Subject: Mathematics
Grade Level: 7th Grade Math Seminar
Materials: This lesson asks students to
have the following each day: a centimeter ruler, pencil, and a
"Mathematics Today!" magazine packet. Students are also given
time to explore fractals and Pascal's Triangle (later in the
lesson) on computers. They are sent to these sites:
http://shodor.org/interactivate/activities/SierpinskiTriangle/
and
http://shodor.org/interactivate/activities/ColoringMultiples/
This is done through wireless Internet and a laptop cart. A
projector is used to have students come up to present patterns
of the multiples in Pascal's Triangle. The goal is to have
students find that the multiples of 2 strongly resemble
Sierpinski's Triangle.
About: Students create Sierpinski's
Triangle using the centimeter ruler and a pre-designed
equilateral triangle with sides of 16cm. On the projector, I
show the classroom examples of other fractals. I ask the
students to look for similarities between all fractals and see
if they can create a definition. We discuss the definition of
a fractal and how this is the process that creates one. We
talk about what an iteration is and find the relationship
between the area of each smaller triangle and the largest
triangle. We then ask about the areas after the second
iteration. The students color their fractal in after 3
iterations and then find the area of each color. Students use
fractions and the powers of fractions to find the areas. They
also learn about Pascal's Triangle and how to create it. Using
the ideas of multiples, they see the relationship between
Pascal's Triangle and Sierpinski's Triangle.
The final project culminates in a "Mathematics Today"
magazine. The magazine includes a cover containing
Sierpinski's Triangle, directions on creating a fractal,
examples of fractals, definition of a fractal, Pascal's
Triangle, written observations of Pascal's Triangle and its
relevance to the project, and the final sheet that includes
answers to the areas of each color on the cover.
This project covers many of the topics that need to be
learned in mathematics. In the New York State curriculum,
students are required to use exponents with fractions, find
the area of a triangle, identify patterns, make comparisons,
create definitions from observations, be creative with colors,
and learn to use the computer to aid in instruction. I feel
that the computer/Internet adds vital practice to the project.
Students are allowed to "play". This strengthens instruction
and aids in conversations about what Pascal's Triangle has to
with Sierpinski's Triangle. This connection is a key part of
comparisons between Pascal's Triangle and Sierpinski's
Trangle.
One of the great things about this project is that it can
be run in any mathematics classroom. Teachers should arrange
time in the computer lab if portable laptops are not
available. Markers or crayons should also be available.
Students tend to rush through the measurements for midpoints.
These poorly measured points affect the cover sheet for the
project. Students may not see a connection between Pascal's
Triangle and Sierpinski's Triangle. A few of the students
struggle with the idea of 1/4 raised to a power. The nice
thing about this project is that once that proper measuring is
stressed, teachers can work with these students to make sure
areas work out correctly.
http://eev3.liu.edu/oc0507/bm/fracfracs.htm
| For this lesson I would like the
students to understand the idea of a fractal. They may
use the Internet to look at various fractals and see how
multiple versions are all similar to each other. All
students have a chance to propose a definition during
class. |
| Students will understand how to create
a fractal called Sierpinski's Triangle. They use the
midpoints of each side and realize that connecting them
is called an iteration. Each iteration will create
triangles that are one-fourth of the triangle that they
are drawn from. |
| Students will understand how Pascal's
Triangle is formed. They fill in missing numbers and see
that there are some interesting patterns that appear
within the triangle. |
| Students continue with Pascal's
Triangle on the Internet and look at the multiples of
numbers within the pattern. In their attempt to discover
the relationship between Pascal and Sierpinski, students
look at what happens when the multiples of 2 are filled
in on Pascal's Triangle. We first look at this on the
Internet. Then we look at it on the overhead. When the
two are overlapped, they match up (roughly). Students
continue to explore patterns of multiples in Pascal's
Triangle. |
| Students color in Sierpinski's Triangle
and find the area's of each color. They use the area
formula once and, like the problem states on the
magazine cover, find the area of each color. This is
done by multiplying the total areas by powers of 1/4.
Students begin to find patterns in their answers and use
those to get answers for other parts of the problem.
They are able to check their answers by adding all of
the area together to get the original area. |
| The project will finish with the
comparison, observation, and conclusion about Pascal's
Triangle, Sierpinski's Triangle, and the relationship
that the two figures have. Students will have to write
coherent sentences that outline this information. |
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| This link provides the
inspiration to the lesson. It is a great way for
students to understand how to create a fractal
called Sierpinski's Triangle. The students can
look at the steps involved and understand what is
included in an iteration. |
| http://shodor.org/interactivate/activities/SierpinskiTriangle/ |
| This link shows students that
Sierpinski's Triangle does not have to be an
equilateral triangle. Sierpinski's Triangle can
actually be created on any type of triangle.
Students can "play" to see this. Again, this play
will increase comprehension. |
| http://serendip.brynmawr.edu/complexity/sierpinski.html |
| This site enables students to see
other types of fractals call the Mandelbrot Set
and the Julia Set. This will drive students
towards a definition of the word "fractal." |
| http://facstaff.unca.edu/mcmcclur/java/Julia/ |
| This is another site that lets
students explore different types of fractals.
Students should not only make a definition for the
word fractal but try to think of a reason for
creating a fractal. |
| http://nlvm.usu.edu/en/nav/topic_t_3.html |
| This is another site that allows
students to see fractals being created. These
other fractals are called the "Jurassic Park"
fractal, Koch's Snowflake, and the Anti-Snowflake.
These sites offer even more opportunities for
students to come up with a definition and decide
the purpose of a fractal. |
| http://math.rice.edu/~lanius/frac/index.html |
| This site lets students explore
Pascal's Triangle. The students can see how the
triangle is created and what happens when specific
multiples are colored in. This enables students to
see the similarity between Pascal's Triangle and
Sierpinski's Triangle. |
| http://shodor.org/interactivate/activities/ColoringMultiples/ |
| Students see other patterns that
exist in Pascal's Triangle. The creation of the
triangle is a small part of the entire figure.
This identifies items called natural numbers,
triangular numbers, tetrahedral numbers, and
Fibonacci numbers. There are other patterns that
students may research on their own. |
| http://mathforum.org/workshops/usi/pascal/mo.pascal.html |
| There are a host of other links
that are important to the project. I have compiled
them into one link that encourages the students to
conduct further investigation and study. On the
link, sites that provide students a chance to
further explore fractals and Pascal's Triangles
are listed under the explorations link on the left
menu. |
| http://eev3.liu.edu/oc0507/bm/fracfracs.htm |
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| Students understand numbers, ways
of representing numbers, relationships among
numbers, and number systems. They work with
fractions, exponential expressions, and multiples,
and understand what an exponent is and how to find
the answer to an exponential expression that
contains fractions. Students must also develop and
analyze algorithms for computing with fractions,
decimals, and integers and develop fluency in
their use. |
| NCTM 6-8 |
| Mathematics |
| In this project, students
practice creating and critiquing inductive and
deductive arguments concerning geometric ideas and
relationships such as congruence and similarity.
They analyze characteristics and properties of
two- and three-dimensional geometric shapes and
develop mathematical arguments about geometric
relationships and recognize and apply geometric
ideas and relationships in areas outside the
mathematics classroom such as art, science, and
everyday life. |
| NCTM 6-8 |
| Mathematics |
| Students apply appropriate
techniques, tools, and formulas to determine
measurements. They select and apply techniques and
tools to accurately find length, area, volume, and
angle measures to appropriate levels of precision.
They learn how to check their answers by adding
results to obtain the original area. They also
check their observations through sharing them with
the class. |
| NCTM 6-8 |
| Mathematics |
| Besides the mathematics that is
involved in the Sierpinski vs Pascal project,
there are many parts of the New York State
curriculum that are really discussed and developed
within the student. There are few times where
students are asked to defend their ideas, present
new ideas, make comparisons, conclusions, use
physical objects as representations, apply
strategies, etc. These are a big part of the
middle school mathematics curriculum. |
| New York State Math Curriculum
(6-8) |
| Mathematics |
| Students understand and use the
elements and principles of art (line, color,
texture, shape) in order to communicate their
ideas. They use their colors in Sierpinski's
Triangle. This brings mathematics into the
Creative Arts classroom and gives evidence that
they have developed an emerging personal style. |
| The Arts ( K-12 ) |
| Visual Arts |
| How were such ideas created so
long ago? Blaise Pascal lived in the emergence of
the first global age. Waclaw Sierpinski lived in
the age of revolutions. Students in upper grades
may benefit from some discussion about whose time
was harder to live through. Who had the harder
life? |
| Social Studies (6-9) |
| Social Studies |
| Students read, write, listen, and
speak for critical analysis and evaluation. As
listeners and readers, they analyze experiences,
ideas, information, and issues presented by others
using a variety of established criteria. As
speakers and writers, they present, in oral and
written language and from a variety of
perspectives, their opinions and judgments on
experiences, ideas, information, and issues. |
| English Language Arts
(7-10) |
| English Language Arts |
| Students are asked to look at
pictures and make written observations about what
they see. This asks them to organize their
thoughts and put them into clear sentences.
Students are also asked to make comparisons
between two different pictures. In their written
observations, they will see that Pascal's Triangle
and Sierpinski's Triangle have a
relationship. |
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| Day 1: Creating a
fractal |
| Students are shown how to create a fractal on
the Internet. They follow the process and look at
the other types of fractals. |
| Students create their own definition for a
fractal and then discuss it with the class. We end
with one final definition for the word
fractal. |
| The class discusses the use of fractals and
what the point is of creating such things.
Students should discuss the artistic form of a
fractal and how they create shapes that are
pleasing to look at. |
| Students finish the day with coloring the
fractal with at least four different colors. If
the coloring is not completed, they must complete
it for homework. |
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| Students need all of the items required for
all math classes. They need laptop computers,
rulers, calculators, pencils, pens, erasers,
computers, graph paper, lined paper, "Fractions,
Fractals, and Pascal" packet, and a positive
attitude. |
| Students need a laptop computer and links to
the Shodor.Net site. They will use this to see the
steps on how to create a fractal. |
| Students continue to use the laptops to see
the different types of fractals. They use these to
create a definition for the word fractal. They
need the second sheet of the packet as well. |
| The class also requires markers to creatively
color the fractal that they created with
instructions given at the beginning of class and
modeled on the Internet. Students will be able to
play with the fractal on sites provided to
increase
comprehension. |
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| Students are given a "Fractions, Fractals, and
Pascal" packet. We start with the definition of
the word "midpoint." The example given on the
Internet is a great way for students to see the
idea of the midpoint. Students are asked how to
find the midpoint using the supplies that they
have with them. |
| We draw the midpoint on each side of the
triangle and connect them. This is called an
iteration. I ask the students to complete a second
iteration on their own. A second iteration
includes three more triangles that point down, not
just one. |
| We then look at other types of fractals.
Students are asked to look at all of the fractals
and create a definition in groups of three or
four. We discuss the word and then give the
definition. |
| We start a discussion about the reason for
fractals. Fractals are important in art and in
determining reason in the randomness of life. Many
think that there are events that can be predicted
in the picture of the Mandelbrot set. It is
similar to the digits in Pi. |
| Students begin to create their covers. Once
they are done with creating the fractal called
Seirpinski's Triangle to four iterations, they can
begin coloring. They must use at least four
different colors. |
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| Students complete coloring in their
Sierpinski's Triangle and construct a theory on
how to solve the "Mathematics Today" question.
Answers will be discussed at the beginning of the
next
class. |
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| As students are working, I walk around to see
what they are writing on their papers and
inspecting their use of the Internet. The goal of
today's assessment is to correct any
misunderstandings so that more students can reach
the large goal at the
end. |
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| Day 2: How do we find the
area? |
| Students need to find the area of each color.
In the evening prior, students were to find ways
to get their answer. Students will all have a way
to get the answer in the first 10 minutes of
class. We will see how to express that way in
terms of exponents and compute it using a
calculator. |
| Students continue to color, explore, find the
areas of each color. Those who are finished will
come up with a method to check their area answers.
They present their answers in the last five
minutes of class. This gives the class time to
catch up to each other so that we can move
together as a group. |
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| Students need markers, laptop computers,
rulers, pencils, "Fractions, Fractals, and Pascal"
packet, pencils, pens, erasers, calculators, and a
positive attitude. |
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| Students are asked to take out a sheet of
paper and write "Method of Area Calculation" at
the top. They present their methods to the rest of
the class so that every student will see that each
iteration breeds triangles that are 1/4 the size
of the bigger triangle. |
| Students look at the Explorations sites from
the web page and make observations about types of
fractals, Pascal's Triangle, or the relationship
between the two. |
| Students present methods that will prove area
answers are correct. They should see the method
where you add all of the colors areas together and
end up with the original area. |
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| Students take home the Pascal Triangle sheet
and look for ways to fill in the missing numbers.
They should also create one pattern that exists
within the triangle. |
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| Students are assessed on how they find the
areas of triangles in their Sierpinski's Triangle.
Make sure that they only use the formula for area
one time. All of the colors must be calculated
using the powers of
1/4. |
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| Day 3: What is Pascal's
Triangle? |
| Students share how they filled in the missing
numbers. They should see that each number is the
sum of the two numbers directly above it. |
| Students see if they can find any other
patterns in Pascal's Triangle. We will present
them to the class. |
| Students again travel to Shodor.net and look
at Pascal's Triangle. We are looking for
similarities between Pascal and Sierpinski. The
students see that Pascal's triangle is made up of
patterns of multiples. Students explore this and
eventually see that when the multiples of 2 (even
numbers) are colored in, Pascal's Triangle turns
into Sierpinski's Triangle. |
| Students make observations about multiples in
Pascal's Triangle. They also make conclusions
about the relationship between Pascal and
Sierpinski. |
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| On a sheet of lined paper, tudents write down
different patterns that they see in Pascal's
Triangle. Of course, the students need lined
paper, pencils, "Fractions, Fractals, and Pascal"
packet, calculator, laptop computer, etc. |
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| We spend time discussing missing spaces on
Pascal's Triangle. Students look at the triangle
and see that each number is the sum of the two
numbers above it. |
| We see other patterns like the second diagonal
is the list of Natural Numbers. Other students
will know what triangular numbers are, or even
tetrahedral numbers. Students may even find the
sum of each row to be the powers of two. |
| We see some great patterns when the students
color the multiples of specific numbers. Before
they start, they must extend the triangle in the
picture to its largest size. |
| We end with interesting patterns the students
found. Students share their findings with the rest
of the class at the end of the period. |
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| Students must make sure that all parts of the
"Fraction, Fractals, and Pascal" packet are filled
in. |
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| Each student choses a multiple and colors
their Pascal Triangle on the computer. It is great
practice with factors and multiples. Discussions
are a bit harder to assess. However, all students
benefit from the class discussions. If not seen
during classtime, students will know that their
grade will be lower because of the attached
rubric. |
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| Students are asked to complete work on
Sierpinski's Triangle. All students should have
completed and colored their Sierpinski Triangle.
Students that are completed will sit with students
who are not and aid in their completion. |
| Students are asked to complete the area for
each color in Sierpinski's Triangle. Their work
page will include the use of "Powers of 1/4".
Students who have completed should help. |
| Students are asked to have Pascal's Triangle
completed. All numbers should be filled in and all
patterns recognized in class should be written
down. Students who have completed should
help. |
| Students should have observations about the
similarities between Pascal and Sierpinski written
down. They should have observations about what
happens when they color in multiples of any
number--specifically, what happens when they color
in the multiples of 2. Students who have completed
should help. |
| If all students are finished, they should work
on comparisons between the two triangles and set
up the magazine according to instructions. They
may use the rubric to grade their own project
before it is turned in. |
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| Students should bring in all materials from
the previous three days of work. |
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| Students are divided into groups. The teacher
works with students who have not yet completed
Sierpinski's Triangle. Those who have finished
will work with students who have yet to complete
the areas of each color. Once all students are
finished creating the triangle, I will begin
working with my students in that group. Once they
are done, I will work with the last group that has
to create observations about the project. |
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| Students are asked to complete the project for
homework. The next day is spent on discussion and
fine-tuning before the project is turned in. All
parts of the magazine must be
completed. |
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| While helping students, they are assessed on
whether they are behind for lack of effort or from
being absent. |
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| Day 5: Discussion - Fractions, Fractals,
and Pascal |
| We discuss the reason for fractals one last
time. We then talk about the fractal called
Sierpinski's Triangle and its relationship to
Pascal's Triangle. |
| We also discuss Pascal's Triangle and the
groups of multiples within the triangle. We
discuss why this happens. |
| We talk about other parts of the project that
the students liked or did not like. |
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| Students need to bring all of the supplies
necessary to complete the "Mathematics Today"
magazine. They also need to bring the ideas and
supplies to share with other students about the
project. |
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| Students are asked questions from the packet.
When the answer is given, students make sure their
answer is correct. |
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| The project is collected and graded according
to the rubric that was
provided. |
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Brian McCarty
bmccarty@hhh.k12.ny.us
West Hollow Middle School
250 Old East Neck
Road
Melville, NY 11747
I have been teaching for many years and have noticed that
students are more attentive when they can play. Oddly enough,
they learn more, too. I have made it my job to find places on
the Internet where students can play and learn. I try to
incorporate these "games" in my lessons as often as I can. The
students appreciate it and even offer examples of interactive
thinking games that they find on the Internet. I feel that
this attitude has helped me in my career. I have won Redwood
Middle School's "Viking of the Year" and "Teacher of the Year"
awards. I look forward to continuing my career in the same
fashion.
Important documents for this lesson plan.
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