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Length of the Sides in a Sierpinski Triangle
I. Information from the triangles – Data Tables
We will use information from the last lesson about the number of upward pointing
triangles at each stage in the construction of a Sierpinski triangle and the lengths of the
sides of these triangles. The first 5 stages, (0 – 4), of the construction are shown below so
you remember what they look like. The upward pointing triangles are shaded in each
stage.
0
1
2
3
Stages in constructing a Sierpinski triangle
4
The table below has values filled in up through stage 4 for the number of upward pointing
triangles and the length of each side. There are three columns in this table.
1. Look for patterns for the numbers in the second and third columns and fill-in the rest
of columns 2 and 3 through stage 8.
Stage
Number of small upward
pointing triangles
Length of a side in each
upward pointing triangle
0
1
4 inches
1
3
2 inches
2
9
1 inch
3
27
½ inch
4
81
¼ inch
5
6
7
8
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Sometimes exponents are useful for representing numbers. They can let us write the
number in a simpler way and sometimes allow us to see patterns we would normally
miss.
2. Using what you know about exponents, go back and rewrite the numbers in the
second and third columns of the table using exponents. You will have to determine
what base to use for each column. The base will be different for the two different
columns. The third column is tricky. It starts off nicely but soon gets into fractions.
For the fractions just represent the denominator as a power of some base.
Stage
Number of small upward
pointing triangles
Length of side of each
upward pointing triangle
0
1
2
3
4
5
6
7
8
From the data you have gathered, you can make predictions about the number of upward
pointing triangles and the length of their sides without drawing pictures or counting
triangles. Use your reasoning to answer the following.
3. How many upward pointing triangles will there be at stage 18? ________ Represent
this number using exponents and then use a calculator to actually find the value of
this number.
Written as an exponent ___________
Calculator value ___________
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4. Explain the reasoning you used to get your answer in question 3.
II. Predictions - Using graphs of data
A graph can be useful to see what is happening with data. We can make a graph using
columns 1 and 2 in the table by doing the following.
Number of
upward
pointing
triangles
0
1
2
3
4
Stage
Look at the graph above. It has a horizontal and a vertical axis. The horizontal axis has
the stage numbers written on it for stages 0 through 4. You will plot a point directly
above each stage number representing the number from column 2 that goes with the stage
from column 1. You will have to make your own scale for plotting numbers on the
vertical axis. These numbers come from column 2 and represent the number of small
upward pointing triangles. Be careful, look at all the numbers in column 2 first. They
increase quickly. Think about how to scale the vertical axis so you can fit all of them on
the graph. You may have to estimate the position of some of the numbers.
5. Plot the numbers from column 2 for stages 0 through 4 on the graph above.
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When the points are all plotted for stages 0 through 4, connect the points with line
segments from dot to dot. You now have a graph representing how the number of
upward pointing triangles change from stage to stage.
6. As you draw more and more stages, what happens to the number of upward pointing
triangles? Is it increasing, decreasing, staying the same? ___________
7. Use the tables and graph you made to justify why you answered question 6 the way
you did.
8. If you were able to continue drawing the triangles forever, how many upward
pointing triangles would you get? ______________________
We will now look at numbers in column 3. Thinking about the pattern you see for the
numbers in column 3 written as exponents, answer the following questions.
9. What will the size of a side be, written as an exponent, for stage 18? Use your
calculator to actually find the decimal value of this number.
Written as an exponent __________
Calculator value _______________
10. Explain the reasoning you used to get your first answer in question 9.
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We are next going to make a graph using columns 1 and 3. But first it will be easier if
you change all the fractions in column 3 to decimals. Since we will be plotting these
decimals on a graph, round all the decimals off to hundredths.
11. Fill-in the decimals values for column 3.
Stage
Number of small upward
pointing triangles
0
1
1
3
2
9
3
27
4
81
5
243
Length of side of each
upward pointing triangle
(written as decimals)
Now we will make a graph as before with the number of the stage on the horizontal axis.
But this time, the decimal value, rounded to hundredths, from column 3 is graphed
above each stage number. Here you have the opposite problem from before, the numbers
from column 3 get very small. Look at them carefully and determine the scale you need
on the vertical axis. You may again have to approximate their position on the graph.
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12. Plot the side lengths from column 3 for stages 0 through 5.
Length of
sides
0
1
2
3
4
5
Stage
13. As you draw more and more stages, what happens to the length of the sides of the
upward pointing triangles? Is it increasing, decreasing, staying the same?
_______________
14. Use the tables and graph you made to justify why you answered question 13 the way
you did.
15. If you were able to continue drawing the triangles forever, the lengths of the sides of
the triangles get very close to what number? __________