Modern Algebra For Grade 5

Report
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
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.

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