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FACULTY OF EDUCATION AND LANGUAGES
SEMESTER MAY / 2011
HBMT 4203
TEACHING MATHEMATICS IN FORM FOUR
MATRICULATION NO : 770218015450002
IDENTITY CARD NO. : 770218-01-5450
TELEPHONE NO. : 013-7018071
E-MAIL : znas77@yahoo.com.my
LEARNING CENTRE : JOHOR BAHRU
INTRODUCTIONS
SETS
What is sets in mathematics? A set is a collection of distinct objects, considered as an
object in its own right. Sets are one of the most fundamental concepts in mathematics.
Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics,
and can be used as a foundation from which nearly all of mathematics can be derived. In
mathematics education, elementary topics such as Venn diagrams are taught at a young age,
while more advanced concepts are taught as part of a university degree.
SETS THEORY
Set theory is the branch of mathematics that studies sets, which are collections of
objects. Although any type of object can be collected into a set, set theory is applied most
often to objects that are relevant to mathematics. The language of set theory can be used in
the definitions of nearly all mathematical objects.
The modern study of set theory was initiated by Georg Cantor and Richard Dedekind
in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems
were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with
the axiom of choice, are the best-known.
Concepts of set theory are integrated throughout the mathematics curriculum in the
United States. Elementary facts about sets and set membership are often taught in primary
school, along with Venn diagrams, Euler diagrams, and elementary operations such as set
union and intersection. Slightly more advanced concepts such as cardinality are a standard
part of the undergraduate mathematics curriculum.
Set theory is commonly employed as a foundational system for mathematics,
particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its
foundational role, set theory is a branch of mathematics in its own right, with an active
research community. Contemporary research into set theory includes a diverse collection of
topics, ranging from the structure of the real number line to the study of the consistency of
large cardinals.
SETS HISTORY
Mathematical topics typically emerge and evolve through interactions among many
researchers. Set theory, however, was founded by a single paper in 1874 by Georg Cantor:
"On a Characteristic Property of All Real Algebraic Numbers".[1][2]
Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the
West and early Indian mathematicians in the East, mathematicians had struggled with the
concept of infinity. Especially notable is the work of Bernard Bolzano in the first half of the
19th century. The modern understanding of infinity began in 1867-71, with Cantor's work on
number theory. An 1872 meeting between Cantor and Richard Dedekind influenced Cantor's
thinking and culminated in Cantor's 1874 paper.
Cantor's work initially polarized the mathematicians of his day. While Karl
Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of
mathematical constructivism, did not. Cantorian set theory eventually became widespread,
due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, his
proof that there are more real numbers than integers, and the "infinity of infinities" ("Cantor's
paradise") the power set operation gives rise to.
The next wave of excitement in set theory came around 1900, when it was discovered
that Cantorian set theory gave rise to several contradictions, called antinomies or paradoxes.
Bertrand Russell and Ernst Zermelo independently found the simplest and best known
paradox, now called Russell's paradox and involving "the set of all sets that are not members
of themselves." This leads to a contradiction, since it must be a member of itself and not a
member of itself. In 1899 Cantor had himself posed the question: "what is the cardinal
number of the set of all sets?" and obtained a related paradox.
The momentum of set theory was such that debate on the paradoxes did not lead to its
abandonment. The work of Zermelo in 1908 and Abraham Fraenkel in 1922 resulted in the
canonical axiomatic set theory ZFC, which is thought to be free of paradoxes. The work of
analysts such as Henri Lebesgue demonstrated the great mathematical utility of set theory.
Axiomatic set theory has become woven into the very fabric of mathematics as we know it
today.
SETS DEFINITION
Georg Cantor, the founder of set theory, gave the following definition of a set at the
beginning of his Beiträge zur Begründung der transfiniten Mengenlehre.
A set is a gathering together into a whole of definite, distinct objects of our perception and of
our thought - which are called elements of the set.
The study of algebra and mathematics begins with understanding sets. A set is
something that contains objects. To be contained in a set, an object may be anything that you
want to consider. An object in a set may even be another set that contains its own objects. In
everyday language, a set can also be called a “collection” or “container”, but in mathematics,
the term set is preferred. The objects that a set contains are called its members.
The elements or members of a set can be anything: numbers, people, letters of the
alphabet, other sets, and so on. Sets are conventionally denoted with capital letters. Sets A
and B are equal if and only if they have precisely the same elements.
As discussed below, the definition given above turned out to be inadequate for formal
mathematics; instead, the notion of a "set" is taken as an undefined primitive in axiomatic set
theory, and its properties are defined by the Zermelo–Fraenkel axioms. The most basic
properties are that a set "has" elements, and that two sets are equal (one and the same) if and
only if they have the same elements.
CONCEPTS
A set A consists of distinct elements :
If such elements are characterized via a property E, this is symbolized as follows:
satisfies property E}.
The following notations are commonly used:
notation meaning
is element / member of
is not element / member of
is a subset of
is a strict subset of
number of elements in
empty set
If ( ), is called a finite (infinite) set.
Two sets are called equipotent, if there exists a bijective map between their elements (
for finite sets and ).
The set of all subsets of is called power set, i.e. . . In
particular, we have and . Moreover, .
The following sets are standardly denoted by the respective symbols:
natural numbers:
integers:
rational numbers:
real numbers:
complex numbers:
The following notations are also commonly used and as
as well as , , , , , , respectively.
OPERATIONS ON SETS
The following operations can be applied to sets and :
Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of
A, or B, or both. The union of {1, 2, 3} and {2, 3, 4} is the set {1, 2, 3, 4}.
Intersection of the sets A and B, denoted A ∩ B, is the set of all objects that are
members of both A and B. The intersection of {1, 2, 3} and {2, 3, 4} is the set {2, 3} .
Set difference, complement of U and A, denoted U \ A is the set of all members of U
that are not members of A. The set difference {1,2,3} \ {2,3,4} is {1} , while,
conversely, the set difference {2,3,4} \ {1,2,3} is {4} . When A is a subset of U, the
set difference U \ A is also called the complement of A in U. In this case, if the choice
of U is clear from the context, the notation Ac is sometimes used instead of U \ A,
particularly if U is a universal set as in the study of Venn diagrams.
Symmetric difference of sets A and B is the set of all objects that are a member of
exactly one of A and B (elements which are in one of the sets, but not in both). For
instance, for the sets {1,2,3} and {2,3,4} , the symmetric difference set is {1,4} . It is
the set difference of the union and the intersection, (A ∪ B) \ (A ∩ B).
Cartesian product of A and B, denoted A × B, is the set whose members are all
possible ordered pairs (a,b) where a is a member of A and b is a member of B.
Power set of a set A is the set whose members are all possible subsets of A. For
example, the power set of {1, 2} is { {}, {1}, {2}, {1,2} } .
Some basic sets of central importance are the empty set (the unique set containing no
elements), the set of natural numbers, and the set of real numbers.
The so- called Venn diagrams illustrate the set operations.
union:
Intersection: difference: symmetric difference:
If , some of the above diagrams are identical to one another:
Union: intersection: complement set:
LESSON PLAN:
Day : Thursday
Date : 30 June 2011
Class : 4 Bukhari
Subject : Mathematics :
Time : 10.00 am – 11.20 am
Duration : 80 minutes
Learning Area : 3) Sets
Learning objectives : Students will be taught to :
3.1 Understand the concept of sets
Learning outcomes : Students will be able to :
(i) represents sets by using Venn diagrams
Teaching aids : Manila cards ((Closed Geometrical shaped), activity sheets,
mahjong papers, quiz papers, LCD, worksheets.
Attitudes and Values : Patient, self-confident, concentrate, cooperative, follow instructions,
honesty, careful
Thinking Skills : Categorize, recognize the main idea, making sequence to represent
sets by using Venn diagram, locate and collect relevant information,
analyze part / whole relationships, reflection.
Previous Knowledge : i) sort given objects into groups
ii) define sets by description and using set notation
iii)identify whether a given object is an element of a set
STEPS CONTENTSTEACHING AND LEARNING
ACTIVITIESNOTES
Step 1 Parts of “Definition of Set” Revision
Example:
A = {Factors of 30)
A = {1,2,3,5,6,10,15,30}
Teacher shows an example Categorize set A
on whiteboard.
Students pay attention on the example
written on the whiteboard.
Teacher wants the students to look at the Set
A and try to list out all the elements by using
set notation.
Students try to list out all the elements of Set
a by using set notation.
Teacher calls a student to give an answer on
the whiteboard.
A student tries to give an answer.
Teacher checks the answer.
Teaching Aids:
Manila Cards
Values:
Self confident,
honest, patient.
Thinking Skills:
Categorize
STEPS CONTENTS TEACHING AND LEARNING
ACTIVITIES
NOTES
Step 2 Venn Diagram
Besides the methods of description
and set notation, sets can be
representing by using Venn
Diagrams.
Closed Geometrical Shapes
Circle Oval Rectangle
Square Triangle Hexagon
Teacher introduces Venn diagram to the
students and explains that it is easier to see
which group each element belongs to in a
Venn diagram.
Students pay attention on explanation and
the given examples of Closed Geometrical
Shapes
Teacher stress that rectangle are usually used
to represent the set which contain all the
elements that are discussed and the circles or
enclosed curves to represent each set within
it.
Teacher gives examples of Closed
Geometrical Shaped.
Method:
Explanation
Teaching Aids:
Manila cards
(Closed
Geometrical
Shapes)
Values:
Concentrate
Thinking Skills:
Recognize the
main idea,
making
sequence to
represent sets by
using Venn
diagram
STEPS CONTENTS TEACHING AND LEARNING
ACTIVITIES
NOTES
Step 2
(continued)
Examples:
(1)A={1, 2, 3, 5, 6, 10, 15, 30} Each ‘dot’A •1 •2 represents •3 •5 •6 one element •10 •15 •30
Then, teacher uses the example from
induction set and teaches the students the
steps to draw a Venn diagram to represent
set A as listed below:
1. Draw a circle.
2. Represent set A by
labeling the circle as A.
Vocabulary:
Set
Element
Description
Label
Set notation
Denote
(2)B={a, b, c}
B •a •b •c
(3)Q={Multiples of 3 between 8 and 18}Q={9, 12, 15}
Q •9 •12 •15
3. Determine the number of
elements in set A, and
represent each of them
with a dot inside the circle.
Students concentrate on showing set in Venn
diagram.
Teacher reminds students to put a “dot” to
present one element and label the set.
Students remember the important point.
Teacher shows another two examples - (2)
and (3).
Teacher asks a student to put elements into
diagram.
Venn Diagram
Empty set
Equal sets
Subset
Universal set
Complement of
a set
Intersection
Common
elements
STEPS CONTENTS TEACHING AND LEARNING
ACTIVITIES
NOTES
Student tries to put elements into diagrams.
For example no. (3), teacher asks a student
to list out elements first then put the
elements into Venn diagram.
Another student has to do example no. (3).
Teacher gives time to copy notes.
Students copy the notes.
Step 3 Teaching Progression
Teacher conduct the group activity to
further enhance student’s
After copy the given notes, teacher enhance
students understanding with conduct group
activity calls “fast and correct”.
Values:
Cooperative,
Follow
Instructions
understanding about the lesson learnt
today.
Group Activity
1. Given that W is a set
representing days of a week,
draw a Venn diagram
representing the elements of
W.
Teacher explains the rules of the group
activity:
“Group that can answer all from 4 questions
fast and correct, will be the winner”.
Students follow the rules in the activity.
Teaching Aids:
Activity sheets,
Mahjong Paper
STEPS CONTENTS TEACHING AND LEARNING
ACTIVITIES
NOTES
2.
If,
s= { x : x is an odd numbered
20<x<3 o } , Can you
draw a Venn diagram to
represent the elements in
this set?
3. Draw a Venn diagram to
represent the set given
below.
P= {x : x is a primenumber ? } i) 1≤x≤10
ii) 41≤x≤50
4. Given that B is the set of
common factors of 24 and
36.
Draw a Venn diagram of set
B.
Teacher observes students to do the activity.
Students ask questions if not understand.
Teacher wants one representative from each
group to come up randomly to check the
solution from other group on the mahjong
paper.
Example: every first member of the group
will check number 1, second member from
each group to do number 2 and so on.
Each group writes down their solution on
the mahjong paper, depends on the problem
solving.
Finally, teacher discusses the solution with
the students.
Students do the corrections if they make any
mistakes.
Thinking Skills:
Locate and
collect relevant
information.
STEPS CONTENTS TEACHING AND LEARNING
ACTIVITIES
NOTES
Step 4 Quiz
(1) Given that P = {1, 2, 3, 4, 5}.
Represent set P by using Venn
diagram.
Solution:
P •1
•2 •3 •4
•5
(2) If R = {x : 30 ≤ x ≤ 40, x is a
multiple of 3}, can you draw a
Venn diagram to represent the
elements in this set?
Solution:
R = {30, 33, 36, 39}
R •30
•33 •39
•36
Teacher distributes each student a quiz
paper (with two questions).
Students get a piece of quiz paper.
Teacher asks students try to draw a diagram
without asking friends or teacher.
Students try to draw a Venn diagram by
themselves.
(Teacher do not forget gives guide line to
the students)
Teacher observes students to do quiz.
Teacher collects papers after three minutes.
Students pass up quiz papers.
Teacher discusses the answers for the quiz
with the students and guides them.
Students respond to the teacher and listen to
the answer.
Teaching Aids:
Quiz Paper,
LCD
Values:
Honest, Careful
Thinking Skills:
Analyze
relationships.
STEPS CONTENTS TEACHING AND LEARNING
ACTIVITIES
NOTES
Step 5 Summary and exercises Teacher asks students to make a summary of Teaching Aids:
Conclusion
on Venn diagram. the day’s lesson.
Students make summary of the day’s lesson.
Teacher reminds students what they have
learnt how to draw a Venn diagram to
represent the elements of a set.
Teacher will stress that to put a “dot” to
present one element.
Students remember the important points.
Teacher distributes the activity sheets.
Students do the worksheets and exercises
given.
Worksheets
Values:
Self Confident
Thinking Skills:
Reflections
TEACHING AIDS
WORKSHEETS 1
NAME:_____________________________________ DATE: ______________
FORM: 4 _________________
Answer all questions.
1. Given that Z = {multiple of 3}, determine if the following are elements pf set Z. Fill in the following boxes
using the symbol or .
a) 52 Z b) 18 Z c) 69 Z
2. Y is a set of the months that start with the letter “M”. Define set Y using set notation.
3. Determine whether 8 is an element of each of the following set.
a) {2,4,6,8}
b) {Multiples of 4}
c) {LCM of 2 and 4)
d) {HCF of 4 and 8}
4. State whether each of the following sets is true or false.
a) 4 {Common factors of 8 and 12}
b) 1 {Prime number}
5. State the number of elements of each of the following.
a) X = {Cambodia, Singapore, Malaysia, Indonesia, Thailand}
b) Y = {3,6,9, …21)
c) Z = {Integers between -3 and 4, both are inclusive}
WORKSHEETS 2
NAME:_____________________________________ DATE: ______________
FORM: 4 _________________
Answer all questions.
1. Draw a Venn diagram to represent each of the following:
a) F = {1,3,5,7}
b) M = {Pictures of durian, mangosteen, mango, starfruit}
c) N = {The first 5 prime numbers}
2. Use the notation {} to represent set A, B and C.
a) A
b) B
c) C
3. State n(A) when
a) A = {The letters in the word ‘SCIENCE’}
1 4 9
h j k l m n p r
100 250 150 200
b) A = {x : 5 x < 30 where x is not an odd number}
c) A is a set of perfect square numbers between 0 to 90.
CONCLUSION
This assignment shows us the introduction that explain on what is sets, the sets theory
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