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SACRED HEART SYMMETRY
A CONCEPTUALLY-INTEGRATED MINI-UNIT
INTRODUCTION--Information for teachers
The following activities all address the concept of symmetry. Each activity is designed for the Grade 2/5 levels.
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.
The children will be working today for about 2 hours to explore symmetry--through a variety of different and, hopefully,
exciting activities. In total, we have designed 12 activities. However, we would like to have at least two 'empty'
stations to start with to allow children to move freely from station to station, once they have completed their
work. We would like the children to be grouped such that we have two 'empty' stations. Thus, if we have 12 stations,
we start with 10 that are occupied, requiring the children to be in groups of 4 or 5 (we're assuming there are
about 40-50 children). We can place the groups of children at any initial starting station. [For every group to
be able to visit all 12 stations would mean that a 10 minute limit be placed on each station. We feel that 10 minutes
is not sufficient time to explore some of the stations. We do not feel that we should be 'pushing' the children
through all the activities]. Maybe for today the groups of children could be given a starting station and then,
when the group is finished that station, they could move as a group to an 'empty' station. Each group would have
to show us some 'evidence' of work they had completed while at the station and we could help them select a different
station. It might be difficult to have the groups rotate (i.e., move from station to station in order) throught
each station. Whatever method we use, it will be very unlikely that ALL groups will explore ALL stations.
If we had time, and if we knew the children really well, the following
is an idea of what we might do. We would have a symmetry schedule card something like the following for each child.
Note that the children get to select (with us) their starting station--we would have a large chart like the one
below on the magnetic board. Children would discuss and negotiate where they would start. If we thought their initial
choices would fit into the maximum numbers allowed at each station, we 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 Iwewould
then ask them to give a 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 go 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.
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ORGANIZING WHAT THE CHILDREN DO
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.
Station #
Station Title
What I Learned
What I Would Like to Know
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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 explore words and try to decide if the words are symmetircal--or not. You will also try to think of words that are symmetrical.
What you need:
What to do:
written horizontally
flipped horizontally
written horizontally
flipped horizontally
written vertically
flipped horizontally
written vertically
flipped vertically
10. 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. Make sure you try to DRESS
THE COWBOY; to do this you need to place the MIRA beside each item of
clothing and move it around (rotate the MIRA) to move the clothing to the cowboy's body.
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:
Everyone:
Take a hinged mirror and place each pattern block in turn in the hinge part of the mirror--right where the two parts of the mirror meet, and then close the mirror around the block so that the hinger mirro touches the block. 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?
Place about 4-6 blocks in the hinge of the mirror and close the mirror as best you can around the blocks. Now look
into the mirror, but DO NOT move the mirror. Try to construct AROUND THE
OUTSIDE OF THE MIRROR what you seeh
IN the mirror.
Do this activity again but this time, after you have placed about 4-6 blocks in the hinge part of the mirror and have examined the reflection in the mirror, REMOVE the mirror and try to recreate the reflected image.
Grade 5:
Children, make sure that your teacher takes a picture of your work so that it can be saved to show others.
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 with a partner. Each of you should have your own geoboard. Sit beside your partner and place your geoboard right beside your partner's geoboard. The axis or line of symmetry will be in between the two geoboards. If you are sitting BESIDE your partner the line of summetry will be vertical. Later, you can sit ACROSS from your partner and have a horizontal axis. You can mark this 'line' with a ruler if you wish. 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
At this station you will go onto the Internet and explore a website that has an interactive geoboard.
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. Find an adult or older child to read this story to you and then do the following:
What To Do:
1. Find an available computer and log-on.
2. Visit the following website:
http://www.explorescience.com/activities/activity_list.cfm?categoryID=4
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: Symmetrical
Moves
Each person take a task card and prepare to lead the group in the activity on the card.
Grade 2 Leader: Mirror Imagery
Play Simon Says. You be Simon and do actions that involve hands and legs, staying in the same place. If
you say "do this" and the others follow, you say "I didn't say Simon says" and continue the
game.
Grade 2 Leader: Marching in Place
Face the group and ask the others to march with you in place. Say "do
what I do." Turn around so that your back is to the group. When you step with your right foot, swing your
left hand forward. Then do the opposite
with your left foot and right hand. Alternating hands and feet (left and right) is called "cross extension."
Marching Poem
A marching we will go
A marching we will go
Left and right, we sail tonight
A marching we will go.
Stop!
Grade 2 Leader: Ask
your group to gently get down on their hands and knees in crawling position. You stay standing and say: "See
how you change hands and feet when you crawl."
Ask the crawlers to crawl while you say the poem.
Crawling Poem
Crawling is good for your brain.
Crawling is good for your brain.
Crawl and walk,
Then learn to talk
Crawling is good for your brain.
Stop!
Grade 2 or 5 Leader: Rotational or Turning Symmetry
This is the kind of symmetry used when you see yourself in a video monitor in a store: you move your right arm
and your TV image moves its right arm. Have a pair of people stand in front of the video camera. One person is
the leader and the other is the follower. Do five hand/arm moves and follow. Take turns.
Mirror Symmetry in Conversation
Introduction
When you are listening to a friend talking, you can give them messages that you are listening by mirroring some of their gestures. How you mirror may give the person a non-verbal message about your interest in the conversation.
Copy Cat: If you do exact and precise mirroring of a person's movement and posture, while they are talking, it often causes embarrassment as it makes fun of the person talking. This is not such a good idea.
Do your own thing: If you take positions that are totally unlike what the speaker is doing, they are likely to think that you are not really interested. This might make them think that you are rude.
Good listener: Slowly and gradually move to include some of the talker's movements using a smaller size. Do not copy all of the movements, just some of them.
Grade 5 Task Card:
In pairs, have one person be the talker and listener. Have them sit in chairs facing each other.
Tell the talkers that they are to talk for one minute (12+ sentences) to their listener. See the list of topics
below.
Students can take turns being the listener, and choose one of the three possibilities: copy cat, do your own thing,
good listener.
See if the group can guess which you are doing. Talk about how it feels and what it looks like. Have someone else
be the talker and listener.
Topics for Conversations:
Funny pets
The problem with peanut butter.
What to pack before a long trip
Explain to an adult why you need an increase in allowance.
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 a mirror combing and gelling your hair.
You are standing at the kitchen counter making a sandwich (and pouring juice).
It is a very hot day. You are giving your dog a bath.
You are making your bed (or helping to make your bed).
You are playing on the monkey bars.
You are climbing a ladder to save a kitty in a tree. You need to come down the ladder holding onto the ladder with
one hand and the kitty with the other hand.
You are brushing and flossing your teeth.
You are watering a row of plants (house, classroom, or garden)
You are washing and drying dishes.
You are sharpening a new box of pencils.
You are cleaning a dirty window.
You are erasing a blackboard
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
Note: It is important for students to experience other symmetry stations before
this one because musical symmetry adds new meanings. This station expands the concepts of symmetry that students
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 Music Note Pad that is used for this station is available on the Internet. If you have an e-mail
address, you can download this music notation program for free from http://www.codamusic.com/coda/
A. Note-by-Note Symmetry in Music
Examples of note-by-note 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 to B. Play the
piece. Press stop. Look at the melody to see if you can find the center, that is, where the mirror image begins.
Task 2: Open the file called note-note,
that gives you the first half of a new melody. Compose the rest of the piece so that it is symmetrical. Play your
piece. Can you hear the pattern in reverse?
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 guess why this title might fit?
Grade 5 students --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.
B. Symmetrical Forms (Auditory)
Musicians use ideas of symmetry to describe larger musical forms where listeners can hear large sections of music
that are repeated. Rather than do the analysis note-by-note, the music is chunked into phrases/sentences or sections.
Open the file "Twinkle, Twinkle". Press play and listen to it. Notice that "Twinkle, Twinkle"
has three phrases with the first and third being identical. It is not the words (called lyrics), but the musical
line that makes it symmetrical.
This is called A B A form, a symmetrical form.
Some familiar objects can remind you of this ABA form, for example, Oreo cookies [cookie, filling, cookie] and
sandwiches [bread, jam, bread].
Task 1: Listen to "Hot Cross
Buns." It has a repetition of the first A, so it is AABA. This is one of the most common song forms.
Listen to "The More We Get Together". Which form is it? ABA or AABA
Enrichment
Task 2: 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.
Station 6: Symmetrical
Poetry
The materials at this table will help you to write a short poem. It is fine to use both symmetrical and non-symmetrical
words in your poem. The following is a list of 6 kinds of symmetry. Do not try to use them all. Read through these
ideas for symmetry before deciding on your poem title. There are specific instructions for Grade 2 and Grade
5 students.
1. Pairs of words where one is the reverse of the other:
deer reed
draw ward
leg gel
live evil
not ton
no on
pets step
rat tar
rats star
saw was
sleep peels
ten net
trap part
war raw
won now
2. Words that are symmetrical, with each letter being symmetrical: MOM, EYE, WOW, WOW, EVE, EVE, OTTO.
3. Words that are symmetrical (palindromes): dad, did, level, kayak.
4. Two word combinations that are symmetrical:
A Toyota
A nut (tuna)
Wonton (not now)
Race fast safe car (the line of symmetry is through the middle of the t)
Don't nod (the line of symmetry is through the middle of the t)
5. Three words where the first and last are the same (ABA form). Examples:
Laugh and laugh
Warm cold warm
6. Reflection. See the Peace Poem, the words "peace" and "now".
Grade 2 Students:
Grade 5 Students:
If you would like to review what symmetry is you can visit this website:
http://pbskids.org/cyberchase/classroom/games/symmetry/index.html
Station 7--A Visual Arts -- Alphabet Symmetry
Materials: plastic letters, playdoh, plastic knife, miras.
What to do: Place a letter of the alphabet in front of you. Put a mira beside the letter. With your playdoh make a mirror image of your letter. When you look into the mirror what you see IN the mirror should be identical to the letter you placed in front of you. Is it?
Station 7--B Visual Arts -- Building Symmetry
Materials - photographs, Lego, pattern blocks, black construction paper, scotch tape
People who design buildings (architects) often use symmetry. Can you find the line of symmetry in the photographs
of these buildings?
What To Do:
1. Choose a photograph.
2. What is symmetrical about the picture you chose?
3. Can you find the line of symmetry.
4. Cover half of the photograph with a piece of black paper.
5. Tape the paper down so it doesn't move.
6. Using Lego or pattern blocks, try to make the covered half of your picture by looking only at the uncovered
side. Remember it is symmetrical!
West Wing of Saskatchewan Legislature
Entrance to Sacred Heart School
Northern Bank Building
Church of the Blessed Sacrament
West Wing of Legislature
House near Sacred Heart School
Station 7--C Visual Arts -- Remembering Symmetry
Materials - half-photographs, paper/pencils, crayons etc, scotch tape
What To Do:
1. Choose half a photograph.
2. Each picture is symmetrical. If I give you one side, can you draw the other side?
3. Tape the photograph to a piece of paper, and try to draw the other side.
Blessed Sacrament Door
Fountain at the Legislature
Facade at the Legislature
Changing of the Chief Office Ceremony
Brick Pattern on Scott Collegiate
Resources
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