Title: Investigating Area Using Tangrams
By: Michael P. Yeager
Mathematics Teacher, Pitman High School, Pitman, NJ
Student Level: Plane Geometry (Low to Average Ability Students)
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Overview
This lesson provides students with a connection between history and geometry while
developing concepts and formulas for area of geometric figures. After introducing
Euclid's Elements and some of his Propositions (which deal with area of figures),
students will follow directions to create their own tangram pieces with construction paper.
After a problem solving activity which involves deciding what fractional part each figure is of the whole, activities will be completed which will lead to informal and/or formal proofs of
three of Euclid's Propositions. The proof of the Propositions will lead to the development
of the formulas for the area of parallelograms and triangles.
Objectives
1) Students will gain a historical perspective of Euclid's Elements that pertain to area of
figures.
2) Students will construct their own tangram and make observations about the pieces as
they relate to each other piece and to the original tangram square.
3) Students will write directions as to how to reconstruct the tangram.
4) Students will calculate the area of each piece in two ways: a) using the smallest figure as
the base unit of area and b) using the original tangram square as the base unit of area.
5) Students will use their observations and calculations to informally and/or formally prove
three of Euclid's Propositions as they relate to area of figures.
6) Students will develop from this work the formulas for the area of parallelograms and
triangles.
Historical Perspective
It has been my experience, that the concept of Area is often presented as a series
of formulas. Many times students have no grasp of what the variables mean, where they came from, etc. Therefore, they memorize a formula and plug numbers into it to get answers come test
time. This lesson is designed to help students get a better understanding of area and the
origination of the formulas that are used to calculate it.
When the area of a figure is calculated, the area is the number that results from the formula.
Since we study "Euclidean Geometry," we use Euclid's Elements (circa 300 BC) as a
foundation for our work.
Historical Perspective (continued)
In Euclid's Elements, lengths of segments and formulas for area are not indicated by
numbers. He uses what are commonly referred to as the "Euclidean Tools." They are the
collapsible compass and the straightedge.
If Euclid wanted to show that two figures had equal areas, he did so by demonstrating that
one figure can be divided into parts so that when the parts were put back together, they
produced the other equal figure. Euclid never stated formulas the way they are presented in
today's textbooks.
The following Propositions are from Euclid's Elements, Book I:
Prop. 34: A parallelogram is divided by a diagonal into two triangles equal in area.
Prop. 35: Two parallelograms with the same base and lying between the same parallel
lines are equal in area.
Prop. 37: Two triangles with the same base and lying between the same parallel lines
are equal in area.
Students will create a Tangram, which is a Chinese puzzle developed in the early
1800's. The puzzle is an original square broken down into seven smaller geometric shapes.
Through investigation of the properties and the areas of the puzzle pieces, students will prove
Propositions 34, 35, and 37.
The proofs of these propositions will help students to develop area formulas for
parallelograms and triangles on their own.
By conducting this lesson, the teacher is really moving across a timeline from
Euclid's Elements (300 BC) to Chinese Tangrams Puzzles (1800 AD) to Area Formulas
in today's classroom (2000 AD).
Prerequisites
Some basic knowledge as to history of Geometry (I do this in the beginning of the year).
Properties of figures and classification Angle Classification
Problem Solving Fractions
Congruency and similarity of figures Concept of area without formulas
Materials needed to teach the lesson:
9" by 9" squares of construction paper,
sets of plastic tangrams (for overhead demonstration, if needed)
scissors
Procedures Day One
1) Introduce the lesson by having students give the formulas for area of a square and
a rectangle. You may want to ask if the students know any other area formulas.
Write results on the board. If a student does not ask, write the following question
on the board "Where did these formulas come from?" Have students write down their
thoughts in their notebook (I use the journal section for this type of assignment).
2) Refer back to the Historical Perspective section to introduce Euclid, the Elements and
the Propositions that are going to be studied.
3) Ask students if they have ever heard of, or used a tangram before. Discuss its history
and what it is. Begin to hand out the 9" by 9" squares. Have students discuss the
characteristics of a square mentioning that it is a quadrilateral, its vertices, has four right angles,
the measures of all the sides are equal, there are two pairs of parallel sides.
STUDENTS SHOULD MOVE INTO GROUPS OF 2 OR 3.
THE FOLLOWING STEPS FOR CONSTRUCTING A TANGRAM CAN BE SHOWN
ON THE OVERHEAD SO THAT NO STUDENT MISSES A STEP OR THE
QUESTIONS THAT PERTAIN TO THE STEP (Directions could also be printed and
distributed to students. However, if you want to keep everyone at the same step, I suggest
the overhead.)
4) Creating 2 triangles: Fold the square along one of the diagonals so that two triangles
are formed. Have the students fold the paper back and forth so that the crease to cut
will be visible. Once two triangles are formed, have students make observations in their
notebooks about the triangles and then discuss them. Point out that the triangles are
right triangles because of the right angles. Rotate the triangles so that students realize
right triangles can be in many positions. Establish that these right triangles are isosceles
right triangles because two of the sides have equal measure. You might also mention
the hypotenuse, the legs and the measures of the acute angles. Have students discuss
and make notes about the congruency of the two triangles as well.
Have students use the two triangles to recreate the original square.
Students' Observations will be titled "Investigating Area Using Tangrams" in
their notebook for organizational purposes.
5) Using one right triangle to create two right triangles:
Have students take one of the right triangles. Find the two acute angles or base angles
of the triangle and fold the triangle so the base angles are on top of each other forming
a crease down the center of the triangle from the right angle to its opposite side or the
altitude of the triangle.
After folding the crease, have the students cut the figure into two triangles. Discuss and
note the characteristics of these right triangles and why they are right triangles as well
as their congruency with each other and their similarity with the larger right triangle
that remains.
Have students take the three existing triangles and form the original square.
6) Creating a right triangle and trapezoid from the large right triangle:
Have students fold the two base angles on top of each other and crimp the bottom of the triangle so that a crease is formed not across the whole triangle but only on the side
so that the side is bisected thus forming a midpoint. Tell students to fold the right angle
down so that it meets the midpoint and to form a crease across the triangle. This will
form a trapezoid. Have students discuss and note the characteristics of the trapezoid.
(Students should identify the properties of the trapezoid and classify it as an isosceles
trapezoid because two of its sides are equal.).
Cut the figure so that a triangle and the trapezoid are separate. Discuss and note the
attributes of this right triangle. Is it similar to the other? Why?
After discussing students' answers, have the students "mess up" all existing pieces and
reconstruct the four pieces to form a square.
7) Creating two right trapezoids from the trapezoid:
Discuss the base angles of the trapezoid being equal in measure as well as the obtuse
angles of the trapezoid. Have students fold the trapezoid so that the base angles are on
top of each other and form a crease which acts as the line of symmetry for the two
trapezoids that are formed. Cut the figure to make the two trapezoids. Hold these up.
Are these congruent? (Yes). Are these trapezoids? (Yes). Are these isosceles
trapezoids? (No, they are right trapezoids). Why are they right trapezoids? (They
contain right angles).
Have students recreate the 5 pieces so that it forms the original square.
8) Creating a square and a triangle from one right trapezoid:
Have the students fold the figure so the vertex of the acute angle of the trapezoid
touches the vertex of the adjacent right angle and form a crease. This will show a
square and a triangle. Cut the figure to form the square and the triangle and have
students discuss and note their attributes. Why is it a square? Why is it a right
triangle? Is it congruent or similar to the other right triangles they have just made?
Students should record answers in their notebook and discuss answers.
"Mess up" all the pieces and reconstruct the original square.
9) Create the sixth and seventh pieces:
Take the right trapezoid and ask students to fold the figure so the vertex of the obtuse
angle is on top of the vertex of the right angle which is diagonally opposite it. When the
crease is made a triangle and a parallelogram are formed. Cut the figure to form the two
new figures. Discuss and note the attributes of this "new" quadrilateral. It is a
parallelogram because it has two pairs of parallel sides which are equal in measure.
Discuss angle classification as well. Is the triangle which is formed congruent to any
other triangle which has been created? (Yes, the other small triangle). Is it similar to
any others? (Yes).
Have students "mess up" the pieces from someone else in their group and then
reconstruct the original square.
10) HOMEWORK ASSIGNMENT:
Hand out Tangram Worksheet #1: Putting It Together (Appendix A)
Student will list steps to create the tangram square from the seven pieces. On day 2
groups will exchange their steps and see if they work as a warm-up activity. All
worksheets will be collected.
Day Two
11) After completing the warm-up activity, let students know that the focus of today's
work is to show relationships between the pieces and the whole tangram square. Have
students order the seven pieces from smallest to largest and explain what criteria they
used for their arrangement. Students should be able to verify their arrangement with
the other groups. Focus on the arrangement of the pieces based on area. Use the small
right triangle as the basic unit of area = 1 triangular unit. What are the areas of each of
the pieces in triangular units? Students will record their answers in their notebooks
and be able to explain their reasoning.
Answers: 2 Small Right Triangles = 1 triangular unit each
Medium Sized Right Triangle = 2 triangular units
Square = 2 triangular units
Parallelogram = 2 triangular units
2 Large Right Triangles = 4 triangular units
12) After going over the solutions, hand out Tangram Worksheet #2: A Closer Look at What Makes It Up (Appendix B). Students will determine the fractional value of all the
parts of the tangram if the original square has a value of one unit. Students will write
the numerical answer and their reasoning behind it in the space provided.
Answers: 1) Large Right Isosceles Triangle = ¼ each or 25%
2) Medium Right Isosceles Triangle = 1/8 each or 12.5%
3) Small Right Isosceles Triangle = 1/16 each or 6.25%
4) Small Square = 1/8 each or 12.5%
5) Parallelogram = 1/8 each or 12.5%
All worksheets will be collected.
13) Ask students to create different geometric figures with any number of the tangram
pieces as the worksheet is collected. Then tell students to write down the following
three statements:
A parallelogram is divided by a diagonal into 2 triangles equal in area.
Two parallelograms with the same base and lying between the same parallel lines are equal in area.
Two triangles with the same base and lying between the same parallel lines are equal
in area.
(You may have to demonstrate "lying between the same parallel lines" on the board.
Use a rectangle or square for the example.)
14) HOMEWORK ASSIGNMENT:
Using the tangram pieces, draw a picture to represent each statement and explain
how the information in the picture "proves" the statement to be true. Remind students
that the statements are actually Propositions 34, 35, and 37 from Euclid's Elements.
So their homework is really a 2300 year old set of problems!
Day Three
15) Ask for a couple of volunteers to draw the picture to represent Proposition 34 on the
board. Students could use the parallelogram piece and add a diagonal to it. They also
could use the 2 small right triangles to form a parallelogram where the shared side
of the triangles is also the diagonal of the parallelogram. Once students agree on the
pictures, ask for how they could show this to be a true statement.
*If you want to use a formal proof, I suggest using the one at this web address:
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI34.html
Depending on what properties of parallelograms your students use, you can prove
(informally) that the 2 triangles are congruent by SAS or SSS.
Ask the following questions: (Students will write their answers in their notebook)
1) What is the relationship between the area of the parallelogram and the area of the
2 triangles?
2) What is the relationship between the area of 1 triangle and the area of the parallelogram?
Answers: (Phrasing may vary)
1) Area of Parallelogram = Sum of the Areas of the 2 Triangles
2) Area of 1 Triangle = ½ of the Area of the Parallelogram
Have students note the importance of both answers since they will be using them later to develop the area formulas.
16) Ask for volunteers to draw a picture that represents Proposition 35 on the board.
(The one problem students may have is that the parallelograms should overlap -
"with the same base"). They could use the parallelogram piece and recopy it or
use the 2 small triangles that form a congruent parallelogram. Make sure that the
parallel lines are drawn and labeled.
* If you want to use a formal proof, I suggest using the one at this web address:
http:// aleph0.clarku.edu/~djoyce/java/elements/bookI/propI35.html
Or you can prove it informally by using subtraction and addition of the triangles
formed by the overlapping parallelograms.
Once you have proven the proposition, ask students to draw the 2 parallelograms
lying between the same parallel lines but not overlapping.
Some students may keep the orientation (vertical/horizontal) of the original
parallelograms from the proof while others may rotate them onto the smaller/larger
side. Some students may have one parallelogram slant to the left and the other
slant to the right. Remind students that as long as they have the same base and are
lying between the same parallel lines then they have equal area.
Put different "correct" versions of the picture on the board.
Ask the questions: If Proposition 35 guarantees that in each case the 2 figures will
have equal area, what measurements (lengths) remain constant in each case?
Using those constants can you come up with a formula for area of a parallelogram?
Answers: The bases of the parallelograms remains the same (call it b).
The distance between the parallel lines remains the same
(it is the perpendicular height, so call it h).
The Area of a Parallelogram = base x height.
Which, of course, is the textbook formula for area of parallelogram.
17) Ask for volunteers to draw a picture representing Proposition 37 on the board.
Just as in the previous Proposition's picture, the triangles should overlap with the
same base. Most students will use the small right triangles to draw the picture.
Make sure the parallel lines are drawn and labeled.
* If you want to use a formal proof, I suggest using the one at this web address:
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI37.html
Or you can prove it informally by adding parallel lines which form overlapping
parallelograms (Proposition 35). Then relate the area of the parallelograms, divide
by 2 and prove the area of the 2 triangles are equal.
Once you have proven the proposition, ask the students to draw the 2 triangles using
the hypotenuse as the base lying between the same parallel lines and not
overlapping.
Put correct version(s) on the board.
Ask the following question: What measurements (lengths) remain constant since
Proposition 37 guarantees that the 2 triangles have equal area ?
Answer: The bases are the same.
The distance between the parallel lines stays the same.
Ask the questions: So then can we assume that the formula for the area of a
triangle = base x height (The same as Parallelogram)?
Answer: No
Questions: Why not? What else do you know about triangles and parallelograms?
Answer: Area of a triangle = ½ of the area of a parallelogram from Proposition 34.
Question: What do you think is the formula for the area of a triangle?
Answer: Area of a triangle = ½ of (base x height).
Again, the textbook formula for the area of a triangle.
References
1) Sample Tangram Lesson Grades 4-12
http://www.col-ed.org/cur/math/math12.txt
2) Constructing Tangrams
www.smls.org/tour/math.html
3) Constructing Tangrams
http://forum.swarthmore.edu/trscavo/tangrams/construct.html
4) Other Tangram Resources
http://forum.swarthmore.edu/trscavo/tangrams/resources.html
5) Euclid's Elements
http://aleph0.clarku.edu/~djoyce/java/elements/elements.html
6) Formal Proofs of Euclid's Propositions I34, I35 & I37
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI34.html
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI35.html
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI37.html
7) Lawrence, Paul. "Getting Great With GEPA." L.L. Teach. 1999.
APPENDIX A
NAME:________________________ DATE:_________________
Tangram Worksheet #1: Put It Together
Read directions and be prepared to use your work tomorrow in class.
1) Create a list of steps that could be used by anyone to arrange the seven pieces of a tangram puzzle into the tangram square. Be as specific as possible in your directions. Assume that the person can already identify the seven shapes in a tangram puzzle.
2) Record the steps below making sure to allow enough detail so the square can be formed following the directions.
3) Try the steps out yourself and make modifications as needed.
4) A group will be chosen at random to read their directions while a student models the directions in front of the class using overhead tangram pieces. Be prepared to do both !
5) All sets of students' directions will be collected.
The Perfect Way to put a Tangram Together
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APPENDIX B
NAME: ________________________________ DATE:__________________
Tangram Worksheet #2: A Closer Look at What Makes It Up !
If the tangram square is defined to have an area of one square unit, tell what fractional part and what percent of the tangram square each of the unique shapes are. Explain how you know your
conclusion is right.
1) Large Right Isosceles Triangle:__________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
2) Medium Right Isosceles Triangle:________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
3) Small Right Isosceles Triangle:__________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
4) Square:_____________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
5) Parallelogram:_______________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________