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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.

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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.

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Directions

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

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Directions

• As you create each stage, you will fill in the information in the chart below the triangle.

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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

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Try Again

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Switch to the second person for Stage 1

The second person will now control the mouse.

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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.

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midpoint

Stage 1• Find the midpoints of the other two sides

of the triangle, and color them red.

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Stage 1• Connect the midpoints with line segments,

using a straightedge if you would like.

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Stage 1

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• How many small triangles do you have?– On your chart look at the first box in Stage 1 to seeif 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.

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Stage 1• With a red crayon shade in the

center triangle on your paper.

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Stage 1

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

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

1 2

3 4

Stage 1

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Try Again

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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

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sizethisoftrianglesofnumbertotal

trianglesuncoloredofnumber

Stage 1

• Are the new small triangles the same shape as the original big triangle?

Select your answer and click when a hand appears.

Yes No

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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.

Yes No

Stage 1

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Try Again

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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.

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Now it’s time to switch back to the first person to

control the mouse for Stage 2

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Each person in the group should put away their red pencil and grab a purple

pencil.

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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.

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Stage 2• How many new uncolored

triangles do you have altogether? Put your answer in the first box in Stage 2 on your chart.

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Stage 2Are the new uncolored triangles

congruent?Click on the correct answer where a hand

appears.

Yes No

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Try Again

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Stage 2

• Shade in the middle triangle in each group.

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Stage 2

How many new triangles are still uncolored?

Place answer in the 2nd box in Stage 2.

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• What fraction of the new triangles is still uncolored?

• Put your answer in the 3rd box in Stage 2 in your chart.

• Challenge: Try to reduce your answer to lowest terms.

Stage 2

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sizethisoftrianglesofnumbertotal

trianglesuncoloredofnumber

Each person in the group should put away their

purple crayon and grab a green crayon.

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It’s time for person

number two tocontrol the mouseonce again.

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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.

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Stage 3

• Count the number of new uncolored triangles

• Compare to the answer in your table for stage 3.

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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.

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Stage 3Compute and record in your

table:The fraction of uncolored

triangles to total new triangles.

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sizethisoftrianglesofnumbertotal

trianglesuncoloredofnumber

Sierpinski Triangle Stage 3

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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!

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Four applets!

• Any three points

• See stages again

• Here’s a puzzle

• Musical gasketTry them all before you move on

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There is more!

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There is more!

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There is more!

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This is the last slide.