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.
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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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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
different
lengths
Sides are
all same
length
Try Again
Click on the return button.
Click on the forward button.
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.
midpoint
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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
• 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.
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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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Stage 1
What did you write?
(select your answer and click when a hand appears)
1
2
3
4
Try Again
Click on the return button.
Yes, there are
4 triangles of
this size, but
only 3 are
uncolored.
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Stage 1
• What fraction of the original triangle
is now still uncolored?
number of uncolored triangles
total number of triangles of this size
• Place your answer in the
3rd box in Stage 1 on your chart.
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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
Hint:
Try Again
Click on return button.
Are all of
the
angles
the
same?
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Stage 1
• Are all the triangles in your picture the
same size? Select your answer and click when a
hand appears.
Yes
No
Try Again
Click on return button.
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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 2
Are the new uncolored triangles
congruent?
Click on the correct answer where a hand
appears.
Yes
No
Try Again
Click on the return button.
Click on forward button.
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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Stage 2
• What fraction of the new triangles is still
uncolored?
number of uncolored triangles
total number of triangles of this size
• Put your answer in the 3rd box in Stage 2 in
your chart.
• Challenge: Try to reduce your answer to
lowest terms.
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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 to
control the mouse
once 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 3
Compute and record in your
table:
The fraction of uncolored
triangles to total new
triangles.
number of uncolored triangles
total number of triangles of this size
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Sierpinski Triangle
Stage 3
Does your
design look
like this?
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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 gasket
Try 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