# 1. GROUP THEORY?

The back story

Peter Taylor (Queen's University) has been the inspiration for this idea. He has been exploring geometric transformations and group theory fundamentals at the secondary school level, so the question arose: what can we do with this in elementary school?

Two connections came quickly: (1) transformations and (2) operations with numbers. Both of these topics run across all grades in elementary school.

In thinking about group theory it was helpful to also chat with Graham Denham (Western University) and discuss examples.

Plus Magazine and the NRICH site has also given us some concrete ideas to try with young children.

Chris Yiu (Computer Science, Western University) did the initial coding for the Scratch programs with transformations below.

The initial activities below were first tried with students by George Gadanidis and Janette Hughes at Janette's UOIT STEAM-3D Research Lab.

See the latest work on this topic: lesson plan, story + coding environment.

Group theory?

Group theory falls within the field of abstract algebra and is typically studied in second year university.

Before we define what groups and group theory are, let's see some examples from a grade 2/3 class that will set the stage.

# 2. WORKING WITH GRADE 2/3 STUDENTS

Here's what we tried with a class of grade 2-3 students, working with 4 different groups of 3-5 students each ... spending about 40 minutes with each group ...

2.A. ROTATE 30 DEGREES AND STAMP

Children used the Scratch simulation below:

• to rotate and stamp
• and to discover all the unique positions could be stamped

Try it yourself!

Children concluded that there will be 12 unique positions stamped.

Here they are:

Then we asked: what does this remind you of?

2.B. ADDING HOURS ON A CLOCK

So we said: let's see what this looks like on a clock.

We showed the image below and asked: what time will it be after 5 hours pass?

Children explained that it would be 2 o'clock.

• 9 + 5 = 14 = 2 o'clock

That's weird, we noted: 9 + 5 = 2?

That's how clocks work, said the children.

Note: By the way, this is what we call modular arithmetic, where numbers wrap around when they reach a certain value. The arithmetic of hours on a clock uses modulo 12.

We asked children to make up a different addition problem for the clock. We solved a couple of these together. No problem.

We looked at the clock again, and in chorus we counted back the hours from 6 o'clock: 6, 5, 4, 3, 2, 1 ... 0

We agreed that 12 o'clock could also be written as 0 o'clock.

So we asked children to complete the addition table below for hours on a 12-hour clock. Get the handout.

"This is fun!" said one of the students. "It is," we agreed, although a little surprised at first that they said that. There is a pleasure in creating and identifying patterns!

Here is one of the children completing the table, and noticing diagonal number patterns.

2.C. ROTATE 90 DEGREES AND STAMP

Then we asked children to use the Scratch simulation below:

• to rotate and stamp
• and to discover all the unique positions could be stamped

Try it yourself!

Children concluded that there will be 4 unique positions stamped.

We asked: what does this remind you of on a clock?

Some children answered: the hours 3, 6, 9 and 12

Others said: 15 past, half past ...

2.D. ADDING QUARTER HOURS ON A CLOCK

We showed the image below and asked: where would we be on the clock of we started at 0 and added 30 minutes and then 45 minutes?

Some children said 75. Hmm. BUt there is no 75 on the clock. Children rotated 75 minutes on the clock above to get 15 minutes.

We solved a couple more problems like this.

We then asked children to complete the addition table below for quarter hours on a clock. Get the handout.

Here is one of the completed tables. Notice the diagonal patterns appearing again.

2.E. RENAMING THE CLOCK POSITIONS

We looked at the 60 minute clock again.

We asked children to come up with their own names for each of the four clock positions (0, 15, 30, 45).

One of the groups we worked with came up with the following names:

• unicorn = 0
• pony = 15
• zebra = 30
• mango = 45

So we asked, what would be the sum of: unicorn + pony + zebra + mango?

We calculated that: unicorn + pony + zebra + mango = zebra

Can you see why?

Using the first letter of each new name, children created and solved their own addition problems.

Here is another example from a different group of children.

Why did we do this?

• the 4 quarter hours on clock and the 4 geometric shape rotations have the same underlying structure - they are isomorphic
• by letting children name the quarter hours as unicorn, pony, zebra and mango, we can abstract beyond the clock example
• that is, we take their labels unicorn, pony, zebra and mango, and use them on the geometric rotations and see that they work exactly the same way when they made up and solved their own addition problems (like unicorn + zebra + mango = ?)
• group theory is part of abstract algebra, which is concerned with the underlying structures that tanscend specific instances, and we wanted children to get a glimse of this idea

We didn't get to do this. Time ran out and children moved to different centre.

Forty minutes is such a short time. But we actually accomplished more than we anticipated!

3.A. WHAT DID WE DO?

Covered some curriculum:

• time: hours, minutes
• geometry: transformations (rotations)
• patterning: patterns when adding numbers on a clock

• between geometric rotations and adding hours on a clock
• 9 hours + 5 hours = rorate 30 degrees 9 times + rotate 30 degrees 5 times
• although we did not give this relationship a name (isomorphism), students did identify that the underlying structure is similar (which is the essense of isomorhism)
• isomorphic means equal form or equal structure
• there is a one-to-one correspondence between each of the 12 hours on a clock and the 12 30-degree rotation positions (see images above)
• and there is a one-to-one correspondence between each of the quarter hours on a clock and the 4 90-degree rotation positions (see images below)

Used a Scratch simulation:

• students used a Scratch simulation to investigate the possible rotation positions

3.B. WHAT ELSE COULD WE DO?

Extend the conceptual connections:

• we already mentioned that we din't have anough time to get children to apply their labels unicorn, pony, zebra and mango to the geometric rotations and see that they worked the same way when they made up and solved their own addition problems (like unicorn + zebra + mango = ?)
• we could also engage students in a conversation about the similarities between the hour clock, quarter hour clock, 90 degree rotation and 30 degree rotation (all of which are finite cyclic groups, but we don't need to name them yet: finite because they have a finite number of elements; cyclic because applying a single operation over and over creates and repeats all the elements; and more about groups later)
• would children see that all of these have a similar structure?
• could they create other examples that also had the same structure?

Edit the Scratch simulation:

• students used a Scatch simulation to rotate and stamp a geometric shape in various positions
• they could edit the code to explore different rotations
• here is what the code looks like for the 90 degree rotations

Here is what a grade 6 student did in this situation:

• He wanted to explore the case where the rotation was 0.1 degrees
• he decided that it would be too tedious to rotate and stamp all the different locations manually
• so he added a repeat block so the simulation would rotate and stamp on its own
• the result looks like a "disk" he said

# 4. WHAT IS A GROUP?

Group theory has been useful in solving a wide variety of problems, including how the Rubik's Cube.

Let's see what group theory is all about.

Our minds are really good at making abstractions. For example, if you look at dogs you will find that they come in a variety of sizes, body types, colours, temperments and so forth. Yet despite all their differences, when you see a dog you recognize it as a dog. To be able to do this, your mind has created an abstract image of "dog" that captures the underlying structure of the myriad or individual dogs. So when you see a table, a mouse or a cat with 4 legs, you know it's not a dog even though dogs have 4 legs.

We have seen in Example 1 and Example 2 in Section 1.A, that the underlying structure is the same. 1 o'clock corresponds to a rotation of 30 degrees, 2 o'clock corresponds with two rotations of 30 degrees, and so forth. This one-to-one correspondence is called an isomorphism.

The same relationship was also the case for Example 3 and Example 4. A quarter after corresponds to a rotation of 90 degrees, two quarters (or one half) after corresponds with two rotations of 90 degrees, and so forth.

Seeing the underlying structuren and recognizing that different looking examples are isomorhic, is a form of abstraction.

Historically, group theory developed with mathematicians looking at problems that they could not solve, mathematicians noticed that a number of these unsolved problems had the same underlying structure, which they came to call a group.

A key idea in group theory is that of symmetry. If you look around you - in nature, in art, in architecture, and so forth - you will see many beautiful examples of symmetry.

Group theory is the 'algebra' of symmetry.

"The understanding that symmetries are best viewed as transformations arose when mathematicians realized that a set of symmetries on an object is not just an arbitrary collection of transformations, but has a beautiful internal structure. ... the fact that the symmetries of an object form a group is a significant one. However, it's such a simple and 'obvious' fact that for ages nobody even noticed it; and even when they did, it took mathematicians a while to appreciate just how significant this simple observation really is. It leads to a natural and elegant 'algebra' of symmetry, known as Group Theory." [see pages 37-40 of Stewart, I. & Golubitsky, M. (1992). Fearful symmetry. Cambridge, MA: Blackwell Press]

When group theory mathematicians talk about symmetry they refer to things that stay the same when transformed: they are invariant. For example, take a table: it can stand up on its 4 legs, you can rotate it 90 degrees, another 90 degrees, and another 90 degrees, as shown in the image at right. No matter how many times you rotate the table 90 degrees, it remains invariant: it is still the same table.

The four different positions of the table (shown at right) form a set, and along with the the operation "rotate 90 degrees" they form a group.

• there is an inverse operation: "rotate -90 degrees"
• there is an identity operation: "rotate 0 degrees"
• applying the operation, its inverse or its identity results in an element that is part of the original set
• the operations are associative

A group is a set of elements, together with an operation which allows us to combine two elements to get a third, with the following properties:

1. there is an identity element in the group that leaves elements in the set unchanged
• for example, for the set of Integers under addition, the identity element is 0
• we can add 0 to any integer and it does not change the integer
2. every element has an inverse
• for example, for addition, 2 and -2 are inverse elements
• when we combine an element and its inverse using the operation, we get the identity element
• 0 is the identity element for addition, and 2 + (-2) = 0
• rotate 90 degrees and rotate -90 degrees are inverse elements for rotation
• can you think of other pairs of operations and inverses?
3. using the the operation to combine any two elements of a set result in elements that are part of the original set
• let's use the operation addition
• if we take any 2 Integers and add them, we always get an answer that is an Integer
• the identity element (0) is also an Integer
• the inverses of all elements (like 2 and -2) are also Integers
• so, we say that the set of Integers is closed under addition
• for another example, let's start start again with the set on Integers
• but now let's use the operation multiplication
• if we take any 2 Integers and multiply them, we always get an answer that is an Integer
• the identity element (1) is also an Integer
• however, not all elements have inverses that are Integers (like 2 and 1/2)
• so, we say that the set of Integers is not closed under multiplication
4. operations are associative: (a x b) x c = a x (b x c)
• for example, for the set of integers under addition
• we can add (2+3)+4, that is 2+3 and then 4
• or, we can add 2+(3+4), that is 3+4 and then 2
• either way we get the same answer

In Section 2 below we see how we can form groups with objects and transformations.

Then we explore groups based on sets of numbers in Section 3 below.

# 5. GROUP THEORY, TRANSFORMATIONS + CODING

Give these a try!

5.A. ROTATIONS

Go to https://scratch.mit.edu/projects/108197367/#editor to see and edit the code.

This is what you will see:

Use the operation RotateCW and its inverse RotateCCW to rotate the object to various positions.

• Use Stamp to leave an image of the various positions.
• How many different positions are there?

Edit the code so that each rotation is 45 degrees.

• What changes?

Edit the code so that each rotation is 30 degrees.

• What changes?

Edit the code so that each rotation is 60 degrees.

• What changes?

Is it a group?

Is the following a group?
• SET: the set contains the 4 elements below

• OPERATION: Rotate clockwide 90 degrees
• INVERSE: Rotate counterclockwide 90 degrees
• CLOSED: The set is closed under the operation of rotate clockwide 90 degrees
• IDENTITY: We can easily create the identity of rotate clockwide 0 degrees (or rotate clockwide 360 degrees)
• ASSOCIATIVE: PIck any 3 operations in sequence: does it matter whether you do the first 2 first or the last two first?

5.B. X/Y MIRRORS

This is what you will see:

Use MirrorX and MirrorY to flip the object to various positions.
Use Stamp to leave an image of the various positions.
How many different positions are there?
Draw and number all the possible positions on the grid above.

Is it a group?

• SET: What is the set?
• OPERATION: What is (are) the operation(s)?
• INVERSE: What is (are) the inverse operation(s)?
• CLOSED: Is the set closed under the operation(s)?
• IDENTITY: Is there an identity?
• ASSOCIATIVE: PIck any 3 operations in sequence: does it matter whether you do the first 2 first or the last two first?

5.C. SLIDES

This is what you will see:

Use Up, Down, Left and Right to transform the object to various positions.
Use Stamp to leave an image of the various positions.
How many different positions are there?
Draw and number all the possible positions on the grid above.

Is it a group?

• SET: What is the set?
• OPERATION: What is (are) the operation(s)?
• INVERSE: What is (are) the inverse operation(s)?
• CLOSED: Is the set closed under the operation(s)?
• IDENTITY: Is there an identity?
• ASSOCIATIVE: PIck any 3 operations in sequence: does it matter whether you do the first 2 first or the last two first?

# 6. GROUP THEORY, SETS OF NUMBERS + CODING

Give these a try!

6.A. {-1,1} & MULTIPLICATION

Set

Let's start with the set {-1, 1} and the operation of multiplication.

Operation

Multiply as many different pair combinations of numbers from the set as possible.

-1 x -1 = ___
1 x 1 = ___
-1 x 1 = ___
1 x -1 = ___

Are all the products in the original set {-1, 1}?

Inverse

What if we use the inverse operation of division?

-1 ÷ -1 = ___
1 ÷ 1 = ___
-1 ÷ 1 = ___
1 ÷ -1 = ___

Are all the quotients in the original set {-1, 1}?

Associative?

Is (a x b) x c = a x (b x c)?

Identity?

Is there an identity that leaves elements of the set unchanged?

Set

Let's start again with the set {-1, 1} but this time let's use the operation of addition.

Operation

Add as many different pair combinations of numbers from the set as possible.

___  + ___ = ___
___  + ___ = ___
___  + ___ = ___
___  + ___ = ___
___  + ___ = ___
___  + ___ = ___

Are all the sums in the original set {-1, 1}?

Inverse

What if we use the inverse operation of subtraction?

-1 ÷ -1 = ___
1 ÷ 1 = ___
-1 ÷ 1 = ___
1 ÷ -1 = ___

Are all the differences in the original set {-1, 1}?

Associative?

Is (a + b) + c = a + (b + c)?

Identity?

Is there an identity that leaves elements of the set unchanged?

6.C. INTEGERS & MULTIPLICATION

Set

Operation

Are all the products in the original set?

Inverse

What if we use the inverse operation of division?

Are all the quotients in the original set?

Associative?

Is (a x b) x c = a x (b x c)?

Identity?

Is there an identity that leaves elements of the set unchanged?

Set

Operation

Are all the sums in the original set?

Inverse

What if we use the inverse operation of subtraction?

Are all the differences in the original set?

Associative?

Is (a + b) + c = a + (b + c)?

Identity?

Is there an identity that leaves elements of the set unchanged?

# 7. GROUP THEORY, SETS OF NUMBERS, MOLULUS + CODING

Give these a try!

7.A. WHAT IS MODULUS?

The programs below explore sets with operations that involve modulus.

You use modulus daily when you tell time.

• you use modulo 12 when you tell time
• for example, if it's 11 am and 3 hous pass, the time is 2 pm
• what if it's 6 am and 20 hous pass?

If we divided our day into 16 hours and our clocks used 8 hour intervals instead of 12, then we would use modulo 8

• for example
• if it's 5 am and 7 hous pass, the time would be 4 pm
• what if it's 6 am and 20 hous pass?

Generally speaking, modulus means find the remainder. So,

• 15 modulo 12 = 3
• 15 modulo 10 = 5
• 15 modulo 6 = 3
• 15 modulo 7 = 1

7.B. USING MODULUS WITH OPERATIONS ON SETS

The examples below are availabe at https://scratch.mit.edu/projects/108288452/#editor

Do they form groups?

Set = {0,1,2,3}
Operation = addition with mod 4
The code below randomly picks 10 pairs of numbers from 0, 1, 2, 3 and finds mod 4 of their sum

Set = {2,4,6,8}
Operation = addition with mod 10

The code below randomly picks ten pairs of numbers from 2, 4, 6, 8 and finds mod 10 of their sum

Set = {1,2,3,4}
Operation = multiplication with mod 5

The code below randomly picks 10 pairs of numbers from 1, 2, 3, 4 and finds mod 5 of their product

Set = {1,3,5,7}
Operation = multiplication with mod 8

The code below randomly picks 10 pairs of numbers from 1, 3, 5, 7 and finds mod 8 of their product

# 8. MAKE A GROUP!

Time to invent your own group!

Can you make one that none else might think of?

What would it be?