SYMMETRY
A CONCEPTUALLY-INTEGRATED MINI-UNIT
INTRODUCTION
The following activities all address the concept of symmetry. Each activity is designed for the Grade 3/4 level. Some of the directions in your task cards are worded more for your adult level, but the actual activity is still appropriate for the Grade 3/4 level.
Each activity is based on a few connected concepts in relation to symmetry, as follows.
Children need to be exposed to and explore 'objects' that are symmetrical and others that are not. From these comparisons they can abstract the distinction between symmetical and asymmetrical 'objects.' They will be able to label an object as either being symmetrical or not. Children need time to explore, both in the classroom and outside the classroom, the symmetrical (or not) features of 'objects.' We need to bring into the classroom authentic examples of things that are symmetrical. Some objects like preserved butterfiles can serve us quite well. However, we know that these objects are not completely symmetrical, but they almost fit the concept. Likewise we can explore other (almost) symmetrical objects (e.g., the human body, leaves, insects). We can use pictures of objects as well as real objects. The purpose in all of this is for the children to locate objects which are symmetrical and others that are not and to be able to label them as such. We would also like children to create symmetrical shapes. They may write their name and then a mirror image of this name; they may make symmetrical shapes out of cut paper, or they may draw a shape that is symmetrical. In mathematics we would also like the children to know if a shape is symmetrical on one axis (reflection), or on many axes (rotation). Generally, we want the children to have a deep understanding of symmetry that can be transferred across all subject areas and we want them to be able to use the term and the concept in intelligent ways in their critical and creative thinking.
The following activities connect with different subject area curricula. However, as the overal purpose of these
activities is to enable children to make sense of symmetry, we have deliberately NOT stated up front which subject
area each activity comes from (except one that is obvious). You can think about the most likely subject area connection
as you proceed through the activities. IF the concept is generally the same in each subject area, then children
should be able to 'see' the concept and work with this concept in interesting ways across the subject areas. The
subject area can become the medium or the vehicle for enabling the concept to be understood; likewise the concept
can become a vehicle for making sense of a subject area (sort of a symbiotic relationship). Many different materials,
resources, and tasks are incorporated into the following activities. Technology, being a resource, and vehicle
of communication and interaction, is also employed in some of the following activities--where it makes sense to
do so. We are working on the understanding that many classrooms have or will have a four-computer networked mini-lab;
hence you will find that EACH activity below may not use the computer, or an activity may have the option of using
a computer if one is available, but the activity could also be accomplished using other resources and materials.
We have structured the classroom learning environment to explore symmetry in a series of stations. This will provide
what Seymour Papert would term a "Microworld" of exploration, a very carefully-structured 'playground'
where the goals and intentions are clear, where children can make choices in their learning, and where, most likely, children will make sense of symmetry.
Because we only have about 50 minutes today to explore these activities we will start you in groups of about 4
at one station, and you can proceed in your own time, as a group, to another station (probably a 10 minute limit
at any one station would be appropriate today). When I work with children I would not manage the activities in
this way. I would have a symmetry schedule card something like the following for each child. Note that the children
get to select (with me) their starting station--I would have a large chart like the one below on the magnetic board.
Children would discuss and negotiate with me where they would start. If I thought their initial choices would fit
into the maximum numbers allowed at each station, I would have children post their magnetic names on the board
inside their starting station number. If too many children wanted to go to any one station I would then ask them
to give me their second choice and so on. The main thing is to get all names on the board as quickly as possible
and get the children started on their work. From that pont on they are on their own. They usually like to work
in pairs, but some children prefer to work alone. This kind of organizational system enables children to work at
their own pace, with no stress or pressure to complete an activity at a certain time (like you might experience
today), and to make choices about which station to got to next (and with whom). You may organize the stations in
such a way that any ONE station would be sufficient. However, if different stations are 'drawing' from different
subject areas, you will probably want the children to visit a few stations to experience the concept in different
ways.
Your Name: ______________________
About Symmetry
In the following stations you will learn about symmetry. You can visit and explore EACH station if you wish, because each station has interesting things for you to do. However, you may not have time to visit each station. We would like you to make sure that you do at least one activity at station 1, all of station 2, all of station 3, all of station 4, one activity at station 5, and all of station 6. After you have completed each activity please write a few sentences about what you learned at that station. We would also like you to record any questions you still have, or simply write about something else you would like to know about symmetry. Please place any work you complete into your symmetry booklets. The following stations have a maximum number of students who can be there at any one time time. Please look through this card and let me know which station you would like to begin with. Once you have completed your first activity you can then go to a station that has space. Please check the magnetic board to see how many people can be at the station you want to go to.
We will be studying symmetry for the next two days. At the end of each day please write in more detail about what
you have learned about symmetry in your journals. You can refer to the notes you have made in your symmetry schedule.
Station #
Station Title
What I Learned
What I Would Like to Know
1 A--4 students
1 B--4 students
1C--4 students
1D--4 students
1E--1 student
2--2 students
3--5 students
4--4 students
5--4 students
6--4 students
ASSESSMENT
Some strategies for assessing symmetrical understanding will by now be pretty obvious to you.
- The children will be DOING stuff at each station, most of which will result in some sort of product. You may need to run around with a polaroid or (digital) camera to capture some of their creations before the next child uses the material. These products are placed in a folder or booklet of some kind for later examination--make sure they have the child's name, that day's date, and a title of what it is. At the end of the day you can look through these products and make your own notes on satisfactory completion (or not). From these samples you may select one (with the child) for the child's portfolio collection.
- As you are going around the room you will no doubt be taking anecdotal records on what the children are doing and saying (to you and to each other).
- You may also have a checklist with all the children's names and the specific symmetry concepts you are intending for the children to learn.
- You may also become part of the learning activity/experience at one of the stations--that way you will really know if the children are understanding symmetry.
- You may also have short interviews with the children and question them about what they know. I like to do these interviews after the first day or two, when I have had a chance to look at the children's work and observe them in groups.
- You can, of course, read their schedule notes and their journals and get a sense of any difficulties that the children are experiencing.
In all of the above there is the assumption that you know what you are looking for as satisfactory knowledge or understanding of the concept, in some outward demonstrable format. You are listening carefully to the children's conversations--what might they say that would lead you to know that children understand symmetry? You are watching what they do, the actions they make, the way they fold, or manipulate the materials, the way they compose, etc. etc.--what might some of these actions be? You are also reading what they write in the schedule cards and in their journals--what might they write that would help you to know they understand symmetry? Lastly, their products--what kind of information are you looking for in the children's creations that would help you make sense of their understanding?
We have developed the following activities for
you to do.
Station # 1A: Geometric Flips
At this station you will first quickly read through an article entitled "Geometric Flips." Then you will write words that are symmetrical.
What you need:
1. The article "Geometric Flips"
2. Paper to write on
3. The chart below to record words in
What to do:
written horizontally
flipped horizontally
written horizontally
flipped horizontally
written vertically
flipped horizontally
written vertically
flipped vertically
11. What is the longest word that you can find that will reflect itself when
flipped horizontally? Flipped vertically?
Station 1B: Mira Math
This station uses a concrete manipulative called a MIRA (like a mirror, only you can see through it). A MIRA will help you know if something is symmetrical. You place the MIRA in the center of the shape and if a shape is symmetrical then what you see on one side of the MIRA should be identical to what you see through the MIRA.
What you need:
1. A MIRA
2. A tub of butterflies, insects, books and pictures
3. A handout of MIRA activities
What to do:
First of all use the Mira to examine the butterflies and also to explore some pictures in the books in the tub.
Are the butterflies completely symmetrical? Can you find some shapes in the books that are completely symmetrical?
In the tub with the Miras is a four-page handout with Mira activities. Go
through this handout, using the Mira to do the activities. Try to complete this handout. Make sure your name is
on it. Take the handout with you when you go to the next station.
Station 1C: Pattern Blocks
At this station you will explore symmetry with pattern blocks. You will explore two kinds of symmetry: reflection symmetry and rotation symmetry.
What you need:
1. Pattern Blocks
2. Hinged mirrors
What to do:
Take a hinged mirror and place each pattern block in turn in the hinge part of the mirror. What do you see when you look into the mirror? How many reflections of the original shape are there? If you count the shape that you have placed in the center, AND the reflections of your shape, how many of your shape can you see?
(with older children I would have them predict how many reflections they would see, given any one shape, or more than one shape; older children would know the angle size of each vertex of each pattern block, and if you placed two blocks together, they could add the angle sizes. If this angle size is known and also if this angle size can be divided into 360, then a whole number of reflections can be predicted. For example, the green triangle is equilateral, which means that each angle is 60 degrees. Sixty can be divided into 360 6 times. Thus there would be 6 triangles in total--one actual block and 5 reflections).
Now take the hinged mirror and some pattern blocks and build a small design inside the hinge. Make sure all the
blocks touch the sides of the mirror. Now you need to create what you see
in the mirror around the mirror. If it is easier you can remove the mirror and create the repetitions of your
design without the mirror to help you.
(This is one activity that I would capture on camera. It takes
a long time for children to re-create the picture they have made with Pattern Block stickers and the process of
doing this may not be very mathematical)
Station 1D: Geoboards
At this station you will use geoboards to explore symmetry.
What you need:
1. Geoboards
2. Elastics
3. Geoboard dot paper
What to do:
Work in pairs, but each student should have a geoboard. Place your geoboards side by side; the axis of symmetry should be in between the two boards. One of you should make a shape with elastics on your board. Your partner should then try to create the reflection of that shape. If you are not sure if you have truly made the reflection then place a MIRA between the two boards and look into the MIRA to check on your reflection. Each of you should create 3 shapes and your partner should make the reflections. On your geoboard dot paper draw one of the shapes and its reflection (use two of the small geoboards on the dot paper).
Station 1E: Interactive Symmetry
What you need:
1. Access to a computer with an Internet connection.
At this station you will visit websites where you can explore symmetry. At the first site you select the type of
symmetry you are interested in and the color you want to use and before your eyes you will see examples of this
type of symmetry. At the second site you need to use the online interactive geoboard to create your own symmetrical
shapes.
What to do:
1. Go to the following website:
http://www.gwydir.demon.co.uk/jo/learn/symmetry.htm
Play inside the white square for a few minutes and then try to complete the activities under the white square.
2. The new Principles and Standards for School Mathematics has an interactive geoboard at the following site: http://standards.nctm.org/document/eexamples/chap4/4.2/
Use one band to make a verical 'line' down the center of the goeboard. Create a shape on one side of the band and mirror it on the other.
Station 2: Snowflakes
Has anyone ever managed to catch a snowflake on their mitten and taken a very close look? Wilson Bentley spent
his life looking at snowflakes and photographing them. Take some time and look at the book Snowflake Bentley by
Jacqueline Briggs Martin.
What To Do:
1. Find an available computer and log-on.
2. Visit the following website:
http://www.explorelearning.com/index.cfm?method=cResource.dspView&ResourceID=45
3. Choose the shape of snowflake you'd like to make.
4. Follow the instructions and experiment with the symmetry found in snowflakes!
Station 3: Flocking
Most definitions of symmetry refer to the ways in which a pattern repeats, is balanced, or is similar in shape
to itself. There are many different kinds of symmetry: reflection (mirror), translation (sliding), rotational (turning),
and glide. In this activity, we will explore these types of symmetry.
What To Do:
1. Each person is given a Symmetry card.
2. Read it quietly to yourself, and prepare to teach.
3. Taking turns, demonstrate and practice the symmetry described on your card with your group.
Station # 4: Miming and Mirroring
Mirroring: Looked in a mirror lately? What does your reflection do when you move? In this activity we will explore
how your reflection moves. In groups of two, one person (the leader) will move in front of a mirror or window and
the other person (the follower) will be that person's reflection.
What To Do:
You are sitting in front of the television eating your lunch.
You are standing in front of the mirror getting ready for a night out with your friends.
It is a very hot day. While walking on the sidewalk, you notice a dog trapped inside a car with its windows closed.
The dog is in distress! You must get the dog out!
You have slept in and are late for school! The alarm clock goes off - and you realize that you are late and rush
to get dressed.
Player's choice! Can you think of something that you do in front of a mirror or window? (That has not already been
performed!)
Player's choice! Can you think of something that you do in front of a mirror or window? (That has not already been
performed!)
Player's choice! Can you think of something that you do in front of a mirror or window? (That has not already been
performed!)
Player's choice! Can you think of something that you do in front of a mirror or window? (That has not already been
performed!)
You are a hockey player skating towards the boards. Another player body-checks you, and you crash into the glass.
Station 5: Symmetry
in Music
It is important for students to experience other stations dealing with symmetry before this one because musical
symmetry is somewhat unconventional. These applications of the idea of symmetry expand the notions of symmetry
that you have learned so far.
Many musicians have a strong interest in Mathematics, so it is no surprise that concepts such as mirror-image and
symmetry occur in music. There are several kinds of symmetry in music.
The music program for this station was downloaded free from
http://www.codamusic.com/coda/
A. Symmetrical Forms (Auditory)
Musicians use ideas of symmetry to describe larger musical forms where listeners can hear/notice the similarities.
Rather than do the analysis note-by-note, the music is chunked into phrases/sentences or sections.
Open Twinkle, Twinkle from the CD. Play it. Notice that Twinkle, Twinkle has three phrases with the first and third
being identical. It is not the words, but the musical line that makes it symmetrical.
This is called A B A form, a symmetrical form.
Task 1: In pairs, go to a lap
top where music notation has been down-loaded for you. As a group you are going to compose a short tune in ABA
form. You need to start a new page and give it a title and composer. Select your instrument and add this instrument.
Start composing. Play it back. Can you hear the repetition of the A section? Save it with your names + ABA. (Saving
may be a problem on university computers.)
Enrichment
Task 2: Listen to Hot Cross Buns from the CD. It has a repetition of the
first A, so it is AABA. This is one of the most common song patterns.
The More We Get Together is longer. Which form is it? ABA or AABA
B. Visual Symmetry in Music
When one is dealing with a musical score (printed page) then the meanings of symmetry are similar to those in other
visual domains like mathematics and dance. Examples of visual symmetry in music are very rare, but a few composers
played with mathematical ideas and worked them out in music.
Nancy has composed a short tune called "Forwards and Backwards" to show how a melody can be symmetrical.
When the melody is very short you can sometimes hear the symmetry. In a long symmetrical piece, you can not tell
unless you look at the score.
Task 1: Open F & B from the
CD. Play the piece. Press stop. Look at the melody to see if you can find the center, where the mirror image begins.
Task 2: Open a new file, give
it a title and composer, choose a recorder, flute or other instrument. Compose a short piece that is visually symmetrical.
Play your piece. Save with your names + FB.
Enrichment
The Crab Canon by J. S. Bach is a duet for two instruments where one instrument plays a line of music that is the
mirror image of the other instrument. The Crab Canon is on a paper at the station.
If you were to listen to the Crab Canon, you would not be able to tell that the music is symmetrical.
Task 3: Compare the beginning
of Instrument 1 and the end of Instrument 2. Can you see the mirror image? Check a little further, does the pattern
continue?
What shape is a crab? How does a crab move? Can you describe why this title might fit?
Do you have the patience to check all the way through the piece to find out if the two parts are an exact mirror
image?
If you have musical background and/or can think hard about this piece you might figure out that it is symmetrical
in the auditory sense (through the ears) because if you could imagine or calculate the middle of the piece, you
would find that the pattern of notes on either side is the same. This is a mental/technical kind of symmetry.
Reference Source for Bach examples:
http://www.math.iastate.edu/mathnight/activities/modules/music/musicmiddle.pdf
Station 6: Symmetrical
Poems
Included with this activity card are examples of concrete poems. Notice how the image formed in the poem is related
to the subject of the poem.
In this activity, we are going to make a more specific form of concrete poem, a symmetrical poem.
If you would like to review what symmetry is you can visit this website:
http://pbskids.org/cyberchase/classroom/games/symmetry/index.html
What To Do:
Resources used in the preparation and implementation of this mini-unit.
http://www.gwydir.demon.co.uk/jo/learn/symmetry.htm
http://standards.nctm.org/document/eexamples/chap4/4.2/
http://www.explorescience.com/activities/activity_list.cfm?categoryID=4
http://www.codamusic.com/coda/
http://www.math.iastate.edu/mathnight/activities/modules/music/musicmiddle.pdf
http://pbskids.org/cyberchase/classroom/games/symmetry/index.html
www.pbs.org/teachersource
www.mathdance.org
www.scottkim.com
www.explorescience.com