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Chamberlin – Capstone Project
A. Purpose
I chose my “Looking for Pythagoras” unit for my project. It is an 8th grade book in the
Connected Math series. During the unit, students develop a conceptual understanding of square
roots, reason through a proof of the Pythagorean Theorem, and use the Pythagorean Theorem to
find missing side lengths of right triangles. There were several hands-on activities embedded
throughout the unit as well as a project at the end.
Our driving question is: What is the Pythagorean Theorem? Students discovered the
Pythagorean Theorem through a series of activities, are able to explain it in their own words, and
are able to apply it to problem situations. The Pythagorean Theorem has many real-world
applications and my students had the opportunity to understand these applications through the
completion of the Looking for Pythagoras unit.
B. Objectives
1. Studentcentered – I believe the nature of Connected Math is a student-centered curriculum.
Students work together and with the teacher to develop an understanding of unit concepts. My
classroom is very student- and learning-centered.
2. Learning and doing – I am a big proponent of hands-on learning. My students learned about
things such as the relationship between square roots and the side lengths of squares and the area of
the square on leg a plus the areas of the square on leg b equals the area of the square on the
hypotenuse through discovery.
4. Facilitator – Being a facilitator of learning is necessary to teach Connected Math effectively. I
have my students work together with a partner/small group to discover the main ideas of each
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investigation. As they work, I walk around the room to make sure students understand the
questions and are drawing accurate conclusions.
5. Flexible grouping configuration based on individual student needs – Students work in groups
on a consistent basis. I typically allow students to choose their partner so they are comfortable to
share their ideas, ask questions, and make mistakes. I have a couple of students who like to work
alone, but they know they can ask their neighbor or me for help.
6. Multiple instructional and learning modalities to include all students – I use many hands on
activities, verbal explanations, handouts, drawings on the board, etc. to meet the needs of all
students. Students used dot paper and the overhead throughout many investigations to develop a
relational understanding of the square root being the side length of a square’s area. The unit
project included many different products, enabling students to choose the one they found most
appealing.
8. Interdisciplinary – There are several reading and writing activities and some history
incorporated throughout the unit. Students were required to respond to Math Reflection questions
at the end of each investigation to make sure they were developing the key concepts of the unit.
We also did a KWL chart and a wiki entry. We discussed the “Did you know?” sections about
famous mathematicians in our book. (Some students chose to research them further for their
project.)
9. Collaboration – My students regularly work together in cooperative groups. Sometimes I
choose them and sometimes they get to choose. It may be for class work or it may be for a project.
10. Performancebased assessments – In past years, I have had some students who did really well
on projects, but not so well on quizzes and tests. I try to offer projects at the end of each unit to
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give students the opportunity to demonstrate their understanding in a different way. I created a
Webquest project, including a rubric, for the Looking for Pythagoras unit.
12. Technology fully integrated into the classroom – Students use TI-84 graphing calculators on a
daily basis. I use the laptop cart periodically. In the Looking for Pythagoras unit, students
complete a double-entry journal on the class wikispace, a task on the Webquest (unit project), and
an interactive quiz through Study Island.
13. Teachers addressing the learning styles of all learners – I tried to include a variety of
instructional strategies throughout the unit that required writing, hands-on-activities, and
reasoning. I think the best place for students to express their learning preferences is the end of the
unit project. Students had a choice between creating a mathematician poster (linguistic, spatial),
explaining a proof (logical/mathematical, spatial), creating a crossword puzzle (spatial,
naturalistic?), composing an original song (musical, linguistic), or creating a podcast lesson
(bodily-kinesthetic). They also got to choose whether they worked alone (intrapersonal) or with a
partner (interpersonal).
14. Learning how to learn – Every time I teach something new, I model what I am thinking out
loud for students. After I have done it a few times, I have student volunteers present problems on
the board and describe their thinking process.
15. Using a variety of types of information to complete authentic projects – Depending on the task
students chose to complete for the unit project, they may have had to research some historical
information, find a song to guide them in creating their own, or practice using webcam technology
in recording a lesson.
C. Timeline
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Unit Overview (about 5 weeks – see calendar at end of document):
Investigation 1
1.1 City of Euclid - TSWBAT find distances on a coordinate grid.
1.2 Planning Parks - TSWBAT connect properties of quadrilaterals to coordinate
representations.
1.3 Finding Areas - TSWBAT develop strategies for finding areas of irregular figures on a
grid.
Investigation 2
2.1 Squares - TSWBAT draw squares on 5 dot-by-5 dot grids and find their areas.
2.2 Square Roots - TSWBAT understand square root geometrically, as the side length of a
square with known area.
2.3 Line Segments - TSWBAT use geometric understanding of square roots to find lengths
of line segments on a dot grid.
Investigation 3
3.1 Pythagorean Theorem - TSWBAT discover the Pythagorean Theorem through
exploration. TSWBAT use the Pythagorean Theorem to find unknown side lengths of right
triangles.
3.2 Proof - TSWBAT reason through a geometric proof of the Pythagorean Theorem.
3.3 Distance – 2 points - TSWBAT use the Pythagorean Theorem to find the distance
between two points on a grid. TSWBAT relate areas of squares to the lengths of the sides.
3.4 Right Triangle? - TSWBAT determine whether a triangle is a right triangle based on
its side lengths.
Investigation 4
4.1 Theodorus - TSWBAT estimate the values of square roots that are irrational numbers.
TSWBAT estimate lengths of hypotenuses of right triangles.
4.2 Baseball - TSWBAT apply the Pythagorean Theorem to a problem situation (to find
unknown side lengths).
4.3 30-60-90 triangles - TSWBAT describe the properties of 30-60-90 triangles.
4.4 Special Right Triangles - TSWBAT describe the properties of 30-60-90 triangles.
TSWBAT use the properties of 30-60-90 triangles to solve problems.
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D. Project Components
Through the instruction, activities, and assessments, I expect students to meet the following
learning targets: 1) Relate the area of a square to its side length. 2) Develop strategies for finding
the distance between two points on a coordinate grid. 3) Understand and apply the Pythagorean
Theorem. 4) Use the Pythagorean Theorem to solve everyday problems. 5) Distinguish between
rational and irrational numbers. 6) Describe the properties of special right triangles. 7) Describe
the contributions of famous mathematicians.
Inquiry – Students work through investigations with their partner/group. The Connected Math
series is designed as a sequence of problems based on a specific topic. Students sequentially build
upon their understanding of the Pythagorean Theorem and related concepts as they worked through
the book. My students are always encouraged to ask each other and/or myself questions as they
develop their understanding of new topics.
Projects – Students chose a task from among the following: write a song, create a mathematician
poster, demonstrate a proof, create a crossword puzzle, or create a podcast that demonstrated
understanding of the Pythagorean Theorem or Looking for Pythagoras unit.
Technology – The unit project is designed as a Webquest. Students also responded to a double-
entry journal on our class wikispace and completed an interactive quiz on Study Island. Students
used graphing calculators and the overhead projector throughout the instructional unit. I also used
a Discovery Education clip that another teacher shared with me.
Dynamic, Flexible Grouping – Students worked on investigations in groups and students reasoned
through a proof of the Pythagorean Theorem with me. I have much more success with engaging
students in the learning process when I have them work with one or more other students to find
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solutions to the problems we are presented with. After the students respond to all of the questions
in the investigation, we come back together as a class and summarize what we have discovered.
Authentic Teaching and Learning Experiences – One of the major unit goals is for students to
understand square roots as the side length of a square with given area. We did a variety of
activities with dot paper and the overhead to help students develop a concrete understanding of this
abstract concept. We also used construction paper squares and right triangles to reason through a
proof of the Pythagorean Theorem.
E. Resources
“Looking for Pythagoras” books assigned to each student
Lappan, G., Fey, J.T., Fitzgerald, W.M., Friel, S.N., & Phillips, E.D. (2006). Looking for
Pythagoras: The Pythagorean Theorem. Boston: Pearson.
Overhead projector and transparencies
Graphing calculators are available during class or may be signed out overnight
Laptop cart for Webquest project/internet research – signed out through the 8th grade Math Dept.
head
Webquest – http://teacherweb.com/WQ/MiddleSchool/PythagoreanTheorem1/
Wiki – www.essentialsofalgebra.wikispaces.com
Study Island – www.studyisland.com
F. Desired Outcomes
I hope my students will understand the Pythagorean Theorem and how to use it.
Specifically, I want them to be proficient in the unit learning targets: 1) Relate the area of a square
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to its side length. 2) Develop strategies for finding the distance between two points on a
coordinate grid. 3) Understand and apply the Pythagorean Theorem. 4) Use the Pythagorean
Theorem to solve everyday problems. 5) Distinguish between rational and irrational numbers. 6)
Describe the properties of special right triangles. 7) Describe the contributions of famous
mathematicians.
Success will be based on student performance. I will observe students as they work
through the investigations and during in-class activities. I will also give and evaluate several
assessments to determine student understanding of the material. If students do well, I will see that
they understand the material and we can continue developing the unit concepts.
My students did well with the Looking for Pythagoras unit. They demonstrated their
understanding of the unit goals by thoroughly explaining problems on the board throughout the
unit, appropriately answering questions during class discussions, challenging each other during the
Jeopardy review, and performing well on the quizzes and test. They also demonstrated the depth
of their understanding by successfully completing the unit project.
8.
January/February 2009
Sun Mon Tue Wed Thu Fri Sat
22 23 24 25 26 27 28
Prob. 1.1 Prob. 1.2 Prob. 1.3 Prob. 2.1 Prob. 2.2
Wiki DEJ-1 due Think-Pair-Share
Inv. 1 Math Refl.
1 2 3 4 5 6 7
Admit-Slip Inv. 2 Math Refl. Inv. 1/2 Quiz Prob. 3.1 Prob. 3.2
Prob. 2.3 8 square quiz start KWL “Pyth. Thm. 1”
“Operating with
Square Roots”
8 9 10 11 12 13 14
Prob. 3.3 Prob. 3.4 Inv. 3 Math Refl. Inv. 3 Quiz Prob. 4.1
“Pyth. Thm. 2” Review Real Number
Original problem graphic organizer
due
15 16 17 18 19 20 21
Prob. 4.2 Prob. 4.3 Prob. 4.4 Inv. 4 Math Refl. Inv. 4 Quiz
“Is That Rational?” Exit-Slip finish KWL Review
Rational/irrational “4.3 Skills”
game
22 23 24 25 26 27 28
Look. for Pyth. Look. for Pyth. Jeopardy Review Look for Pyth. Test
Project Project Wiki DEJ-2 due
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