Math Lesson Notes

  
P.6 TOPICAL BREAK DOWN
TERM ONE TOPICS
1. Set concepts
2. Whole numbers
3. Operation on whole numbers
4. Number patterns and sequence
5. Fractions 
TERM II & III TOPICS
1. Data handling (Graphs and interpretation of information)
2. Money
3. Distance, Speed and Time
Length, Mass and Capacity
4. Lines, angles and geometrical figures
5. Integers
6. Algebra  
TOPICAL ANALYSIS
1. SET CONCEPTS
Types of sets
Equal sets /un equal sets
Equivalent sets/Non Equivalent sets
Intersection of sets / union sets 
Universal sets, Difference of sets, complements
Describing parts of a venn diagram
Application (interpreting set statements, probability)
Sub sets
Listing and forming subsets
Finding number of subsets and proper subsets
Application of proper subsets
Representing information on the venn diagram



2. Whole numbers (Numeration systems and place values)
Place values and values of numbers
Expanded forms
Writing numbers in figures/in words
Rounding off whole numbers
Roman numerals and Hindu-Arabic numerals
Bases (Grouping in Base five) – place values & values
Naming non decimals to decimals and vice versa

3. OPERATIONS ON WHOLE NUMBERS
Addition/ its application
Subtraction and its application
Multiplication and its application
Division and its application
Mixed operation on whole numbers (BODMAS)

4. NUMBER PATTERNS AND SEQUENCES
Divisibility tests of 2,3 and 5
Number patterns and sequences (developing number pattern) Number systems
Whole numbers, counting numbers, Even, Odd, Triangular, Square, Composite, Cube numbers
Consecutive numbers
Prime factors and its application
Application of L.C.M and G.C.F (bells, number with or without a remainder)
Comparison (product) L.C.M and G.C.F)
Factorizing using powers of whole numbers and its applications
Squares and square roots of numbers
Cubes and cube roots of numbers
Magic squares (Revision)

5. FRACTIONS
Definition (meaning of fractions)
Types of fractions (proper/common/vulgar fractions, Improper, Mixed number/fraction, Decimals)
Operations and fractions (BODMAS)
Application fractions
Converting from common fractions to decimals and vice versa
- Place values and values of decimals
- Expansion of decimals
Writing decimals in words and figures
Rounding off decimals
Operations on decimals (Reciprocals)

6. Ratios and Proportions
a) Ratios
- Forming Ratios (Expressing quantities as Ratios)
- Expressing fractions as Ratios and vice versa
- Comparing ratios (Comparison of ratios)
- Increasing and decreasing quantities using the given Ratios
- Finding the Ratio of increase or decrease
- Sharing quantities using ratios
- Finding shared quantities
- Application of rations
b) Proportions
- Direct proportions
- Inverse (Indirect) proportion)
- Constant proportion
c) Percentages
- Definition
- Percentages as fractions and vice-versa 
- Percentages as decimals and vice-versa
- Expressing ratios as percentages and vice-versa
- Percentage increase and decrease
- Percentage profit and loss
- Application of percentages
- Simple interest/amount




TERM II AND III
1. DATA HANDLING
Collection and organization of data
Line graphs
Finding mean; median, mode and Range
Grouped data
Application of mean
Pie charts (circle graphs)
Drawing/constructing pie charts
More about pie charts
Co-ordinate graphs (plotting, drawing, forming shapes and naming shapes)
Bar graphs
Simple statistics
Probability

2. MONEY
Identifying different currencies
Finding number of notes
Local and foreign currencies (Exchange rates)

3. DISTANCE, SPEED & TIME
Conversion of time
Finding duration
Finding Speed, Distance and Time
Conversion of speed
Average speed
Travel graph
Drawing/plotting travel graphs 

4. LENGTH, MASS AND CAPACITY
Conversion of metric units
Perimeter of polygons
Difference of areas
Comparing areas of shapes
Area of combined shapes
Circumference of a circle and its sectors
Area of a circle and its sectors
Total surface area, prisms (cube, cuboid)
Volume of cuboids, cubes and cylinders
Capacity
Conversion of capacity
Finding capacities of different containers (tanks)

5. GEOMETRY
Complementary and supplementary angles
Construction of angles
Bisecting of angles and lines
Constructing of parallel lines
Angles formed on parallel lines
Construction of regular polygons
Pythagoras’s Theorem and its application
Properties of prisms and pyramids
Nets of cubes, cuboids and other prisms
Net of pyramids 
6. INTEGERS
Plotting integers on Number lines
Addition, subtraction and multiplication of integers on number line
Application of integers
Solution sets
Inequalities


7. ALGEBRA
Algebric expressions and phrases (statements)
Simplification in algebra
Solving simple equations
Forming and solving simple equations
Substitution
Word question statements involving algebra
e) 2    3 four f)    6   5 seven
       - 2    2 four             - 4   6 seven

g)  464 eight - 237eight
h) 463 nine - 155 nine
i)  354 six - 245 six  
LESSON TWO
CHANGING DECIMAL TO NON DECIMAL BASE
Changing from decimal to non-decimal bases
1. Change 25 to base seven.

Base  No.  Rem
7       25      4
          3
Therefore: 25 = 34seven

2. Change 38 to base eight.

Base  No.  Rem
   8     38     6
           4   

Therefore: 38 = 46eight

LESSON THREE
MULTIPLICATION OF BASES
Example: 
1. Multiply: 232five x 3

2 3 2five
x    3
    1201five
2. Multiply: 214four x 3

2 1 4four
x    3
         1320four
Activity
Multiply the following bases
1. 214five by 3
2. 432four x 4
3. 320five x 4
4. 354six x 5
5. 122three x 3
6. 464eight x 5
ACTIVITY
1. Change the following to base three.
a) 19ten                    b)  31ten                c) 26ten
2. Convert the following to base four.
a) 34ten                   b) 42ten               c) 75ten
3. Change the following to base six.
a) 31ten                  b)  46ten               c) 94ten
4. Convert the following to base seven.
a) 96ten           b) 68ten               c) 536ten
5. Change the following to base eight.
a) 73ten b) 26ten c) 431ten




LESSON FOUR
CHANGING FROM A NON DECIMAL BASE TEN.
1. Change 234six to base ten.
2 3 4six
(2x62) + (3x61) + (4x60)
2x6x6 + 3x6 + 4x1
72      +  18   + 4
 =     94ten

2. Change 41five to base ten
4  5 five
4  1 five
41 five = (4x51) + (1x50)
= (4x5) + (1x1)
= 20 + 1
= 21ten
Change the following to base ten.
a) 23four f) 214six
b) 131four g) 63seven
c) 55eight h) 62seven
d) 413five i) 1011two
e) 122three    j)  144five




LESSON FIVE
CHANGING FROM NON DECIMAL TO A NON DECIMAL BASE
Examples
1. Change 413five to base two

4 1 3 five
(4x52) + (1x51) + (3x50)
4x5x5 + (1x5) + (3x1)
(4x25) + 5 + 3
100 + 8
108 ten



2. Change 43five to base six
4 3 five
(4x51) + (3x50) 
4x5 + 3x1
20   +  3
Therefore  43five=  35six

ACTIVITY
Change the following as instructed.  
a) 413five to base three
b) 101two to base five
c) 203five to base six
d) 144five to base two
e) 341six to base five
f) 1110two to base four

LESSON SIX
DETERMINING THE UNKNOWN BASES
1. If 17y = 15ten, find base y
17y =    15ten
(1xy1) + (7xy0) + 15
1xy + 7x1  = 15
y   +  7  =  15
y   +  7-7   =  15-7
y =   8
=   y   is  eight
2. If 24z  =  42five
Find base z
24z = 42five
(2xz1) + (4xz0)  = (4x51) + (2x50)
2xz + 4x1     = 4x5 + 2x1
2z + 4     =    20 + 2
2z + 4     = 22
2z+4 – 4     =   22 - 4
2z     =   18
2z     =   18
2     =    2    
z =        9

LESSON SEVEN
Addition of bases in table from
Completion of bases in table form.

Base five
+ 1 2 3 4 10
1 2 3 4 10 11
2 3 4 10 11 12
3 4 10 11 12 13
4 10 11 12 13 14
10 11 12 13 14 20

1 + 1 = 2 3 + 4 = 7
1 + 2 = 3 7÷5 = 1 rem 2
1 + 3 = 4
1 + 4 = 5 4 ÷ 4 = 8
5÷5 = 1 rem 0 8÷5 = 1 rem 3

Activity:
1. Complete the base six additions in the table below.
+ 1 2 3 4 5 10
1
2
3
4
5
10





MULTIPLICATION OF BASES IN TABLE FORM
x 1 2 3 4 10
1 1 2 3 4 10
2 2 4 11 13 20
3 3 11 14 22 30
4 4 13 22 31 40
10 10 20 30 40 100

TOPIC:   OPERATIONS ON WHOLE NUMBERS

Examples
1. Add      1 2 3 4 6 7 8
          +   2 9 7 8 6 8
            1 5 3 2 5 4 6

2. Add      1 8 4 8 6 9 4
         + 3 3 0 2 5 2 0
            5 1 5 1 2 1 4

3. Subtract   5 2 3 3 1 8 6
              - 1 3 4 5 1 0 2
                 3 8 8 8 0 8 4 

4. Subtract     8 4 0 0 0 7
                -    3 4 6 5 2
                  8 0 5 3 5 5
Activity: 
Ref. to MK bk 6 pg 55-57
Ref to fountain bk 6 pg 32-34

Solving word problems involving addition and subtraction of whole 
Examples
1. Find the sum of 67802  and  14007
Soln
6 7 8 0 2
       + 1 4 0 0 7
          8 1 8 0 9

2. A diary processed 6500 litres of milk, if 5650 litres of milk of milk were sold, how many litres remained?

6 5 0 0 litres
                 - 5 6 5 0 litres
    8 5 0 litres of milk

Note:
Sum    is the result of adding any given numbers.
Difference:   Is the result got after subtracting numbers

Activity:   Ref. to Mk bk6 pg 58-60     
      Ref. to fountain maths bk6 pg 32-34 

3. Multiplication of whole numbers
Examples
1. Multiply       3 2 4
                    x 1 8

2. Multiply 465 by 472
3. Use lattice method of multiplication to work out
    9 4 7
    x 6 5
                                 9    4    7    x
6   5    2    4    6
       4     4    2
0   3    1    2    4
       6     6    8   
      
        60608
Activity
Ref. to Mk bk6 pg 59-60
Ref. to fountain bk6 pg 35-36





Division of whole numbers using long division
Examples
1. Divide
     152
   1976
       13-13
              67
            - 65
                 26
              -  26
                 - -
Activity:
Ref to Mk bk6 pg 62
Ref to fountain bk6 pg 37

Solving word problems involving division
Examples
1. A petrol station manager bought 22,000 litres of petrol. If she put equal amount of oil in 440 drums. How many litres of oil were in each drum?

Soln: 
22,000 ÷ 440 = 50
              50
  440 22000
    5x440 2200
                      0
 0x440        -  0
                -

There were 50 litres of oil in each drum.
Activity:
Ref to Mk bk6 pg 62-63
Mixed operations (BODMAS)
Examples
1. Simplify 3x4+5
2. Workout 5-2+6
3. Work out 3+9÷3-1
4. Workout 4x7+9x3

Activity: 
Ref to Mk bk6 pg 61
Ref to fountain bk6 pg 38-39

TOPIC:  PATTERNS AND SEQ
LESSON I
Sub topic:     Divisibility tests
Content:    Divisibility tests of 2, 3 and 5

Examples 
(a) By 2
A number is divisible by 2 when it is an even number e.g 2, 4, 6, 8, 10, 12, 14 
b)    By 3
A number is divisible by 3 when the sum of its digits are a multiple of 3 e.g. 612
=  6+1+2
=  9 ÷ 3 = 3
Therefore 612 is divisible by 3
c) By 5
A number is divisible by 5 when the last digit is either 0 or 5. e.g 10, 15, 20, 770, 405 etc
Activity
Mk new edition pg 34-36
Fountain pg 41-42
Understanding pg 60-61

LESSON II
Sub topic: Developing number patterns
Content: Odd numbers, even numbers, Triangular numbers, Square, numbers, cube numbers, composite numbers, counting numbers, whole numbers
Examples
i) List down the following
a) Counting numbers/natural numbers less than 15
b) Whole numbers lessthan 10
c) Even numbers between 10 and 20
d) Odd numbers lessthan 20
e) Triangular numbers lessthan 36
f) Square numbers lessthan 49

NB:
Triangular numbers are obtained by adding consecutive counting numbers.
Square numbers are obtained by adding the consecutive odd numbers starting with one as the first square number
Or by squaring natural numbers

Activity: 
Fountain pg 43-48
Mk new edition pg 37
Understanding pg 62-65

LESSON III
Subtopic: Prime numbers and composite
Content:   -   List prime numbers
- List composite numbers
Examples
1. What is the sum of the 3rd and the 7th prime numbers?
2, 3, 7, 11, 13, 17, 19, 23
sum = 5 + 17
     =  22
2. Workout the sum of the first five composite numbers.
4, 6, 8, 9, 10, 12, 14, 15
sum = 4+6+8+9+10
     =   37

Activity:  New Mk bk6

LESSON IV
Subtopic: Consecutive counting numbers, odd and even numbers
Content: Finding consecutive counting, odd and even numbers
Examples:
The sum of 3 consecutive whole numbers is 36. What are these numbers?
Let the 1st number be n.
2nd number n+1 1st number = n
3rd number n+2
n+n+1+n+2 = 36 2nd number  = n+1
3n + 3  =  36 =  n+1
3n + 3 -3  =  36 =   12
3n = 33 3rd number   =  n+2
3 3 =  11+2
n  =  11 =   13
Even and odd numbers increase in intervals of 2
The sum of three consecutive even numbers is 24.
List down the 3 numbers.

Let the 1st number be x 1st number = x
      2nd number  x+2 =  6
3rd number  x+4 2nd number = x+2
x+x+2+x4  = 24 = 6+2
3x+6 =   24 =  8
3x+6-6    =   24 3rd number = x+4
3x =   24-6 = 6+4
3x = 18 =  10
3 3
x    =   6    

Activity:
Mk old edition pg 77-78
Mk new edition 43

LESSON V
Subtopic:  Prime factorization
Content: Methods used in prime factorization
- Multiplication,  Subscript method/set notation
- Powers of ten/exponents
Examples
a) Find the prime factors of 60     b)
2   60   =  2x2x3x5 (multiplication)
2   30
3   15  =   22x31x51 (powers/exponents)
5    1   =   21, 22, 31, 51 (subscript form)


Activity:
Mk old edition pg 82
LESSON VI
Content:    - Finding prime factorized number
- Finding the missing prime factors
Examples:
i) What number has been prime factorized
ii) Prime factorise and find missing factors
The prime factorization of 30 is 2xyx5, find y
2xyx51 = 30
10y = 30
10      10
  y = 31

If 144 = a4 x b2 find “a”  and  “b”

2 144 =    24 x 32
2   72
2   36 =    a4 x b2
2   18
3    9 ÷ a = 2
          3   3 b = 3
               1

Given that 22x x 2 = 32  find the value of x
              22xx2 = 25
    2x+1  =  5
    2x+1-1  = 5-1
    2x  =  4
    2       2
     x = 2
Activity:
Mk old edition pg 83

LESSON VII
Content:   Finding L.C.M and G.C.F using venn diagram
Examples
Show the prime factors of 30 and 36 on the venn diagram




Venn diagram
F30              F36



GCF  =  F30  ∩  F36
=   21, 31
=    6




Activity 
Mk old edition pg 86-87

LESSON VIII
Subtopic: Unknown values/prime factors
Content:     1)   Find the missing numbers
2)   Find the unknown factors
3)   Workout the H.C.F and L.C.M
Example:
Find x  and  y
Fx              Fy Fx  =  21, 22, 23, 31
      =  2x2x2x3
      =    24
Fy =  21, 22, 31, 32, 33
              =   2x2x3x3x3
     =     108

G.C.F = Fx ∩ Fy L.C.M  =  21, 22, 23, 31, 32, 33
=   21, 22, 31 =   2x2x2x3x3x3
=   2X2X3 =     216
=   12

Find the unknowns
   F20            F30   i)   21, 51 x  = F20
=  x+2x5 = 20
10x = 20 2
                                               10      10 1
X = 22
ii)   =   21, 51, y = F30
=   2x5xy = 30
=   10y   =  30
     10        10
y = 31
Activity:
Mk old edition pg 88-89

LESSON IX
Sub-topic: Application of G.C.F/L.C.M
Content:   - Relationship between G.C.F and L.C.M
- Other problem related to H.C.F/G.C.F
Examples
1. The LCM of two numbers is 144 their GCF is 12 and one of these numbers is 48. Find the other number.
Solution
2nd no.  =  LCM x GCF
1st no. 
=    144 x 12
48

What is the largest possible divisor of 24 and 36?
2  24  36
2  12  18
2   6    9        =   2x2x3
3   3    9   = 12
3   1    3
1 1

Activity
- Oxford primary MTC bk6 pg 34-41

LESSON X
Subtopic:  Application of LCM
Content:    Find the smallest number which when divided by 9 and 12 leaves
a) no remainder?
b) Remainder of 1?
c) Remainder of 5?

2 9  12
2    9   6
3    9   3
3    3   1
      1   1




ii) Kelvin has an article of 40cm and his father has a stride of 60cm. What is
    the width of the narrowest path and they both cross in a whole number of
    strides.
M40  =  40, 80, 120, 160 …
M60  =   60, 120, 180, …
L.C.M  = 120
Therefore: The width is 120cm
Activity:
Oxford primary MTC pupils bk 6 pg 34-36
LESSON XI
Subtopic: Working with powers of whole number
Content:  - Find a number from powers
- Express numbers as a product of powers of a given numbers
- Operation on powers
Example    i)   What is 73
    = 7x7x7
    =  343
ii) Express 64 using powers of 4.
4  64
4  16 Therefore 64 = 4x4x4
4   4 =  43
                 1
Activity
A new Mk pupils bk 6 pg 86 and 85
LESSON XII
Subtopic = Square numbers and square roots
- Square of :-
a) Whole numbers
b) Fractions
c) Mixed fractions
d) Decimals
Example
1. What is the square of 12?
122  = 12x12
      =   144
2. Workout the square of ¾ 
(¾)2  = ¾ x ¾
=    9
      16
3. What is the square of 0.6?
(0.6)2 = 0.6 x 6    36   =  0.36
            10    10   100

4. Find the square root of?
a) 144
                144





b)     9
   16



Activity
- A new primary MTC bk6 pg 90-91
- Fountain pg 49
LESSON XIII
Subtopic :   Cubes and cube roots
Content: Finding cube numbers and cube roots
Examples
1. What are the cubes of the following
a) 2 = 23
= 2x2x2
=   8

b) 2 = (2)3
3     3
= 2x2x2
   3  3 3
=    8
     27
Find the cube root of 8.
  



Activity 
A new MTC bk 6 pg 90-91




THEME:  NUMERACY
TOPIC:  FRACTIONS
Review on addition and subtraction of fractions.
Examples
1. Add:  3/5   +  1/3
Solution
3/5  +  1/3  =  (3 x 3) + (5x 1)
                                    15
                    =  9 + 5
                           15
                    =  14
                         15
2. Subtract 5/7 – 1/4 

Solution 
5/7 - 1/4  = (4 x 5) – (7 x 1)
                                  28

                  =  20 – 7
                          28

                  =  13
                       28
Activity 1:1
1. Add the following
a) ¾ + ½                                  b)  1/3 +1/5                              c)  2/3 + 3/5
d)  1 ½ + 2 ¼                           e)  3 1/5 + 1/3 
2. A boy was given 5/12 of the bread in the morning and ½ of the remainder in the evening. What fraction of the bread did he get altogether? 
3. Mugisha divided his land as follows:
¼ occupied by cows 1/3 occupied by crops and 1/8 is occupied by goats while the rest of the land is occupied by cash crops.
Find the piece of land occupied by animals.
Activity 1:2
1. Subtract the following:
a) ½ - 1/3                   b)  ¼ – 1/6                 c)  1/3 – 1/5                d) 5/7 – 2/5 

2. Natasha had 7/10 of the bread. She gave out 2/5, what fraction of the bread remained?

3. I ate ¾ of the chapatti, what fraction of the chapatti remained?
4. The teacher instructed pupils to read 4/20 pages of the book. What fraction of pages remained un-read?
5. A bus covered 7/15 of the journey before breaking down. Find the part of the journey it had remained with.

EVALUATION
Strong Points: __________________________________________________________________
Weak Points: ___________________________________________________________________
Way forward: __________________________________________________________________
Multiplication of fractions:
Finding reciprocal:
Reciprocal is also known as multiplicative inverse. This is because, a number multiplied by its reciprocal, the result is 1.
Example 1                                                                     Example 2
Find the reciprocal of 3.
Number x reciprocal = 1
3 x reciprocal = 1
3 x r = 1
  3         3
      Reciprocal of 3 = 1/3 
Find the reciprocal of 2/5 
Number x Recip = 1
2/5 x Recip =1
5 x 2 x R  =  1 x 5         2 x R  =  5
       5                                   2          2
            Recip of 2/5 = 5/2


Activity 1:3
Find the reciprocal or multiplicative inverse of the following:
a) 3/3                          d)  10                        g) 2 ½                         j)  0.2

b) ¼                            e)  5                          h)  1 1/3                      k)  4.5

c) 4/5                           f)  4                           i)  3 ¼                         l)  3.2

EVALUATION
Strong Points: __________________________________________________________________
Weak Points: _________________________________________________________________________
Way forward: ________________________________________________________________________

Multiplying fractions
It should be noted that, it is only in multiplication of fractions where L.C.M is not applied.
Examples;
1. 2 x 1
3     4
        = 2 x 1
3 x4
        =   2 (by reducing)
12
1
6

2. 1 1 x 1
    2    3
3 x 1
2    3
3 x 1
2 x 3
3 (by reducing)
6
        =  1
 2


Activity 1:4
Simplify the following.
1. 1 x 1                                           4.  2 x 3                                      7.  2 x 4
3    2                                                 5    4                                           9   10

2. 1 x 1                                           5.  3 x 1                                      8.  3 x 4
3    4                                                 8    3                                           7     8

3. 2 x 1                                           6.  1 x 2                                      9.  3 x 4
3    4                                                10  5                                            8    5

Simplify:
a) 1 1 x 1                                                      d)  2 1 x 3
    5    3                                                               10  9

b) 1 1 x 1                                                      e)  4 1 x 4
    4    5                                                               2     9

c) 2 1 x 1                                                      f)  9 1 x 10 1
    3   14                                                             7          5
Division of fractions:
Division is always done in two ways;
i) Use of reciprocal
ii) Use of L.C.M
Examples;
1. Workout 
8 ÷ 2

8/1 x ½ 

8 x 1
1 x 2

  8/2
  
= 4
2. Simplify 
2/3 ÷ 2

2/3 ÷ 2/1 

2/3 x ½ 

    2/6

= 1/3 

3. Divide 
¼ ÷ 2/3 (len =12)

312 x ¼ ÷ 2/3 x 124

(3 x 1) ÷ (2 x 4)

3 ÷ 8

3/8

 

4. How many small bottles of ¼ litre can be obtained from a 20litre jerrican 
                        Soln
                we divide 20 ÷ ¼
                              20 ÷ ¼ 
                             20 x 4 
                                      1
                  80 quarter litre bottle  

Activity 1:5
1. Workout the following.
a) 20 ÷ 4                   b)  3/10 ÷ 6                  c)  ¼ ÷ 2/3               d)  9 1/5 ÷ 2/10

2. Simplify
a) 2/3 ÷ 1/12                          b)  4/9 ÷ 3 ¾                            c)  5/9 ÷ 3 1/3

3. Divide the following
a) ¾ ÷ 7/8              b) 2/3 ÷ 7/9                c) 2 4/5 ÷ 1 1/3                 d)  8/9 ÷ 2/3
Activity 1:6
1. How many ½ liter cups of water can be got from a 5 litre container?
2. How many small spoons of 1 ½ ltr can be obtained from 2 ¼ ml?
3. By what fraction can 6 ½ be divided to get 2 ½?
4. How many pieces of ¾ m of cloth can be cut from a long piece of 9m?
5. A bag contains 5 ¼ kg of posho. How many ¼ kg packets can be got from the bag?
Mixed operations of fractions:
Examples;
2/3 x 4/9 ÷ 1/3

2/3 x 4/9 x 3/1
2 x 4 x 1
3 x 3 x 1

8/9

1/3 x ¼ + ½

Using Bodmas, Multiplication comes first.

(1/3 x ¼) + ½
1 x 1 + 1
12       2 1/12 + ½

(1 x 1) + (6 x 1)
12
1 + 6
12
7/12


Activity 1:7
Workout the following:
½ x 3/5 ÷ ¾ 

4/9 x 2/3 ÷ 1/3 

3/8 x 1/9 ÷ 2/3 
1 2/3 x 3 ½ ÷ 1/5 

(1/3 x ¼) + ½ 

¾ x 2/3 – ½ 5/6 ÷ (5/4 of 3)

1/3 x 1/8 + ¼ ÷ 1/7 

1/3 + ½ of 1/7 x 1/5

EVALUATION
Strong Points: __________________________________________________________________
Weak Points: ___________________________________________________________________
Way forward: __________________________________________________________________
Converting fractions to decimals
Rational fractions
Examples;
Convert 2/5 to a decimal 
                0.4
2/5 = 5    2
                 .0
                 20
                -20
                 
  2/5 =  0.4 Change ¼ to a decimal 
             0.25
¼ = 4   1
              .0
              10
               -8
               20
              -20                  ¼  = 0.25


Activity 1:8
Convert the following fractions to decimals.
a) ½ b)  1/5 c)  4/5 d)  2/8
e)  ¼ f)  2/5 g)  3/5 h)  ¾ 

Operation on decimals
Addition and subtraction
Addition and subtraction is done after arranging digits vertically in their correct place values.

Examples:
a) Add:  0.23  +  2.3 b)  Subtract:  7 – 0.34
Soln        Soln
2 . 3 7 . 0 0
       +  0 . 2 3         -  0 . 3 4
           2 . 5 3 6 . 6 6
Activity 1:9
1. Add the following. 2.  Subtract the following.
a) 8 . 24   +  0 . 16        a)  8 – 0. 94
b) 0 . 25  + 2.5        b)  7 . 00 – 2 . 34
c) 8  + 2.3  + 1 .54        c)  0 . 23 – 0 . 13
d) 0 . 11  + 1.1  + 11        d)  2 . 5 – 0 . 25
e) 17.04 + 1.3                    e)  48.6 – 0.057
Activity 1 :10
Simplify the following.
a) 1.64  + 5 – 4.42
b) 9 – (2.31 + 1.69)
c) 2.71 – 1.88 + 7.24
d) 3.82 – 7.34 + 9.68
e) 2.78 – 6.06 + 10.77
Multiplication of decimals
Note:  The product of decimals must reflect the number of decimal places in the question.
a) 0 . 4 x 0 . 3 OR  0 . 4 x 0 . 3
   =     0 . 4 4/10   x 3/10
      x  0 . 3 =  12/100
          1 . 2 =  0.12
   + 0 0
      0. 12   
=  0.12
Activity 1 : 11
Work out the following.
a) 0 . 17 x 0.3 b)  9.6 x 0.2
c)  0.36 x 0.4 d)  7.5 x 0.5
e)  1.2 x 0.5 f)  4.5 x 3.2
g)  3.21 x 0.4 h)  0.45 x 0.12

EVALUATION
Strong Points: __________________________________________________________________
Weak Points: ___________________________________________________________________
Way forward: __________________________________________________________________
Division of decimals
(Decimals by whole numbers and whole numbers by decimals)


Examples:
a) Divide:  8  ÷ 0.2 b)  Work out:  0.32 ÷ 8
8  ÷  2 32     ÷   8
1     10 100        1
4  8    x  10          4  32    x   1
    1         21 100       8 1
4 x 10 4 x 1
=  40 100
=  4/100   
=  0.04

Activity 1:12
Work out the following.
a) 10 ÷ 0.2 b)  0.24 ÷ 8 c)  5.6 ÷ 0.07
d) 56 ÷ 0.7 e)  0.3  ÷ 9 f)  8.1 ÷ 0.027
g) 27 ÷ 0.3 h)  1.2 ÷ 48 i)  9.6 ÷ 0.08
j)  36  ÷ 0.12 k)  0.06  ÷ 36 l)  1.44 ÷ 0.12
m)  55 ÷ 0.11 n)  1.2 ÷ 0.6 o)  19.6 ÷ 0.07

EVALUATION
Strong Points:__________________________________________________________________
Weak Points: __________________________________________________________________
Way forward: _________________________________________________________________


Division and multiplication
Examples:
a) Simplify:   0.24 x 0.2 b)  Work out:    1.44 x 3.6
                         0.08       0.12  x 0.4
Soln Soln
0.24  x 0.2 1.44 x 3.6
     0.08 0.12 x 0.4
24/100  x 2/10  ÷ 8/100 1.44 x 3.6 x 1000
24/100  x 2/10  x 100/8 0.12 x 0.4 x 1000
 24 x 2 144 x 36
    10     12 x 4
 3 x 2 12 x 9
   10
=  0.6 =  108
Activity 1:13
Work out the following.
a) 0.9 x 0.8 b)  0.72 x 0.2 c)  0.09 x 0.6
       0.3            0.036 0.18
      d)  0.72 x 0.96 e)  0.36 x 0.4 f)   4.5 x 1.6
     0.014            0.018       4.8 x 1.5
      g)  9.6 x 1.25 h)  2.4 x 0.54
            4.8 x 0.5             0.36

EVALUATION
Strong Points: __________________________________________________________________
Weak Points: ___________________________________________________________________
Way forward: __________________________________________________________________
Rounding off decimals
To round off is to estimate a number to the nearest value.
Note:  In rounding off decimals, the decimal digits cancelled are not replaces with zeros.
Examples:
a) Round off 5.72 to the nearest whole number.
O Tth Hth
5    . 2
                   +  1 
    6    .
Therefore; 5.72  ≈  6
b) Round off 29 .  97 to the nearest tenths.
T O Tth Hth
2 9     . 9 7
        + 1
3 0     . 0
Therefore; 29.97 = 30.0
Zero after the decimal point represents the tenths place value required.
Activity 1:14
1. Round off the following to the nearest tenths.
a) 1.32 b)  6.85 c)  2.41
d) 7.96 e)  3.93 f)  5.49
g)  8.54 h)  8.985

2. Round off to the nearest hundredths.
a) 12.623 b)  20.841 c)  6.829
d)  8.728 e)  7.936 f)  0.483
g)  12.998 h)  3.452

3. Round off the following to the nearest whole number,
a) 36.7 b)  0.736 c)  9.39
d) 6.94 e)  142.83 f)  11.52
g)  68.77 h)  68.259 i)  4.930

EVALUATION
Strong Points: _________________________________________________________________
Weak Points: ___________________________________________________________________
Way forward: __________________________________________________________________
Square and square root of fractions and decimals
Square and square roots of fractions.
Examples:
1. Find the square of;
a) 2/9 b)  1 1/5 
Square of 2/9        6/5
=  2/9  x 2/9        square  = 6/5 x 6/5
=  4/81            = 36/25

2. Find the square root of 1/9
√1/9  
= √1 x 1
       3 x 3
=  1
                 3
b)   √36/81 3    81 2    36
=  √36 3    27 2    18
     √81 3    9 3     9
=   6 3    3 3     3
     9       1                    1
3 x 3 x 3 x 3 2 x 2 x 3 x 3
3 x 3 2 x 3
9 6

Activity 1:15
1. Find the square of the following.
a) 1/6 b)  ¾ c)  3/8 d)  1 ½ 
e)    ½ f)  4 ½ g)  2/3 h)  4/5
i)    1/5 j)  1 2/3 k)  3 2/3 l)  5 2/3 

2. Find the square root of the following.
a) ¼ b)  9/16 c)  4/25 d)  6 ¼ 
e)   7 1/9 f)  5 1/16 g)  4/9 h)  9/25
i)  36/49 j)  1 9/16 k)  1 11/25 l)  5 4/9


3. Find the area of the figure.

3 ½ cm


4. Find the area if a square land whose side is 4 ¼ km.
5. The area of a square is 144cm2.  Find one side of that square.
6. A square land has an area of 3 6/25 m2.  How long is one side of that land?

EVALUATION
Strong Points: __________________________________________________________________
Weak Points: ___________________________________________________________________
Way forward: __________________________________________________________________
Square and square roots of decimals
Examples
1. Find the square of 0.3 2.  Find the square root of 0.25
Soln       Soln
Sq. of 0.3 = (0.3)2        √0.25  = √25
=  0.3 x 0.3    100
=  3/10 x 3/10        √25
=  9/100          100
= 0.09 =  5/10
=  0.5




Activity 1 :16
1. Find the square of the following.
a) 01 b)  1.10 c)  0.18 d)  0.72
e)   0.5 f)  0.12 g)  0.34 h)  1.2
i)    0.7 j)  0.16 k)  0.42 l)  2.5

2. Calculate the square root of the following.
a) 0.09 b)  0.81 c)  2.25 d)  0.0025
e)   0.49 f)  1.44 g)  0.04 h)  0.0064
i)    0.64 j)  3.24 k)  0.0016 l)  0.0169

EVALUATION
Strong Points: __________________________________________________________________
Weak Points: ___________________________________________________________________
Way forward: _________________________________________________________________
Converting fractions to recurring decimals
Note:  Fractions are easily changed to decimals by use of long division.
Examples:
1. Change 5/9 to a decimal.
Soln:
        0.55
         9    5
             - 0
                50
             -  45
                    50
                 - 45
                      5
Therefore; 5/9  =  0.55 …




2. Convert 1/6 to a decimal

0.166
                  6   1
                      -0
                        10
                         -6
                           40
                          -36
                              40
                             -36
                                 4
Therefore 1   =  0.166 ….
                      6
Activity 1:17
Express each of the following fractions as a decimal.
a) 1/3 b)  3/11 c)  4/11 d)  5/7
e) 2/3 f)  2/9 g)  1/11 h)  1/7
i)    1/9 j)  5/11


EVALUATION
Strong Points: __________________________________________________________________
Weak Points: ___________________________________________________________________
Way forward: __________________________________________________________________
Converting recurring decimals to fractions
Note:
- Recurring decimals are of two types.
- Pure recurring decimals whose digits all repeat. E.g:0.22…
- Mixed recurring decimals whose digits don’t all repeat. E.g: 0.122…


Examples:
a) 0.77… b)  0.1515… c)  0.233…
Let p be the fraction       Let the fraction be y   Let the fraction be p
p  = 0.77…                 y = 0.1515…              x    = 0.233…  ___1
10p  = 7.7…      10y  =  1.515…            10x   = 2.33…  ___2
We subtract       100y =  15.15            100x  = 23.3… ___3
10p  = 7.7       100y =  15.15            100x    =  23.3
 - p  = 0.7                   -   y =     0.15 - 10x         2.3
  9p = 7.0         99y =  15        90x       =  21
9p = 7      99y =  15       90x        =  217
9            9      99      99        90              90 30
p = 7/9          y =    5/33              x   = 7/30

 Activity 1:18
Convert the following fractions as common fractions
a) 0.33… b)  0.1212… c)  0.7272…
d)  0.153153… e)  0.55… f)   0.2424…
g)  0.3636… h)  0.133… i)  0.255…
j)   0.66… i)   0.1010… j)  0.123123…
k)  0.255… l)  0.11… m)  0.2121…

EVALUATION
Strong Points: ___________________________________________________________________
Weak Points: ____________________________________________________________________
Way forward: ___________________________________________________________________
Application of fractions
Examples:
1.   ¾ of my father’s age is 36 years.  How old is my father?
Let the father’s age be y
¾ of y = 36 yrs
¾ x y = 36yrs
4 x 3y  = 36 yrs x 4
       4
3y =  1236 yrs x 4
3y 3
y =  12yrs x 4
y =  48yrs
Therefore father’s age is 48years
2. 1/3 of my age is equal to ½ of John’s age.  If John is 24 yrs old.  How old am I?
Let my age be m.
1/3 of m  = ½ of 24
1/3m = ( ½  x 24)
1/3m = 12
3 x 1/3 m = 12 x 3
m = 12 x 3
m = 36yrs

3. On Kahima’s farm, ½ of the animals are cows, 1/3 of the remaining animals are goats.  If the rest 12 animals are pigs, how many animals are on the farm?
Soln
Whole farm =  1
Cows = ½
Remaining animals =  1 – ½ 
=  ½ 
Goats =  1/3 of rem
=  1/3 of ½
=   1/3 x ½ 
=  1/6
Total (cows + goats)
½  + 1/6
3 + 1 
   6
4/6
Fraction for pigs  =  6/6 – 4/6
=  2/6
Or   1/3

1/3 is equivalent to 12 pigs
Let the total be y.
1/3 of y = 12 animals
3 x 1/3 y = (12 x 3) animals
y = (12 x 3) animals
y = 36 animals
Therefore; there are 36 animals on Kahima’s farm.

Activity 1:19
a) 2/3 of my salary is 6,000/=.  What is my salary?
b) ¼ of my grandmother’s age is 20yrs.  How old is my grandmother?
c) ¾ of the Askari’s salary is 3,000/=.  How much does he get?
d) 2/5 of Kadodi’s weight is 10kgs.  Find Kadodi’s weight?
e) 1/3 of the distance from Iganga to Jinja is equal to ¼ of the distance from Jinja to Kampala.  If it is 80km from Jinja to Kampala, how far is Iganga to Jinja?
f) 1/3 of my salary is equal to 5/6 of Musa’s salary.  If my salary is 12000/=, what is Musa’s salary?
g) 1/6 of the area of a triangle is 7/12 that of the area of the rectangle.  If the area of the triangle is 420m2.  Find the area of the rectangle.
h) John spends ¼ of the salary on food and ¼ of the remainder on rent and is left with 600/=.  What is the salary?
i) A teacher spends ½ of the salary on rent and ¼ of the remainder on fees and is left with sh. 8000.  Calculate his salary.
j) ½ of the pupils in a school like maths and 2/5 of the remainder like English.  If there remains 180pupils who like other subjects, how many pupils are there in the school? 

Ratios and proportions
- Ratios are comparisons between two or more quantities by division.
- Ratios are always expressed in the simplest form.
Forming ratios
Examples
In a class, there are 24 girls and 18 boys.  Express the number of boys as a ratio of girls.
Soln:
Ratio Boys  to Girls
   18     :   24 (divided by 6)
318 :    24 4
                            6            6
                           3 :       4
Girls to  Boys
  24 :     18
  24 :     18
    6             6
   4 :       3
Activity 1:20
1.   In a music club, there are 12 singers and 8 instrumentalists.
a) What is the ratio of singers to instrumentalists?
b) Find the ratio of instrumentalists to singers.
2. There are 120 pupils in a school.  If 40 are girls;
a) Find the ratio of girls to boys.
b) Find the ratio of boys to girls.
c) Express the number of boys as a ratio of the whole class.
3. Express 500g as a ratio of 1 hour.
4. Express 500g as a ratio of 1 kg.
5. In a village, 40 farmers grow beans, 30 grow maize and 60 grow cabbages.
a) Find the ratio of farmers who grow maize to those who grow cabbages.
b) Find the ratio of those who grow maize to the total number of farmers.
c) What is the ratio of those who grow maize to those who grow beans and those who growing cabbages?
EVALUATION
Strong Points: ___________________________________________________________________
Weak Points: ____________________________________________________________________
Way forward: ___________________________________________________________________

Increasing and decreasing in ratios
Increasing
For ratio increase, the new amount is bigger than the old amount.
Example
 Increase 300/= in the ratio 5 : 2
New old
5 2
? 300/=
2 parts =  300/=
1 part =  300
        2
3 parts =  (300/2 x 5)/=
     150 x 5
      750/=
OR:  New x   Amount
          Old
5   x  300/=
2
5 x 150/=
750/=

Decreasing
For ratio decrease, the new amount is smaller.


Example I
Decrease 400kg in the ratio 3:4
New Old
3 4
? 400kg
4 parts =  400kg
1 part =  400kg
        4
3 parts =  (400/4 x 3)kg
=  (100 x 3)kg
300kg

OR: New x Amount
Old
3  x 400kg
4
3 x 100kg
300kg

Example II
A class had 60 pupils, the number reduced to 50 pupils.  In what ratio did it reduce?
Old number  = 60
New number  = 50
New :   Old
50 :    60
5 :    6
Activity 1:21
1. Increase 300 in the ratio 2:1
2. Increase 600 in the ratio 3:2
3. Increase 4800/= in the ratio 5:4
4. Increase 360 in a ratio 5:4
5. The school fees was 18,000/- and increase in the ratio 11:10.  What is the new amount of school fees?
6. The school had 800 pupils last year.  This year they have 1000 pupils.  In what ratio has the number increased?
7. What is the ratio increase from 800 to 960?
8. Decrease the following in their respective ratios.
a) 600 in ratio 2:3
b) 400kg in the ratio 2:5
c) 1000/= in the ratio 3:5
d) 800litres in the ratio5:8
e) 700bags in the ratio 7:10
9. A class had 72 pupils, the number reduced in the ratio 7:8.  What is the new number of pupils?
10. The marked price of a radio is 90,000/=.  The man bought a radio at a 
            reduced price of the ratio of 2:3.  How much did he buy the radio?

EVALUATION
Strong Points: ___________________________________________________________________
Weak Points: ____________________________________________________________________
Way forward: ___________________________________________________________________
Sharing in ratios
Examples:
1. Divide 4200kgs of sugar in the ratio of 2:5
Ratio = 2:5
Total =  2 +5
=  7
1st share = 2  x  4200kg 2nd share = 5  x 4200 kg
               7 1                 71
=  2 x 600kg =  5 x 600kg
=  1200kg =  3000kg

2. Share 200/= in the ratio 2:3
All parts = 2 + 3
=  5 parts
5 parts =  200/=
1 part =  200
2 parts =  (200 x 2)/=
        5
=  40 x 2
     80/=

3 parts =  (200 x 3)/=
         5
=  (40 x 3)/=
     120/=
Therefore, the shares are 80/= and 120/=

Activity 1:22
1. Share 360 in the ratio 2:3
2. Divide 72 in the ratio of 5:3
3. Divide 4500/= in the ratio 7:8
4. Share 90kg of sugar between two people in the ratio of 7:3
5. John and Diana shared 3000/= in the ratio 2:3.  How much did each get?
6. A man and his wife had 200kg of coffee.  They decided to share it in the ratio 7:3 respectively.
a) How many kilograms did the man get?
b) How many kilograms did the wife get?
7. Amos, Andrew and Allan shared 24,000/= in the ratio 1:2:3 respectively.  How much did each get?
8. A B and C contributed money for a business in the ratio 3:4:5.  If C contributed 10,000/=.  How much did they contribute altogether?
9. Share 480 in the ratio 4:5
10. Dan and Mike shared money in ratio 3:5 respectively.  If Mike got 3000/=, 
     how much did Dan get?

EVALUATION
Strong Points: ___________________________________________________________________
Weak Points: ____________________________________________________________________
Way forward: ___________________________________________________________________
Proportions
There are three types of proportions.
- Simple direct proportions
- Inverse proportions
- Constant proportions

Simple proportions
In simple proportions, the more the number of items, the more the amount.
Examples
1. 4 girls have 8 breasts.  How many breasts have 7 girls.
Soln:
4 girls = 8 breasts
1 girl = (8/4) breasts
7 girls = (8/4 x 7) breasts
= (2 x 7) breasts
14 breast
2. 4 pen costs 2000/=.  Find the cost of 9 pens.
Soln:
4 pens = 2000/=
1 pen = (2000/4)/=
9 pens = (2000/4 x 9)/=
9 pens =  500 x 9
=  4500/=
Activity 1:23
1. One book costs 600/=.  Find the cost of 5 books.
2. 2 bags weigh 70kg.  What is the weight of 5 bags?
3. 1200/= can buy 2kg of maize flour.  How many kgs can you get from 3600/=
4. A bag of coffee weighs 65kg.  How many kilograms will 12 bags weigh?
5. 3 dresses can be made from 6 metres of cloth.  How many metres of cloth can I use for 9 dresses?
6. 5 pieces of timber are used to make 2 tables.  How many tables can you make from 15 pieces of timber?
7. The bus fare for 3 people is 25000/=.  What is the fare for 2 people?
8. 5 jerrycans cost 75000/=.  How many jerrycans can one buy with 105,000/=?

Inverse proportions
In this type of proportion, the more the number employed to do work, the less the time they will take and vice versa.
Examples
1. 3 men can do a piece of work in 6 days.  How long will 9 men take to do the same piece of work?
Soln:
3 men = 6days
1 man = (3 x 6) days
9 men = (3 x 6) days
     9
9 men = (18) days
                9
9 men 2 days
Therefore; 9 men can do the same piece of work in 2 days.
Activity 1: 24
1. It takes 4 days for 12 women to dig a shamba.  How long will it take 8 women to do the same job?
2. 25 girls can construct a road in 8 days.  How many girls will construct a road in 10 days?
3. A carpenter takes 2 hrs to make a chair.  How many chairs will be made in 6 hours?
4. 12 technicians can paint a school building in 10 days.  How long will 15 technicians take?
5. 6 porters can dig a piece of land in 5 days.
a) How many porters can do the same work in 10 days?
b) How many days will 15 porters take?
6. 5 children take 4 days to slash the school compound.  How many days will 10 children take?
Constant proportions
This is a type of proportion whereby time taken to complete a task remains the same though the number of parties change.
Examples:
1. 6 men can sing a song in 10 minutes.  How long will 10 men take to sing the same song?
2. 5 shirts take 20 minutes to dry.  How long will 20 shirts take to dry?

Percentages
- Percent means out of 100.
- The symbol for percent is %
That is; 
6 %  = 6/100
20%  = 20/100

Changing percentages to fractions and vice versa

Examples
1. Express 25% as a fraction in its lowest terms.
25% =   25
100
=               51 x 51
    2 x 2 x 51 x 51
1
      4

2. Express 12 ½ % as a fraction.
12 ½ =   25%
        2
=  25/2
     100
=  25 ÷  100
      2        1
=  25 x     1
      2      100
=  25
   200
=  1/8


3. Express 4/5 as a %
4/5
4     x  100 20
51
=  (4 x 20)%
=  80%

4. Convert 0.25 as a percentage.
0.25
=  25
    100
=  25  x 1001
    1001
=  25 x 1
=  25%

5. Change 20% as a decimal.
20%
=  20
    100
=  2
    10
=  0.2

Activity 1:25
1. Convert the following percentages to fractions in their lowest terms.
a) 50% b)  30% c)  21% d)  33 ½ %
e)   12% f)  40% g)  65% h)  90%
i)    62 ½% j)  16 2/3% k)  37 ½%
2. Change the following fractions to percentages.
a) ½ b)  7/20 c)  3/20 d)  3/8
e)   ¼ f)  12/25 g)  2/25 h)  5/8
i)   9/20 j)  4/10 k)  13/50 l)  2/3

3. Change the decimals to percentages.
a) 0.7 b)  0.45 c)  0.003 d)  0.9
e)   0.001 f)  0.12 g)  0.75 h)  0.025
i)  0.067


4. Change these percentages to decimals.
a) 10% b)  32% c)  2.5% d)  75%
e)   25% f)  82% g)  1.2% h)  62%
i)    30% j)  6 ¼% k)  12 ¼%

EVALUATION
Strong Points: ___________________________________________________________________
Weak Points: ____________________________________________________________________
Way forward: ___________________________________________________________________
Expressing percentages as ratios
Examples
1. 2%
=   21
   100 50
=  1
    50
=  1:50

2. 33 1/3%
=  100%
        3
=  100  ÷  100
        3            1
=  100 1   x     1
        3           100 1
=   1
     3
=  1:3
Express as percentages
Examples
1. 3:10
=  3    x 100 10
   10 1
=  3 x 10
=  30%
2. 1/5  : 1/3  
=  1    ÷   1
     5         3
=  1  x  3
     5      1

=  3
     5
=  3   x  100 20
     5 1
=  3 x 20
=  60%

Activity 1:26
1. Express the following percentages as ratios
a) 10% b)  56% c)  125%
d)   25% e)  76% f)  80%
g)   50% h)  98% i)  144%

2. Express the following ratios as percentages.
a) 1:2 b)  3:5 c)  5:16
d)   2:3 e)  3:8 f)  3:4
g)   2:5 h)  1/5 : ¼ i)  1/8 : 1/5
j)    1/20 : 1/7 k)  2/5 : ½ l)  ¾  : 1/5
Activity 1:27
1) A child paid 55% of the school fees.  What percentage is left for him to pay?
2) A buyer paid 85% of the cost of a radio.  What percentage is left for him to pay?
3) 30% of the people in Uganda are male, 50% are female and the rest are children.  What percentage is for children?
4) Henry had 40 cows, he sold 15.
i) What percentage was sold?
ii) What percentage of the cows was not sold?
5. A boy got 8 marks out of 20.  What percentage is this?
6. If 40 out of 120 pupils in a class passed their exams.
i) What percentage of the pupils passed?
ii) What percentage of the pupils failed?
7. Write 40 as a percentage of 200.
8. Express 200gm as a percentage of 1 kg.
9. What percentage of 1 hour is 150 minutes?
10. Express 60 as a percentage of 80.

EVALUATION
Strong Points: ___________________________________________________________________
Weak Points: ____________________________________________________________________
Way forward: ___________________________________________________________________
Percentage of numbers
Examples
1. What is 20% of 200/=?
  20    x  200 2/=
100 1 
(20 x 2)/=
 40/=
2. What is 12 ½ of 800 people?
25 % of 800 people
  2
  25  ÷  100  of 800
    2          1
25   x     1  x  800 4
              2 1      100 1
25 x 4
=  100 people

Activity 1:28
What is;
a) 10% of 200/=?
b) 15% of 240 books?
c) 20% of 400 kgs?
d) 11% of 8000 books?
e) 60% of 2400 cows?
f) 25% of 1200/=?
g) 12 ½ of 3200 kg?
h) 35% of 640?
i) 90% of 360cm?
j) 80% of 1500?
k) EVALUATION
l) Strong Points: ___________________________________________________________________
m) Weak Points: ____________________________________________________________________
n) Way forward: ___________________________________________________________________
Application of percentages
Activity 1:29
1. In a class 10% of the pupils are absent.  If there are 60 pupils.
a) How many pupils are absent?
b) How many are present?
2. There are 280 animals in the zoo.  10% are birds, 50% are primates and 40% consists of others.
How many animals of each type named are in the zoo?
3. A school has 360 books.  30% are English books, 20% are science books and the rest are maths books
How many books of each category are in the school?
4. 20 % of the pupils in a school are girls.  If there are 35 girls in the school.  How many pupils are in the school?
5. 10% of a number is 40.  What is the number?
6. 25% of a number is 80.  What is the number?
Percentage increase and decrease
Examples
1. Increase 200/= by 20%
Get 20% of 200/=
20  x  200 2/=
100 1
(20 x 2)/=
      40/=
New amount  = 200/=
                               + 40/=
                               240/=


OR
100%  =  200/=
1% =  200/=
     100
100% + 20% =   200 2  x 120 /=
      100 1
=  2 x 120/=
      240/=



2. Decrease 300 by 10%
  10    x  300  3
 100  1
 10 x 3
 30
New amount = 300
    - 30
    270


OR
100% - 10 = 90%
90  x 300 3
            100  1
90 x 3
270

Activity 1:30
1. Increase  80 by 10%
2. Increase 240/= by 20%
3. Increase 400 by 15%
4. Increase 15000 by 30%
5. Okidi’s pay was 1200 dollars.  It was increased by 30%.   What is his new pay?
6. The number of pupils in a class were 50 but they increased by 10%.  What is the new number?
7. A shirt was priced at 9000/= last year.  This year its price increased by 40%.  What is the new price of the shirt?
8. Last year, there were 30,000 cars in Uganda.  This year there are 20% more cars imported.  How many cars are in Uganda this year?
9. Decrease 400 by 20%
10. Decrease 500kg by 10%
11. Reduce 80kg by 20%

EVALUATION
Strong Points: _________________________________________________________________
Weak Points: ___________________________________________________________________
Way forward: ___________________________________________________________________
Profit and Loss
- Profit is realized when the selling price of a commodity is more than the buying price.
- Loss is suffered when one sells a commodity at a lower price than it was bought.

Formulae
Profit = Selling Price (S.P) – Buying Price (B.P)
Loss =  Buying Price (B.P) – Selling Price (S.P)

Examples
1. Nakajjumbe bought a phone at 80,000/= and sold it at 110,000/=.  What profit did he make?
Soln:
Buying price =  80,000/=
Selling price =  110,000/=
Profit =  S.P – B.P
=  110,000/= - 80,000/=
=  30,000/=


2. Makonde bought a pair of trouser at 19,000/= and later sold it at 15,500/=.  What loss did he suffer?
Buying price =  19,000/=
Selling price =  15,500/=
Loss =  Buying price – Selling price
=  19,000/= - 15,500/=
=  3,500/=

Activity
1. A cattle dealer bought a cow at 135,000/= and sold it at 147,000/=.  Calculate the profit he made.
2. A land agent bought a piece of land at 750,000/= and sold it at 870,000/=.  Find the profit made.
3. A trader bought 50kg of maize flour at 600/= per kg and sold it at 800/= per kg.
a) Find the profit per kilogram.
b) Find the total amount of money used to buy all the maize flour.
c) Calculate the total amount earned as profit.
4. A soda agent bought 50 crates of soda at 23,250 each and sold at sh. 24,250 each.
a) Calculate the profit made on each crate.
b) Calculate the total profit made.
5. Akram bought a radio at 75,000/= and later sold it at 69,000/=.  Calculate the loss he made.
6. A business lady bought a box of cosmetics at 240,000/= and collected 226,000/=.  Calculate the loss.

EVALUATION
Strong Points:_______________________________________________________________________
Weak Points: ____________________________________________________________________
Way forward: ___________________________________________________________________


Percentage profit and loss
Formulars
Percentage profit =  Selling price – Buying price    x 100
                        Buying price

OR: Profit      x 100
                       Buying price

Percentage loss =  Buying price – Selling price    x 100
Buying price

OR: Loss   x 100
Buying price 

Examples
1. A boy bought a bicycle at 120,000/= and sold it at 130,000/=.
a) Express the profit as a fraction of the buying price.
Fraction =  Profit
     Buying price
=  130,000/= - 120,000 /= 
120,000
=  10,000/=
     120,000/=
=  1/12
b) Calculate the percentage profit.
%age profit =  profit     x 100
     B.P
=  10,000/=   x  100
     120,000/=
=  100 25
     12  3
=  25
      3
=  8 1/3%
2. Obiina bought a bicycle at sh. 70,000.  Two years later, he sold it at a loss of 15%.
a) Calculate the loss.
%age loss =  loss    x  100
     B.P
15 =  Loss  x  100
  1      70000
100 loss =  15 x 70000
100 100
Loss =  15 x 700
=  10500/=
Activity
1. Namuwaya bought a dress at 10,000/= and sold it at 12,000/=.
 Calculate the percentage profit. 
2. Okello bought a blanket at 30,000/= and sold it at 350,000/=.
a) Express his profit as a fraction of the cost price.
b) Find his percentage profit.
3. A business man bought a 50 kg of G-nuts at 800/= per kilogram.  He paid 5000/= as transport.  If he sold each kg at 1000/=.
a) Find the total amount collected from the sales.
b) Find his profit.
c) Calculate the percentage profit.
4. A passenger bought an air ticket at 500,000/=, he later sold it at 550,000/=.
a) Express the profit as a fraction of the cost price.
b) Calculate the passenger’s percentage profit.
5.   A marial vendor bought 20 bunches of matooke at sh. 8000/=.  She sold them at a loss of 500/= per bunch.
a) Find the amount got from the sales.
b) Find the total loss.
c) Calculate the percentage loss.
6. After selling a bed at 60,000/=.  Mulonde made a percentage profit of 20%.
a) Find the price at which Mulonde sold the bed.
b) Find Mulonde’s profit.


EVALUATION
Strong Points: _________________________________________________________________
Weak Points: ___________________________________________________________________
Way forward: __________________________________________________________________
Simple Interest
In banking
- The money banked, borrowed or lent is called Principal (P)
- The percentage used to calculate interest is called Rate (R).
- The period that the principal is invested is called Time given in years or months.
- The additional amount paid back is Interest
- Total Amount = Principal  + Interest
Simple interest  = P X R X T

Examples
1. My father deposited sh. 120,000 in the bank that offers an interest rate at 10% per year.
a) Calculate the interest got after 2 years.
S I =  P x T x R
=  120000 x 2 x 10/100
=  1200 x 10 x 2
=  24,000/=

b) Calculate the amount collected after 2 years.
Amount = principal  + Interest
=  120,000/=  +  24,000/=
=  144,000/=

2. A trader borrowed 400,000 from a bank at an interest rate of 5% per annum for 6 months.
a) Calculate the simple interest.
S I =  P x T x R
=  40,000 x 6 /12 x 5/100
=  (100 x 2 x 5)/=
= 10,000/=

b) What amount will the trader pay altogether?
Amount =  Principal  + Interest
=  400,000   +  10,000
=  410,000/=
Activity
1. Calculate the simple interest on 50,000/= at a rate of 15% per year for 2 years.
2. Calculate the simple interest on 150,000/+ at 5% per annum for 3 years.
3. What interest is paid on a loan of 70,000/= at a rate of 20% per annum for 2 years?
4. A school kept 800,000/= in a bank at a rate of 15% per year for 1 ½ years.
Calculate the simple interest.

EVALUATION
Strong Points: _________________________________________________________________
Weak Points: _________________________________________________________________
Way forward: __________________________________________________________________



1. The Headteacher gave out balls to the classes to make practices for football.
Scale:  =  10 balls

P.4

P.5

P.6

P.7


a) Which class got the least number of balls?
b) How many less balls did P.6 get than P.7?
c) Find the total number of balls given to all the classes.
d) If each ball was costing sh. 20,000, how much money did the H/M pay for the balls of P.6.
2. In Maya village, five men planted three in order to trap strong winds.
Name Number of trees
Moses
Peter
Paul
Chris 300
500
450
150


a) Who planted the highest number of trees?
b) How many trees did Paul and Chris plant altogether?
c) Construct a pictograph for the information above.
d) If each seedling costs sh. 300, how much did Chris spend?
e) EVALUATION
f) Strong Points: ___________________________________________________________________
g) Weak Points: ____________________________________________________________________
h) Way forward: ___________________________________________________________________
Bar graphs
Definition:
- These are graphs constructed using bar like structures.
- The bars can be vertical or horizontal depending on the information of wish of the drawer.
Scales
- It has the vertical and horizontal scales.  The scales used depends on the broadness of the quality.

Bars
- All bars should be proportional and bars are always shaded.

Example
The table below shows the number of pupils who like different food types.
Type of food Rice Matooke Yams Millet Cassava
Number of pupils 8 6 8 4 10

We can put the data above in a bar graph.



10

8

6

4

2

0
         Rice          Matooke       Yams    Millet Cassava
Types of food
Exercise
1. The bar graph below shows the number of pupils in different classes at Mother Mony p/s.
100

80

60

40

20

0
P.1             P.2           P.3        P.4       P.5            P.6

a) Which class has the highest number of pupils?
b) What is the total number of pupils in the classes?
c) Which two classes have the same number of pupils?
d) Find the average number of pupils in the school/per class.

2. Below is a bar graph showing the number of vehicles that park at different hotels per day.

Africana

Mambo Bado

Speke Hotel

Equatorial
     0 100 200 300 400 500
Number of vehicles
i) Find the total number of vehicles that park at Speke hotel and Mando Bado.
ii) If each vehicle pays sh. 2000 for parking, how much money is collected at Equatorial hotel?
iii) Which Hotel is the least busy?
iv) How many more vehicles park at Africana that Speke hotel?

3. The table below shows the number of time each type of food is prepared in a term at a school.
Food Matooke Posho Rice Cassava Millet
Number of times 10 20 30 20 25

a) Which food type is prepared 30 times?
b) Which two food types are prepared the same times?
c) Find the total cost matooke takes in a term if each time the Headteacher spends 400,000/=
d) Draw a bar graph to show the above information.
EVALUATION
Strong Points: ___________________________________________________________________
Weak Points: __________________________________________________________________
Way forward: __________________________________________________________________
Line graph
- This type of graph is interpreted using a straight line drawn on the graph.  The line can also be zig-zag
- The graph has two scales i.e the horizontal and the vertical.
It is mostly used for;
- Distance against time used.
- Quantity of goods sold against cost.
- Temperature change against time.


Example:
A motorist covers 40km in 2 hrs.
40

Distance 20

          1 2     Hours

Exercise
1. The graph below shows the number of litres of petrol consumed by a car through a distance.

60
40
                Km
20

   0
    1    2    3    4    5    6   7
Number of litres 

a) How many kilometres does the car travel on 1 litre?
b) What distance can the car cover on 6 litres of petrol?
c) How many litres does the car need to cover 45km?

2.   The weight of Nderema’s children is shown on the graph below.

Napyo
Apio
Kaanu
Adong
Mugoli
10 20 30 40 50 60 70 80
Weight in kg
a) Who is the heaviest child?
b) Which of the children weighs 30kgs?
c) Which two children have the same weight?
d) Find the average weight of all the children.

3. The graph below shows the cost of units of different items in a supermarket.

8000

6000 Nido       Rice 

4000
  Maize flour
2000

        1kg           2kg           3kg           4kg

a) What is the cost of 1kg of rice?
b) Find the cost of 2 ½ kg of maize flour.
c) How many kg of maize flour can I buy with 3000/=
d) What is the cost of ½ kg of Nido?
e) Find the total cost of 1 kg of Nido, 4 kg of rice and 3 kg of maize flour.


EVALUATION
Strong Points: __________________________________________________________________
Weak Points: __________________________________________________________________
Way forward: ___________________________________________________________________



Pie – chart
- Pie - charts are also called circle graph.  
- The chart is circular in form.

It constitutes one of the following;
a) 1 whole
b) 10%
c) 3600


½  or 50% or 1800




¼ or 25% 0r 900



Exercise
1. A man spends his salary as shown by the pie chart below.  He earns 180,000/= per month.



                   Food             Rent
                    1500              X
                                     Fees

a) Find the value of x.
x  + 900 + 1500  = 3600
x  + 2400 = 3600
x + 2400 – 2400 = 3600 – 2400
x = 1200

b) How much does he spend on fees?
1200   x  180,000
3600
=  1 x 60,000
=  60,000/=

c) Express the expenditure on food as a fraction of the total.
1500
3600
=  5
   12

2. The pie chart below shows the types of books in the school library of 1440 books

MTC
                                        1080             SST
   yo       
                           Others           580
                                  yo         500          ENG
                                                SCI

a) Find the value of y.
b) How many books are there for MTC?
c) How many more books are for English than Science?
d) What fraction of the book represents SST books?

3. Mrs Bogoyo spends her salary of sh. 72,000 as follows;


          Fees               Food
             30%             25%
              

                       X         15%
            Rent              Medical
a) Find the percentage for rent.
b) How much does she spend on medical?
c) How much more does she spend on fees than on food?

EVALUATION
Strong Points: ________________________________________________________________
Weak Points: __________________________________________________________________
Way forward: __________________________________________________________________
Constructing pie charts
Steps:
- Change the amount given (in the sectors to degrees)
- Check and see if the degrees got add up to 3600.
- Draw a circle (with a reasonable radius)
- Use the outer scale reading of the protractor and count the degrees clockwise.
Example
1. Alupo spends his money as follows.
¼ on food, 1/3 on rent and 5/12 on fees.  Construct a pie chart of radius 3cm.
Changing fraction parts to degrees.
Food =  ¼ x 360
=  900

Rent = 1/3 x 3600
=  1200

Fees =  5/12  x 3600
=  1500

Pie chart (3cm radius)


Exercise
1. On a farm, 1/10 of the animals are goats, 1/5 are sheep, 2/5 are cattle and 3/10 are chicken.  Use the information to draw a pie chart.
2. In a village meeting, 35% of the people were women, 40% men and the rest children.
a) What percentage represented children.
b) Construct a pie chart.
3. A man got sh. 60000 from the sales of beans, sh. 80000 from peas, sh. 50000 from tomatoes and sh. 60000 from others.  Use the above information to draw a pie chart.
4. Mwebe’s land is divided into plots as shown below.
Land for; Sheep Goats Cows
Percentage 30% 50% 20%

If he had 270 hectares;
i) How many hectares are for sheep?
ii) Use the information to draw a pie chart.

STATISTICS  (Mean, Median, Mode and Range)
Definition:
Mean: Is the total/sum of scores divided by the number of scores.
Median: Is the midway mark between the highest and lowest.
Mode: Is the score which appears more than others.
Range: the difference between the highest and lowest marks.
Frequency: The number of times an event occurs.

Example:
Given  9 , 2 , 6 , 3 and 4, Work out;
a) The mean
Mean = Total of scores
    Number of scores
=  9 + 2 + 6 + 5 + 6 + 3 + 4
7
=  35
      7
Mean =  5

b) The mode
Scores freq
9 1
2 1
6 2
5 1
3 1
4 1
6 is the mode
c) The range
Range = Highest – Lowest
=  9 – 2
Range =  7


d) Median
Median =  2  ,  3  ,  4  ,  5  ,  6  ,  6  ,  9
Median =  5

Activity
1. Given that 18 , 12 , 6 , 24 , and 30.  Find the;
i)         Mean
ii) Range
2. Work out the median of 6 , 7 , 4 , 9 and 8.
3. Given  20, 4 , 8 , 8 , 4 , 7 and 14, find;
i)          The mode
ii) The median
4. Okello scored the following marks in a series of maths tests; 55% , 60% , 40% 60% , 55% , 80% and 60%.
i) Find his modal mark.
ii) Work out the average

5. Calculate the mean of 40 and 42.

EVALUATION
Strong Points: ____________________________________________________________
Weak Points: _______________________________________________________________
Way forward: _______________________________________________________________

MORE ABOUT STATISTICS
Examples
The table below shows a pupil’s marks in a science test.
Scores 80 70 60 12 30 40
Frequency 1 2 3 1 2 5

Find;
a) The modal mark
Modal mark was 40
b) The modal frequency
Modal frequency was 5

c) The mean
Mean =  80 + (70x2) + (60x3) + (30 x2) + (40 x 5)
      14
=  80 + 140 + 180 + 60 + 12 + 200
14
=  672
      14
=  48




             Activity
1. A group of boys was given a test.  They scored as follows;
Scores 20 80 60 70
Number of boys
i) How many boys did the test?
ii) Find the modal mark
iii) Calculate the mean mark.

2. The following are the marks in their raw form.
20, 15 , 10 , 35 , 40, 10 , 30 , 30 , 20 , 40 , 30 , and 40.
i) Make a table to show the scores, tallies and frequency.
ii) What is the modal frequency
iii) What is the mode?
iv) Find the range of scores.


MORE ABOUT AVERAGE
Examples
1. The mean age of 5 pupils is 14 yrs.  The age of 4 of the pupils are 16 , 12 , 13 and 15.
a) What is the age of the fifth pupil?
Total age of 5 pupils =  5 x 14  
=  70 yrs

Total age of the 4 pupils = 12 + 16 + 13 + 15 
=  56 yrs
Age of the fifth pupil =  70 – 56
=  14 years

b) Find the median.
Ages in order =  12 , 13 , 14 , 15 , 16

Median age = 14 years
2. The average weight of 4 men is 55kg. If one of the men weighs 70kg, what is the average weight of the other 3 men?
Total weight of all (4 x 55) =  220kg
Weight of one man   -   70kg
Weight of 3 remaining men =  150kg
Average weight for the 3 men =  150
              3
=  50 kg

Exercise
1. The mean age of 4 girls is 16 years.  If three of them are aged 17, 12 and 15, find the age of the 4th girl.
2. The average number of books for 4 classes is 20.  If one class has 23 books, what is the average number of books does the other class have?
3. The average of 3 numbers is 15. Of which one of them is 21, find the average of the other 2 numbers.
4. The average length of 8 sticks is 9cm.  If three sticks of them measure 6cm, 10cm and 6cm, find the average length of the remaining 5 sticks.

EVALUATION
Strong Points: _________________________________________________________________
Weak Points: ___________________________________________________________________
Way forward: ___________________________________________________________________
PROBABILITY
Probability is the measure of chances for an even to occur.
Some things happen - certain
 Some things don’t ever happen - impossible
Some things may happen sometimes likely



The probability scale
0% 50% 75% 100%

0  very unlikely      ¼ unlikely    ½ likely       ¾                                   very                   
  (impossible)                           likely(certain)

If something can’t happen then the probability is 0.
If something can happen certainly, then the probability is 1.
Therefore:  0 <  p(x)  <  1       
Probability  = Desired chance
                            Sample space  

Example
1. Amos has five cards numbered; {3 , 4 , 5 , 6 , 7 }
If the cards are mixed and put in a box, what is the probability that he randomly chooses;
a) a card marked 5?
Prob (5) = 1/5

b) a card of an even number?
Even numbers {4 , 6 }
Prob (even)  = 2/5

c) a counting number?
Counting numbers = {3 , 4 , 5 , 6 , 7}
Prob (counting numbers)  = 5 /5  =  1

Exercise
1. A die is tossed once.  What is the probability that;
a) An odd number appears on top.
b) A number less than 4 appears on top.
2. Okello will visit his mother next week.  What is the probability that he will visit her on;
a) a day that begins with letter “T”
b) Sunday
c) on a day that ends with letter “Y”
3. 4 red pens and 6 blue pens are mixed and put in a box.  If a pen is picked at random, what is the probability that it is;
a) a red pen
b) blue pen
c) black pen
4. Letter card are as below.
F L O W   E    R
If the cards are mixed and put in a bucket, what is the probability of picking at random;
a) a vowel
b) a consonant

EVALUATION
Strong Points: _________________________________________________________________
Weak Points: ___________________________________________________________________
Way forward: _________________________________________________________________
MEASURES  (TIME)
Note:  1 hr =  60mins
1 min = 60sec
1 hr = (60 x 60) sec
= 3600 seconds 



Changing hours to minutes
Example
Change 4 hrs to minutes.
1 hr = 60 mins
4 hrs = (4 x 60) mins
= 240mins
=  240 minutes

Activity
Change the following hours to minutes.
a) 3 hrs b)  11 hrs c)  4 ½ hrs
d)   6 hrs e)  1 ½ hrs f)  20 hrs 

Changing minutes to seconds
Example
Change 5 minutes to seconds
1 min = 60 seconds
5 mins = (5 x 60) seconds
=  300 seconds

Activity
Change the following minutes to seconds
a) 20 minutes b)  30 minutes c)  15 minutes
d)   72 minutes e)   60 minutes f)  25 minutes
g)   10 minutes h)  3 minutes i)  50 minutes

Changing hours to seconds
Example
How many seconds are there in 1 hr.
1 hr = 60 mins
1 min =  60 seconds
Therefore:  60 mins = (60 x 60)
=  3600 seconds

Activity
Change the following hours to seconds.
a) ½ hrs b)  4 hrs c)  6 hrs
d)   5 hrs e)  20 hrs f)  ¼ hrs
g)  3 ½ hrs h)  ¾ hrs i)  1/10 hrs

EVALUATION
Strong Points: __________________________________________________________________
Weak Points: ____________________________________________________________________
Way forward: __________________________________________________________________
MEASURES (DISTANCE , TIME & SPEED)
Distance , Time and speed go can be realized concurrently.
ITEM UNITS
Distance km, metres
Time hours, minutes , seconds
Speed (the rate of moving) m/sec , km/hr
 
Examples
1. A car moves 40km every hour.  Find its average speed.
Its speed  = 40km/hr
2. A tax covers 40km in 30 mins
Therefore; 40km take = 30m
80km take =  (30 +30) min
Therefore its speed =  80km/hr
Finding distance
Distance  = speed x time





Example
Find the distance covered by a cyclist at 14km/hr for 3 hrs.
Distance = speed x time
=  14km  x 3 hrs
                                  hr
=  (14 x 3)
=  42 km

Activity
1. A bus moved at a speed of 60km/hr for 4 hrs.  What distance did it cover?
2. The speed of a train is 20km/hr.  What distance does it cover in 5 hrs?
3. Mayanja drove his car for 45km at a speed of 40km/hr.   What distance did he cover?
4. A motorist makes a journey from 8:30am to 10:50am at  70km/hr.  What distance does he cover?
5. What distance is covered by a lorry that moves for 6 hrs at a speed of 35km/hr.

EVALUATION
Strong Points: _________________________________________________________________
Weak Points: __________________________________________________________________
Way forward: ___________________________________________________________________
MORE ABOUT DISTANCE (Given time points)
1. Moving at 60km/hr, a bus completed the journey from 10:30am to 1:20pm.  How long was the journey?
2. At a speed of 54km/hr a cyclist left Katonga at 9:00am and arrived at Kamapala at 12:30pm.  How far is Kampala from Katonga?
3. A bus moves at an average speed of 90km/hr from 8:15am to 11:15am.  What distance does it have?

Finding Time
Example:
How long will it take a car to cover a distance of 120km at a speed of 40km/hr?
Speed =  Distance
         Time
Time =  Distance 
Speed
=  100km    ÷  40km
hr
=  120km    x     hr
        40km
Time =  3 hrs

Activity
1. How long will it take a car to cover a distance of 80km at a speed of 20km/hr?
2. A bus travels at a distance of 200km at a speed of 40km/hr.  Find the time it takes.
3. The speed of a cyclist is 70km/hr.  How long will he take to cover a distance of 350km?
4. Moving at a sped of 70km/hr, Lule covered a distance of 490km on his bicycle.  How many hours did he take?
5. It is 220km from Masaka to Kampala.  How long will a car take to cover that distance at a speed of 40km/hr?

EVALUATION
Strong Points: _________________________________________________________________
Weak Points: __________________________________________________________________
Way forward: _________________________________________________________________


Finding speed
Speed = Distance
Time
Example 
A car travels for 2 hrs to cover a distance of 210km.  Find its average speed.
Speed =  Distance
Time
=  210hr
3hr
=  70km/hr

Activity
1. A bus travelled for 2 hrs to cover a distance of 160km.  At what speed was the bus travelling?
2. Muguma took 10 minutes to run a distance of 100m.  What was his speed (m/min).
3. Find the speed at which a driver should drive to cover a distance of 240km in 5 hrs.
4. Find the average speed of a train that covered 144km in 4hrs.
5. It is 150m from Kampala to Malaba.  At what speed should a taxi run to cover the journey in 2 ½ hrs.

Changing km/hr to m/sec
Example
Change 180km/hr to m/sec.
1 km = 1000m
1 hr =  3600 sec
Therefore; Distance in metres
Time in seconds
=  180 x 1000
(3600 x 1) sec
=  50m/sec


Exercise
Express the speed below in m/second.
a) 36km/hr b)  72km/hr c)  216 km/hr
d)  162km/hr e)  144km/hr f)  360km/hr
Changing m/sec to km/hr
Example
Convert 20m/sec to km/hr.
1000m = 1km
Therefore;  1m =   1 km
   1000
20m =    20    km
                   1000
=  20/1000    ÷  (1/3600)  hr
=  20 km    x  3600
    1000             1hr
=  2 x 36
=  72km/hr

Activity
Convert to km/hr.
a) 40m/sec b)  30m/sec c)  90m/sec
d)  100m/sec e)  25m/sec f)  70m/sec
g)  60m/sec h)  150m/sec i)  50m/sec


EVALUATION
Strong Points: ________________________________________________________________
Weak Points: ___________________________________________________________________
Way forward: __________________________________________________________________

Finding Average speed
Example
A car takes 3 hrs to cover a certain journey at 60km/hr but it takes only 2 hrs to return through the same distance.  Calculate the average speed for the whole journey.
Distance = S x T
= 60km   x  3 hr
                              hr
=  180km
Total distance =  (180 x 2)
Total time = 3 + 2
=  5
Average speed = 180 x 2
          5
=  72km/hr

Activity
1. A car takes 2 hrs to cover a certain distance at 60km/hr but it returns in 3 hrs.  Calculate its average speed for the whole journey.
2. A lorry takes 4 hrs to travel from Kampala to Lyantonde at 45km/hr but returns in 6 hrs.  Find its average speed for the whole journey.
3. Ali took 4 hrs to cover a journey at 60km/hr but it takes only 2 hrs to return though the same distance.  Calculate its average speed for the whole journey.
4. Bosco ran for 3 hrs at a speed of 6km/hr and another 2 hrs at a speed of 5km/hr.  Find the average speed for the whole journey.
5. A bus takes 6 hrs to cover a distance of 80km/hr but it returns in only 4 hrs.  Calculate its average speed for the whole journey.

EVALUATION
Strong Points: _________________________________________________________________
Weak Points: ___________________________________________________________________
Way forward: ___________________________________________________________________
TRAVEL GRAPHS
1. Town A and C are 100km apart.  A motorist travelled from A to B for 2 hrs at a speed of 30km/hr, rested for 30mins and continued to C in only 1 ½ hr.

  100 C

80

60 B
Km
40

20

A A
9am 10am 11am 12noon

i) What is the distance from A to B.
ii) How far is C from B?
iii) At what speed did he more from B to C.
iv) What time did he take from A to B.
v) Find the average of the motorist for the whole journey.

EVALUATION
Strong Points: __________________________________________________________________
Weak Points: ___________________________________________________________________
Way forward: ___________________________________________________________________





NAMUWAYA EDUCATION CENTRE
TERM III LESSON NOTES 2019 BY CHAMBE IBRA.
MEASURES
CIRCUMFERENCE
Finding Radius
Radius = Diameter
         2
Example
1. Find the radius of a circle of diameter 9cm.
Radius = Diameter
2
=  9cm
     2
=  4 ½ cm    or 4.5cm

Activity
Find the radius of a circle whose diameter is;
a) 6cm b)  12cm c)  24cm
d)   20cm e)  45cm f)  100cm
g)   17cm

Finding diameter
Example
Find the diameter of a circle whose radius is 15cm.
Diameter = Radius x 2
=  15cm x 2
=  30cm

Activity
Find the diameter of a circle whose radius is;
a) 4cm b)  10cm c)   16cm d)  6cm
e)  10 ½ cm f)   11cm

EVALUATION
Strong Points: ________________________________________________________________
Weak Points: _________________________________________________________________
Way forward: ________________________________________________________________

Calculating the circumference of a circle
Example
1. Find the circumference of a circle whose diameter is 10cm.  (Use Ο€ = 3.14)
C = Ο€ D
=  3.14 x 10cm
=  31.4cm

2. Calculate the circumference of a circle whose radius is 3 ½ .(Use Ο€ = 22/7)
C =  Ο€ D  or Ο€ r
=  2 x 22 x 3 ½ cm
                         7
=  2  x  22  x  7  cm
                1        7       2 
C =  22cm

Activity
1. Find the circumference of a circle whose diameter is 5cm.  (Use Ο€ =  3.14)
2. A circular bottom of a mug has a radius of 50mm.  Find the circumference. 
(Use Ο€ = 3.14)
3. Find the circumference of a circle whose radius is 7cm.  (Use Ο€ = 22/7)





Area of squares
Examples
1. Find the area of a square whose side is 6cm.

6cm

Area =  side x side
=  6cm x 6cm
=  36cm2
2. Find the area of a square whose side is pcm.

Pcm

Area =  side x side
=  pcm x pcm
= p2 cm2

Activity
1. Find the area of the following squares.
a) b)
                                                                                                            10cm

7cm
2. The side of a square is 8cm.  Find its area.
Finding the side of the square
Example
1. The area of a square is 64cm2.  Find the length of each side of the square.
Let each side be y.

ycm

y x y = Area
√y2 =  √64
√y x y =  √2 x 2 x 2 x 2
y  =  8
Therefore each length = 8cm
64
     2       32
2      16
2 8
      2       4
2 2
       2        1
Activity
1. Find the length of each side of the square whose area is;
a) 25cm2 b)  36cm2 c)  81cm2
2. Find the area of a square piece of paper whose side is 12cm.  Find the value of x2.
3. If 3x2  = 27.  Find the value of x.
4. The area of a square is 900cm2.  Find the length of each side of the square.


EVALUATION
Strong Points: ___________________________________________________________________
Weak Points: __________________________________________________________________
Way forward: ___________________________________________________________________







Area of rectangle
Example
A rectangle is 10cm long and 5cm wide.  Find the area of the rectangle.

                                          5cm
                   10cm
Area of a rectangle =  Length x Width
=  10cm x 5cm
=  50cm2

Activity
1. The length of the field is 700m long and 600m wide.  Find its area.
2. Find the area of a rectangle whose length is 40cm and width 30cm.
3. A rectangle measures 25m by 20m.  Find its area.
4. The length of a rectangle field is 120m by 80m.  Find the area of the field.



Finding the side of a rectangle when area is given
Example
The area of a rectangle is 56cm2.  The length is 8cm.  Find the width of the rectangle.

   Area = 56cm2    w
              8cm
Length x width = Area
8cm x w =  56cm2
8cm x w =  56cm2
8cm      8cm
Width =  7cm

Activity
1. A rectangular piece of paper is 4800mm2, its width is 60mm.  Find its length.
2. The area of a rectangular field is 96cm2, its width is 8cm.  Find the length.
3. The area of a rectangle is 42cm2.  The length is 7cm.  Find its width.  
4. A rectangular garden is 50m2, its width is 5cm.  Find its length.

EVALUATION
Strong Points: __________________________________________________________________
Weak Points: __________________________________________________________________
Way forward: __________________________________________________________________
Finding area when given perimeter
Example
1. The perimeter of the rectangle is 24cm and the width is 5cm.
a) Find length
b) Find area

P = 24cm      w (5cm
                      Length 
L + W + L + W =  Perimeter
L + L  + W + W = p
2L + 5cm + 5cm =  24cm
2L + 10cm – 10cm  =  24cm – 10cm
2L =  14cm
2       2
L =  7cm
The length = 7cm
Area = Length x Width
=  7cm x 5cm
=  35cm2




Activity
1. The perimeter of a rectangle is 40cm and the width of the rectangle is 8cm.
a) Find the length of the rectangle.
b) Find the area of the rectangle.
2. The perimeter of a rectangle is 36cm, its length is 10cm.
a) Find its width.
b) Find its area.
3. The perimeter of a rectangular garden is 80cm, its width is 18cm.
a) Find its length.
b) Find its area.

EVALUATION
Strong Points: ______________________________________________________________
Weak Points: ________________________________________________________________
Way forward: ________________________________________________________________
Finding sides, area & perimeter
Example
ABCD is a rectangle.
A       (2x – 5)cm B
     (x – 1)cm

D        (x + 3)cm C
a) Find the value of x.
2x – 5 =  x + 3
2x – 5 + 5 = x +3 + 5
2x – x = 8
x = 8



b) Find the width and the length.
Width Length
=  (x +3)cm =  (x -1)cm
=  (8 + 3)cm =  (8 – 1)cm
=  11cm =  7cm
c) Find the area and perimeter.
Area Perimeter
A = L x W P = 2(L x W)
A = 11cm x 7cm P =  2(11 + 7)
A = 77cm2 P = 36cm

Activity
1. Work out the following.
             (2x)cm
5cm

(x + 6)cm
The diagram is a rectangle.
a) Find the value of x.
b) Find the length.
c) Find the area.
d) Work out the perimeter.
2.            x + 9cm
x
                                   2
         2x + 1cm
a) Find the value of x.
b) Find the length and width of the rectangle.
c) Find the perimeter of the rectangle.
d) Find the area of the rectangle.




3.     (2y + 5)cm

            (y+1)cm

            (4y + 3)cm
a) Find the value of y.
b) Find the width and length of the rectangle.
c) Work out the area
d) Find the perimeter of the rectangle.

EVALUATION
Strong Points: _________________________________________________________________
Weak Points: ___________________________________________________________________
Way forward: ___________________________________________________________________
Area if trapezium
Example
Find the area of the trapezium below.
                      8cm

     7cm

                      10cm
Area of trapezium =  ½ h (a + b)
=  ½ x 7cm (8 + 10)
=  ½ x 7 x 18 cm
=  ½ x 7cm x 9cm
=  7cm x 9cm
=  63cm2



Activity
1. Find the area of the shapes below.
a)                  5cm

6cm
                                  
                                    8cm

b)                                               6cm

   8cm

10cm

c)                                4cm
                                  
  9cm
  
                          8cm
2. Find the area of a trapezium of height 10cm and the parallel sides are 12cm and 18cm.
3. If A = ½ h (a + b).  Find A if h = 8cm, a = 14cm and b = 15cm.


EVALUATION
Strong Points: __________________________________________________________________
Weak Points: ____________________________________________________________________
Way forward: ___________________________________________________________________



Finding one side of a trapezium
Example
The area of a trapezium is 60cm2, height is 4cm and one of the parallel sides id 10cm.  Find the length of the second parallel side.
    10cm

    4cm

  a
½ h(a + b)  = Area
½ x 4 (a + b)  A
2(a + 10)  =  60
2a  + 20  =  60
2a + 20 – 20 = 60 – 20
2a = 40
2        2
a  =  20cm

Activity
1. Find the length of the second parallel side of a trapezium if the area is 56cm2, the height 8cm and one of the parallel sides is 4cm.
2. A = ½ h(a + b).  Find the value of A, if b = 6cm, h = 9cm and a = 10cm.
3. The parallel sides of a trapezium are 10cm and 12cm.  Find the height of the trapezium if the area is 77cm2.



EVALUATION
Strong Points: ________________________________________________________________
Weak Points: __________________________________________________________________
Way forward: __________________________________________________________________

Area of parallelogram
Example
Find the area of the parallelogram below.
15cm

9cm 10cm

Area of parallelogram  = b x h
=  15cm x 9cm
=  135cm2

Activity
Find the area of the shapes below
1.  
      7cm                     8cm

12cm
                                                                                                                    
2.  
11cm        14cm

                                    18cm
3.         A                                     B

7cm                                                  8cm

      E              D           13cm              C
Area of triangle
Examples
1. Find the area of the 

         8cm
                            6cm
=  ½ b h
=  ½ x 6cm x 8cm
=  3cm x 8cm
=  24cm2
2. Workout the area of the triangle below.
A  = ½ x b x h
8cm A = ½ x 7cm x 8cm
                                     7cm A = 7cm x 4cm
A = 28cm2
Activity
Find the area of the triangle below.
1.  
5cm
                  9cm
2.  
     9cm                      7cm
                                           6cm
    8cm

3.  
7cm                    11cm

10cm


Find the area of the shaded triangle
4.  
6cm

5cm            4cm

5.  

                              10cm             
                                                8cm 6cm
                        5cm                3cm

6.  
    14cm                          8cm
                                           11cm
                 8cm                  4cm
EVALUATION
Strong Points: _________________________________________________________________
Weak Points: ___________________________________________________________________
Way forward: __________________________________________________________________









Finding one side of a triangle when area is given
Example
Find the base of a triangle whose area is 60cm2 and height is 12cm.


                       12cm
               
                  base
½ x base x height = Area
½ x b x 12cm = 60cm2
b x 6cm =  60cm2

6b cm =  60cm x cm
6 cm        6cm
b =  10cm

Note:  Base  =  2 x Area
                 Height
Height =  2 x Area
        base

Activity
1. Find the height of a triangle whose area is 36cm2 and its base is 12cm.
2. Find the base of a triangle whose area is 20cm2 and height 8cm.
3. The area of a triangle is 40cm2.  Find the height if the base is 10cm.
4. The height of a triangle is 9cm and its area is 36cm2.  Find its base.

EVALUATION
Strong Points: ________________________________________________________________
Weak Points: ____________________________________________________________________
Way forward: ___________________________________________________________________
Finding base or height by comparing area
Example
ABD is a triangle.  AC and BE are heights of the same triangle.  BD  = 12cm AC = 10cm BE = 8cm.  Find length AD.
A

   10cm
E
8cm
              B C D
12cm

½ x AD x 8cm    =  ½ x 12cm x 10cm
AD x 4cm =  6cm x 10cm
4ADcm =  60cm

4ADcm =  60cm x cm
  4cm         4cm
AD =  15cm

Activity
Find the value of the unknown in the figure below.
1.  
16cm
12cm      h

20cm
2.         A                   32cm                      B
                                             h
                              30cm                                 C 48cm

D
3.   
15cm Y
                         X
       12cm
                                     W 20cm
  15cm
Z
EVALUATION
Strong Points: ___________________________________________________________________
Weak Points: ___________________________________________________________________
Way forward: ___________________________________________________________________
Finding sum of area
Example
Find the total area of each figure below.
a)     

8cm       A
3cm                 4cm
  B
                    6cm                    9cm

Area of fig. A Area of fig. B
=  ½ x b x h =  ½ x b x h
=  ½ x 4 x 8 =  ½ x 7 x 6
=  2 x 8 =  7 x 3
=  16cm2 =  21cm2
Area A + Area B
=  (16 + 21)
=  37cm2

b)  
                           7cm
             
                             A                B 8cm
3cm
Area of fig. A Area of fig. B
=  ½ x b x h =  L x W
=  ½ x 3 x 8 =  8 x 7
=  12cm2 =  56cm2
Total area  = (12 + 56)cm2
=  68cm2
Exercise
Find the area of the shapes below.
1.  


20cm
        36cm
15cm

2.            
            9cm

 
    5cm
    7cm

                                                                8cm
EVALUATION
Strong Points: ___________________________________________________________________
Weak Points: ____________________________________________________________________
Way forward: ___________________________________________________________________
Finding differences of Area
Examples
1. Find the area of the shaded part.
11cm

8cm                                      4cm
                                  9cm             
Area of outer rectangle =  L x W
          =  11cm x 8cm
=  88cm2

Area of inner rectangle =  L x W
=  9cm x 4cm
=  36cm2

Area of shaded part =  88cm2 – 36cm2
=  52cm2

2. Study the diagram below carefully and find the area of the shaded part.
         16cm
             2cm

        2cm      3cm     10cm
                                           
                                                1cm


Finding length of inner figure.
=  16 – (2 + 3)
=  16 – 5
=  11cm

Finding width of inner figure.
=  10 - (2 + 1)
=  10 – 3
=  7cm

Area of outer fig. =  L x W
=  16 x 10
=  160cm2

Area of inner fig. =  11cm x 7cm
= 77cm2


Exercise
Find the area of the shaded parts.
1.     2.                    15cm

  5cm                    9cm

8cm        6cm
        12cm                 10cm


            

                              7cm   

       3. 

14cm
                                               10cm

Finding total surface area of cubes and cuboids
Examples
1. Find the Total Surface Area of the cube below.
T.S.A =   (S x S) + (S x S) + S x S)
=  2 (S2 + S2 + S2)
=  2 x 3S2
=  6s2
                                                     5cm =  6 x 52
=  6 x 25m
=  150m2
2. A cuboid measure 8cm long 4cm wide and 5cm high.  Find its volume.

                                                      5cm
4cm   
                        8cm
T.S.A = 2 (lw  + wh + lh)
=  2(8x4 + 4 x 5 + 8 x5)
=  2 (32 + 20 + 40)
=  2 x 92
=  184cm3

Exercise
Find the area of the figures below.
a)     b)  
2cm
                 3cm 8cm
   4cm

4cm
                       2cm
      c)  d)  
3cm
16cm
Finding the measures of each side of a cube given its total surface area
Examples
The total surface area of a cube is 384cm3
T.S.A  =  6s2
6S2 =  384
6S2 =  384
6                6
S2 =   64
√S2 =  √64
S =  8cm
Each side measures 8cm

Activity
Find the measure of each side of a cube whose total surface area is;
a) 96cm2 b)  150cm2 c)  726cm2
d)   600m2 e)  2646m2 f)  2904m2
g)   486cm2 h)  6m2


EVALUATION
Strong Points: __________________________________________________________________
Weak Points: ____________________________________________________________________
Way forward: ___________________________________________________________________
Finding volume of cubes and cuboids
Volume is the amount of space occupied by an object.
Units can be m3 , cm3 , mm3

Examples
1. Find the volume of cuboid below.
                                     3cm

                                     2cm
           3cm
Volume = l x w x h
=  3 x 2 x 3
= 6 x 3
=  18cm3

2. A cube measures 4cm a side.  Find its volume.
Volume = S x S x S
  =  S3
=  4 x 4 x 4
=  16 x 4
                 4cm =  64cm3

Activity
1. Find the volume of the figures below.

5cm
             8cm
                                              4cm
                         6cm         2cm
                                                                                                          
5cm
        2. Calculate the volume of a cuboidal tank whose length is 11m, width 4m and 
             height 6m.
         3.  A cubical milk tank measures 6m a side.  Find the volume of the tank.
          4.  A soap box measures 40cm by 10cm by 5cm.  What is its volume?

EVALUATION
Strong Points: ___________________________________________________________________
Weak Points: ____________________________________________________________________
Way forward: __________________________________________________________________
Finding one unknown side of a cuboid given the volume and the other sides.
Examples
1. Find the height of the rectangular prism below whose volume is 180cm3.

h
                                                        4cm
                              9cm
Volume  = l x w x h
  9 x 4 x h = 180
  36h =  180
  36 h = 180
  36      36
h =  5cm
height  = 5cm
2. Find the width of a rectangular prism whose volume is 480cm3, length 15cm and height 8cm.

8cm
wcm
15cm
Vol = l x w x h
15 x 8 x w = 480cm3
120cm w =  480cm3
120cm2w = 480cm3
120 cm2     120cm3
w =  4cm




Exercise
1. Find the side marked by the letter.
Volume = 36cm3
3cm
                                                          wcm
    6cm
      
          h The volume is 120cm3
5cm

       6cm
     3.    The volume of a rectangular prism is 135m3.  If the height and length are
             3m and 5m respectively, what is the width?
3. The square base area of a cuboidal prism is 36cm2, and the volume is
 360cm2.   Find the height of the tank.

Finding volume of rectangular prism/cubes in litres
Examples
1. A rectangular tank is 30cm by 60cm by 90cm.  Find its volume in litres.

                                                 90cm
                                                          60cm
                                  30cm
          Vol. of the tank  = l x w x h
=  (30 x 60 x 90
But 1 litre  = 1000cm3
No of litres in the tank  = (30 x 60 x 90)cm3
         1000 cm3
=  3 x 6 x 9
=  18 x 9
=  162 litres



2. A tank in form of a cube measures 50cm a side.  Find the number of litres at full capacity.
Vol  = S x S x S
        =  (50 x 50 x 50)cm3
But 1 litre = 1000cm3
Number of litres = 50 x 50 x 50
        50cm                  1000
=  5 x 5 x 5
=  25 x 5
=  125 litres
Activity
1. Calculate the volume in litres of a rectangular tank 80cm by 70cm by 20cm.
2. Below is a tank.  Find its volume in litres.

70cm
4cm            
                  100cm   
3. How many litres of petrol are in a rectangular tank measuring 50m by 100m by 20m?
4. The bottom area of a rectangular tank is 1480cm2.  Find its volume in litres if the height is 300cm.

EVALUATION
Strong Points: _________________________________________________________________
Weak Points: __________________________________________________________________
Way forward: _________________________________________________________________





MORE ABOUT VOLUME IN LITRES
Examples
1. The tank below is holding 72 litres of water.
i) Calculate the volume of h.
Volume of tank = 72litres
But 1 litre =  1000cm
Therefore; 72 hrs = 72 x 1000
40 x 60 x h = 72000
h = 72000
  40 x 60
h =  120
       4
=  30cm
ii) How many litres are needed to fill the tank?
Height needed  = (80 – 30)
=  50cm
Volume needed =  40 x 60 x 50
=  2400 x 50
=  120000cm3
Number of litres =  120000
     1000
=  120 litres
2. Container has a volume of 108000cm3.
i) Find the capacity of the container.
ii) If the container is ¾ full of water.  How many litres are needed to fill it?

Changing litres to millitres
Conversion table
L dl cl ml
1 0 0 0
1 0 0
1 0
1 litre = 1000ml
Example
1. Change 7 litres to millitres
L  to ml
1 l  = 100ml
7l    = (7 x 1000)ml
=  7000ml

2. How many ml are in 4 ½ l?
L to ml
1 l =  1000ml
9l =  9  x 1000 ml
2              2  
=  4500ml

Exercise
1. Change the following litres to millitres
a) 5 litres b)  6 ¼ litres c)  0.8 litres
d)   7 litres  300 lillilitres
More about litres and millitres
2. A cow gives 14 litres of milk daily.  Express its daily milk production in millitres.
3. Aber fetched 15 litres of water and Mugura 18 litres of water.  How many millitres of water did the two fetch?

Changing millitres (ml) to litres (l)
Examples
1. Change 3500 millitres to litres
Ml to l
1000ml =  1l
1ml = 1  l
1000
3500ml =  (1/1000  x  3500)l
=  1/1000  x  3500
=  3 ½ litres
OR  =  3.5 litres

Exercise
1. Express the following millitres as litres.
a) 2000ml b)  6000ml c)  12000ml
d)   8500ml e)  870ml f)  5600ml
    2.  A baby takes 250 millitres of milk every feeding.
         i) How many litres does it take in 1 feeding.
         ii)  How many litres does it take in 4 feedings.


EVALUATION
Strong Points: ___________________________________________________________________
Weak Points: ____________________________________________________________________
Way forward: ___________________________________________________________________

GEOMETRY
Construction of regular polygons
- Equilateral triangle
- Square
- Pentagon
- Hexagon
- Septagon/heptagon
- Octagon
- Nonagon
Construction of parallel lines
Angle proportion parallel lines
A line which intersects a set of parallel lines is called transversal line
 


Transversal line
When a transversal line intersects a pair of parallel lines, 8 angles are formed.
Examples
   p      x
  m    a1
n     b1                q       y
<a  + <b1  = 1800 <x  + <y = 1800
<m  + <n  =  1800 <p  + <q  = 1800
Finding the unknown angles
Examples
1.  
X0
                                     mo
c + 1110 = 1800 (co-interior <)
x + 1110 – 1110 = 1800 – 1110
x  = 690
2.       750


                                         p +600
p + 600 + 750  = 1800 (co-ext <)
p + 1350 – 1350 =  1800 – 1350
p = 450

Activity
1.  
                                 x
                                   1000
2.  
                  300

     p
3.  
                                        500

   p
Alternate angles
1.    
x
                                   x1
<x  = <x1 (alternate <s)

2.  
b
b1
<b = <b1 (alternate <s)
3.  
600
p
<p = 600 (alternate <s)

4.  
2x
     1000
2x = 1000 (alternate <s)
2x = 100
2       2
x  = 500




Activity
1. Find the size of the marked angles below.

2m                                                          850

                 600 p



EVALUATION
Strong Points: _______________________________________________________________
Weak Points: _________________________________________________________________
Way forward: __________________________________________________________________
Finding angles formed by parallel lines
Examples
1. Find the values of the unknowns below.

3p                     p+20

                            600                       1200
3p + 600 = 1800
3p + 600 – 600 = 1800 – 600
3p = 120
3p = 120
3          3
p  = 400






2.     
    1100
  100
x
a          b
a + 1000 = 1800
a + 1000 – 1000 = 1800 – 1000
a = 800

b + 1100 = 1800
b + 1100 – 1100 = 1800 – 1100
b  = 700


a  + x + b = 1800
800 + x + 700 = 1800
1500 + x = 1800
1500 – 1500 + x = 1800 – 1500
x  = 300

1. Find the value of the unknowns
x0                 x – 400

                   700                 1100

2.  


             500                             x   x



3.  
   720     y      2x

            p                           z

CONSTRUCTION
- Constructing perpendicular lines on a given point.
- Constructing perpendicular bisectors on a given line.
- Constructing perpendicular bisectors from a given point.
- Constructing of angles (300 , 600 , 900, 450 ….) using a pencil, ruler and a pair of compasses.
- Constructing triangles using;
S.S.S
S.A.S
A.S.A
- Constructing quadrilaterals
Rectangles
Squares
Vertically opposite angles
P R
b
a   c
d
             T Q

Lines TR and PQ have a common point O.
O is the point of intersection
<a  and <c are vertically opposite angles and are equal.
<b and <d are vertical <s and they are equal.



Activity
1. Find the size of the angles marked by letters giving reasons.

b
a 300
c
i) <a =  _______
ii) <b + 300 =  ______
iii) <b =  ________
iv) <a + <b = ________
v) <a ++ <c =  _________

2.  
     p          700

3.  
x 1000

4.  
               x
              x                   x

5.  
                                                           400
   y


900
An angle with 900 is called a right angle.
Any two angles that add up to 900 are complementary angles.
Complementary angles add up to 900.
400 and 500 are complementary angles because 400 + 500  = 900
400 is a complement of 500.

Finding complementary angles
What is the complement of 300?
Let the complement be y.
y + 300 = 900
y + 300 – 300 = 900 – 300
y = 600

Activity
What is the complement of each of the following?
a) 150 b)  890 c)  750
d)   500 e)  270 f)  250
    g)  (x + 200)              h)  (x – 200)

EVALUATION
Strong Points: ___________________________________________________________________
Weak Points: ____________________________________________________________________
Way forward: ___________________________________________________________________
Angle descriptions
- Acute angles
- Obtuse angles
- Right angles
- Angles on a straight line
- Angles at a point


Interior and exterior angles of a triangle
- Right angled triangle
- Equilateral triangle
- Scalene triangles
- Isosceles triangles

Angles in a triangle
Find the size of the angles marked by letters.
1. 2.  
x             400                 c

                      400           600                                              300

      3.         
                            m
                       300

4.  
              600

                      600                                x






Angles of Isosceles triangles

          b Two base <s of an isosceles triangle are equal
 
             700                   700
      b + 700 + 700  = 1800
b + 1400 – 1400 = 1800 – 1400
b = 400

Activity
a) b)  
    40 d

       x        m                             p
c) 
      e

      m                  450
MORE ABOUT INTERIOUR AND EXTERIOR ANGLES
  a
                               500
       b       600                700
c
Using angles on a straight line
<a  = 1800 – 500
=  1300
<b = 1800 – 600
=  1200
<c = 1800 – 700
=  1100

1300

           600          700
<a  = 600 + 700
=  1300
Finding centre angles given sides
1. Find the size of each centre angle of a regular polygon of 3 sides.
All centre angles = 3600
Each centre angle = 3600
       3
=  1200
2. A regular polygon has 12 sides.  What is the size of each centre angle?
No of sides  = 12
All centre angles = 3600
Each centre angle = All centre
    No of sides
=  3600
       12
Centre angle = 300
Activity
Find the size of each angle of the following regular polygons whose number of sides are;
a) 4 sides b)  8 sides c)  6 sides d)  10 sides
e)   5 sides f)  12 sides g)  7 sides h)  20 sides




Finding the number of sides when the centre angle or exterior angle is given
1. Find the number of sides of a regular polygon whose centre angle is 600.
Number of sides = All centre <s
Each centre <s
=  3600
      600
=  6 sides

2. Find the number of sides of a regular polygon whose exterior angle is 720.
Number of sides =  All centre <s
     Each Ex <z
=  3600
       720
=  5 sides
Activity
Find the number of sides of regular polygons whose centre are?
a) 100 b) 600 c)  300 d)  900
e)   400 f)  1200 g)  450 h)   360


Exterior angles and interior angles of a regular polygon

600   1200     1200
                                                    600
              1200                         1200

          600     1200            1200 600
                             600

Find the size of each interior angle of a regular polygon whose exterior angle is 1200.

Let the interior be y.
y + ext <  = 1800
y + 1200 = 1800
y + 1200 – 1200 = 1800 – 1200
y = 600

Find the size of each interior angle of a regular polygon whose exterior angle is;
a) 100 b)  800 c)  300 d)  700
e)   400 f)  200 g)  500 h)  600

Find the size of each exterior angle of a regular polygon whose interior angle is;
a) 1200 b)  1600 c)  1080 d)  1300
e)   1500 f)  1000 g)  1700 h)  1400

Angles on a compass
N

900    900
W E
900     900

S

N
NW NE
W E

SW         SE
S
1. What is the larger angle between N and E?
2. What is the smaller angle between W and S?
3. What is the larger angle between W and S?
4. What is the angle between N and W?
Turns
N N
Right hand turn clockwise        Left hand turn ant-clockwise



1. Through what angle do I turn if I turn from North to SE clockwise?
2. Through what angle do I turn if I turns clockwise;
a) From NE to South?
b) From West to SE
c) From SN to SE
3. Through what angle do I turn if I turned anti-clock wise.
a) From NE to West.
b) From West to East.

PYTHOGRAS THEOREM
 Short
            Side         Hypotenuse

                            Short side
Finding the longest side (hypotenuse) of a right angled triangle.
1.  

4cm                       hcm

                       3cm
Find the value of h.
a2 + b2  = c2
32 + 42 = h2
(3 x 3) + (4 x 4) = h2
     9     + 16 = h2
         √25  = √ h2
√5 x 5 =  √h x h
5  =  h
Therefore h  =  5cm
2.  

12cm                            hcm

5cm
Find the value of h
a2  + b2  = c2
52 + 122 = h2
(5 x 5)+ (12 x 12)  = h2
25  + 144 = h2
√169 =  √h2
√12 x 13 = √h x h
13cm = h
Therefore  h = 13cm

Pythogram theorem
1. Given that PS = PQ = 10cm, PR = 6cm and bisect <P
  P


10cm
  6cm
Q                      R                     S
i) Find the length of QS.
PR = 6cm
PS = 10cm
RS2  + 62 = 102
RS2 + 6 x 6 = 10 x 10
RS2 + 36 = 100
RS2 + 36 – 36 = 100 – 36
√RS2 = √65
√RS x RS = √8 x 8
RS = 8cm
The length of QS = 8cm x 2
   =  16cm
ii) Calculate the perimeter of the figure.
 P = sum of all sides
=  16cm  + 10cm + 10cm
=  26cm + 10cm
=  36cm
iii) Calculate the area of the figure.
base  = 16cm
height = 6cm
Area = ½ bh
=  ½ x 16cm x 6cm
=  8cm x 6cm
=  48cm2
iv) Find the area of the triangle PRS
=  ½ bh
=  ½ 8cm x 6cm
=  4cm x 6cm
=  24cm2





2. PQS is an isosceles triangle.  PQ = 13cm
    P


   (2x + 1)cm
12cm

 Q                      R                   S
i) Calculate the value of x.
20pp sides equal
(2x + 1)cm =  13
2x + 1 =  13
2x + 1 – 1 =  13 – 1
2x =  12
2x =  12
2       2
x =  6
ii) Find the length of QS using pythogras theorem
       P
                             6cm                           13cm
      R                            S
RS2 + 122 = 132
RS2 + (12 x 12) =  13 x 13
RS2 + 144 =  169
RS2  + 144 – 144 = 169 – 144
√RS2 =  √25
√ RS x RS = √5 x 5
RS =  5cm
The length of QS =  RS  + RS
=  5cm + 5cm
=  10cm
iii) Find the perimeter of the figure
P = sum of all sides
     =  13cm + 13cm + 10cm
               =  26cm +10cm
   =  36cm

iv) Calculate the area of the triangle.
A  = ½ bh
=  ½ x 10cm x 12cm
=  5cm x 12cm
=  60cm2

Calculate the height, the perimeter and the area of each of the figures below
a) b)                       P

                            h       10cm      7cm  13cm

        Q                                     R
                      12cm 10cm

     c) d)  
A               16cm               C
                                   h
                                          10cm
    B
Find x, height of lines, the area and the perimeter
       P

                   2cm                   (3x + 2) cm
                 
                Q               32cm           R


                   25cm                          (3x+1)cm
                                             hcm  
                                       
                                       48cm
Lines of folding symmetry
- Equilateral triangle
- Isosceles triangle
- A rectangle
- A kite
- A square
- A trapezium
- An isosceles triangle
- A circle
- A regular pentagon
- Letters of alphabet

Length
Converting km to metres.
1. Change 3km to metres
1km = 100m
3km  =  3 x 1000m
           =  3000m

2. Change 0.4km to metres
1km  = 1000m
0.4km = 0.4 x 1000m
              =      4   x 1000  m
                      10
=  4 x 100 m
=  400m
Activity
Change cm to km.
a) 40000cm b)  110,000cm c)  160,000cm
d)   48,000cm e)  32,000cm f)  490,000cm



EVALUATION
Strong Points: __________________________________________________________________
Weak Points: ____________________________________________________________________
Way forward: ___________________________________________________________________
Changing metres to cm
1. Change 4m to cm
1m  = 100cm
4m  =  4 x100cm
         =  400cm

2. Express 0.9 m to cm
1m  = 100cm
0.9m   =  0.9 x 100cm
              =  9  x 100cm
                 10
              =  9 x 10 cm
              =  90cm
Activity
Change the following m to cm.
a) 5m b)  1.2m c)  25m d)  9.6
e)   36m f)  0.18m
EVALUATION
Strong Points: _________________________________________________________________
Weak Points: ___________________________________________________________________
Way forward: _________________________________________________________________
Changing cm to metres
1. Change 40cm to metres.
100cm =  1m
40cm =  40     m
                                       100
=   4   m
    10
=  0.4m
2. Change 600cm to metres.
100cm =  1m
600cm =  600  m
     100
=  6 cm


Activity
Change the following cm to metres.
a) 120cm b)  18cm c)  360cm
d)   80cm e)  700cm f)  90cm
Activity
Change the km to metres.
a) 5km b)  0.6km c)  7km d)  24km
e)   8km f)  36cm g)  9km h)  0.93km
i)  13km j)  11km

EVALUATION
Strong Points: __________________________________________________________________
Weak Points: ____________________________________________________________________
Way forward: ___________________________________________________________________
Changing metres to km
1. Change 9000m to km.
Since 1000m =  1km
9000m =  9000   km
    1000
=  9km
2. Change 800m to km
1000m =  1km
800m =  800   km
    1000 =  8   km
               10      
=  0.8km

Activity
Change the following metres to km.
a) 6000m b)  700m c)  8000m d)  80m
e)   900m f)  130m

Changing km to cm
1. Change 7km to centimeters
1km = 10000cm
7km =  7 x 10000cm
=  70000cm

2. Change 0.4km 10 cm
1km =  100000cm
=  0.4 x 100000 cm
=    4 x 100000cm
        10
=  4 x 10000cm
=  40000cm

Change km to cm
a) 4km b)  0.06km c)  11km d)  48km
e)   14km f)  69km g)  18km h)  53km

Change cm to km
Since 100,000cm   =  1km
800,000cm   =  800000
       100000
  =  8km
Changing square metres to cm2
1. Change 2m2 to cm2
1cm =  100cm
1m x 1m =  100cm x 100cm
1m2 =  100,000 cm2
2m2 =  (2 x 10000)cm2
=  20,000cm2

2. Express 1.2m2 to cm2.
1m  = 100cm
1m x 1m =  100cm x 100cm
1m2 =  10,000cm2
1.2m2 =  1.2 x 10,000cm2
=  12  x  10,000)   cm2
                             10
=  (12 x 1000)cm2
=  12000 cm2
Activity
Change the following to square centimeters. (cm2)
a) 3m2 b)  8.2m2 c)  5m2 d)  10.5m2
e)   4m2 f)  20m2 g)  6m2 h)  12m2

EVALUATION
Strong Points: _______________________________________________________________
Weak Points: _________________________________________________________________
Way forward: _______________________________________________________________
Expressing km2 to m2
1. Express 4km2 as m2
1km  =  1000m
1km x 1km  =  1000m x 1000m
1km2 =  1000,000m2
4km =  (4x 1,000,000)m2
=  4,000,000m2

2. Change 2.5km2 to m2
1km2    = 1,000,000m2
2.5km2 =  (2.5 X 1,000,000) m2
=  25  X 1,000,000    m2
                            10
=  2,500,000m2

Activity
Change the following to square metres.
a) 2km2 b)  0.02km2 c)  4km2
d)   0.03km2 e)  0.25km2 f)  3.6km2
g)   0.03km2 h)  8km2
EVALUATION
Strong Points: _________________________________________________________________
Weak Points: ____________________________________________________________________
Way forward: __________________________________________________________________
Finding perimeter of geometry shapes.
1. Find the perimeter of the figure below.
               10cm
   5cm                           13cm

                     15cm
   P      = 15cm  + 13cm + 10cm + 5cm
=  28cm  + 15cm
=  43cm
Activity
Find the perimeter of the following figures.

  8cm 10cm 12cm

6cm
10cm
EVALUATION
Strong Points: ___________________________________________________________________
Weak Points: ___________________________________________________________________
Way forward: ___________________________________________________________________

NUMERACY (THEME)
TOPIC 5:  INTEGERS
Intergers are positive and negative numbers including zero plotted equidistantly on a number line.
Negative Positive
   + + + + + + + + + + + + + + +
  -7 -6 -5  -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7
A set of positive integers  =  {1 , 2 , 3 , 4 , 5 , 6 , …}
A set of negative integers = { … -4 , -3 , -2 , -1 }
N.B 1: Zero in neither positive nor negative.
N.B 2:  Any integer without a sign is a positive integer.
Review the following;
- Inverse
- Integers on a number line/arrows on a number line.
- Ordering integers

Addition and subtraction of integers
Addition of integers
Examples
1. Add:  +3  + +2
+3 + 2
Therefore +3  + + 2  = +5

2. Add:  +2  + +6
+2 + 6
+2 + +6  = +8


3. Work out:  -2  + -5
-2 – 5
-2 + -5 = -7

4. Add:  -6 + -1
-6 – 1
Therefore -6 + -1 = -7

Subtraction of integers
Examples
1. Subtract:  +3 - +2
+3 – 2
Therefore +3 - +3 = +1

2. Work out:  +5 - +4
+5 – 4
Therefore; +5 - +4 = +1
Activity
1. Add the following integers
a) +2 + +3 b)  +5 + +1 c)  -6 + -2
d)   -3  + +2 e)  +6 + +6 f)  -3 + -3
g)  +7 + -2 h)  -2 + +6

2. Subtract the following integers.
a) +3 - +2 b)  +4 - +6 c)  -4 - -4
d)   -7 - +8 e)  2 – 4 f)   -1 -1
g)  +3 - -3 h)  +5 – 3

EVALUATION
Strong Points: ___________________________________________________________________
Weak Points: ____________________________________________________________________
Way forward: __________________________________________________________________





Integers on a number line
Addition and subtraction of integers on a number line.
Examples
1. Add:  +3  + +2
+2
            +3
   + + + + + + + + + + + + + + +
  -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7
+5
Therefore:  +3 + +2  =  +5

2. Simplify:  +2 + -6 

-6
+2
   + + + + + + + + + + + + + + +
  -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7
                                              - 4

Therefore; +2  + -6  = -4

3. Subtract:  -2 - -3
- (-3)
-2
   + + + + + + + + + + + + + + +
  -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7
                                                                           +1

So; -2 - -3 = +1




Activity
1. Simplify the following using a number line.
a) +4 + +2 b)  -3 + -4 c)  -2 – 5
d)   -3 - -3 e)  4 – 6 f)  +5 + -3
g)   -4 + 6 h)  3 - -6 i)  4 - +3
2. Show +3 + -6 on a number line.

EVALUATION
Strong Points: _________________________________________________________________
Weak Points: ____________________________________________________________________
Way forward: ___________________________________________________________________
Mathematical statements (Addition and subtraction statement) and algebraic statement
Examples
Study the number line below and answer questions that follow.

b
   a
   + + + + + + + + + + + + + + +
  -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7
                               c

a) Write an algebraic statement for the above number line.
An algebraic statement  = (+a) + (-b) = -c

b) Write the mathematical statement for the above number line.
Mathematical statement consists of either addition or subtraction statement.
i) Addition statement
a = +3
b = + (-10)
c = -7
Addition statement = +3 + -14 = -7

ii) Subtraction statement
a = +3
b = - (+10)
c = -7
Subtraction statement = +3 - +10 = -7

Activity
1. Write the subtraction statement for the number line below.


   + + + + + + + + + + + + + + +
  -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7

2. Write the addition statement for the given number line below.


   + + + + + + + + + + + + + + +
  -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7

3. Study the number line below and answer questions that follows.

       y
                                                x
   + + + + + + + + + + + + + + +
  -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7
Z
a) What integer is respected by letter x , y  and z?
i) X  ______________ ii)  Y  ________________ iii)  z  _______________
b) Write the mathematical statement for the above number line.

APPLICATION OF INTEGERS
Questions (some)
i) Kato borrowed shs. 5,000.  If he paid back shs. 3,000. Find Kato’s financial status.
ii) The temperature was 200F but has dropped by 230F.  What is the temperature now?
iii) Peter was born 20AD and died 15BC.  How old was he when he died?
iv) A motorist moved 100m forward and reversed 150km.  How far is she from the starting point?
v) A patient’s temperature dropped by 20C and by another 30c,  find the patient’s present temperature.

SOLUTION SETS AND INEQUALITIES
 Inequality symbols are > , < , > , <
Solution set is a set of all possible values of unknown letter from the given inequality like;
x > 2


Finding solution set from the given inequality.

Examples
1. Given that; x > 2.  Find all possible values of x if x is a positive integers.
X > 2 means all positive integers greater than 2
x
   + + + + + + + + + + + + + + +
  -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5………..
Solution set for x = {3 , 4 , 5 ,  ….}






2. If x > 2.  Find the solution set for x (x is a positive integer)
x >2 means all positive integers greater or equal to two.

        x
   + + + + + + + + + + + + + + +
  -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5………
The solution set for x = { 2 , 3 , 4 , 5  ………….}
N.B:  The circle is shaded because of equal sign

3. Given that; p > -5, find the solution set for p if p is a negative integers.
P > -5 means p is a set of negative integers greater that or equal to -5


p
   + + + + + + + + + + + + + + +
  -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7
The solution set for p = { -5, -4 , -3 , -2 , -1 }

NB:  Zero is neither negative nor positive integer, so all negative integers greater or equal to -5 stops in -1.  Hence no dot after negative one (-1)
 
Activity
1. Find the solution set for x < 5 if x is a positive integer.
2. Given that p > - 6.  Find the possible values of p is a negative integer.
3. Find the solution set for x < -2.
4. Given that; -4 < x < 4
5. Find the solution set for -3 < p < 6

EVALUATION
Strong Points: ___________________________________________________________________
Weak Points: ____________________________________________________________________
Way forward: ___________________________________________________________________
TOPIC:  ALGEBRA
Writing phrases for algebraic expressions.
Examples
Add b to a  = a + b
Subtract b from a  =  a – b
Multiply b by a = ab
Divide b by a =  b
     a
2x + 3 =  multiply x by 2, then add 3 to the result
2(x +3) =  Add 3 to x then multiply the results by 2

4y – 7 = multiply y by 4, then subtract 7 from the result
4(y – 7) = subtract 7 from y then multiply the result by 4

x   +  5  =  Divide x by 4, then add 5 to the result
4
x + 5 Add 5 to x then divide the result by 4
    4

Exercise
Write phrases for the following.
a)   x + y b)  x y c)  x + 2 d)  x – y
e)   x f)  3n +5 g)  10x + 4 h)  12a + 10
       y
i)  2a – 1 j)  6p – 6 k)  (m + 10) l)  t + 3
        2
m)    7 n)  b  +  20
       x – 1       5

Expressing phrases as algebraic expressions
2 more than p =  p + 2
2 less than x =  x – 2
Twice x =  2x
Three times of q =  3q
Half a =  a
    2

5 years younger than x  =  x – 5
7 years older than p = p + 7
Twice as old as n =  2n
Four times k’s age =  4k
Average of a and b =  a + b
        2
Square p =  p2
Multiply the square of a by 3 =  3a2
Multiply n by 3 and square the result =  (3n)2
Multiply the sum of m and 9 by 7 =  7(m +9)

Exercise
Express the following as algebraic statements.
a) a more than 5
b) b divide by 4
c) m times 3
d) 13 more than d
e) X subtracted from 8
f) Thrice the sum of x and y
g) A third the difference between 9 and m
h) Half the sum of 2b , 3b , 8t and 6

Meaning of algebraic term
Example
2p means 2 x p  or p x p
3qp means 3 x q x p
4q2 means 4 x q x q
(4q)2 means 4q x 4q
Exercise
Expand the following.
a) 3x b)  3mn c)  2y2 d)  (5x)2
e)   n3 f)  6y g)  7ab h)  5x3
i)   (ab)2 j)   (5x)3 k)  xy2 l)  a5
m)  2a5 n)  4ab2 o)  (4ab)2

EVALUATION
Strong Points: ___________________________________________________________________
Weak Points: ____________________________________________________________________
Way forward: ___________________________________________________________________
Substitution
Examples
1. Given b = 6
Find b + 8
6 + 8
=  14

2. If = = 8, q = 6, what is pq9
p x q x 9
8 x 6 x 2
=  96


3. Given b = 6 , c = -3 , a = 2
Find  b c  
            9
=  b x c
        9
=  3 x -3
=  -9
Exercise
If p = 8, q = 6 , r = 4 , a = 2, b = 6
Find the value of the following;
a) 3p b)  p + 3 c)  qr d)  2p
e)   ¼ pc f)  pqr g)  q – r h)  2q + r
i)   6ac j)  4c2 k)  abc l)  ab
m)  abc





Collecting like terms
Examples
1. Simplify:  x + y  + 2x + 4y
x + 2x + y + 4y
3x + 5y

2. 3x + 6y – x – 2y
3x + 6y – x – 2y
3x – x + 6y – 2y
2x + 4y

Exercise
Simplify;
a) p + q + p b)  8x + 3 + 4y – x c)  x + y + 2x + 3y
d)  6t + 5 – 2t + 5 e)  m + p + p + m f)  3x + 4 + 4x + 5x
g)  q + 4p + 3q + 2p h)  2b + 4 – b i)  m + 2b + 3m + 5
j)  b + 3k – 4b – k

EVALUATION
Strong Points: __________________________________________________________________
Weak Points: __________________________________________________________________
Way forward: _________________________________________________________________
Removing brackets
+ (2x + 3) = 2x + 3
-  (4x + 6) = -4x - 6

Simplify;  ½ (8a + 4b)
½ x 8a + ½ x 4b
1 x 4a  + 1 x 2b
4a  + 2b


Exercise
a) Remove the brackets
i)         2 (a + 3)
ii) 2 ( 6 + b)
iii) 5 (4b – 2)
iv) 4 (a – 2b)
v) 3 (8x + 5y)

b) Substitute if a = 4 , b = 1 and c = 3
i)         3b – c
ii)        2 (a + b)
iii) 2 (c – b)
iv) 5 (a – b)
v) 7 (b + c) 

Change of positive and negative signs involving brackets.
i) -2 (x – 2y)
ii)  -2 (3x + 5)
iii)  +4 (x + 1)
iv)   +3x (y – 1)
v)   +5 (x – 6)

Simplify
a) ½ (2a + 4b) b)  1/3 (6x – 9y) c)  2a + 4b
            2
      d)   1/3 (+2xy – 15x) e)  1/5 (15x – 20y)

More about removing brackets
Examples
1. Remove the brackets
a)   3 (2 + x)  + 2(x + 4)
b)   3x 2 + 3x y + 2 x y + 2 x 4
c) 6 + 3x + 2x + 8
d) 3x + 2x + 8 + 6
e) 5x + 14

2.  
a) 3 (x + 3) – 2(x – 1)
b) 3 x y + 3 x 3 – 2 x y – 2x – 1
c) 3x + 9 – 2x + 2
d) x + 11


Exercise
Remove the brackets and simplify.
a) (x + 1) + (2x + 3)
b) (2x + 3) + (4x + 4)
c) (9x – 4) – (x – 1)
d) 5 (q + 3) + 3 (q – 1)
e) 4 (m + 3) + 3 (m + 1)
f) 5 (q + 3) – 3 (q – 1)

Finding the products of powers with the same base
Examples
Simplify
a) m x m b)  y2  x y3
 m x m y2 x y3
 m2 y x y x y x y x y
=  y5
OR
m1 x m1
=  m 1 + 1
=  m2

x2 x x3
= x 2 + 3
=  x5
Simplify; 4y2  x 3y4
=  4y2  x 3y4
=  (4 x 3)  (y2  x y4)
=  12(y2 +4
=  12 y6

Exercise
Simplify the following.
a) m x 3 x m b)  q x q x q x q c)  y2  x y3
d)   w3 x w x w2 e)  5y1 x 4y5 f)  2x2 x 4x3
g)  8m2 x 3m6 h)  7m2 x 6m3


Dividing powers of the same base
Examples
1. Simplify:  p5 ÷ p3
P5 ÷ p3 =  p x p x p x p x p
P x p x p
= p2

2. Simplify: y4  ÷ y3
y x y x y x y 
y x y x y
=  y


3. Simplify: 12 t4  ÷ 3t2
=  12 x t x t x t x t
           3 x t x t
=  4 x t x t
=  4t2

Exercise
a) y4 ÷ y3 b)  m7 ÷ m2 c)  d9 ÷ d5
d)  q7 ÷ q4 e)  t7 ÷ t4 f)   10m9 ÷ 2m5
g)  18p3 ÷ 9p2 h)  48x5 ÷ 16x2 i)  36m6 ÷ 6m4

EVALUATION
Strong Points: ___________________________________________________________________
Weak Points: ____________________________________________________________________
Way forward: ___________________________________________________________________
Finding unknown
Examples
1. Solve: p + 4 = 12
p + 4 = 12
p + 4 – 4 = 12 – 4
p = 12 – 4
p = 8

2. h – 15 = 12
h – 15 + 15  = 12 + 15
h = 12 + 15
h = 27

3. Solve 2x = 8
2x  = 8
2        2
x = 4
Exercise
Find the value of the unknown
a) y + 3  = 7 b)  a – 2 = 7 c)  m + 11 = 24
d)  p – 11 = 12 e)  x + 51 = 84 f)  n – 57 = 63
g)  r + 49 = 75 h)  q – 43 = 41 i)   p + 24 = 42
j)  x – 28 = 36 k)  3a  = 21 l)  8w  = 72
m)  13m = 260 n)  21h = 168 o)  20p  = 400


Word problems in algebra
Example
Katamba bought some eggs.  On his way home 4 eggs broke and he was left with 8 eggs.  How many eggs did he buy?
Let the eggs bought be n.
Eggs bought n
Eggs broken 4
Therefore n – 4 = 8
n – 4 + 4 = 8 + 4
n = 12

Exercise
1. A farmer had some cows.  She paid 8 cows as dowery.  Altogether she had 15 cows.  How many cows had she before?
2. I think of a number, add 7 to it, my answer is 12.  What is the number?
3. A tube brewed y litres of local beer.  He sold 17 litres and was left with 4 litres.  How many litres did he brew?
4. Bulya had y mathematics numbers to work out.  She worked out 23 and was left with 2.  How many numbers was she given altogether?

5. A number multiplied by 13 gives 52.  Find the number.


Solving equations
Example
1. Solve:  m +4m = 20
5m  = 20
5 5
m  = 4

2. Solve:  3g + g + 2g = 30
6g  =  30
6          6
g    =  5
Activity
Solve the equations
a) 2y + y = 12 b)  2x + 2x + x = 25
c)   t  + 4t  = 45 d)  p + 5p + 2p  = 40
e)  n + 7n = 32
f)  A mother is 4 times as old as her daughter.  Their total age is 30 years.  Find the 
     daughter’s age.
g) A father is 3 times as old as his daughter.  Their total age is 48 years.  How old is the daughter?


EVALUATION
Strong Points: ___________________________________________________________________
Weak Points: ____________________________________________________________________
Way forward: ___________________________________________________________________

Finding the unknown involving fractions
Examples

1. Solve:  a  = 4
              3
=  3 x a  = 4 x 3
3
a = 12
2. Find the number of oranges that can be divided among 5 boys so that each gets 6 oranges.
Let the number of oranges be p.
So;  p   =  6
        5
5 x p  = 6 x 5
       5
p = 30
Exercise
Find the value of the unkown.
a) a  =  13 b)   m   =   8 c)   x   =  5
2         4        7
      d)  k  =  8 e)  d  =  20
            9       4

      f) 4 pupils shared x books equally.  Each pupil received 16 books.  How many books  were there?
      g)  8 homes shared p litres of paraffin equally.  Each home received 64 litres.   How much paraffin was shared?

Forming and solving equations
Example
The perimeter of a rectangle is 24cm.  Find x.

4cm
                             x
l x w x l x w = perimeter
x + 4 + x + 4 = 24
x + x + 4 + 4 =  24
2x + 8 = 24
2x + 8 - 8 = 24 – 8
2x = 16
2      2
x =  8cm
Exercise
Find the values of the unknown letter.
1. The perimeter of the triangle below is 60cm.  Find x.

3x                    25cm

                        4x
2. The perimeter of the trapezium below is 44 cm.  Find x.
                   (2x)cm

(x + 3)cm                                          (2x)cm

                                                  (3x + 1)cm
3. The perimeter of an isosceles triangle below is 36cm.  Find n.

n

     10cm
Solving equations involving brackets
Examples
1. Solve:  3 (y +4) = 21
3 x y + 3 x 4 =  21 
3y + 12 =  21
3y + 12 – 12 = 21 – 12
3y =  9
3      3
y =  3

2. 5(y +1) -3(y – 1)
5y + 5 – 3y + 3 = 14
5y – 3y + 5 + 3 = 14
2y + 8 =  14
2y + 8 – 8 = 14 – 8
2y =  6
2      2
y =  3

Exercise
Solve;
a) 2(x + 2) = 10 b)  3 (x -2) + 2(x – 1) = 12
c)   5(x + 1) = 15 d)  4 (x +2) + 3(x – 1) = 12
e)   7(x – 3) = 7 f)  7(n + 3) – (2n – 4) = 35
g)   6(p – 4) = 30 h)  (p – 2) + (p – 4) = 0
i)    6(x + 3) = 30 j)  5(t – 1) – 3(t – 7) = 0

More of equations
Examples
1. Solve:  4x – 3  = x + 6
4x – x = 6 + 3
3x =  9
3      3
x = 3
Exercise
a) 2x + 4 = x + 11 b)  2x – 7  = x + 1
c)   5x + 1  = 4x + 4 d)  11x + 3  = x + 33
e)   7x – 4 = 3x + 8 f)  5(x – 2) = 2(x – 2)
g)  6(x + 4) = 3 (x – 2) h)  (x – 1) =  4 (x – 12)
i)   3(x – 2)  =  2(x – 1) j)  7(x -2) = x + 10

Forming equations and finding the unknown
Examples
1. Find the value of x in the figure.
        (2x – 1)cm

(x + 1)cm

Opposite sides of a rectangle are equal
Therefore; 2x – 1 =  x + 3
2x – x =  3 + 1
x = 4




Exercise
Form equations and find x.
a) b)  
         5x
2x                             8cm 2x

3x 10x

     c)    d)   
3x cm                   15cm                                                                                  x      8cm
                                                                                                                                    6cm
2x + 5cm     12cm

e)      3x - 3
                        
                                        2x + 4
Basic algebraic symbols
= equal to
= not equal  
> greater than
< less than
< less than or equal to
> greater than or equal to

Examples
a) n = 4 means n is equal to 4
b) n  = 4 means n is not equal to 4
c) p > 5 means p is greater than 5




Exercise
1. Write in words.
a) x = 3
b) x = 3
c) p < 7
d) p <  7
e) y > 6
f) y > 6
g) y < 10
h) t < 10
i) x = 30
j) a > 13

2. State each of these using symbols.
a) m is less than 9
b) w is equal to 4
c) 8 is greater than x
d) h is less than or equal to 11
e) ½ and 2/4 are equal

Finding the solution set for the inequality
Examples
1. Given the solution set for; x < 5
(If x is a whole number)
x is a whole number less than 5
members of x = {0 , 1 , 2 , 3 , 4}

2. Given the solution set for; x > -5
(if x is a negative integer)
x > -5 means
x is a negative integer than -5
members of set x = { -4 , -3 , -2 , -2}


Exercise
Find the solution set for the inequalities
1. a)  x < 3 (If x is a counting number)
b)   x < 3

2. n > -4 (If n is a negative integer)
b)  n > -4

3. a)  p < 8 (If p is a whole number)
b)   p < 8

4. m > 7 (If m is a counting number)
 m > 7


Forming and solving equations
Example
The perimeter of a rectangle is 24cm.  Find x.

           4cm
0
  x
L + W + L + W =  Perimeter
x + 4 + x + 4 =  24
x + x + 4 + 4 =  24
2x + 8 =  24
2x + 8 – 8 =  24 – 8
2x =  16
2       2
x =  8cm




Exercise
Find the values of the unknown letter.
1. The perimeter of the triangle below is 60cm.  Find x.


3x                         25cm

              4x

2. The perimeter of the trapezium below is 44cm.  Find x.
(2x)cm

   (x +3)cm                                         (2x)cm

                               (3x +1)cm

3. The perimeter of an isosceles triangle below is 36cm.  Find n



Finding the solution set for the inequalities
Give sets of numbers and number lines for the following.
Examples
1. -1 < x < 3
x represents integers between -1 and 3

+ + + + + + + + + + + +
-4 -3 -2 -1 0 1 2 3 4 5 6 7

     The solution set for x = {0 , 1 , 2}

Find the solution sets and their number lines
a) -2 < y < 3
b) -2 < y < 3
c) -2 < y < 3
d) -1 < p < 4
e) -1 < p < 4
f) -5 < q < 2
g) -5 < q < 2
h) -3 < x < 4

Solving inequalities and finding solution set.
Example
Solve the inequality and find the solution set for x.
When 3 is added to x, the result is greater than 5
x + 3 > 5
x + 3 -3 > 5 – 3
x > 2

If x is a counting number less than 10
Therefore; x > 2 = a set of number greater than 2 but less than 10
=  { 3 , 4 , 5 , 6 , 7 , 8 , 9 }


Exercise
Find the solution set for the inequalities below.
a) y + 2 > 4 b)  p + 6 > 10 c)  x + 3  < 9
d) n + 5 < 12 e)  y + 7  > 11
f)  When 4 is added to an integer (x) the result is less than 5.  Find all possible values of n. 
Measures (Money)
Uganda currency
The Uganda currency consists of the following;
Coins, bank notes
a) Coins are usually or a smaller value.
b) Bank notes are usually or bigger denomination than coins but also differ.
Each bank note has a value and an identifying number which is different from that on any other bank note.
The central bank writes new coins, and prints bank notes.  The bank notes in each bundle are numbered.
Example
1. If  bank notes are numbered consecutively from AP 003782 to AP 003881.  How many notes are there?
First note AP003782
Last note AP003881

Number of notes
   AP = 003881
- AP =003782
       99
This means there are 99 notes without the last note.  Therefore, the notes are (99 + 1) = 100 notes
Subtract and later add 1 note because during subtraction one note is left out.
(99 + 1 = 100)




2. Amos has bank notes numbered from AP 004300 to AP 004399
a) How many bank notes does Amos have?
Number of notes
    AP   004399
-  AP    004300
                         99  + 1
            100 notes
b) If each note is worth 1,000shillings in value, how much money does Amos have?
Amount of money in the bundle   =  100notes x 1000/=
=  sh. 100,000

Activity
1. Find how much money is in a bundle of sh. 1000 bank notes if they are numbered from UH 627400 to UH 627499.
2. A cashier is paying salaries to workers.  How many 1000 shilling notes will give to a worker who gets a salary of sh. 90,000?
3. How many 500 shilling coins are equivalent to one ten thousand shilling notes?
4. Ali deposited some money.  He has bank notes numbered from AZ 00360 to AZ 00389.  The cashier told him that bank notes from AZ 000372 to AZ 00382 are counterfeit (forged).  If the bank notes were of 10,000 shillings denomination;
a) How much money did he deposit in the bank?
b) How many bank notes were forged?

EVALUATION
Strong Points: __________________________________________________________________
Weak Points: ___________________________________________________________________
Way forward: ___________________________________________________________________
Uganda and other currencies
Different countries use different types of money.  The money used by a country is its currency.
Example
Country Currency
Uganda Uganda shilling (Ush)
Kenya Kenya shilling (Ksh)
Rwanda Rwanda Francs (RWF)
South Africa Rand (ZAB)
Zambia Kwacha (Kch)
USA US Dollars ($)
Britain Pound sterling (€)
Japan Japanese Ven (¥)
European union Euro (Euro)
German Deutsh mark (DM)

CURRENCY BUYING SELLING
1 pound sterling (€) 3730 3780
1 US Dollar (US $) 2355 2370
Kenya shilling 28 29
1 Rwanda Francs 2 2.4
1 Tanzania shilling (TZ) 1.57 1.58

Activity
1. Convert Ush. 34,000 to Kenya shillings.
2. I have Ush. 860,000. Find how much money I have in US Dollars.
3. A Tanzanian trader arrived in Uganda with TZ sh. 40,000.  Find how much money she had in Uganda shillings if 1 T.Z sh. = 1.6

EVALUATION
Strong Points: ___________________________________________________________________
Weak Points: ____________________________________________________________________
Way forward: ___________________________________________________________________