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