Michelle Felis CU

Covering Geometry with Quilts

Michelle A. Felis

Academic Setting

This unit is being written for use in teaching in a fourth and fifth grade multi-age inclusion classroom at S.R. Marmon Elementary School which is located on the West Mesa of Albuquerque, New Mexico. Marmon is located in a quiet suburban neighborhood adjacent to I-40.   The school is in the West Mesa High School Cluster. Students feed into either John Adams or Jimmy Carter Middle Schools. The school is built entirely of portable buildings, some of which have been “permanently” located in asphalt pavement. The school was opened 14 years ago.

Marmon is a year round multi-track school serving approximately 850 students. Students are enrolled in one of six tracks. The tracks start in waves with each track having twelve weeks, on followed by a three week off track time. The entire school is closed for three weeks in the beginning of July. The academic year begins the third week in July and ends the last week of June. The majority of classes are required to rotate due to a shortage of classroom space. As one track goes off, they put their materials into storage. The class returning from being off track moves into the newly vacant room. Because of this restriction, (as well as typical budget constraints) teachers tend to have fewer supplies and mateìrials on hand than in a standard classroom.

The student population at Marmon consists of students from the surrounding neighborhood as well as from several low income apartment complexes located both north and south of I-40. The majority of students come from economically challenged households. The school has an extremely mobile population. It is not uncommon for a teacher to have more than half of the class turn over during the course of a year.    However, daily attendance at the school is very good at 95.4%. The ethnic make up of the school for the 2000 - 2001 school year was as follows: Hispanic 66.8%, Anglo 20.1 %, Native American 6.8%, African American 5.7%, Asian American 0.1%, and other .5%. During this same school year, approximately 55% of the students participated in the free-lunch program. The school is designated as a Title I and Title VI school. The school is currently rated as “Meets Standards” by the New Mexico Department of Education. Over the course of the past ten years, the Terra Nova scores of the school have progressed from the low to mid teens, to slightly above the 45%.

My classroom is a multi-age inclusion class. The basic make-up of the class is 24 general education students and 10 special needs students. The special needs students have a range of disabilities including specific learning disabilities, hearing impairments, motor impairments, behavioral and emotional impairments, and Autism Spectrum Disorders. There are two teachers and an educational assistant for the program. The reading ability level of students in the classroom ranges from pre-primer to 9th grade. The general education students tend to perform one to two years below the predicted math achievement range for fourth grade. There are generally one or two students who are identified as being in the gifted range each year. The mathematical abilities of the students are uniformly below their literacy skills. The majority of the class will enter the fourth grade unable to multiply or divide. Many are struggling to master addition and subtraction which requires regrouping. The school has recently adopted the Trailblazer series as the math text.

Goals

I am developing this unit in order to introduce the students to geometric terms and concepts in a unit which can be integrated into the literacy and social studies units throughout the year. The majority of the students have already acquired a dislike for math. While we must address their need for basic computation skills, one can not allow this need to overwhelm the need to develop their math reasoning skills. I am attempting to introduce geometric concepts in a fun, interesting, and most importantly engaging manner. The students will work on mind expanding geometric exercises while they think they are doing their favorite thing, art. The unit will address the competencies covered under the District’s Math Strand III for Grade 4

The students should also develop their reasoning skills as they determine why certain configurations of geometric shapes are possible, and why others are not possible. In addition, a primary goal is that the students will see themselves as competent mathematical decision makers. Students will explore a variety of methods for changing a simple square into complex patterns. I hope that some issues will spark the students’ creative logic as they try to figure out ways to make patterns using the fewest number of cuts, the fewest number of colors, the greatest number of shapes, etc. As they study traditional quilt patterns, and create designs of their own, students will expand their abstract thinking skills. They will do concrete work with hands on materials that they can physically flip, slide, rotate, and cut. These projects will all demonstrate one or more basic geometric concepts. In addition, the students can progress at their own pace. A student with advanced spatial abilities may produce complex designs while a struggling student can be proud of a simple, yet elegant, design.

Context and Background

All students, in fact all people, need to be able to assess and manipulate items in space. At some point, they will look to fit objects of a certain shape into a defined space. In order to do this efficiently, they will need to have developed spatial relations skills. In addition, many of my special needs students exhibit abilities in these areas which they are unable to demonstrate in typical classroom math (basic arithmetic). By offering something in addition to the standard arithmetic lesson, I can hope that some of these students will find their own path to success. I want to offer all of the students a chance to develop or expand upon their abilities to think, reason, and problem solve in the mathematical realm.

            This unit will offer the students an opportunity to explore an area of math without the overwhelming concern of getting the right answer.   Students at this age are beginning to develop an extreme awareness of their peers. Getting the wrong answer in front of the entire class can be a paralyzing event to some. By permitting them to work on a series of concepts which encourage collaboration with almost guaranteed beautiful results, I can help them to develop mathematical confidence. In addition, they will have an opportunity to display their work in eye-catching ways that will invite discussion and exploration by their peers.

Quilting is an ancient craft practiced by the common people of many cultures. While many people think of quilts as being simple bed covers, the skill of quilting has been used by people to make and adorn household objects, clothing, and even armor (fact which should help engage any reluctant boys in the group). The oldest quilted object found to date was a first century BC rug found in a tomb in Siberia. The oldest quilted objects from Europe are armor and bed coverings from Sicily.   Quilting, like most everyday math, developed from necessity. Housewives had to find ways to keep their families warm and make their often bleak surroundings more attractive without wasting materials.   As a result, they managed to find ways to make large projects from small scraps of fabric. By developing the technique of creating a fabric sandwich, they were able to make objects stronger, warmer, and more durable than any single piece of cloth. All quilted objects have three layers: the quilt top which can be plain, or extremely ornate, the batting which can be from a number of materials classically including cotton, fabric, wool, and even horsehair, and the backing.

            Perhaps the most common type of quilt tops are those which use the patchwork technique. This permitted frugal housewives to piece together scraps from other items into functional “new” items. Girls would begin learning to sew together small squares of cloth as soon as they could hold a needle without hurting themselves.

            As generations passed, the quilt tops became more than ways to stay warm without spending money on new fabric. Quilters began to use patterns and variations in order to show off their skills. Certain patterns began to appear which told stories or marked certain events in life. A girl would work to develop a supply of quilted objects (bedcovers, curtains, and petty coats) into her hope chest before she married. As her family developed, new quilts would be made to mark the birth of a child or perhaps a journey to a new home in a new land. As old items wore out, they would be cut up and reassembled into a new, useful object.

Quilting also served an important social function. There wasn’t any leisure time in “the old days.” Women had precious little time to get together and socialize. However, if they were spending the day together sewing a practical and necessary item like a quilt, the time couldn’t be considered to have been wasted. The “quilting” portion of the process refers to stitching together all three layers of fabric. These stitches can be done in very simple or extremely complex patterns. The actual quilting is most often done in a pattern which will compliment the color patterns of the quilt top.   These stitches must be done carefully or the quality of the quilt will suffer. As one can imagine, stitching over an entire bed size quilt alone would be extremely time consuming. By coming together, the hand quilters of the days before electric sewing machines could finish a quilt in a day or two. Quilting bees remain a popular way to get a lot of sewing done on a project in a short amount of time. The “many hands make light work” may well have been spoken after a successful quilting bee.

Modern quilting guilds often meet to share techniques or to create special quilts for fundraisers or commemorative purposes. Perhaps the most famous quilt of the modern age is the AIDS quilt. This ongoing project which pieces together squares which commemorate AIDS victims has grown so large that on its last public display, it covered the entire mall in Washington DC. Many quilt guilds are currently working to make quilts for the families of victims of the attack on the World Trade Center. Activities like quilting for charity or to commemorate people and events are not specifically mentioned with the unit lessons but can easily be tailored to the needs and personality of specific classes.

            It has been said that a listing of common traditional patchwork quilt patterns reads like a history of the settlement and exploration of America: “London Roads, Ocean Waves, Lost Ship, Star and Compass, Charter Oaks, IL Lafayette Orange Peel, Tail of Benjamin’s Kite, Turkey Tracks, Bear’s Paw, Indian Hatchet, Washington Pavement, Dolly Madison Block, Whig Rose, Democrat Rose, Philadelphia Beauty, Virginia, Star, Georgetown Circle, Horn of Plenty, Free Trade Blocks, 54-40 or Fight, Tippecanoe and Tyler too, Clay’s Choice, Little Giant, Mexican Rose, Lincoln’s Platform, The North Wind, Arkansas Traveler, Oklahoma Boomer, Kansas Troubles, World without End, Wagon tracks, Road to California (101 Patchwork Patterns, Ruby McKin).

One way to incorporate literacy into the lessons would be to have the children research a specific pattern, or progression of patterns that are associated with a specific region of the country or a particular period in history. If the students have access to technology, they would even be able to view a wide range of patterns online.

It is worthy of noting that many native cultures have their own quilting patterns. The Seminole tribe of southern Florida have used a specific style of quilting to decorate their clothing since they were introduced to the sewing machine.    As is so often the case with quilting, the art developed out of a necessity to make do with little pieces of fabric. The left over bits of cloth from the ends of bolts were very affordable. The women of the tribe found a beautiful way to make decorations for their clothing using the cheapest cloth available. Seminole strip patterns are deceptive in that they appear to be extremely complex while being made with two or three strips of cloth manipulated in a variety of ways.   This tradition combines small bands to make larger very complex designs. These designs provide a wonderful opportunity to introduce students to the concepts of horizontal, vertical, flips and segments.

In addition, there is a strong tradition of quilting in the Orient and Near East. Some evidence even suggests that the ancient Egyptians may have quilted some objects. Traditional patchwork quilts are made by cutting fabric pieces into a few basic geometric shapes. These shapes are sewed together into more complex geometric patterns. The patterns can be made of very simple color and shape designs or can become extremely complex. Traditional designs can all be made with squares, triangles, and rectangles. Some modern designs include curves and more complex shapes like hexagons and even octagons. Most quilt patterns offer examples of several of the basic geometric shapes. For example, the pattern known as the Seven Sisters uses hexagons, trapezoids, and diamonds. Starlight Nights shows octagons, equilateral triangles, and squares. Some patterns use only one shape in a variety of configurations. The traditional Broken Dishes pattern used only equilateral triangles.

There are simple utilitarian patchwork patterns that combine one simple shape in rows. “Grandmother’s Flower Garden” is row after row of hexagons. I find it interesting that some of the names of the more complex designs indicate that they were made by ladies with more time on their hands. “ Old Maid’s Puzzle” is a fine example. Many traditional patterns also carry a teaching or message in their title. The convoluted curves of Drunkard’s Path became popular during the 1930”s prohibition era.

With thousands of traditional patterns easily available in books and on-line, a teacher can identify a quilt pattern to demonstrate almost any elementary geometric term. Working with these patterns can lead students to develop a familiarity with a vocabulary that will serve them well in their future math studies.

Prerequisites

Implementation

The students will only need to have the most basic knowledge of shapes, and a few supplies (graph paper, pencils, scissors, colored paper, and crayons or colored pencils). I have found that many of my most economically challenged students love to draw. I hope to tap into this interest and get them to develop their special and logic skills while they design their own quilt patterns.

The activities in this unit can be done over a two week time span or can be done as individual activities throughout the school year.   I would encourage teachers to introduce the lessons as an introductory unit prior to, or in conjunction with, a geometry section in the text. Because the work product of most lessons will be so visually pleasing to the students, one might even consider teaching these lessons as art lessons. The students may or may not make a conscious association with the geometric concepts that are being taught from a math text.

Introductory Activities: Set Up a Quilting Center

Provide an area in room that contains several quilting books (available at local libraries).   In addition, put out any fabric samples you may have. If possible, include a few quilts. You will probably be able to borrow one from a room parent who quilts or make arrangements through a local quilt shop. Most quilting guilds would be happy to make arrangements for you to contact local quilters who will bring samples in to your room. As you progress through the lessons, you may wish to add items to the center in order that students can go to the area during free time and create their own paper and fabric quilts. Giving students an opportunity to explore the materials beforehand will help them to feel more comfortable and confident during the actual lessons.

District Standards Addressed:

Strand I: Global Mathematical Processes”

Content standard The student understands and uses mathematical processes.

K-12 Performance Standards

1. Develops resourcefulness and perseverance in problem solving mathematics and other disciplines.

3. Develops and uses strategies for solving given problems.

4. Monitors, discusses and reflects on the process of mathematical problem solving.

8. Works in teams to share ideas, to develop and coordinate group approaches to problems, and to communicate findings.

10. Analyzes and evaluates mathematical thinking and strategies of others.

12. Identifies and connects functions with real world applications.

            Strand III Geometry, Spatial Sense and Measurement

Content Standard: The student demonstrates an understanding of concepts, properties, and relationships of geometry and measurement through experiences with meaningful math problems that focus on identifying, describing, classifying, visualizing, comparing, estimating, and measuring various aspects of shapes and objects.

Grade 3 Performance Standards:

1. Demonstrates transformations of geometric figures (e.g. flips, slides, rotations) and their effects on shapes and objects.

2. Describes and compares symmetrical and congruent shapes.

Grade 4 Performance Standards:

3. Defines the terms perpendicular, congruent, symmetrical, parallel, angles, and right angles.

5. Describes how objects look from different views (e.g., top, front, side).

11. Solves problems involving perimeter and area using a variety of techniques.

Materials required:
Class set of scissors
Several colors of construction paper
Poster boards for each student
Graph paper
Colored pencils or crayons
Glue sticks

It should be noted that these lessons provide a wonderful opportunity for students to practice recycling and materials management. After all, most traditional quilters used whatever scraps they had on hand. Encourage the students to use small pieces of paper left from other projects, or to share pieces they don’t use completely.

Vocabulary

Acute angle, adjacent, angle, closed figure, common, complementary angles, consecutive, equilateral triangle, intersection, isosceles triangle, obtuse, parallel, perpendicular, polygon, protractor, quadrilateral, rectangle, rhombus, right angle, right triangle, segment, square.

Lesson one - Strip Patterns

Provide each student with a sheet of construction paper. Ask them to cut the paper lengthwise into strips that are ½ inch wide. This will give them an opportunity to practice measurement skills. Once they have finished, they should trade strips with several other students in order that each child has several colors of paper.

Give each student a second sheet of construction paper. This sheet will be used as a background for their work. Demonstrate how the strips of different colors can be glued next to each other to create a wide strip that is sort of like a rainbow. Once the students have created their own pattern, have them use their scissors to cut vertically across the rainbow strip they have made. At this point, students should be encouraged to experiment with the little colored segments they have created. Demonstrate how it is possible to create a pattern by flipping the segments and arranging them next to each other. Once the students find a pattern they are pleased with, they can glue it on a background piece. These can be displayed in the classroom. Encourage the students to describe the patterns in their math journals. This would be a great time to review the use of adjectives by having the students take turns reading a description of a classmates pattern and letting the class try to figure out which one they were describing.

Assessment - observe the students as they create the patterns, see how well they can match descriptions to patterns.

Extension: Have the students create strip patterns on the computer using a drawing program. They can practice cutting and pasting skills while they create new patterns.

Lesson two - How many patterns can you make?

Have the students grouped into pairs. Provide each student pair with a sheet of graph paper. Demonstrate how to create a small square by outlining the squares across and three down. This should produce a square which contains 9 smaller squares.   Give the pairs each 18 tiles. Nine in one color and nine in a different color. Ask the students to make an estimate of how many different patterns they could make to fill the squares using only the two colors they have. Encourage them to actually build a square from the tiles. Discuss the initial patterns made by the student pairs. Define rules for determining if one square is different from another. It is often very helpful to have students actually build the pattern out of tile and then color them on the graph paper to record them. Discuss flips, rotations, and slides.   If one pattern can be turned or flipped over, or rotated, and then it looks exactly like another square, they are not really different.    Give the students 20 minute to create as many different patterns as they can. Have student pairs report on how many patterns they made. The teacher can act as moderator in discussions about patterns being the same or different. This is a wonderful time to introduce the concept of similarity.    Ask the students to revise their estimates if that is appropriate.   Post a record of the patterns created by the students and challenge them to continue to add new patterns. There are at least 84 different patterns.

Assessment: Provide a student with two nine block patterns and ask them to determine if they are similar. Ask them to create a drawing of a nine block pattern and then a drawing of that same pattern flipped or rotated

Extension: Ask the students to predict how many more patterns they could make if the square was extended to be four squares by four squares. Give them an opportunity to work on creating the patterns (This can be done during read aloud or during free time).

Lesson Three - Quilt Letters

Provide each student with a sheet of one inch graph paper. Make available construction paper made into one inch squares. You can pre-cut them or have the students measure and cut them at the beginning of the lesson. Ask the students to make the first letter of their name using the paper squares.   Tell them that they can use whole squares or they can cut the square diagonally in half. Have them glue the letters on to the graph paper. Give the students 15 to 20 minutes to make the letters. Once they have completed the letters, have them trade papers with another student. On the board, or on an overhead, review different shapes (Square, rectangle, trapezoid, equilateral triangle,...). You may wish to write the names of some of the newly introduced shapes. Ask the partners to make a list of all the shapes they can identify in their partner’s letter.     Have them draw the shapes they see on their own piece of paper. Have them cut out the shapes and label them. Give the students an opportunity to discuss the shapes they have found. As the students finish, have them paste the pages with the initials onto a large sheet of butcher paper. This creates a class quilt. I also use this quilt as a word scramble by having the students find all the different words they can make using the letters on the quilt. This makes a great display for open house or conferences. The children love showing their families their initials on the quilt.

Extensions: Offer the students a chance to make a pattern from the shapes they identified in their partner’s initial.   These shapes can be glued down and displayed with the class quilt. Students can also make puzzles for each other by cutting their letters into shapes and putting the pieces into bags. These can be kept in a math center for use during free time. Have students identify the types of angles found in each letter. More advanced students could even measure the angles

Assessment: Provide the students with an opportunity to identify the various shapes by matching them with the names. Ask the students to identify the shapes verbally or to point to various shapes as you name them. Have the students list a letter from the quilt that contains a specific shape or shapes.

Lesson four - Class quilt

Provide the class with an opportunity to review a wide range of quilt patterns. Discuss them and them vote on a pattern or design for a class quilt. Give each student an opportunity to create their own paper block for the quilt. Remind the students that it is important to measure accurately if the final quilt is to look right. Once all the members of the class have finished their squares, ask the students how all the squares could be arranged to have a rectangle as the end result. Have the students measure their own squares and then predict how big the final quilt will be when it is assembled. If your class size does not break into even rows, it may be necessary to invite “guests” (the resource room teacher, room parent, or principal) to create a square to join the quilt. Once the squares are all finished, have the students glue them onto a butcher paper backing. Measure the quilt to see if it matches their predictions.

Extension: Introduce or review the concept of perimeter. Remind students that real quilts have borders around them. Ask them to devise a strategy to accurately predict how much paper would be required to put a six inch border around the quilt they have created. Record student predictions.   Cut six inch wide strips from a complementary color of butcher paper. Place the pieces around the quilt like a border. Once the pieces are trimmed to the right length to border the quilt, place them end to end and have the students measure the length. Record this length as the perimeter of the quilt. Replace the pieces around the quilt and tape or glue them into place. The final product can be displayed in the school or even mailed to a pen pal class.

Lesson Five – Transformations

Review a variety of quilt patterns from the pattern books in the quilt center. Discuss the manner in which patterns can be either simple or complex. Ask the students to think about how they could make a simple design into a more complex one without using more colors.

Provide each student with a large sheet of construction paper to be used as a background for their work. Give them six equal squares all of the same color. Have the students each draw one straight line cut all the way across their square. They should cut along the line they have drawn and then form it into a shape by rearranging the pieces. They should glue this shape onto their background piece. Move to the second square and have them make two cuts all the way across the square. Count the number of pieces they have created with two cuts. Have them predict how many pieces they will have if they make six cuts.   Repeat the process increasing the number of cuts until they have made six cuts across the last square. The patterns they have glued onto their background piece should be fairly complex. The students can compare their patterns with those of their classmates. Ask them to identify the various shapes they have created. Some students may wish to try and duplicate the patterns made by classmates.

Extension: Post a picture of a traditional quilt pattern ( found in one of the resource books in your quilt center). Ask the students to predict how many cuts they would need to make and which shape they would start with in the beginning. Give them an opportunity to actually construct the pattern using colored paper. Remind them to compare the product with their predictions.

Bibliography

Argone, Wendy. Geometry for Dummies. New York, NY: Hungry Minds, 2001.

This is a very readable text which offers a quick review of basic geometric concepts. There are lots of definitions and several very humorous cartoons which reinforce the basic concepts.

Ascher, Marcia. Ethnomathematics‚ A Multicultural View of Mathematical Ideas. Florida: CRC Press Boca Raton, 1998.

This text contains a very informative section on strip patterns. Much of the book is devoted to games and number systems from various parts of the world. It is a great place to find new and engaging activities for intermediate students.

Fall, Cheryl. Quilting for Dummie:, A Reference for the Rest of Us. Foster City, CA: IDG Books Worldwide, 1999.

This highly readable text offers most of the basic information one might need to discuss quilts in a knowledgeable way. Much of the text is devoted to sewing skills, but there is also a good deal of information about the history and cultural context of quilting.

Hanisko, Dorothy. Simply Seminole Techniques & Designs in Quilt Making. Chicago, IL: The              Quilt Digest Press, 1997.

This beautifully illustrated book offers examples of traditional and modern strip patterns. There are some techniques, but it is primarily a great picture book for students to investigate strip patterns.

Hinson, Dolores. A Quilter’s Companion, New York, NY: Arco Publishing Inc.1984.

This extremely comprehensive text offers information on most of the traditional quilt patterns. The illustrations are full sized which make it easy for students to analyze and compare them. The illustrations are in black and white which can make it easy for some students to recognize the geometric components of the patterns.

Kavaya, Karol, and Vicki Skemp. Community Quilts: How to Organize, Design & Make a Group Quilt, Asheville, NC: Lark Books, 2001.

This is a charming book which details the way a small rural community developed a tradition of creating a quilt to mark each major event (birth, death, marriage...) Which takes places in the community. The illustrations are beautiful. The book would provide a good starting point for a class unit on quilts.

Peck, Amelia. American Quilts and Coverlets in the Metropolitan Museum of Art. New York, NY: The Metropolitan Museum of Art, 1990.

This book offers a sample of quilting techniques which are more art than utility. It provides samples of some of the most complex patterns and styles of traditional and modern quilting

Van Cleave, Janice. Geometry for Every Kid: easy Activities that Make Learning Geometry Fun. New York, NY: John Wiley & Sons, Inc., 1994.

This kid friendly text offers many simple definitions for geometric terms. In addition, it   offers activities suitable for independent or small group practice, or center activities.
This book offers a very “can do” attitude for students and teachers alike.