LEARNING UNIT
Polygon Search

CONTENT KNOWLEDGE

ESSENTIAL QUESTIONS

• What is geometry?
• How does geometry affect our everyday life?
• How are geometric figures classified in our world?
• What are the characteristics of a geometric figure?

"Children who develop a strong sense of spatial relationships and who master the concepts of language and geometry are better prepared to learn number and measurement ideas, as well as other advanced mathematical topics." (NCTM 1989a, P.48)

INITIATING ACTIVITY

Students are asked to write the answer to the question: What is geometry? This enables the teacher to have a pre-assessment to the students knowledge of geometry. The question is asked again at the conclusion of the unit. A KWL graphic organizer can be used to elicit knowledge from the students prior to answering the question: What is geometry? The chart can be displayed in the classroom through out the unit and as a reminder at the conclusion of the unit.

Connection to State Learning Standards

Content Area: Mathematics

Level: Upper Elementary

Unit Theme: Geometry - A Polygon Adventure

LEARNING EXPERIENCES

Describe what will be done…

GEOMETRY VOCABULARY

Chart paper labeled Geometry Vocabulary is posted on the chalkboard or in an easily accessible area for students and teacher. The teacher explains that the chart is used throughout the unit to record geometry vocabulary words. As new words are presented they are added to the chart. The can be posted on a word wall or on the ceiling to refer to all year.

Describe what will be done…

POLYGON SEARCH

Each students is given a three inch square piece of construction paper. The class describes the properties of the square and the vocabulary is recorded on the chart. I.e. A square, has: four edges, four sides, four corners, four angles, four right angles, four vertices, two faces or sides, it is: two dimensional, a polygon.

The students are instructed to fold the square in half on the diagonal. Students are asked to predict how many pieces the square would show when opened. Students open the square and discuss, then students cut the square on the folded diagonal line. Discuss that the cut forms two triangles that are equal in size or congruent. Describe the triangles and record new vocabulary words on the chart. Three edges, three sides, three corners, three angles - one right angle and two acute angles, three vertices, two faces, two dimensional, a polygon and congruent triangles.

Angles and types of triangles can be discussed at this time: acute, obtuse, and right angles, equilateral, isosceles, and scalene triangles. The degrees in an angle can be discussed depending on interest and abilities of students.

Ask the students to remake the square with the two triangles. Ask the students to find all the possible ways to put the two triangles together following the directions that edges of the same length must touch and match to make a new polygon. Examples of this rule should be modeled.

  After the students discover the three possible polygons, have three students tape their triangles together to make the square, larger triangle, and parallelogram. Post these on the vocabulary sheet and discuss the properties of each polygon and record new vocabulary on the chart. (Give the three students a new three inch square to replace the one they taped together)

A three minute pause can be used to allow students the opportunity to summarize the information and/or vocabulary words recorded at this point by discussing the meaning or the new words on the chart with a partner. The teacher can also ask students to show, using the squares and triangles, the vocabulary.

Describe what will be done…

Read aloud the story The Greedy Triangle by Marilyn Burns. Discuss the story while reading. At the end of the story discuss why the triangle was dissatisfied with his shape and what happened as a result of the triangle's visits to the shape shifter. Discuss the triangle many shapes and how the fit in the world around us. Encourage students to name the different polygons the triangle becomes. Tell the students that in a few days, they will choose a polygon and write an adventure story about their polygon.

Describe what will be done:

PRESENTING THE PROBLEM

Give each student a new three inch square of a different color. Ask them to fold and cut on the diagonal. Now they have four triangles to use. Have the students work with partners or in small groups. The problem should be presented: Find all of the polygons that can be made using the four triangles, two of one color and two of another. The rule should be followed that edges of equal length must match and they must use two triangles of one color and two of the other color. The groups should tape the triangles together when they have made a new polygon. (There are 14 possible polygons that can be made. Do not share this information with the class at this time. Give them the opportunity to discover the polygons. Many students will make only the commonly named polygons like the trapezoid, parallelogram, square, triangle, etc. Encourage them to look at pentagons and other many sided shapes.)

Move among the groups encouraging the members. When a group says that they have found all of the polygons, ask them how they know they have all of them. If duplicate polygons are found, discuss this with group, what happens if you flip or rotate the polygon, are they congruent. Encourage them to check each polygon. After a specified time, have the students collect their polygons and place them in a manila envelope for the next class period. Have groups label the manila envelope. Not all groups will discover all of the polygons, but they should be given the opportunity to search, as long as there is interest. Allow two or three class periods for the students to continue their search depending on the need of the group and interest level to continue to search. Depending on age levels, some are ready to move on after one class period. Other age groups are interested in finding all of the possible polygons before the group discussion begins. Usually older students want to continue to search.

Describe what will be done…

SORTING THE POLYGONS

The next activity uses the polygons that the students made. The polygons are sorted onto a chart labeled POLYGONS. The procedure for this lesson is for the teacher to sort the polygons on the chart in a specific way. The students are to determine how the polygons are sorted.

Have one student in one group choose a polygon from the groups' envelope and give it to the teacher. That student should remake that polygon for his group and place it inside the groups manila envelope. The other groups find that polygon in their pile and discuss the polygon with the whole class, naming its properties and perhaps giving it an object name, i.e.: a rocket ship or a fox's head. ) If a group does not have that shape, a member of the group should make it and place it inside their manila envelope.

The teacher tapes the polygon to the chart paper. (The polygons should be sorted according to the number of sides, or edges and angles it has. The first row should be for triangles, the second for quadrilaterals, the third for pentagons, and the fourth for hexagons.) Continue posting the polygons. Encourage the students to predict which row each new polygon should be posted. Some groups will not be sure they have a specific polygon. Encourage them to bring their polygon to the chart and check for congruency by rotating and flipping their polygons.

After the 14 polygons have been posted, or while in the process of posting the polygons, discuss how they have been sorted. Students can be encouraged to look for similarities and differences among the polygons. Label the rows triangle, quadrilateral, pentagon, and hexagon. Label the polygons in the quadrilateral row as square, parallelogram, trapezoid, and rectangle. Many students become interested in finding the names for other sided polygons like heptagon - 7, octagon - 8, nonagon- 9, and decagon - 10. Encourage them to look for the answers in encyclopedias or other sources.

Describe what will be done:

EXTENSIONS FOR SORTING:

Future class sessions could include sorting of polygons by other geometric terms. Other charts can be made sorting the polygons into concave/convex shapes. This introduces the vocabulary and allows groups to critically look at each polygon to determine the placement on the chart. Further suggestions include the number of right angles each polygon has, four to zero; Which polygons have lines of symmetry?; or Which polygons be folded on the cut lines to make a square or a triangle?

Extending and Refining

Describe what will be done:

COMPARISION OF POLYGONS

The students will work as partners to compare and contrast two polygons chosen from the chart. Attributes of each polygon are compared and contrasted then recorded on chart paper.

I.e. : Two students will each select one of the polygons from the polygon sorting chart. Then each child will go to their groups packets and remove the selected polygon. The child will remake that polygon and replace it back into the packet. The newly made polygon will be used to compare and contrast with the partners polygon. The two polygons will be pasted at the top of chart paper. The students will draw a T table on the chart and label one side ALIKE and the other side DIFFERENT. The students will then discuss and record how the two different polygons are alike and how they are different. The partners will present their chart to the class at the completion of the chart.

Describe what will be done:

ROTATING GEOMETRIC DESIGNS

Each student chooses a polygon from the groups packet. They will trace their polygon onto oak tag paper and then return the polygon to the packet. The oak tag polygon will be used for this art activity.

Each student is given a large sheet of construction paper (18x24). The students determine the center of the paper by folding the paper in half horizontally and vertically making a small crease in the middle of each fold. Where the two creases meet should be the center of the paper. A pencil dot is put at the center of the paper. Each student then must decide which vertex of the polygon he/she will use to rotate the polygon on. A x is marked on the vertex. The student then places the polygon on the construction paper with the x and the center dot matching and traces the polygon with a pencil. The student then rotates the design, maintaining the match of the dot and x, and again traces the polygon. The student continues to rotate and trace the polygon until a minimum of 8 rotations and tracings have been made. More rotations can be made, this is just a minimum. The student then can use markers to color the designs on the paper. The oak tag polygon and the colored rotating design can be displayed on the bulletin board. The bulletin board can become a math center where students match the polygon with the correct rotating design.

Technology Component:

Describe what will be done:

DRAW PROGRAM ON COMPUTER

Students will utilize the Microsoft Word Program or any other draw program on the computer to practice drawing a variety of polygons. Completed designs can be printed. Some designs maybe used in Geometry story as illustrations. Suggestions to use the CAD or Computer Assisted Drawing program for older students.

Internet site bookmarked on the computer for students to practice geometric vocabulary during math time or other designated time. This site provides flash card images of geometric shapes and properties. The student is given choices for the correct answer. If the student gives the correct answer they move on to the next flashcard. If a student gives an incorrect response the answer is given and an explanation provided. Then a new problem or flashcard is presented.

Internet site: www.aplusmath.com

Describe what will be done:

COLOR ARRANGEMENT

An optional activity to do with the polygons is to consider color placement of the triangle within each polygon. Working with partners, assign a polygon to each group. Their job is to discover other ways to make the same polygon but to change the placement of the colored triangles. Each polygon started with two squares of contrasting colors cut on the diagonal to create four triangles, two of each color. The students can remake the shape using the same colors until they have created all possible ways to make the same polygon. Be sure to discuss flips, rotation, and congruency with the students. How are they sure they have made all of the possible polygons and are they sure that none are the same?

Describe what will be done:

FLIP, ROTATE, AND TURN

The students can use the polygons to practice flips, rotations and turns. Model for the students how to take one shape, trace it on a long sheet of paper, flip it and trace again. Have the students make a minimum of five flips and tracings. Label the row Flips. Then have the students trace and rotate the same shape under the first row and label it rotations. One final time, have the students trace and turn the shape and label the row turns. Give the students the opportunity to write about the discoveries between the three ways to move a shape.

Describe what will be done…

PENTOMINOES

The investigation into polygons continues with the use of five square tiles. Each student is given five square tiles and one inch square paper. The students are shown how one tile is the unit of measure and that only one tile makes a square. Next the students take two tiles and try to create polygons following the rule from the triangle problem, the edges of the same length must match. The students should discover that they can only make two rectangles and that when rotated they are congruent so they are the same. This leaves only one rectangle can be made with two tiles - a domino.

Next have the students use three tiles to make polygons. Record on chalkboard the possible arrangements of tiles. There are only two triominoes. Continue with the exploration of four tiles and record on the chalkboard or cut out of the square paper. There are five possible tetrominoes.

Pose the problem of finding all the possible pentominoes using the five square. Instruct them to cut the pentominoes out of the inch graph paper. Students may work with a partner for this activity or individually. Each person was required to make their own set of pentominoes when working with a partner. This allowed each person to have a paper set to work with on future lessons. Encourage partners to find a way to determine if they have found all of the possible arrangements.

Many math resources are available with the answers to the pentomino problem. Working with the students to determine the correct answer is rewarding for both the teacher and the students.

Once all of the pentominoes are found, other activities can be done with the shapes. Which ones fold into a box, find the area and perimeter of each shape and compare them to each other. Students can make puzzle shape cards to challenge classmates. (A student makes a shape with a number of pentominoes. He/she traces the shape onto oak tag paper or construction paper. Other students try to remake the shape using their set of pentominoes.)

Describe what will be done…

PERIMETER AND AREA

The students should have a lot of information about polygons at this point. Introduce the students to the geoboards. Give each student one geoboard and one geoband. Allow the students opportunity to freely explore the geoboard and the geoband if they have not previously had the experience. You should hear the students using many of the geometric terminology that all the preceding activities presented. After a designated amount of time, instruct the students to make a specific shape. Usually start with a square with the area of one square. With this one square discuss this is the unit on measure for the geoboard. (Explain when you use a ruler you use one inch, a measuring cup you use cups, etc. if they need help understanding that units of measure varies.) Discuss how many sides or what is the perimeter of the square from one nail to the other. Then ask the students to make a rectangle with the area of two and perimeter of six. Be sure to discourage diamonds or creating shapes that cross the diagonal from one nail to the other. Discuss the perimeter and how you determine it and discuss the area. Ask the students to make the largest rectangle on the geoboard and determine its area and perimeter.

At this point introduce the geoboard dot paper. Have students create and draw rectangles and determine their area and perimeter. Further lessons can continue with this activity making triangles and determining the area of the triangle. This can lead to the development of formulas for area of rectangles (b x h )and area of triangles ( b x h \ 2)for the upper elementary grades. For students in third, or those that have not develop multiplication skills, the counting of squares and finding half of that number will be possible. Once the students understand how to find the area of a triangle; pentagons, hexagons, and other sided polygons can be made on the geoboard, drawn onto geoboard dot paper and then the area can be determined.

ASSESSMENT

Assessing students progress is an ongoing process throughout the school year rather than a single event. The following culminating activities can used for assessing growth following the geometry unit or they can be used as reteaching tools for students in need of further instruction to meet the standards. All learning experiences presented in this unit provide evaluative information from the students. Informal evaluation throughout the unit should be used.

CULMINATING PERFORMANCES

1. A GEOMETRY STORY

Students have had many opportunities to investigate polygons. They have had the opportunity to create and compare polygons, make designs with polygons, measure the area and perimeter of some polygons, and develop a geometric vocabulary.

Reread The Greedy Triangle to the class. Allow ample discussion of the various polygons that the triangle changes into. Also discuss the ways the different shapes that the triangle becomes are used in the world.

At the conclusion of the second reading of the book, students are asked to choose a polygon that has been made, either from the triangles or from the squares, and write an imaginative story for that polygon. Illustrations for the story should be included. Drawings created using a draw program on the computer can be used as illustrations. With the completion of the rough draft and the editing process has occurred, students can type the finished story on the computer using Microsoft Work or Creative Writer.

2. PAPER FOLD POLYGONS

Teacher provides students with squares of paper approximately 4 x 4 inches. Each student will need a 12 x 18 piece of newsprint, white construction paper, or manila oaktag. Students are given the following directions for folding the paper:

1. fold paper in half to create two rectangles. Students should refold the square on the same line in the opposite directions. This is to allow easy folding from either direction. Students should do this on each of the following folds. Open the paper to the rectangle.
2. Fold the two upper corners down like you would when making an airplane, matching the top edges to the center fold. This should make a pentagon. Review number of sides and angles. Students should easily be able to provide a description of the pentagon using geometric terms. Refold back on the same lines as in #1.
3. The two remaining corners should be folded as you would when making a paper airplane. If all four corners are folded in it should make a smaller square.

The paper is now prepared for the students to complete the following investigations. Using the paper, the students should investigate the polygons they can make by folding on the existing fold lines. Students can fold the paper on one, two, or more folds, but only on the existing fold lines. They can not make any new folds. The students should trace the different polygons they make on the newsprint and label and describe each polygon using the geometric vocabulary from the vocabulary chart. The students should work until they are sure they have created all of the possible polygons.

There are nine possible polygons.

3. WHAT IS GEOMETRY?

The question should be asked again at the conclusion of the unit to allow students to show their understanding of geometry. Students can compare their first responses to this question to the concluding answers.

The KWL chart can be completed as a whole group at the conclusion of this unit. The KWL chart and the vocabulary chart should be hung in the classroom for the remainder of the year for students to refer to during other areas of study.

Rubrics:

The rubrics used for the Fourth grade English Language Arts test and the Fourth grade Mathematics test were used as guides when developing the following rubrics. The following are student rubric and a teachers scoring rubric.

Teacher Scoring Rubric

1. A GEOMETRY STORY

4 Points: Advanced Achievement

The response

• Fulfill the requirements of the task
• Demonstrates insight and makes connections
• Develops ideas fully with thorough elaboration
• Establishes and maintains a clear focus
• Is fluent and easy to read, with vivid language and a sense of engagement or voice.
• Is stylistically sophisticated, using varied sentence structure and challenging vocabulary.

3 Points: Proficient

The response

• Fulfills some requirements of the task.
• Makes some connections
• May be brief with little elaboration, but is sufficiently developed
• Is generally focused, though may contain some irrelevant details
• Is readable, with some sense of engagement or voice
• Primarily uses simple sentences and grade-level vocabulary

2 Point: Partially Proficient

The response

• Fulfills few requirements of the task
• Makes few connections between ideas
• May contain a few examples or details
• May focus on minor details, or does not establish a focus
• Is often repetitive, with little or no sense of engagement or voice
• Uses minimal vocabulary
• May indicate fragmented thoughts

1 points: Below Proficient

The response is completely incorrect, irrelevant, or incoherent.

Student Rubric:

4 Points: Excellent. Story is enjoyable and imaginative. It relates to geometry and makes use of geometric vocabulary. Very few grammar, punctuation, and spelling errors.

3 Points: Very Good. Story is enjoyable and imaginative. Some geometric vocabulary is used and some relationship to geometry is evident. Some grammar, punctuation, and spelling errors but reader is able to understand the story.

2 Points: Good. Story is difficult to follow. Minimal geometric vocabulary is used. Little relationship to geometry is shown. Grammar, punctuation and spelling errors make the reading difficult.

1 Point: Poor. Story doesn't make sense. Minimal geometric vocabulary is used. Little relationship to geometry is evident. Many grammar, punctuation and spelling errors make the story difficult to understand. Very little work shown. Story shows no thought.

2. PAPER FOLD POLYGONS

Teacher Scoring Rubric

3 Points: A three point response is complete and correct.

This response

• Demonstrates a thorough understanding of mathematical concept
• Indicates that the student completed the task correctly
• Contains clear, complete explanations and/or adequate work when required

2 Points: A two point response is partially correct.

This response

• Demonstrates partial understanding of the mathematical concept
• Addresses most aspects of the task
• May contain an incorrect solution but applies a mathematically appropriate process with valid reasoning and/or explanation
• May contain a correct solution but provides incomplete explanations
• May reflect some misunderstanding of the underlying mathematical concept

1 Point: A one point response is incomplete and exhibits many flaws but is not completely incorrect.

This response

• May contain a correct solution but lacks work when required
• Demonstrates only a limited understanding of the mathematical concept
• May address some elements of the task correctly but reaches an inadequate solution and/or provides reasoning that is faulty or incomplete
• Exhibits multiple flaws related to a misunderstanding of important aspects of the task
• Reflects a lack of essential understanding of the underlying mathematical concepts

0 points: A zero point response is completely incorrect, irrelevant, or incoherent, or a correct response that was arrived at with an obvious incorrect procedure.

Student Rubric

3 Points: All 9 polygons are found. Each polygon is correctly labeled and described using correct geometric vocabulary.

2 Points: All 9 polygons are found. Some are incorrectly labeled or described using incorrect geometric vocabulary. OR At least five polygons are found and are correctly labeled and used correct vocabulary.

1 Point: At least five polygons are correct found and has some correctly labeled and/ used correct vocabulary to describe the polygons.

0 Points: little attempt made to find any polygons. Little or no description or labeling included.

3. WHAT IS GEOMETRY?

2 Points: A two point response is complete and correct

This response

• Demonstrates a thorough understanding of the mathematical concept of geometry
• Contains clear, complete explanations using geometric vocabulary.

1 point: A one point response is only partially correct

This response

• Indicates that the student has demonstrated only a partial understanding of the mathematical concept of geometry
• Addresses some elements of the task correctly but may be incomplete or contain some conceptual flaws minimal use of geometric vocabulary.

0 points: A zero point response is completely incorrect, irrelevant, or incoherent.

No evidence of the mathematical concept of geometry.

Time required

This unit can take from three to five weeks to complete, depending on the grade level and interest of the students. Older students can complete the searches in one to two class meeting with time to discuss the process and learning. Younger students need more time to search and the discussions need more time to develop. The writing activity can carry over into Language Arts time with the final draft completed during computer time. The content learned in this unit carries over into other mathematical units and other content areas.

TEACHER RESOURCES

Burns, Marilyn The Greedy Triangle Scholastic Inc., 1994

National Council of Teachers of Mathematics Addenda Series, Geometry and Spatial Sense NCTM 1993

Rectanus, Cheryl Math By All Means, Geometry Grade 3 Cuisenaire Company of

America Inc., 1994

Internet site: www.aplusmath.com