# Set operations Venn diagram and application up to 3 set

**Subject** :

### Mathematics

**Topic** :

### Logarithms

**Class** :

SS 1

**Term:**

First Term

**Week:**

Week 8

**Instructional Materials:**

- Wall charts
- Online Resources
- Pictures
- Related Audio Visual
- Mathematics Textbooks

**Reference Materials**

- Scheme of Work
- Online Information
- Textbooks
- Workbooks
- Education Curriculum

**Previous Knowledge : **

The pupils have previous knowledge of

### Definition of sets , Set notations and Types of sets

**Behavioural Objectives: At** the end of the lesson, the students should be able to

- define Venn diagram and application up to 3 set
- Set operations

### Content:

*WEEK 8:*

**DATE……………………….**

# TOPIC: SETS

*Content:*

- Set operations
- Venn diagram and application up to 3 set

SUB-TOPIC 5: Operations on Sets

A Venn diagram is a graphical way of representing the relationships between different sets of data. It is made up of a series of overlapping circles, each of which represents a set of data. The degree to which the circles overlap indicates the relationship between the different sets of data.

For example, if we have a set of data that represents the numbers 1 to 10, and another set of data that represents the numbers 2 to 5, we can represent the relationship between these two sets of data using a Venn diagram. The overlap between the two sets of data would indicate that the numbers 2 to 5 are a subset of the numbers 1 to 10.

We can also use Venn diagrams to represent the relationships between more than two sets of data. For example, if we have a set of data that represents the numbers 1 to 10, and another set that represents the numbers 2 to 5, and another set that represents the numbers 6 to 8, we can represent the relationships between these three sets of data using a Venn diagram. The overlap between the three sets of data would indicate that the numbers 2 to 5 are a subset of the numbers 1 to 10, and that the numbers 6 to 8 are a subset of the numbers 1 to 10.

Venn diagrams can be used to represent relationships between any sets of data, no matter how large or small. They can be used to represent relationships between sets of numbers, sets of people, sets of objects, or any other type of data.

There are a few applications of Venn diagrams. One application is to use Venn diagrams to test whether two sets of data are equivalent. For example, if we have a set of data that represents the numbers 1 to 10, and another set of data that represents the numbers 2 to 5, we can use a Venn diagram to test whether these two sets of data are equivalent. To do this, we would draw a Venn diagram with two overlapping circles, one for each set of data. If the overlap between the two sets of data is equal to the size of one of the sets of data, then we can say that the two sets of data are equivalent.

Another application of Venn diagrams is to use them to find the union of two sets of data. The union of two sets of data is the set of all elements that are in either set. For example, if we have a set of data that represents the numbers 1 to 10, and another set of data that represents the numbers 2 to 5, the union of these two sets of data would be the set of all numbers from 1 to 10. We can find the union of two sets of data by drawing a Venn diagram with two overlapping circles, one for each set of data. The area of the overlap between the two sets of data would be the union of the two sets of data.

Finally, Venn diagrams can also be used to find the intersection of two sets of data. The intersection of two sets of data is the set of all elements that are in both sets. For example, if we have a set of data that represents the numbers 1 to 10, and another set of data that represents the numbers 2 to 5, the intersection of these two sets of data would be the set of all numbers from 2 to 5. We can find the intersection of two sets of data by drawing a Venn diagram with two overlapping circles, one for each set of data. The area of the overlap between the two sets of data would be the intersection of the two sets of data.

Venn diagrams can be a helpful tool for visualizing and understanding relationships between different sets of data. They can be used to test whether two sets of data are equivalent, to find the union of two sets of data, or to find the intersection of two sets of data. Venn diagrams can be used to visualize any type of relationship between any types of data.

1. What is a Venn diagram?

2. What are the applications of Venn diagrams?

3. How can Venn diagrams be used to test whether two sets of data are equivalent?

4. How can Venn diagrams be used to find the union of two sets of data?

5. How can Venn diagrams be used to find the intersection of two sets of data?

1. A Venn diagram is a graphical representation of the relationships between different sets of data.

2. Venn diagrams can be used to test whether two sets of data are equivalent, to find the union of two sets of data, or to find the intersection of two sets of data.

3. To test whether two sets of data are equivalent using a Venn diagram, the overlap between the two sets of data must be equal to the size of one of the sets of data.

4. The union of two sets of data is the set of all elements that are in either set. The union of two sets of data can be found by drawing a Venn diagram with two overlapping circles, one for each set of data. The area of the overlap between the two sets of data would be the union of the two sets of data.

5. The intersection of two sets of data is the set of all elements that are in both sets. The intersection of two sets of data can be found by drawing a Venn diagram with two overlapping circles, one for each set of data. The area of the overlap between the two sets of data would be the intersection of the two sets of data.

*The Union of Sets:*

The Union of Sets A and B is the Set that is formed from the elements of the two Sets A and

- This is usually denoted by “A ⋃B” meaning A Union B. Thus A ⋃B is the Set which consists of elements of A or of B or of both A and B.

Using Set notations, the Union of two Sets A and B is represented as follows

### Example 10:

Given that A = {3, 7, 8, 10}

and B = {3, 5, 6, 8, 9} then

(A⋃ B) = {3, 5, 6,7, 8, 9,10}

### Example 11:

If A = {a, b, c, d}, B = {1, 2, 3, 4} and C ={a, 3, θ} Then A ⋃B ⋃C = {a, b, c, d,1, 2, 3, 4, θ}

*Intersection of Sets*

The intersection of Sets A and B is the set of elements that are common to both A and B. This is usually denoted by “A ∩ B” meaning A intersection B. When represented using Venn diagram we have

If A ∩ B = Ø, then the Sets A and B are said to be disjoint. Disjoint Sets are Sets that have no element in common.

ξ

### Example 12:

Given that A = {5, 7, 8, 10} and B = {3, 5, 6, 8, 9}, then A ∩ B = {5, 8}.

### Example 13:

* *

If P = {a, b, c, d, e, f, g}, Q = {b, c, e, g} and R = {a, c, d, f, g} Then, P ∩ Q ∩ R = {c, g}

### Example 14:

If A = {1, 2, 3} and B = {6, 8, 10}, then

A ∩ B = { } or Ø. The Set A and B are disjoint.

**EVALUATION**

- Given thatξ={21,22,23,24,…,29,30},

P = {21, 23, 25, 26, 28},

Q = {22, 24, 26, 27, 28} and

R = {21, 25, 26, 27, 30} are Subsets of ξ.. Find:

(i) P ⋃Q (ii) P ∩Q (iii) Q ∩R

(iv) (P ∩Q) ⋃R (v) P ∩Q ∩R

- (P ⋃Q) ∩(Q ⋃R)
- (P ∩Q) ⋃(Q ∩R)

- If A = {1, 2, 3, 4} and

B = {3, 5, 6}, find:

- A ∩B (ii) A⋃ B

- (A ∩ B)⋃ B
- (A ⋃ B) ∩ A
*(WAEC)*

- If ξ = {a, e, i, o, u, m, n} find the complement of the following Sets

(a) A = {e, o, u} (b) B= {i, o, u}

(c) C = {a, u, m, n} (d) P= {u}

- Q = { } or Ø (f) ξ

- If A = {7, 8, 9, 10}, B= {8, 10, 12, 14}

and C= {7, 9, 10, 14. 15} find the following:

- A ⋃B (b) B⋃ C

(c) A ⋃B ⋃ C (d) A ∩B

(e) A ∩C (f) B ∩C

(g) (A⋃ B) ∩(A⋃ C)

(h) A ∩(B⋃ C)

- Let ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10},

A = {2, 3, 4, 5, 6,}

B = {3, 4, 6, 8, 10} and

C = {2, 4, 6, 7, 10}

Find:

(a) (A⋃ B) ∩(B⋃ C)

- A∩( B⋃ C)
- B ∩(A⋃ C)
- Show that:

(A∩B) ⋃ (A∩C) = A ∩(B⋃ C)

SUB-TOPIC 6: *The Complement of Set*

If A is a Subset of the Universal Set ξ, then, the complement of the Set A are made up of elements that are not in A, but are found in the Universal Set ξ. This is usually denoted by A^{c} or A′. for example

If ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {3, 5, 6, 9} then A^{c} or A′ = {1, 2, 4, 7, 8, 10}

Using Venn diagram, this is represented by the shaded portion below:

**Note that **to find the complement of a Set, the Universal Set must be properly defined.

### Example 15:

Given that

ξ = {11, 12, 13, 14, 15, 16, 17, 18, 19, 20} A = {11, 13, 16, 18, 20} and

B = {12, 14, 16, 18, 19, 20}.

Find the following:

(i) A′ (ii) B′ (iii) (A ⋃B)′

(iv) (A ∩B)′ (v) A′∩B′ (vi) A′⋃B′

- (A′)′

### Solution:

(i) A′ = {12, 14, 15, 17, 19}

(ii) B′ = {11, 13, 15, 17}

(iii) A ⋃B = {11, 12, 13, 14, 16, 18, 19, 20}

(A ⋃B)′= {15, 17}

(iv) A ∩B = {16, 18, 20}

(A ∩B)′ = {11, 12, 13, 14, 15, 17, 19}

(v) (A′∩B′) = {15, 17}

(vi) A′⋃ B′ = {11, 12, 13, 14, 15, 17, 19}

(vii) A′ = {12, 14, 15, 17, 19}

(A′)′ = {11, 13, 16, 18, 20} = A

### NB:

From the example above, observe that from **(iii) **and **(v) **(A ⋃B)′ = A′∩B′ Also, from **(iv) **and **(vi)**

(A ∩B)′ = A′⋃B′ and from **(vii)**(A′)′ = A

### Example 16:

Given that ξ = {a, b, c, d, e, f, g, h, i, j} A = {a, c, e, g, i} and

B = {b, c, d, f, i, j}.

Find the following:

- A′ (ii) B′

(iii) A′⋃B′ (iv) A′∩B′

(v) (A ∩B)′ (vi) (A ⋃B)′

(vii) (B′)′

### Solution:

- A′ = {b, d, f, h, j}
- B′ = {a, e, g, h}
- A′⋃B′ = {a, b, d, e, f, g, h, j}
- A′∩B′ = { h }
- A ∩B ={c, i }

(A ∩B)′ ={a, b, d, e, f, g, h, j}

- A ⋃B = {a, b, c, d, e, f, g, i, j}

(A ⋃B)′= { h}

- B′= {a, e, g, h}

(B′)′ = {b, c, d, f, i, j} = B

From the example above, we also observe thatA′⋃B′ = (A ∩B)′ **— **From **(iii) **and **(v)**

A′∩B′ = (A ⋃B)′**— **From **(iv) **and **(vi)**

And (B′)′ = B **— **From **(vii)**From the last two examples we can clearly see that (A⋃B)′ = A′∩B′,

(A ∩B)′ = A′⋃B′ and (A′)′ = A

Generally, for any two Subsets A and B of a Universal Set ξ, the following are true:

(i) (A ⋃B)′ = A′∩B′

- (A ∩B)′ = A′⋃B′
- (A′)′ = A or (B′)′ = B

These are known as *De Morgan’s Laws of Complementation.*

* *

* *

*Equality of Set*

Two Sets A and B are said to be equal if they have exactly the same elements. This means that every element of the first Set also belongs to the second Set and vice versa. E.g.

If A = {2, 3, 4, 5} and B = {2, 3, 4, 5} then the Set A = the Set B because n(A) = n (B) = 4 and all members of the Set A are also members of the Set B.

If P = {8, 2, 4} and Q = {2, 4, 8} the Set P = Q (order of arrangement does not matter, the elements in both Sets are the same).

If A = {1, 2, 3} and B = {3, 5, 6} then A ≠ B.

*Equivalent Sets*

Two Sets are said to be equivalent if the Sets have equal number of elements. E.g.

If A = {2, 3, 4, 5} and B = {a, b, c, d} then the Sets A ≡ B (A is equivalents to B) since n(A) = n(B).

## EVALUATION

- If ξ={x : 0 < x ≤ 15 xεƵ}

A = {1, 5, 7, 11} and

B = {5, 7, 14}

Find the following:

- A ∩B (b) A′ (c) B′

(d) (A ⋃B)′ (e) A′∩B (f)A ∩B′

(g) Show that: (A∩B)′ = A′⋃ B′

- Given that ξ={x : 0 ≤ x < 10, x εƵ}

P = {1, 2, 4, 5}

Q = {2, 4, 6, 8} and

R = {1, 3, 4, 8, 9}

Find:

(a) P′ (b) Q′ (c) R′ (d) Q∩ R′

(e) P′∩ Q′ (f) P′⋃Q′ (g)(Q∩ R′)′

(h) (P′∩ R′)′ (i) (P ∩ ξ) ′

(j) P′∩ Q′∩ R′

- P, Q and R are Subsets of the Universal Set µ such that µ = {3, 4, 5, 6, . . . , 16, 17, 18} P = {x : x is a number divisible by 3}

Q= {x : x is odd}

R = {x : x is a factor of 35}

Find: (i) P∩Q (ii) P⋃ Q′∩ R

(iii) Q⋃ R (iv) P′ (v) P′∩ R

## SUB-TOPIC 7: __THE USE OF VENN DIAGRAMS IN PROBLEM SOLVING__

A Mathematician by name John Venn was the man to first represent the relationship between sets with diagrams. Ever since sets may be represented by diagrams called Venn diagrams.

The rectangle is used to represent the Universal set, and Circles for other sets, as we

shall see later.

__PROBLEMS INVOLVING TWO SETS.__

** **

For two intersecting sets, the diagram is given below with the labels of what each compartment represents.

## A

** **

** **

** **

** **

** **

** **

__Compartment I:____represent the set of__ elements in A only. i.e. A Ç B/ using set

__ notations.__

**Compartment II: **represents the set of elements common to both A and B i.e. AÇB

**Compartment III: **represents the set of elements in B only i.e. A** ^{/}**Ç B

**Compartment IV: **represents the set of elements that are neither in A nor B i.e. (AÈB) ** ^{/}**or A

**Ç B**

^{/}

^{/}### Example 15:

*In a Survey of 40 Students in a class, 19 have visited Lagos and 17 have visited Benin City.*

*If 13 have visited neither. How many Students have visited:*

*(i) Both Cities; (ii) Benin City but not Lagos (i.e. Benin City only)*

### Solution:

n(x) = 40

n(L) = 19

n(B) = 17 n(L È B)** ^{/} **= 13

Let x represents those that have visited both Cities i.e. n(LÇB) = x

L B

19 – x x 17 – x

13

19 – x + x +17 – x + 13 = 40

49 – x = 40

49 – 40 = x

9 = x

\9 Students have visited both Cities

- Those that have visited Benin City only are = 17 – x

= 17 – 9

= 8

\8 have visited Benin City only.

### Example 16:

*In a Class of 45 Students, if 21 offer Agricultural Science, 25 offer Biology and 6 offer both subjects. Find*

*those that offer**the number that offers Biology but not Agricultural Science (i.e. Biology only)*

### Solution:

(i) n(x) = 45 n(A) = 21 n(B) = 25

n(A ÇB) = 6

Let n(AÈB) ** ^{/} **= x

i.e. Let those that offer neither be x.

21 – 6 + 6 +25 – 6 + x = 45

15 + 6 + 19 + x = 45

40 + x = 45

\ x = 5

x = 45 – 40

\Those that offer neither,

i.e. n(AÈB) ** ^{/}**= 5

## EVALUATION

- In a gathering of 30 people, x speak Hausa and 15 speak Yoruba. If 5 people speak both languages, find how many people that speak (i)

(ii)Yoruba only (iii)Hausa only.

- In a Birthday party attended by 22

people, 10 ate fried rice and 13 ate salad. If x ate both fried rice and salad and (2x–5) ate none of the two. How many ate

- both fried rice and salad?
- salad but not fried rice?

(ii) neither fried rice nor salad?

- The Venn diagram below represents a universal set xof integers and its subsets Pand Q
**.**List the elements of the following sets;

- P È Q
- P Ç Q x

### SSCE, NOV. 1995 Nọ 1. (WAEC)

* *

* *

SUB-TOPIC 8: __PROBLEMS INVOLVING THREE SETS.__

The Venn diagram is made up of eight compartments as shown below: x

**C **VIII

**CompartmentI **represents AÇBÇC (elements commontothethreesetsA,BandC). **Compartment II **represents AÇBÇC** ^{/} **(elements common to both A and B only).

**CompartmentIII **represents AÇB** ^{/}**ÇC

(elements common to both A and C only).

**Compartment IV **represents A**/**ÇBÇC

(elements common to both B and C only).

**CompartmentV **represents AÇB** ^{/}**ÇC

**(elements of A only).**

^{/}**CompartmentVI** represents A** ^{/}**ÇBÇC

**(elements of B only).**

^{/}**Compartment VII **represents A** ^{/}**ÇB

**ÇC (elements of C only).**

^{/}**Compartment VIII **represents (AÈBÈC)** ^{/} **or A

**ÇB**

^{/}**ÇC**

^{/}**elements that are not in any of the three sets but are in the Universal set.**

^{/}

### Example 18:

* *

*The are 80 people in a sports camp. Each play at least one of the following games, volleyball, football and handball. 15 play volleyball only, 18 play football only, and 21 play handball only .If 5 play volleyball and foot ball only, 8 play volleyball and handball only, and 10 play football and Handball only.*

*Represent the above information in a Venn diagram**How many people play the three games?**How many people play football ?*

* *

### Solution:

*List of information given in the question is as follows*

* *

__ Let V be Volleyball __

__ F be Football__

__ H be Handball__

n(x) = 80

n(VÇF**/**ÇH**/**) i.e. Volleyball only = 15 n(V**/**ÇFÇH**/**) i.e. Football only = 18 n(V**/**ÇF**/**ÇH) i.e. Handball only = 21 n(VÇFÇH**/**) i.e. Volleyball and Football only = 5 n(VÇF**/**ÇH) i.e. Volleyball and Handball only = 8 n(V**/**ÇFÇH) i.e. Football and Handball only =10

__Let n(V____Ç____F____Ç____H) = x . i.e. Those that play the three games = x__

(a)

x

__(b) 15 + 5 + 18 + 8 + x + 10 + 21 = 80__

__ 77 + x = 80__

__ x = 80 – 77__

__ ____\____ x = 3__

__ ____\__ __3 ____people play the three games.__

__(C) ____The number that plays football__

__ n(F) = 18 + 5 + 10 + x__

__ = 33 + 3__

__ = 36__

__N.B__

__Since the word ONLY was used all through, the values are written directly in to each__

__compartment without any manipulation as shown in the figure above.____ Suppose the question 18 above is framed as shown in the example 19 below, then the__ __Approach would be different.__

__Example 19:__

__ There are 80 people in a sports camp and each plays at least one of the following games:__

__volleyball, football and handball. 31 play volleyball, 36 play football and 42 play handball. If 8__

__play volleyball and football, 11 play volleyball and handball and 13 play football and handball.__

__(a) Draw a Venn diagram to illustrate this information, Using x to represent the number that__

__play the three games.____(b) How many of them play:__

__ (i) All the three games,__

__ (ii) Exactly two of the three games,__

__ (iii) Exactly one of the three games__

__(iv) handball only?__* *

__Solution:__

** **

__Step__ __1:__ __listoutallinformation__ __giveninthequestion__.

__Let V be Volley ball__

__ F be Football__

__ H be Hand ball__

__(a) n(__

__x__

__) = 80__

__n(V) = 31__

__ n(F) = 36__

__ n(H) = 42__

__ n(V ____È__ __F ____È__ __H) = 80 (Since each play at least one of the games).__

__ n(V ____Ç____ F) = 8__

__ n(V ____Ç____ H) = 11__

__ n(F ____Ç____ H) = 13. __

__Let n(V ____Ç____ F ____Ç____ H) = x.__

__ To fill the Venn diagram we start with the centre Compartment__

__ where n(V ____Ç__ __F ____Ç__ __H) = x.__

__ How we obtained the value for each of the other compartments is shown below.__

__For____ Volleyball and football only__

i.e. n(V Ç F Ç H**/**)

__(Since x is already in the circle of V ____∩ ____F)__

__ = n(V ____Ç____ F) – x__

__ = 8 – x__

__For__ __Volleyball__ __and__ __Handball__ __only__

i.e. n(V Ç F**/**Ç H)

__(Since x is already in the circle of V ____∩ ____H)__

__= n(V ____Ç____ H) – x__

__ = 11 – x__

__For Football and Handball only__

__i.e. n(V____/____Ç__ __F ____Ç__ __H)__

__(Since x is already in the circle of F ____∩ ____H)__

__= n(F ____Ç____ H) – x __

__ = 13 – x__

__For Handball only__

i.e. n(V**/**Ç F**/**Ç H)

__n(H) – (All values already written in the circle of Handball)__

__ = 42 – [ (11 – x ) + x + (13 – x ) ]__

__ = 42 – [24 – x ]__

__ = 42 – 24 + x__

__ = 18 + x x__

__ 11– x 13 – x__

__ H__

## For Football only i.e. n(V/Ç F Ç H/)

__ n(F) – ( All values already written in the circle of Football)__

__ = 36 – [ (8 – x) + x + (13 – x) ]__

__ = 36 – [21 – x ]__

__ = 36 – 21 + x __

__ = 15 + x __

## For Volleyball only i.e.n (V Ç F/Ç H/)

__n(V)____ – (All values already written in the circle of Volley ball).__

__ = 31 – [ (8 – x ) + x + ( 11 – x ) ] __

= 31 – [19 – x] **V**

** **

** **

** **

** **

** **

** **

** **

__(b) To get the value of x, which__

__represent those that play all three games, we____ add all the Compartments of the Venn__

__diagram together and equate it to the total__

__value in the Universal set and solve for x.__

__i.e. 12 + x + 8 – x + 15 + x + x + 11 – x + 13 – x + 18 + x = 80__

__ 77 + x = 80__

__ x = 80 – 77__

__ x = 3__

__\____ 3 people play all three games __

__NOTE THAT__

*If this value, x =3, is substituted into the Venn diagram, the answer obtained in the previous example would be got.*

* *

__b (ii) ____Exactly two of the three games__

= n(V Ç F Ç H**/**) + n(V Ç F**/**Ç H) + n(V**/**Ç F Ç H)

__ = 8 – x + 11 – x + 13 – x __

__ = 32 – 3x__

__ = 32 – 3(3)__

__ = 32 – 9__

__ = 23__

__\__ __23 of them play exactly two of the three games.__

__b (iii) Exactly one of the three games__

= n(V Ç F**/**Ç H**/**) + n(V**/**Ç F Ç H**/**) + n(V**/**Ç F**/**Ç H)

__ = 12 + x + 15 + x + 18 + x__

__ = 45 + 3x__

__ = 45 + 3(3)__

__ = 45 + 9__

__ = 54__

__\__ __54 of them play exactly one of the three games.__

__b (iv) F or Handball only__

n(V**/**ÇF**/**Ç H) = 18 + x

__ = 18 + 3__

__ = 21__

__\____21 of them play Handball only.__

__EVALUATION__

__(1) In a Class of 80 undergraduate Students, 21 took elective Courses from Botany only, 16__ __took from Zoology only, 13 took from Chemistry only. If each of the Students took elective from__ __at least one of the above-mentioned Courses, 7 took Botany and Zoology only, 3 took Zoology__ __and Chemistry only and 8 took Botany and Chemistry only.__

__Draw a Venn diagram to illustrate the information above____using x to represent those that took____the____Find the:__

__ (i) Value of x __

__ (ii) Number that took Botany__

__ (iii) Number that took Zoology and Chemistry.__

__In a Group of 120 Students, 72 of them play Football, 65 play Table Tennis and 53____Play Hockey. If 35 of the Students play both Football and Table Tennis, 30 play both Football____and Hockey, 21 play both Table Tennis and Hockey and each of the Students play at least one of____the three games,__

__(a) Draw a Venn diagram to illustrate this ____information.__ __(b) How many of them play:__

__All the three games;____Exactly two of the three games;____(iii) Exactly one of the three games__

__ (iv) Football alone?__

### __SSCE,____ NOV. 1996, № 6 (WAEC).__

* *

__(3) The set A = {1,3,5,7,9,11}, B = {2,3,5,7,11,15} and C={3,6,9,12,15} are subsets of__

__ε____ = {1,2,3,…,14,15} (a) draw a Venn diagram to illustrate the given information. (b) use your__ __Venn diagram to find__

- C Ç A
**/** - A
**/**Ç (B È C)

### __WASSCE, June 2002, No. 2__

* *

__WEEKEND ASSIGNMENT__

* *

__New General Mathematics for SSS, Book 1 Pages 101 – 104 Exercise 8d__

*Question no. 7,9,11,12,13, 14 and 15*

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__WEEKEND READING__

__New General Mathematics for SSS, Book 1 Pages 97 – 104.____4. Man Mathematics for SSS, Book 1, Pages 39- 60__

## REFERENCES

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**F Macraeetal (2011),New General Mathematics for Senior Secondary Schools 1.**

## 6. MAN Mathematics for senior Secondary Schools 1.

**New school mathematics for senior secondary school et al; Africana publishers limited**

## 8. Fundamental General Mathematics For Senior Secondary School by Idode G. O

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**WEEK 11 Revision.**

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WEEK 12 Examinatio

**Presentation**

The topic is presented step by step

Step 1:

The subject teacher revises the previous topics

Step 2.

He or she introduces the new topic.

Step 3:

The subject teacher allows the pupils to give their own examples and he corrects them when the need arises.

**Conclusion:**

The subject teacher wraps up or concludes the lesson by giving out a short note to summarize the topic that he or she has just taught.

The subject teacher also goes round to make sure that the notes are well copied or well written by the pupils.

He or she makes the necessary corrections when and where the needs arise.