Modern Algebra For Grade 5

What are some words that come to mind when you hear the word ALGEBRA?

• • • • • From the grade 5 class: Math Numbers Operations Algae Measuring

Today, we are going to learn about sets. Does anyone know what a set is?

• • • • From the grade 5s: A question Math set Set of apples Set of shoes

What is a set?

• Mathematicians have agreed not to define a set. They just hope that when people hear “the set of all Canadians” or “the set of all grade five students”, everyone know what that means and who/what it includes. • • A set is a subset of a set if everything in the subset is in the set.

Ex) Set is students at Chedabucto Place, and a subset is grade 5 students

Things to assume about sets:

• • • • A set S is made up of elements, and if a is one of these elements, we shall denote this by a S. There is exactly one set with no elements. It is the empty set and is denoted by .

We may describe a set either by giving a property that characterizes the elements, or by listing all the elements. A set is well defined. If S is a set and a is some object, then either a is in S, or a is not in S (definite).

Subsets

• • • • • A set B is a subset of a set A if every element of B is in A.

For a set A, A and the empty set are subsets of A.

A is the improper subset of A.

All other subsets are proper subsets.

A subset is denoted by: B ≤ A or A ≥ B

Activity

• • • Our set today is the whole grade 5 class. Lets arrange into subsets based on hair colour! So, the whole grade 5 class is the set. Smaller subsets can be formed from parts of the whole set.

Next, lets form into subsets based on your favourite colour out of:

• • • •

Red Blue Pink

Purple

• • • •

Green

Yellow Black Orange

Empty Sets

• • • A subset is still considered a subset even if there is nothing in it. This is called the empty set From our activity, if nobody picked orange as their favourite colour, orange would be an empty subset.

Cycles

• • A cycle is a way to describe the order of elements in an orbit.

Equivalence classes are called orbits. An orbit is just a subset of the original set.

Activity

• • • • • • We are going to learn how cycles work by doing a hands on activity. Everyone will stand in a circle.

I will divide the group into cycles. Each cycle will have an order they must follow. Each cycle will have it’s own ball which is to be passed among the cycle. Students pass the ball only to the person who comes after them.

Cycles

• • • Notice how each ball is passed within the same people the whole time. The ball never goes to anybody else in the circle. For example, someone who started with the blue ball was never thrown the yellow ball.

Some Things Students Learned:

“If you separate all the kids in grade five into groups with the same hair colour, each set is a subset. Ex) brown is a subset, blonde is a subset, black is a subset” “If the set was the NHL team Pittsburgh, the subset might be all the forwards.” “A set can be Chedabucto Place, a subset would be grade 5 class.”

• “Sets can have subsets with nothing in it. Cycles keep repeating.” • “We did a cycle and it was when we got in a circle and we threw the balls. The whole point was that a cycle continues and never changes.”

• “A cycle can be from one thing back to that same thing.” • “ex: A set is a class. A subset is a group of people that play hockey in the class.”

Exam Question Exam Question What are two things to assume about sets?

• • • • A set S is made up of elements, and if a is one of these elements, we shall denote this by a S.

There is exactly one set with no elements. It is the empty set and is denoted by . .

We may describe a set either by giving a property that characterizes the elements, or by listing all the elements. A set is well defined. If S is a set and a is some object, then either a is in S, or a is not in S (definite)

References

• • A First Course In Abstract Algebra by John. B. Fraleigh, Seventh Edition. Grade 5 Class at Chedabucto Place.