P.6 MATHEMATICS LESSON NOTES

TOPICAL BREAKDOWN FOR TERM I

TOPIC / UNIT ONE – SET CONCEPTS

LESSON 1

Sub topic: – Types of sets

Content:

1. Types of sets: (a) Equal sets e.g
   1. Equivalent sets
   2. Unequal

Examples

1. Equal sets

   A B

   1 3

   2 3 2 1

1. Equivalent sets / matching sets

   X Y

   1, 2, 3, 4, a, b, c, d,

3. Non equivalent sets

 P Z

 a,e,i 1,2,3,4

ACTIVITY

The pupils will attempt exercise 1 : 1 page 2 from A new MK primary MTC pupils’ BK 6. / Mk new edition pg 1-2 / understanding mtc pg 1-3/ fountain pf 1-8

REMARKS

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LESSON 2

Sub topic: Types of sets

Content

1. Intersecting sets (Ç) / joint sets

   A set of common members from two or more sets.

2. Union sets ( È)

   A set of all elements in the two or more sets.

3. Universal set ( e )

   The biggest set from which other smaller sets are got.

4. Joint and disjoint sets

Examples

Sets M = {a, b, c, d, e, }

 K = {d, e, f, g, h, }

\ (i) M Ç K = {e, d}

 (ii) K È M = {a, b, c, d, e, f, g, h}

 (iii) Universal set ( e )

The biggest set from sets M and K i.e

e =

 M K

 a, b, c e f, g

d h

e = {a, b, c, e, d, f, g, h}

Disjoint set

A = {1,23,4} B = {p, q, r,s}

e =

 A B

 1, 2, 3, 4 p q r s

Activity

Mk new edition pg 3-4

Understanding mtc pg 4-7

Fountain pg 7-8

Remarks

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LESSON 3

Sub topic : Types of sets

Content:

1. Difference of sets

   i) shading of regions

   ii) describing regions

2. Complement of sets

   i) find complement of sets

   ii) shading regions with complement of sets

Examples:

a) A B

(A-B) (B-A)

b) Complements

Given that e Find:

 (i) P¹ = {b, h,g, j, k}

P Q (ii) Q¹ = {c, e, f, j, k }

c, e, f, a b, h (iii) (P n Q)¹

b g (iv) (P u Q) ¹

j k

1. Difference sets:
   1. P – Q = {c, e, f}
   2. Q – P = {b, g, h}
2. Empty sets e.g

   A = {all goats with wings}

Activity

Mk new edition pg 10

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LESSON 4

Sub topics sub sets (Ì )

Content:

1. Listing / forming subsets
2. Numbers of sub sets
3. Number of proper subsets

Examples:

(i) Representing subsets on diagrams

 i.e All cows (C) are animals (A)

A

C cows C animals

1. Listing/ forming sub sets

   A = {x, y}

   Sub sets are { }, {x}, {y }, {x, y}

2. Find number of subsets;

Formula: 2ⁿ (n stands for number of members)

Eg set R = {1, 2, 3}

No of subsets = 2n

 = 2³

 = 2 x 2 x 2

 = 8

iv) find number of proper subsets

 (2ⁿ-1)

 Set P = {a,b,c,d}

No of proper subsets

(2ⁿ-1)

 2⁴-1

 (2x2x2x2)-1

 16-1

 15 proper sub sets

Activity

Mk new edition pg 6-7

Fountain mtc pg 8-10

Understanding mtc pg 4-6

Remarks

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LESSON 5

Subtopic: Finding number of elements in sets.

Content: (a) listing members of sets

1. Number of elements in sets.

Examples: (i) Find members in set N

N = {prime numbers between O and 10}

N = {2, 3, 5, 7}

(ii) n (N) = 4

1. Use the venn diagram to answer questions

e Find

(a) n (x)

X Y But x = {a, b, c, d, e, f, }

b , d f a g, h, \ n (x) = 6

c, e k, j

p q (b) n (y )

(c) n (X n Y)

(d) n (Y – X )

(e) n (X)¹

Activity

Mk old edition pg 20-22

Remarks

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LESSON 6

Subtopic: Application of set concepts.

Content: (a) Representing information on a venn diagram

Given that set A = {a,b,c,d,e,f,g} B = {a,e,I,o,u}

 A B

 b a i

 c d e o

f u

g

n(A) = 7

n(B) = 5

n(AՈB) = 2

n(A-B) = 5

n(B-A) = 3

n(A∪B) = 10

(b) Interpreting information given on a venn diagram

Examples:

1. Given that n (A) = 7, n (B) = 5 and n (A n B) = 2
2. Draw a venn diagram to represent the above information

 n (A) = 7 n (B ) = 5

5 2 3

Activity

Mk old edition pg 22-25

Remarks

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LESSON 7

SUBTOPIC : Application of sets:

Content : Interpreting word problems using the venn diagram (real life situations)

Examples: (a) In a class, 12 pupils like English (E), 15 pupils like Maths (M) and 5 pupils like both Eng and Maths. Draw a venn diagram to represent the information above.

e

 n(E) = 12 n(M) = 15 (i) The class has 7 + 5 + 10 = 22

\
e = 22 pupils

 (12 – 5) (15 – 5)

7 5 10 (ii) How many like one

subject only?

7 + 10 = 17 pupils

1. In a class of 30 pupils, 20 take Mirinda (M), 15 take Fanta (F) and some take both drinks while 2 take neither of the drinks.

   (i) Show this information on a venn diagram

   e = 30
   

   N(M) = 20 n(F) =15 (ii) How many pupils take

   both drinks?

    20 – y + y + 15 – y + 2 = 30 20 – y y 20– y y 15-y 20 + 15 + 2 + y – y – y = 30

   37 – y = 30

   37 – 37 – y = 30 – 37

    2 -y = -7

   -1 -1

Let y represent those who take both. Y = 7

Activity

1. Understanding mtc pg 13-15
2. Fountain p g 10-13
3. Mk new edition pg 8-9

Remarks

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LESSON 8

Sub topic : Probability

Content : (i) The idea of probability / chance

(ii) Formular

Prob. = n (Expected outcome) or n (EE)

 n(possible outcomes) n (SS)

(iii) Application

Example: If B = {counting numbers less than 10}

\ B = {1, 2, 3, 4, 5, 6, 7, 8, 9}

(a) Find the probability of picking an even number

 Even numbers = {2, 4, 6, 8}

 n (Expected outcomes ) = 4

 n (possible outcomes) = 9

\ Prob = 4

9

(b) In a class of 17 pupils, 11 like Eng (E) and 9 like Maths (M) if a pupil is picked at random from the class, what is the probability of picking a pupil who likes Maths only?

e = 17 Pupils who like both:

 n(E) = 11 n (M) = 9 (11 + 9 ) – 17

20 – 17

3

11- 3 3 9-3

Pupils who like Eng only Maths only

(11 – 3) (9 – 3)

Prob = 8 17 6

17

Activity

Fountain pg 14-16

Mk new edition pg 10-12

Remarks

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LESSON 9

Revision work on set concepts

1. Write equal, unequal or equivalent against each

   P Q R S

   1, 2, 3, 1, 3, 9 8, 9, 11 3, 5, 1, 2, 4

   4, 5 2, 7, 5 7, 2, 1

   (i) P and Q (ii) R and S (iii) Q and R

   (iv) Q and S (v) P and S

2. If P = {even numbers less than ten}
   1. Find n (P)
   2. How many subsets has set P?
3. Study the venn diagram and use it to answer the questions about it.

   e Write down the elements for:

   K M (i) K (ii) M

   (iii) K n M

   a, b i g

   d e, f h (iv) M u K (v) K – M) j (vi) K¹

4. (a) List down all the subsets in A if A = {o, u ,i, s}
   1. A set has five elements how many subsets has set A?
   2. Given that a set has 16 subsets. Find the numbers elements in this set.
5. (a) Draw and shade these sets.

   (i) Rn P (ii) M u N (iii) Z – F

 (b) Describe / name the shaded regions below:

(i) T P (ii) X Y (iii) L K

1. Set P = {2, 3, 5, 7}, Q = {1, 2, 3, 4, 6, 7, 8}
   1. Complete the venn diagram

      P Q

   1. Find n (P n Q) (ii) n (P u Q) (iii) n ( Q – P )

      (iv) n (P ) only (v) n(Q) (vi) n (P)¹

2. In a market 24 traders sell cloth (C), and 30 traders sell food (F). If 16 traders sell both items, draw a venn diagram and find out how many traders sell only one type of commodity.
3. In a class of 30 pupils, 18 eat meat, 10 eat beans and 5 do not eat any of the two types of food.
   1. Show this information on a venn diagram
   2. How many pupils eat meat only?
   3. Find those who eat beans only.
   4. How many pupils eat only one type of food?
   5. Find the number of pupils who eat both types of food.
   6. What is the probability of choosing a pupil at random who eats meat?

TOPIC/ UNIT TWO

THEME: NUMERACY

TOPIC: WHOLE NUMBERS

LESSON 1

Subtopic: Value values

Content : Value of digits in numerals

Examples: (i) Find the place value

(ii) Find the value of each digit

ii) Using operations to find values of digits

Activity

Mk new edition pg 14-15

Fountain pg 20-23

Remarks

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LESSON 2

Subtopic: Expanded form

Content (i) Expand using values / place values

(ii) Expand using powers of ten

Examples:

(a) Expand 6845 using values

Th HTO

 6845 = (6 x 1000) + ( 8 x 100) + ( 4 x 10 ) + (5 x 1)

= 6000 + 800+ 40 + 5

b) Using power exponents

 6³8²4¹5⁰ = (6 x 10³) + ( 8 x 10²) + ( 4 x 10¹) + 5 x 10⁰)

 6845 = 6.845 x 10³

Activity

MK new edition pg 16-17

Understanding mtc pg 25

Fountain pg 23-24

Remarks

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LESSON 3

Scientific /standard form

Content : expanding number using scientific notation

 Example: Express 6845 in scientific form

6845 = 6845 ÷10

 684.5 ÷10

 68.45 ÷10

 6.845 x 10³

LESSON 4

SUBTOPIC: Expressing expanded numbers as single numeral.

Content : (i) Expanded form of values

(ii) Expanded form of place values

(iii) Expanded form of exponents.

Examples: (a) Write in short:

 4000 + 60 + 2

 4000

 + 60

 + 2

 4062

(b) (8 x 10000) + ( 7 x 1000) + ( 5 x 100) + ( 9 x 10) + ( 3 x1)

 80,000 + 7,000 + 500 + 90 + 3

 80000

7000

500

90

+ 3

 87593

(c) (6 x 10³) + (4 x 10²) + ( 2 x 10¹) + ( 3 x 10⁰)

 (6x 10 x 10 x 10) + ( 4 10 x 10) + ( 2 x 10) + ( 3 x 1)

 6000 + 400 + 20 + 3

6000

400

20

+ 3

6425

(d) 6.42 x 10² = 6.42 x 100 = 642

Activity

• Fountain pg 23-24
• Mk new edition pg 16-17

Remarks

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LESSON 5

Subtopic: Reading and writing numbers in words

Content : Expressing numerals in words upto millions.

Examples A

9452

9000 – nine thousand

400 – four hundred

52 – fifty two

Therefore; 9452 = nine thousand four hundred fifty two

Examples: (b) write 1486019 in words

1000000 – One million

486000 – Four hundred eighty six

19 – Nineteen

\ 1486019 = One million, four hundred eight six thousand nineteen

Activity:

MK new edition pg 18-19

Fountain pg 25.

Remarks

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LESSON 6

Subtopic: writing words in figures .

Content: Writing number words in figures to millions

Write in figures.

Examples A

Four hundred thousand, seven hundred sixteen

Solution:

Four hundred thousand 400000

Seven hundred sixteen + 716

400716

ii) One million one hundred one thousand eleven

Activity

MK new edition pg 18-19

Fountain pg 25.

Remarks

LESSON 7

Subtopic: Rounding off whole numbers

Content: Round off to the nearest

1. Tens
2. Hundreds
3. Thousands

Examples: (i) Round 677 to the nearest tens

6 7 7

+ 1 0

6 8 0

(ii) Round 1677 to the nearest hundreds

1 6 7 7

+ 1 0 0

1 7 0 0

iii) Round off 34567 to the nearest thousands

Activity

Mk old edition pg 47-48

Remarks

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LESSON 8

Subtopic: Decimal numbers

Content: Place values of decimal in words and figures.

Examples: (a) 1 One tenth – 0.1

10

Place value of 1 in 0.1 is Tenths.

(b) 8 Eight hundredths – 0.8

100

(c) Find the value of each digit

4 . 6

Tenths – 6 x ¹/₁₀ (6 x 0.1) = 0.6

Ones – 4 x 1 = 4

Activity

Mk old edition pg 42-44

Remarks

LESSON 9

Subtopic: Reading and writing decimals in words and the vice verse

Content: (i) Writing decimals in words

(ii) Expressing decimals in figures from words

Examples: (a) Write 0.125 in words

0.125 = One hundred twenty five thousandths

(b) 18.4

18 Eighteen

0.14 Fourteen hundredths

18.14 Eighteen and fourteen hundredths

(c) Twenty six and four tenths

Twenty six 26

Four tenths + 0.4

26.4

Activity

Mk old edition pg 45- 46

Remarks

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LESSON 10

Subtopic: Expanding decimal numerals

Content: (i) Expand using place values

ii) Expand using values

(iii) Expand using exponents

Examples: (i) Expand 3. 5 4

Hundredths – 4 x ¹/₁₀₀ = 0.04

Tenths – 5 x 10 = 0.5

10

 Ones = 3 x 1 = 3

\ 3.54 = 3 + 0.5 + 0.04

(ii) Expand 4.62 using exponents/

0 -1 -2

4 . 6 2

4.62 = (4 x 10⁰) + (6 x 10^-1) + ( 2 x 10^-2)

(iii) Write as a single numeral

1. 3 + 0.5 + o.04

   3

   0.5

+ 0.04

 3.54

(b) Express in the shortest form

(4×10⁰) + (6×10^-1) + ( 2×10^-2)

4 x 100 = 4 x 1 = 4

6 x -10 = 6 x ¹/₁₀ = 0.6

2 x 10^-2 = 2 x ¹/₁₀₀ = 0.02

4.62

Activity

The pupils will do exercises 8 : 8 and 8 : 9 A New MK 2000 BK 6 pg 59 (old Edn)

Remarks

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LESSON 11

Subtopic: Expressing decimal in scientific notation.

Content: Expend decimals of different place values in standard/ Scientific notation.

1. Tenths
2. Hundredths
3. Thousandths

Examples: (i) 0.4 in standard form

0.4 = 4.0 x 10^-1

(ii) 2.52 = 2.52 x 10⁰

(iii) 23.63 = 2.363 x 10¹

(iv) 464.241 = 4.64244 x 10²

Activity

Express the following to standard form:

(a) 4.8 (b) 3.25 (c) 38.06

(d) 207.4 (e) 4819.2 (f) 23.63

(g) 49 (h) 29.7

(i) 0.006 (j) 120.0

Remarks

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LESSON 12

Content: Finding expanded decimals

Example

a) What number has been expanded

i) 3+0.5 + 0.04

ii) (4×10) + (6×1) + (7×0.01)

iii) (6×10³) +(4×10¹) + (9×10^-2)

Remarks

Ref: MK old edition pg 47-48

LESSON 13

Subtopic: Ordinary decimals

Content: (a) Arrange in ascending and descending order

Example: (i) Arrange the following in ascending and descending order

0.1, 2.0 and 0.04

1 , 2 , 4 (LCM = 100)

10 1 100

Þ 1 x 100 = 1 x 10 = 10 (2^nd )

10 1

2 x 100 = 200 = 200 (3^rd )

1 1

4 x 100 = 4 x 1 = 4 (1^st )

100 1

Ascending order = 0.04, 0.1, 2.0

(ii) Arrange the following in descending order

3.5, 4.05, 0.45, 0.02

35, 405, 45, 2 (LCM = 100 )

10 100 100 100

35 x 100 = 350 45 x 100 = 45

10 100

405 x 100 = 405 2 x 100 = 2

100 100

\ Descending order = 4.05, 3.5, 0.45, 0,02

Activity

The pupils will do exercises below:

1. 1.5, 0.015, 0.015, 15.0 (Ascending order)
2. 0.5, 5.5, 1.5, 5.1 (descending order)
3. 0.33, 0.3, 3.3 (Ascending order)
4. 0.2, 0.75, 0.5 (Descending order)
5. 0.25, 0.5, 0.4, 0.6 (Ascending order)

Remarks

Ref: Trs’ collection

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LESSON 14

Subtopic: Rounding off decimals

Content : Round off to the nearest:

1. Tenths / one place of decimal
2. Hundredths / two places of decimals
3. Thousandths / three places of decimal
4. Ones / whole number

Example: (i) Round off 4.25 to the nearest whole no.

4 . 2 5

+ . 0 0

4 . 0 0 \4.25 d 4

(ii) 29.67 to nearest tenths

29. 6 7

+ . 1 0

29. 7 0 \ 29.67 d 29.7

(iii) 39.95 to nearest tenths

3 9 . 9 5

+ . 1 0

4 0 . 0 0 d 40.0

Note: consider the answer upto the required place value

Ref

MK old edition pg 48

Understanding mtc pg 33-35

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LESSON 15

Subtopic: Roman and Hindu Arabic Numerals

Content: (i) Reading writing Roman numerals to 10,000

(ii) Expressing Hindu Arabic numerals in Roman system.

Example: (i) Basic digits / numerals

(ii) 75 = 70 + 5

 LXX + V

= LXXV

 (iii) 555 = 500 + 50 + 5

 D + L + V

DLV

Activity

• Mk old edition pg 49-51
• Understanding pg 36-39
• Fountain pg 26-30

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LESSON 15

Subtopic: Expressing Roman Numerals to Hindu Arabic numerals

Content: Convert from Roman numerals to Hindu Arabic numerals

Examples: (i) Write LXXV in Hindu Arabic system

LXXV

L = 50

XX = 20

V = 5

75

(ii) CCCXCIX

CCC = 300

XC = 90

IX = 9

399

(iii) CMLXIX

CM = 900

LX = 60

IX = 9

969

Activity

• Mk old edition pg 49-51
• Understanding mtc pg 36-39
• Fountain pg 26-30

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LESSON 16

Subtopic: Operations on Roman Numerals

Content: (a) Addition

(b) Subtraction

Examples: (i) Work out and answer in Hindu Arabic

XL + XV

XL = 40

XV = + 15

55

(ii) Simplify in Roman system

LXXX – XX subtract \60 = LX

LXXX = 80 80

XX = 20 – 20

60

(iii) Peter had LIX goats and sold XIV goats

How many goats remained (answer in Hindu Arabic)

LIX 69

XIV – 14

55 goats

Activity

The pupils will do exercises below.

(1) XI + IX (6) XXV – XV

(2) VII + L (7) XL – VII

(3) CD + XIV (8) XIX – IX

(4) XVI + XIV (9) CM – CL

(6) XX + III (10) Word problems

Remarks

Ref: Mk old edition pg 50-51

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LESSON 17

Subtopic: conversing from base ten to base five

Content: (a) Change from base ten to base five

Examples: (i) Change 23 to base five

5 23

 14 3

\ 23 = 43_five

b) Converting from base ten to binary base

 19 _ten

 BW BT R

 2 19 1

 2 9 1

 2 4 0

 2 2 0

1

19 _ten = 10011_two

Remarks

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LESSON 18

Subtopic: Changing to decimal / base ten

Content:

Examples: (a) express 412 _five to base ten

_{2 1 0}

4 1 2 _five = (4 x 5²) + 1 x 5¹) + ( 2 x 5⁰)

 = (4x5x5) + (1×5)+(2×1)

 = 100 + 5 + 2

 = 107_ten

Examples: (ii) change 1011two to base ten

1011two = (1×2³) +(1×2¹) +(1×2⁰)

(1x2x2x2) + (1×2) + (1×1)

8 + 2 + 1

11_ten

Activity

Trs’ collection

Remarks

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LESSON 19

Subtopic: Operations on bases

Content: Addition of same non decimal base numerals

Examples: (i) 2 3 _five + 21_five

2 3 _five

+ 2 1 _five

4 4 _five

(ii) Add: 1101 + 11two

 1101two

+ 11 two

10000 two

Activity

Trs’ collection

Remarks

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LESSON 20

Subtopic : Subtraction of bases

Content: Subtraction in non decimal bases in the same base.

Examples: (i) Subtract 34_five – 13_five

3 4 _five

– 1 3 _five

2 1 _five

1011 two

(ii) Subtract – 111two

0100two

Activity

Trs’ collection

Remarks

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LESSON 21

Subtopic: Multiplication in Binary system

Content: Multiply (i) 2 by 2

 (ii) 3 by 2

 (iii) to 4 b 3 digit numerals

Examples: (i) 10_two x 11_two

10 _two

X 1 1 _two

1 0

+ 100

110 _two

(ii) 11two x 11two 111_two

 x 11_two

111

+ 111

10101_two

Activity

Trs’ collection

Remarks

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LESSON 22

Subtopic: Operations on finites

Content: Addition in finite/modular system

Examples: (i) Add: 3 + 4 = -(finite 5)

(a) (b) 3 + 4 = – (finite 5)

 3 + 4 = 7

 7 ¸ 5 = 1 r 2

 3 + 4 = 2 (finite 5)

 = 2 (finite 5)

(ii) 6 + 8 = y (finite 12)

Activity

Remarks

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LESSON 23

SUBTOPIC: Multiplication in finite systems

Examples: (i) Work out 3 x 4 = x (finite 5)

3 x 4 means

3 groups of 4

\ 3 x 4 = 2 (finite 5)

So x = 2 (finite 5)

(ii) 3 x 4 = x (finite 5)

3 x 4 = 12

12 ¸ 5 = 2 r 2

3 x 4 = 2 (finite 5)

\ x = 2 (finite 5)

Activity

Ref: MK old edition pg 245-253

Remarks

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LESSON 24

Subtopic: Subtraction in finite system.

Content: (a) Using the dial

(b) By calculation method

Example: (i) Subtract 3 – 4 = – (finite 5)

\ 3 – 4 = 4 (finite 5)

(ii) 3 – 4 = – (finite 5)

(3 + 5) – 4

8 – 4

= 4

\ 3 – 4 = 4 (finite 5)

Activity

Mk old edition pg 245-253

Remarks

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LESSON 25

Subtopic: Algebra in finite system

Content: Solve equations in finite system

Examples: (i) Solve: p – 4 = 3 (finite 6)

P – 4 + 4 = 3 + 4 (finite 6

P + 0 = 7 (finite 6)

P = 7 ¸ 6 = 1 r 1

P = 1 (finite 6)

 (ii) Find x if 2x – 3 = 3 (finite 4)

2x – 3 = 3 (finite 4)

2x – 3 + 3 = 3 + 3 (finite 4)

2x + 0 = 6 (finite 4)

2x = 6

2 2

X = 3 (finite 4)

ii) 2x-3=4(finite 5)

 2x-3+3 = 4+3 (finite 5)

 2x = 7 (finite 5)

 2x = 7 + 5) (finite 5)

2x = 12 (finite 5)

 2 2

 X = 6 (finite 5)

Activity

Trs’ collection

Remarks

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LESSON 26

Subtopic: Application of finites.

Contents: Use ideas on finites to solve everyday life problems: (weeks, months)

Examples: (a) If today is a Friday, what day of the week will it be after 23 days.

Day + 23 = – (finite 7)

5 + 23 = 28

28 ¸ 7 = 4 r 0

0 (finite 7)

\ The day will be Sunday.

(b) If today is Friday, what day of the week was 45 days ago?

Day – 45 (finite 7)

5 – 45 6 r 3

7

5 – 3 (finite 7)

2 finite 7

\ It was Tuesday

(c) It is April now, which month will it be after 18 months

Month – 18 (finite 12)

4 – 18 1 r 6

12

 4 – 6

 (4 + 12) – 6

16 – 6 = 10 (finite 120

It will be October.

Activity

MK old edition 252-253

Remarks

_____________________________________________________________________

REVISION WORK ON WHOLE NUMBERS

1. Given digits 8, 4, 2
   1. Write down all the numerals you can form using the digits.
   2. Find the difference between the highest and the lowest numeral formed.
2. Find the place value and value of the underlined digits.

   (a) 4 6657 (b) 16785 (c) 16345

3. Expand 8739 using

   (a) values (b) place values (c) Powers

4. Write 7432 in standard/ scientific form
5. Express the following in single form
   1. 5000 + 70 + 3
   2. (7 x 10000) + ( 8 x 1000) + ( 3 x 100) + ( 7 x 10 ) + ( 2 x 1)
   3. (7 x 10³ ) + ( 4 x 10²) + ( 3 x 10¹) + 5 x 10⁰)
   4. 8.56 x 10²
6. Write 2592028 in words
7. Write: six million, eight hundred thousand, nine hundred sixteen
8. (a) Round off 4867 to the nearest tens

   (b) Round off 79581 to the nearest hundreds.

   (c) Round off 79581 to the nearest thousands.

9. Write the place value and value of the underlined digits

   (a) 0.784 (b) 3.782 (c) 5.948

10. Write 0.328 in words
11. Write Twenty seven and six tenths in figures.
12. Expand 5.78 using

    (a) place values (b) values (c) exponents

13. Express 0.432 in standard form
14. Arrange 0.44, 0.4, 4.4 in ascending order.
15. Arrange 0.35, 0.5, 0.7, 0.33 in descending order.
16. Round off 39.96 to the nearest tenth.
17. Write 99 in Roman Numerals.
18. Write XLV in Hindu Arabic system.
19. Work out: XI = IX
20. Change 26_ten to base six .
21. Write 346_seven in words.
22. Give the place value of each digit in 243_five.
23. Expand 462 _seven using powers.
24. Change 341_six to base ten
25. Change 124_five to base six.
26. If 17_X = 16_ten find value of x
27. Add 55_seven + 33 _seven = _____ seven.
28. Subtract: 44_five – 12 _five
29. Multiply 10_two x 11_two
30. Change 13 to finite 7.
31. Add: 4 + 4 = ______ finite 5
32. Multiply: 2 x 4 = ______ finite 5
33. Subtract: 2 – 4 = _______ finite 6
34. Divide 5 ¸ 3 = ________ finite 7
35. Solve: x – 4 = 3 finite 6
36. If today is Friday, what day of the week will it be after 22 days?
37. If today is Thursday, what day of the week was it 44 days ago?
38. It is 2.00 pm what time of the day will it be after 400 hours?

TOPIC / UNIT OPERATIONS ON WHOLE NUMBERS.

LESSON 1

Subtopic: Addition of whole numbers up to millions.

Content: Adding large whole numbers up to millions with and without carrying.

_{1 1 1 1 1 1 1}

Examples : (a) 7 8 6 4 7 6 2

+ 1 9 7 9 8 6 8

9 8 4 4 6 3 0

Example: (b) There were 246 240 books in a library and 167 645 more books were donated to the same library. How many books are these altogether?

2 4 6 2 4 0

+ 1 6 7 6 4 5

4 1 3 8 8 5 books

Activity

Understanding mtc pg 40-42

Fountain pg 32-35

MK new edition pg 24-25

Remarks

_____________________________________________________________________

LESSON 2.

Subtopic: Subtraction of whole numbers ot millions.

Content: Subtract large numbers up to millions.

Examples: (a) 4 11 12 13

5 2 3 3 1 8 6

– 1 3 4 5 1 0 2

3 8 8 8 0 8 4

. Examples: (b) A dairy processed 6500 650 litres of milk and sold 5650945 litres. How many litres were left?

6 500 650 litres

– 5 650 945 litres

849 705 litres

Activity

MK new edition pg 27

Fountain pg 33-34

Understanding mtc pg 43-45 .

LESSON 3

Subtopic: Multiplication

Content: Multiplication of large numbers

• By 2 digit number
• By 3 digit number

Examples: (i) 1 4 3

x 1 8

1144

+ 1430

2574

Example: (b) A company has 850 workers who earn sh 5460 each a day.

How much does the company spend on wages everyday?

5 4 6 0

x 8 5 0

0 0 0 0

2 7 3 0 0

+ 4 3 6 8 0

4 6 4 1 0 0 0

Activity

Fountain pg 34-36 / understanding mtc pg 46-49/ MK new edition pg 28

Remarks

_____________________________________________________________________

LESSON 4

Subtopic: Division

Content: Divide large numbers.

• By 2 digit
• By 3 digit

Examples: (i) 152

13 1976

– 13

67

– 65

26

– 26

00

(ii) 53

120 6360

– 600

360

– 360

000

Activity

Mk new edition pg 37-38

Fountain pg 37-38

Understanding MTCpg 49-53

Remarks

_____________________________________________________________________

LESSON 5

Subtopic: Division

Content: Word problems involving division of large numbers.

Example: A petrol station manger bought 2200 litres of motor oil. If she put equal amount of oil in 440 drums. How many litres of oil were in each drum?

 50

440 220000

– 2200

0

-0

0

Activity

Mk new edition pg 37-38

Fountain pg 37-38

Understanding MTCpg 49-53

_____________________________________________________________________

LESSON 6

Subtopic: Combined operations on numbers

Content: Use of BO MAS

Examples: (i) Work out: 9 – 15 + 6

(9 + 6 ) – 15

15 – 15

0

(ii) 8 ¸ 4 x 3 (iii) 18 – ( 4 x 3) ¸ 6

B O D M A S

(8 ¸ 4) x 2

2 x 2

4

iv) Kawoya got 32 mangoes in the morning and ate 28 of them .

½ of 32 was got in the evening. How many mangoes did he have at the end of the day?

Activity

Fountain pg 38-39

MK new edition pg31-32

Understanding mtc pg 54-59

Remarks

_____________________________________________________________________

LESSON 7

Subtopic: Properties of numbers.

Content: (i) Commutative properties

(ii) Distributive property

(iii) Associative property

Example: (i) Commutative

Order of addition or multiplication does not change the results

(a) 3 + 4 = 4 + 3 (b) 4 x 5 = 5 x 4

7 = 7 20 = 20

(ii) Associative property

Order of grouping two numbers in addition or

Multiplication does not change results

e.g 3 + ( 8 + 9 ) = (3 + 8 ) + 9

3 + 17 = 11 + 9

20 = 20

(iii) Distribution property

e.g Work out using distributive property

(2 x 3 ) + ( 2 x 4 )

2 ( 3 + 4)

2 (7)

2 x 7 = 14

Activity

Trs’ collection

Remarks

_______________________________________________________________________________________________________________________________________________________________________________________________________________

REVISION WEEK ON OPERATIONS ON NUMBERS

1. Add: 8 9 7 5 6 3 1

    + 2 8 6 7 5 4 2

   

2. Add: 231 048 + 524 628
3. There were 351 272 books in a library and 189 242 more books were donated to the same library. How many books are there altogether?
4. Subtract: 6 4 3 2 2 7 8

   – 2 3 2 1 1 0 1

   

   

5. Subtract 452 367 from 872 291
6. A dairy processed 5300 450 litres of milk and sold 3450833 litres. How many litres were left?
7. Multiply 145 by 19?
8. Multiply 1238 by 134
9. A bus carries 84 passengers each trip. How many people will it carry if it makes 18 trips?
10. Divide 5984 ¸ 68
11. A farmer has sh 688640 to pay to 32 workers. How much money does each worker get?
12. Work out 18 – ( 3 x 2) ¸ 6

TOPIC / UNIT 4: PATTERNS AND SEQUENCES:

LESSON 1

Subtopic: Divisibility tests

Content: – Divisibility tests of 2, 5, 10

– Divisibility by 3, 6, 9

– Divisibility by 4 and 8

Example: (a) By 3

A Number is divisible by 3 when the sum of its digits 15 a multiple of 3.

E. g 612

 6 + 1 + 2

9 ¸ 3 = 3

\ 612 is divisible by 3

(b) Divisibility by 8:

A number is divisible by 8 when the last three digits form a multiple of eight.

e.g 6248 last 3 are 248

\ 6248 is divisible by 8

Activity

MK new edition pg 34-36

Fountain pg 41-42

Understanding pg 60-61

Remarks

_____________________________________________________________________

LESSON 2

Subtopic: Developing number patterns

Content: – Odd and even numbers

– Triangular numbers

– Rectangular numbers

– square numbers

Examples: (i) Lists down the following:

1. Counting / natural numbers less than 15.
2. Whole numbers up to ten
3. Even numbers between ten and 20.
4. Odd numbers less than twenty

(ii) Triangular numbers E.g

 0 1 0 3 0

0 0 0 0

1 + 2 = 3 0 0 0

1 + 2 + 3 = 6

 N.B Find triangular numbers by adding the consecutive natural numbers

i. e (1, 3, 6, 10, 15, ———)

(iii) Rectangular numbers

2 x 1 2 x 3 2 x 5

2 6 10

(iv) Square numbers

e.g 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

1 x 1 = 1 2 x 2 = 4 3 x 3 = 9 4 x 4 = 16

Activity

Fountain pg 43-48

MK new edition pg 37

Understanding pg 62-65

Remarks

_____________________________________________________________________

LESSON 3

Subtopic: Prime and composite numbers.

Content: – List prime numbers

– Composite numbers

Examples: (i) What is the sum of the 3^rd and the 7^th prime numbers

Prime numbers are:

3^rd 7^th

2, 3, 5, 7, 11, 13, 17, 19, 23

 Sum = 5 + 17

= 22

(ii) Work out the sum of the first five composite numbers

Composite numbers are;

4, 6, 8, 9, 10, 12, 14, 15,

Sum is

4 + 6 + 8 + 9 + 10 =

 37

Activity

The Pupils will do exercise 4 : 13 and 4 : 14 from pgs 79 and 80. A New MK BK 6. .

Remarks

_____________________________________________________________________

LESSON4

Subtopic: Consecutive numbers / natural numbers / integers

Content: Find the consecutive counting numbers

Example: The sum of 3 consecutive whole numbers is 36. What are these numbers

Let the 1^st number be n.

2^nd number = n + 1

3^rd number = n + 2

But: n + n + 1 + n + 2 = 36

n + n + n + n + 1 + 2 = 36

3n + 3 = 36

3n + 3 – 3 = 36 – 3

3n = 33

3 = 3

\ n = 11

1^st number = n 2^nd number (n + 1) 3^rd number is

and n = 11 11 + 1 = 12 (n + 2)

11 + 2

13

Activity

Mk old edition pg 76-78

Remarks

_____________________________________________________________________

LESSON5

Subtopic: Consecutive numbers

Content: Find the consecutive EVEN and ODD numbers

Example: N.B Even and Odd numbers increase in intervals of 2

(i) The sum of three consecutive Even numbers is 24. list down the 3 numbers

Let the 1^st number by (x)

 2^nd number be (x + 2)

 3^rd number be (x + 4)

X + x + 2 + x + 4 = 24

X + x + x + 2 + 4 = 24

3x + 6 = 24

3x + 6 – 6 = 24 – 6

3x = 18

3 3

X = 6

These EVEN Numbers are:

 1^st is 6, 2^nd is , 3^rd

X + 2 x + 4

6 + 2 6 + 4

8 10

Activity

MK old edition pg 77-78

Mk New Edition 43

Remarks

_____________________________________________________________________

LESSON 6

Subtopic: Factors

Content: – Listing factors

– The common factors (CF)

– The HCF / GCF

– The LCF

Examples: (i) How many factors does 18 have?

F ₁₈ = {1, 2, 3, 6, 9, 18}

\ 18 has 6 factors

(ii) Work out the sum of all the F20

F20 = {1, 2, 4, 5, 10, 20}

Sum = 1 + 2 + 4 + 5 + 10 + 20

 = 42

(iii) Work out the GCF of 12 and 18

F12 = {1, 2, 3, 4, 6, 12}

F18 = {1, 2, 3, 6, 9, 18}

CF = {1, 2, 3, 6 }

GCF = 6

N.B (iv) The LCF is always 1

Activity

Mk old edition pg 81

Remarks

_____________________________________________________________________

LESSON 7

Subtopic: Prime factorization

Content: – Using (a) Multiplication

 (b) Subscript method

 (c) Powers/ exponents

– Find number prime factorised.

Examples: (i) Find the prime factors of 60.

(a) By ladder (b) by factors tree

2 60 60

2 30 2

3 15 30

5 5 2

1 15

 3

5

5 1

Pf 60 are (a) 2 x 2 x 3 x 5

Or {₂₁, 2₂, 3₁, 5₁}

Or 2² x 3¹ x 5¹

Activity

MK old edition pg 82

Remarks

_____________________________________________________________________

Lesson 8

Content:

i) Finding prime factorized number

ii) Finding the missing prime factors

Examples

i) What number has been prime factorised

ii) Prime factories and find missing factors

The prime factorization f 30 is 2 x y x 5 , find y

a = {2₁.2₂.5₁}

b = 2² x 3¹ x 5¹

(i) If 2 x 3 x y = 30 find y

2 x 3 x y = 30

6y = 30

6 6

 y = 5

(ii) If 144 = a⁴ x b² find ‘a’ and ‘b’

2 144 \ 2⁴ x 3² = a⁴ x b²

2 72

3 36

2 18

3 9

3 3 \ a = 2 and b = 3

1

(iii) Given that 2^2x x 2 = 32 find the value of x.

(1^st prime factorise 32) 32

i.e 2^2x x 2¹ = 2⁵ 2 16

2x + 1 = 5 2 8

2x + 1 – 1 = 5 – 1 2 4

2x = 4

 2 2 2 2

 X = 2₂

Activity

Mk old edition pg 83

Remarks

_____________________________________________________________________

LESSON 9

Subtopic: Multiples of numbers

Content: – Listing multiples.

– The common multiples

– The LCM

Examples: (i) List the multiples of 4 between ten and 30.

M₄ = {4, 8/ 12, 16, 20, 24, 28/ —-}

M₄ between 10 and 30 are

{12, 16, 20, 24, 28}

(ii) Work out the LCM of 24 and 36

(a) Using multiples

(b) By prime factorization method.

i.e 2 24 36

 2 12 18 LCM = 2 x 2 x 2 x3 x 3

 2 6 9

 3 3 9 = 72

 3 1 1

1 1

Activity

Mk old edition pg 86 .

Remarks

_____________________________________________________________________

LESSON 10

Subtopic: Finding LCM and GCF by prime factorization using a venn diagram

Content: – Representing prime factors on the venn diagrams.

– Find the GCF/HCF and LCM from the venn diagram

Examples: (i) Work out the prime factors of 30 and 36

30 and 36 F ₃₀ {2₁, 3₁, 5₁}

2 15 2 18

3 5 2 9

5 1 3 3 F ₃₆ = {2₁, 2₂, 3₁, 3₂}

3 1

(ii) Complete

F₃₀
Ç F₃₆ = (2₁, 3₁}

F₃₀ F ₃₆

5₁ 2₁ 2₂

3₁

(iii) Use the venn diagram to find the:

1. GCF of 30 and 36

   GCF = F₃₀
   Ç F ₃₆ = {2₁, 3₁}

   = 2 x 3 = 6

2. LCM of 30 and 36

   LCM = F ₃₀
   È F ₃₆ = (2₁, 2₂, 3₁, 3₂, 5₁}

   = 2 x 2 x 3 x 3 x 5 = 180

Activity

Mk old edition pg 86-87

Remarks

_____________________________________________________________________

LESSON 11

Subtopic: Unknown values/ factors

Content: (i) Find the missing number

(ii) Find the unknown factors

(iii) Work out HCF and LCM

Example: (i) Find x and y below

F x F y factors of y are

2₃ 2₁ 2₂ 3₂ {21, 22, 31, 32, 33}

3₁ 3₃ y = 2 x 2 x 3 x 3 x 3

y = 108

Factors of x = (21, 22, 31, 23}

 2 x 2 x 3 x 2

X = 24

 GCF = Fx Ç F y = {2₁, 2₂, 3₁} LCM = Fx È F y

= 2 x 2 x 3 = 2₁, 2₂, 2₃, 3₁, 3₂, 3₃,

GCF = 12 2 x 2 x 2 x 3 x 3 x 3

LCM = 216

 (ii) Find the unknowns

F₂₀ F ₃₀

X 2₁ Y

5₁

F20 = {x, 21, 51} F30 = {21, 51, y} GCF of 20 and 30

20 = x + 2 x 5 30 = 2 x 5 x y GCF = F20 Ç F 30

20 = 10 x
30 = 10 y GCF = {21, 51}

10 10 10 10 = 2 x 5

2 = x 3 = y \ GCF = 10

\ x = 2₂

\ y = 3₁

 LCM = F 20 È F 30

= {21, 22, 31, 51}

= 2 x 2 x 3 x 5

\ LCM = 60

Activity

Mk old edition pg 88-89

Remarks

_____________________________________________________________________

LESSON 12

Subtopic: Application of GCF / LCM

Content: – Relationship between GCF and LCM

– Other problem related to HCF/GCF

Examples: (i) The LCM of two numbers is 144 their GCF is 12 and one of these numbers is 48. Find the other number

Solution: Let 2^nd No be y

 1^st No x 2^nd No = LCM x GCF

48 x y = 144 x 42

48 48

y = 36

(ii) What is the largest possible divisor of 24 and 36.

Largest possible divisor is GCF

2 24 36 2 x 2 x 3 = 12

2 12 18 largest divisor = 12

3 6 9

2 3

Activity

Oxford primary MTC BK 6 pgs 34 – 41

Remarks

_____________________________________________________________________

LESSON 13

Subtopic: Application of LCM

Content: – Find the smallest number which when divided by 9 and 12 leaves

1. No remainder?
2. Remainder of 1?
3. Remainder of 5?

   Get LCM of 9 and 12 i.e

   2 9 12 LCM = 2 x 2 x 3 x 3 = 36

   2 9 6 \ Number is LCM + RCM

   3 3 1 = 36 + 1 = 37

   1 1

(ii) Kelvin has a stride of 40cm and his father has a stride of 60cm. What is the width of the narrowest path that they can both cross in a whole number of strides?

LCM of 40cm and 60 cm

M₄₀ = {40, 80, 120, 160, —-}

M₆₀ = {60, 120, 180, ———}

LCM = 120

\ The width is 120 cm

Activity

– Oxford primary MTC pupils BK 6 pgs 34 – 36 .

Remarks

_____________________________________________________________________

LESSON 14

Subtopic: Working with powers of whole numbers.

Content: – Find a number from powers

– Express number as product of powers of a given numbers

– Operation on powers.

Example: (i) What is 7³.

73 = 7 x 7 x 7 = 343

(ii) Express 64 using powers of fours

4 64

4 16

4 4 \ 64 = 4 x 4 x 4

1 64 = 4³

(iii) Work out: 23 + 32 + 50

(2 x 2 x 2 ) + ( 3 x 3) + 1

8 + 9 + 1

= 18

Activity

A New MK pupils’ BK 6 pgs 84 and 85.

Remarks

_____________________________________________________________________

LESSON 15

Subtopic: Squares of numbers

Content: – Squares of

1. whole numbers
2. fractions
3. mixed fractions
4. decimal

Example: (i) What is the square of 12?

12² = 12 x 12 = 144

(ii) Work out the square of ¾

3
² = 3 x 3 = 9

4 4 4 16

(iii) Calculate the square of 1 1 ½

1 ½ x 1 ½ = 1 x 2 + 1 x 1 x 2 + 1 = 3 x 3 = 9 = 1 1

2 2 2 2 4 4

(iv) Find (0.15)2

(0.15)² = 15 = 15 x 15 = 225 = 0.0225

100 100 100 1000

 (v) In general M x M = M²

Activity

• The Pupils will do exercise 9 on pg 42 from Oxford primary MTC BK 6.
• Exercise 4 : 37 pg 95, 4 : 39 pg 98 and 4 : 42 pg 101 of MK BK 6.
• Mk new edition pg 37

Remarks

_____________________________________________________________________

LESSON 16

Subtopic: Square roots.

Content: Square roots of whole numbers.

Example: Find the square roots of Ö 36

2 36 \Ö36 = Ö x 2 x 2 x 3 x 3

2 18 Ö (2 x 2 ) x ( 3 x 3 )

3 9 2 x 3

3 3 \
Ö 36 = 6

1

(ii) Work out Ö 324

2 324 Ö324 = Ö (2 x 2) x (3 x 3) x (3 x 3)

2 162

3 81 Ö 324 = 2 x 3 x 3

3 27

3 9 \
Ö 324 = 18

3 3

1

Activity

A New MK pupils’ MTC BK 6 pg 38.

Remarks

_____________________________________________________________________

LESSON 17

Subtopic: Square roots of fractions

Content: – Find square roots of fractions

1. Proper fractions
2. Mixed numbers
3. Decimals

Examples: (i) Work out the 4

9

4 = Ö2 x 2 = 2

9 Ö 3 x 3 3

 (ii) What is the square root Ö6 ¼

Ö6 x 4 + 1 = Ö25 = Ö 5 x 5 = 5 2 1

4 Ö 4 Ö 2 2 2

(iii) Find the square root of 1.44

1.44 = 144 = Ö144 = 12 x 12 = 12 = 1.2

100 Ö100 10 x 10 10

Activity

New MK pupils BK 6 pages 39-40

Remarks

_____________________________________________________________________

LESSON 18

Subtopic: Application of squares and square roots.

Content: – Solve problems using square

– Solve problems involving use of square roots.

Examples: 1. A square garden has a length of 3 ½ m. What out its area.

Area of sq = S x S

 3 ½ m 3 ½ m x 3 ½ m

7 m x 7 m = 49m² = 12 ¼ m²

2 2 4

\ Area = 12 ¼ m².

(ii) If a square has an area of 576.

(a) Calculate its side

 Area = side x side 24 = side

 576 = S x S

Ö 576 = ÖS² \ side = 24

2 576

2 288

2 144

2 72

2 36

2 18

2 9

3 3

1 = Ö S2

2 X 2 X 2 X 3 = Ö S x S

(b) Find the perimeter of the square.

P = 4 x side

4 x 24

\ P = 96

Activity

The Pupils will do exercise 4 : 41 and 4 : 43 pages 100 and 102.

A old MK pupils’ BK 6 pages 100 to 102.

New mk pg 39

Remarks

_____________________________________________________________________

LESSON 19.

Subtopic: Cubes and cube roots

Content: – Find the cubes

– Find the cube roots

Examples: (i) What is the cube of: 5?

5³ = 5 x 5 x 5 = 125

(ii) Find the volume of the cube below:

Vol of cube = S x S x S

6 cm V = 6cm x 6cm x 6 cm

V = 216 cm³

(iii) Work out the cube root of

(a) 64 = 2 64 ³Ö64 = ³Ö (2 x 2 x 2) x (2 x 2 x 2)

2 32

2 16 = 2 x 2

2 8

2 4

2 2 ³Ö64 = 4

1

Activity

The Pupils will do exercise below

1. Work out 2³
2. Find the number of cubes in the figure:

(a) (b)

1. Work out the volume of a cube of side.

   (i) side = 4cm (ii) side = 10 cm (iii) side = 5

2. Work out the cube root of each of these numbers

   (a) 8 (b) 27 (c) 64 (d) 216

LESSON 20

Subtopic: Number patterns and sequences

Content: Complete series and sequences

Examples: Find the missing number:

1. 2, 3, 5, 7, ___

11 is the next number

(prime numbers)

1. 4, 9, 16, 25, _____

   2 x 2 3 x 3 4 x 4 5 x 5 6 x 6

   (square numbers)

2. 1, 2, 4, 5, 7, 8, 10, 11

⁺ 1 ⁺2, ⁺2, ⁺1, ⁺2, ⁺1, ⁺2, ⁺1

10 + 1 = 11

1. 22, 16, 20, 14, 18, 12

-6, ⁺4, -6, ⁺4, -6

 18 – 6 = 12

(e) ½ , ¼, ¹/₈ , _____

Activity

A New Mk primary MTC BK 6 pages 90 – 91.

Fountain pg 49

Remarks

_____________________________________________________________________

LESSON 21

Subtopic: Puzzles/ magic square

Content: – Dealing with puzzles

– The magic squares:

Examples: (i) Find the missing numbers

(a) Magic numbers is

8 X 6 8 + 5 + 2 = 15

3 5 Y

W 9 2

(ii) x = 15 – (9 + 5) Y = 15 – (3 + 5) W = 15 – ( 8 + 3)

X = 15 – 14 Y = 15 – 8 W = 15 – 11

X = 1 Y = 7 W = 4

 N.B Vary the squares to 16 squares.

Activity

Work on magic squares from Understanding MTC BKs 5 and 6

Understanding mtc pg 74

Remarks:

__________________________________________________________________________________________________________________________________________

UNIT 5: TOPIC: FRACTIONS

LESSON 1

Sub topic: Operations on fractions

Basic operations (i) Addition (+ )

(ii) Subtraction (-)

(iii) Multiplication (X)

(iv) Division ( ¸)

(v) Mixed operations (BODMAS)

Content: (i) Addition of simple fractions with different denomination (ii) Addition of mixed numbers

Examples: (i) Add: 2 + 1 LCM 12

 3 4

2 x 4 + 1 x 3

3 x 4 4 x 3

8 + 3

12 12

11

12

(ii) Find the sum of 2 ²/₃ and 2 ¼

Solution:

2 + 1 = (2 + 2 ) + 2 + 1 LCM 12

3 4 3 4

+ 2 x 4 + 1 x 3

3 x 4 4 x 3

+ 8 + 3

12 12

+ 11

12

11

12

Activity

• Fountain pg 56-57
• Understanding pg 85

LESSON 2

Sub-topic: Operation on fractions

Content: (i) Subtraction of simple fractions with different denominations

(ii) Subtraction of mixed numbers

Examples: (a) Subtract: 3 – 3 LCM = 20

4 5

15 – 12 = 3

20 20 20

(b) Subtraction: 1 – 7

3 8

13 – 15 = 104 – 45

3 8 24

= 59

24

11

24

1 –7 = ( 4 – 1) + ( 1 – 7 )

3 8 3 8

= 3 1 – 7

3 8

= 80 – 21

24

= 59 = 11

24 24

Activity

Understanding mtc pg 87

Fountain pg 58-60

Remarks

_____________________________________________________________________LESSON 3

Sub-topic: Addition and subtraction of fractions involving word problems

Content: – Addition of fractions involving word problems

– subtraction of fractions involving word problems

Examples (a) A man used three quarters of his shamba to grow groundnuts, a half to grow potatoes and two thirds to grow water melons. Fin total fraction of the whole land used.

Solutions

3 + 1 + 2 LCM 12

4 2 3

3 x 3 + 1 x 6 + 2 x 4

4 x 3 2 x 6 3 x 4

9 + 6 + 8

12 12 12

23 = 12 + 11

12 12 12

11

 = 12

(b) One third of the children in a school are girls. One day a quarter of the girls in the class were absent. What fraction of the girls in the school were absent on that day?

Fraction girls = 1

3

Fraction of girls absent = 1 of 1 = 1 x 1 = 1

4 3 4 3 12 Ans

Activity

Trs’ collection

Remarks

__________________________________________________________________________________________________________________________________________

LESSON 4

Sub-topic: Addition and subtraction

Content: Addition and subtraction by use of BODMAS

B O D M A S – subtraction

Addition

Multiplication

Division

Of

Brackets

Example: Simplify: 1 – 2 + 1

 2 3 5

Solution

1 – 2 + 1 (BODMAS)

2 3 5

Rearrange

1 + 1 – 2 LCM 30

2 5 3

(15 + 6) – 20

30 30

21 – 20

30

1

30

(b) Simplify: 1 + 3 – 5

3 4 6

Solution

1 + 3 – 5 ( Use BODMAS)

3 4 6 LCM = 12

4 + 3 – 5

3 4 6

16 + 9 – 10

12

25 – 10 = 15

12 12

 = 12 + 3

12 12
₄

1

 = 4

Activity

Fountain bk 6 pg 59 .

Remarks

_____________________________________________________________________

LESSON 5

Sub-topic: Multiplication of fractions

Content: – Multiplication of fractions

– Multiplication of simple fractions

Examples: Fraction with whole number.

(i) 1 x 12 = 1 x 12 calculate 3 of 12

3 3 1 4

= 12
₄
¹ 3 of 25 3 x 12

3 1 4 4 1

= 9 36 9

7 1

(b) Fraction by fractions

Multiply: 2 x 3

5 4

2 x 3 = 6 3

5 x 4 20 10

 = 3

10

(c) Multiply: 1 x 1

2 3

= 1 x 1 = 1 x 1 = 1

 2 3 2 3 6

= 1

6

Activity

Fountain pg 60-61

Understanding mtc pg 79-81

New Mk pg 46-47

Remarks

_____________________________________________________________________

LESSON 5

Sub-topic: Operation on fractions

Content: Division of fractions

1. Use of LCM
2. Use of reciprocal

Reciprocals

Product of a number by its reciprocal is 1.

What is the reciprocal of ¾ ?

Let the reciprocal of ¾ be t.

3 x t = 1

4

= ¹4 x 3t = 1 x 4

4

 = ¹3t = 4

₁3 3

t = 4

3

\ Reciprocal of ¾ is ⁴/₃

 What is the reciprocal of 2 ¼ ?

 Let the reciprocal of 2 ¼ be y.

 2 ¼ x y = 1

 9 x y = 1

 4

1 9y = 1 x 4

4

9y = 4

9 9

Y = 4

 9

\ Reciprocal of 2 ¼ is 4

9

1 ÷ 1 = 1 ÷ 4

4 9

 = 1 x 4

9

 = 4

9

Activity

Old edition MK pg 48

Remarks

__________________________________________________________________________________________________________________________________________

LESSON 6

Sub-topic: division of fractions

Content: – Divide fractions using reciprocals

• Divide fractions using LCM

Examples: (i) Divide 2
¸ 2

 3

2
¸
1 Reciprocal of 2 is 1

3 2 1 2

2 x 1 = 2
¹ = 1

3 2 6
₃ 3

 (b) Divide: 2
¸ 2

3

2 ¸
2 LCM = 3

3 1

¹3 x 2
¸
2 x 3

3₁ 1

2 ¸ 6.

2¹ = 1

6₃ 3

Activity

New MK BK 6.

Remarks _____________________________________________________________________

Examples (ii) (a) Divide: 3
¸
1

 4 2

LCM Reciprocal

3 ¸ 1 LCM 4 3
¸
1 reciprocal 2

4 2 4 2 1

¹4 x 3
¸
1 x 4 ² 3 x 2

4 ₁ 2₁ 4 1

3 ¸ 2 3 x 2 = 6
³

 3 = 1 ½ 4 x 1 4
₂

1 ½

(b) Divide 2 ½ ¸ 1 ¼

LCM Reciprocal

2 ½ ¸ 1 ¼ 2 ½ ¸ 1 ¼

5
¸
5 LCM 4 5
¸
5 Reciprocal 4

2 4 2 4

²
4 x 5
¸
5 x 4 ¹ 5 x 4

2₁ 4 ₁ 2 5

(2 x 5) ¸ 5 20 = 2

10 ¸ 5 = 2 10

Activity

New MK pg 50

Fountain pg 62-64 .

Remarks __________________________________________________________________________________________________________________________________________

LESSON 7

Sub-topic: Operation on fractions

Content: Mixed operations with fractions

1. Use of BODMAS

   B – Brackets ( )

   O – Of of

   D – Division ¸

   M – Multiplication X

   A – Addition +

   S – Subtraction –

Examples: 1. Simplify: 5 – 3
¸ 1 ½

6 4

 Rename 1 ½ to ³/₂

5 – 3
¸
3 BODMAS

6 4 2

5 – 3¹ x 2¹

6 4₂ 3₁

5 – 1 LCM = 12

6 2

10 – 6 = 4
¹

12 12 ₃

= 1

3

Activity

Fountain pg 64-66

New mk pg 51

Old mk pg 113

Remarks:

Emphasis should be on the order of BODMAS

__________________________________________________________________________________________________________________________________________

LESSON 8

Sub-topic: Decimals

Content: 1. Addition of decimal up to ten thousandths with carrying

2. Addition of decimals up to ten thousandths with carrying.

Examples (a)

(i) Add: 1. 5 + 0.4 (ii) 7.04 + 1.6 (ii) Add 2.4 + 0.254

1. 5 7. 04 2. 4

+ 0. 4 + 1. 6 + 0. 254

1. 9 8. 64 2. 654

(b)

(i) Add; 1. 5 + 1.6 (ii) Add 0.09 + 0.18 (iii) Add 0.067 +0.057

_{1 1
11}

1.5 0.09 0.067

+ 1.6 + 0.08 + 0.057

3.1 0.27 0.124

Content: -Subtraction of decimals up to ten thousandths without carrying.

– Subtraction of decimals up to ten thousandths with carrying.

Examples (a)

(i) Subtract: 2.5 – 1.3 (ii) Subtract: 0.9 – 0.4 (iii) Subtraction 2.085 – 0.03

2.5 0.98 2.085

– 1.3 – 0.4 – 0.03

1.2 0.58 2.602

Example (b)

(i) Subtract 2.8 – 0.9 (ii) Subtract 1.45 – 0.6 (iii) Subtract 2.7 – 0.098

1 0 6 9

2.18 1.45 2.7
10 10

– 0.9 – 0.6 – 0.0 9 8

1.9 0.85 2.6 0 2

Activity

Understanding mtc pg 91-93

MK old Mk pg 114

LESSON 9

Subtopic: Decimals

Content: Addition and subtraction of decimals (consolidated)

Examples (a) 8 – 5.16 + 2.13

(8 + 2.13) – 5.16

9 10

8. 0 0 10. 1 13 = 4.97

+ 2 . 1 3 – 5. 1 6

10. 1 3 4 . 9 7

(b) 7 . (0.45 + 1.71)

6 9

1. 7 1 7. 10 10 = 4.84

+ 0. 4 5 – 2. 1 6

2. 1 6 4. 8 4

(c) (1.306 – 1.1) + 1.067

1. 306 0.206

_ 1. 1 + 1. 067 = 1.273

0. 206 1. 273

(c) 3.64 + 5 – 2.42

3. 6 4 8. 6 4

+ 5. 00 – 2. 42 = 6. 22

8. 64 6. 22

Word problems involving addition and subtraction of decimals.

Example: (d) Mariko bought 4 . 5 litres of milk. If 0.35 litres got spilled. How many litres were left?

4

4. 5 10

– 0. 3 5

4. 1 5

4. 15 litres were left.

(e) In a Ludo game. Okello scored 7. 5 points in the first round and 3. 8 points in the second round. How many points did he score altogether?

1^st round 7. 5

2^nd round + 3. 8

11. 3

He scored 11.3 points altogether.

Activity

Old edition Mk pg 115-116

Fountain pg 71

Remarks __________________________________________________________________________________________________________________________________________

LESSON 10

Subtopic: Decimals

Content: – Multiplication of a decimal by decimal

– Multiplication of a decimal by a whole number and vice versa.

Example (a) (i) Multiply: 0.9 x 0.5

Method I Method 2

0. 9 1 dp 9 x 5

x 0. 5 1 dp 10 10

4 5

+ 0 0 = 45 .

0. 45 2 dp 100

= 0.45

(a) (ii) Multiply 1. 32 x 2.4

Method 1 Method 2

1. 32 2 dp 132 x 24

x 2. 4 1 dp 100 10

5 2 8

+ 2 6 4 = 3168 .

3.1 6 8 3 dp 1000

= 3.168

 (b) Multiply: 1.4 x 25

Method 1 Method 2

25 1 dp 14 x 25

x 1. 4 1 dp 10 1

10 0

+ 25 = 350 .

35.0 1 dp 10

= 35

Activity

Old edition MK pg 116-118

Fountain pg 72

New mk pg 58-60

LESSON 11

Subtopic: division of decimals

Content: division by decimals

Division by whole numbers

Example: (a) Divide 8 ¸ 0.02

Method 1 Method 2

8 x 100 8 ¸
2

0.02 x 100 100

_{400 4}

= 800 = 8 x 100

2
₁ 1 2
₁

= 400 = 400

(b) Divide: 0.02 ¸ 8

Method 1 Method 2

0.02 x 100 2
¸ 8

8 x 100 100 1

= 2
¹ = 1 = 2 x 1

800 400 100 8

⁴⁰⁰

= 2¹ = 1

800 400

⁴⁰⁰

Example: (c) Divide: 2.4 ¸ 0.03

Method 1 Method 2

2.4 x 100 24
¸
3

0.03 x 100 10 100

= _{80 8}

240 24 x 100

3
₁ 10 ₁
3
₁

= 80 = 80

(d) Divide: 0.072 ¸ 0.8

Method 1 Method 2

0.072 x 1000 72
¸
8

0. 8 x 1000 1000 10

72
⁹ = 9 72 9 x 10

800
₁₀₀ 100 1000 8
₁

= 0.09 = 9

100 = 0.09

Activity

New MK pg 61-65

Fountain pg 73-74

Understanding pg 97-98

Remarks _____________________________________________________________________

LESSON 12

Subtopic: Decimals

Content: Consolidation of all operation on decimals

Example: 1. Work out: 0.7 x 0.6

0.3

Method 1 Method 2

0.7 x 0.6 x 100 7 X 6
¸
3

0.3 x 100 10 10 10

42
¹⁴ = 14 = 7 x 6
² x 10
¹

30
₁₀ 10 10 10
₁
3 ₁

= 1.4 14 = 1.4

 10

2. Work out: 35 x 0.5

Method 1 Method 2

35 x 0.5 x 100 35 x 5
¸
5

0.05 x 100 1 10 100

35
⁷ x 50 35 x 5
¹ x 10 0

5
₁ 1 10 5
₁

= 350 = 350

Activity

Old MK pg 121

Fountain pg 64-65

Understanding pg 73

Remarks __________________________________________________________________________________________________________________________________________

LESSON 13

Subtopic: Decimals

Content: Word problems involving multiplication and division of decimals.

Example: (a) The length of one side of a square is 8.75 cm.

What is the perimeter of the square.

Method 1 Method 2

Perimeter of square = 4S P = 4S

 = 4 x 8.75 = 4 x 875

100

 8.75 = 3500

 X 4 100

 35.00

The perimeter is 35 cm = 35 cm

(b) A parcel weighing 5.5 kg contains packets of salt. How many packets of salt are in the parcel if each packet weighs 0.25 kg.

Method 1 No of packets = total weight

 Weight of one packet

= 5.5 ¸ 0.25

110
₂₂ OR 55
¸
25

Either 5.5 x 100 = 550 = 22 10 100

0.25 x 100 25
5
₁

55
¹¹ x 100 2

There are 22 packets 10 255
₁

 = 22 packets

Activity

New Mk pg 65

Old MK pg 118

Understanding mtc pg 98

Remarks __________________________________________________________________________________________________________________________________________

TERM II

TOPICAL BREAKDOWN FOR TERM II

TOPIC : RATIOS AND PROPORTIONS

LESSON 14

Subtopic: Ratios

Content: (i) Form rations

Examples: Rations are away of comparing similar quantities.

4kg and 5 kg

Mass first quantity = 4

 Mass second quantity 5

Ration = 4:5

 (b) Express 40cm to 2m as a ratio. (c) Write 1 to 1 as a ratio

Compare quantities 3 4

40 cm to 2m LCM = 12 of fractions

Must be in same units 1 x 12
⁴ : 1 x 12
³

1m = 100 cm 3 1 4
₁

2 m = 2 x 10 cm 4 : 3

= 200 cm

40 cm to 200 cm ratio 4 : 3

 Ration 40 : 200

10 10

4 : 20

4 4

1 : 5

Activity

New MK pg 66

Remarks _____________________________________________________________________

LESSON 15

Subtopic: Rations

Content: (i) Expressing rations as fractions

(ii) Expressing fractions as ratios

(iii) Expressing quantities as ratios

Examples: (a) Express 1 : 2 as a fraction

Solution

1 : 2 = 1

2 Ans

(b) Express 1 as a ratio

 3

1 = 1 : 3 Ans

3

(c) Henry has 12 books and John has 20 books.

What is the ratio of Henry’s books to John’s books?

Solution

Henry’s to John’s

12 to 20

12
³ : 20
⁵

4
₁ 4 ₁

3 : 5

NOTE: Ratios must be simplified to its lowest terms

Activity

New MK pg 67

Fountain 77-78

Remarks _____________________________________________________________________

LESSON 16

Subtopic: Ratios

Content: Sharing in ratios

Examples: (i) John and Mary share 27 sweets in the ratio 4 : 5. How many sweets does each get?

Ratios: John : Mary

4 : 5

John’s share: 4 x 27
³ sweets

9
₁

4 x 3 sweets

12 sweets

(ii) A Man and his wife had 200 kg of coffee. They decided to share it in a ratio of 7 : 3 respectively.

(i) How many kg did the man get?

 M : W

 7 : 3

 Total ratio = 7 + 3 = 10

 Man’s share 7 x 200 kg

10

= 140 kg

(ii) How many kg did the wife get?

3 x 200 OR 200

10 – 140

60 kg 60 kg

Example: (iii) A sum of shs 30000 was shared by three brothers Amos, Andrew and Allan in a ratio of 1 : 2 : 3 respectively. How much did each get?

Total ratio = 1 + 2 + 3

 = 6

Ratios by names: Amos : Andrew : Allan

Ratio 1 : 2 : 3

₅₀₀₀

Amos = 1 x 30,000

6
₁

= Shs 5000

₅₀₀₀

Andrew = 2 x 30,000

6
₁

= Shs 10000

₅₀₀₀

Allan = 3 x 30,000

6
₁

= Shs 15000

Activity

fountain pg 80-81/ old MK pg 133-135

Remarks _____________________________________________________________________

LESSON 17

Subtopic: Ratios

Content: Finding numbers when ratios are given

Example: The ratio of boys to girls in a class is 1 : 2. If there are 14 boys, how many pupils are in the class?

Solution

Expressing ratios in terms of t.

B G Total t = 14

t 2t 3t Total = 3t

14 = 3 x t

= 3 x t

= 3 x 14

= 42

\ There are 42 pupils in the class

Activity

Old MK pg 135

LESSON 18

Subtopic: Ratios

Content: – Increasing in a given ratio

– Decreasing in a given ratio

Examples: (a) The prize of an article is increased from shs 1200 in a ratio 3 : 2. Find the new prize.

Solution.

3 x 1200 600

2 1

= 1800/=

(b) The prize of an article costing shs 2500 was reduced in the ratio 5 : 8. Find the new prize.

Solution

3145

5 x 25 000

8 1

Shs 15625

Activity

Old MK pg 129-131

Fountain pg 79-80

_____________________________________________________________________

LESSON 19

Subtopic: Rations

Content: – Finding the ratio of increase

– Finding the ratio of decrease

Examples: (a) A man’s salary was shs 10000. it has been increased to shs 12000 in what ratio has it increased ?

New salary = shs 12000

Old salary = shs 10000

6

Increased ratio = 12
000

10
000

5

Ratio increased = 6 : 5

(b) A bag had 40 sweets, 12 more sweets were added.

(i) How many sweets are in the bag now?

 40 + 12 = 52 sweets

(ii) In what ratio have the sweets increased

 Increase in ratio = New No

 Old No

= 52
¹³

40
₁₀

Ratio increase = 13: 10

Content: Finding the ratio of decrease

Example: The number of pupils in a class has decreased from 40 to 35.

In what ratio has the number decreased?

New No 35

Old No 40

Decrease in ratio = New No

Old No

 = 35
⁷

40
₈

Ratio of decrease 7 : 8

A school had 1200 pupils. This year the number has decreased to 1000 pupils. In what ratio has the number decreased?

New No = 1000

Old No = 1200

Increase = New No

 Old No

 5

= 10 00

12 00

 6

Ratio of decrease 5 : 6

Activity

Old MK pg 132

Remarks __________________________________________________________________________________________________________________________________________

LESSON 19

Subtopic: Ratios

Content: Application of ratios in solving daily life situations

Examples: Mary and John have oranges in the ratio of 2 : 3 respectively. If Mary has 10 oranges, how many oranges does John have?

Solution

Mary to John

2 : 3

 Mary’s oranges 10

 2 parts represents 10 oranges

 1 part represents 10 oranges

2

 3 part represents 10
⁵ x 3 oranges

2
₁

= 5 oranges

Activity

Old MK pg 135

Remarks _____________________________________________________________________

LESSON 20

Subtopic: Proportions

Content: (i) Direct proportions

(ii) Constant proportionality

Example (i) One pen costs 200/=. What is the cost of 5 pens?

Method 1 New ratio : 0ld ratio

1 pen costs 200/= 5 : 1

\ 5 pens cost (200 x 5)/= ? : 200

= 1000/= 1 part = 200

5 parts = (200 x 5)/= 1000/=

Example (b) 4 pens cost 2000/=. What is the cost of 7 pens?

4 pens cost 2000/= New : old 1 part = 2000

500 4

1 pen costs 2000 = 500

4 7 : 4

7 pens cost 500 x 7 = 3500 /= ? : 2000 7 parts = 500 x 7

4 parts = 2000 = 3500/=

Example (c) 1800/= can buy 2 kg of sugar. How many kg of sugar can one get with 3600/=?

1800/= can buy 2 kg

1/= can buy 2 kg

1800

\ 3600/= can buy 2 x 3600
² = 4kg of sugar

1800
₁

Example (d) In constant proportionality, one quantity increases in the same proportion as the other. E.g With a moving body, or car in a given distance, it takes 2 hours to carry 30 people, and takes the same time to carry 10 people through the same distance;

Activity

Fountain pg 82-83

Old MK pg 136-137

Remarks _____________________________________________________________________

LESSON 21

Subtopic: Proportions

Content Indirect/ Inverse proportion

Example (a) 3 men can do a piece of work in 6 days. How long will 9 men take to do the same piece of work at the same rate?

 MEN DAYS

 3 men take 6 days

 1 man takes (6 x 3) days

 9 men take 6² x 3¹ = 2 days

9
3
₁

(b) 2 children can dig a garden in 8 days. How many children will dig the same garden in 4 days?

 DAYS CHILDREN

 In 8 days it requires 2 children

 In 1 day it requires (2 x 8) children

 In 4 days it requires 2 x 8
² = 4 children

4
₁

(c) A car moving at a speed of 80km/hr takes 3 hours to cover a certain journey. How long will the car take if it moves at a speed of 120km/hr for the same journey?

 SPEED TIME

 At 80km/hr the car takes 3 hours

 At 1/km/hr the car takes (3 x 80) hrs

\ At 120km/hr the car take 3¹ x 80
² = 2 hrs

120

40 1

Activity

Fountain pg 82-83

New MK pg 71

Remarks

_____________________________________________________________________

LESSON 22

Subtopic: Percentages

Content: – Meaning of percentage

– percentage as fractions

– Fractions as percentages

Examples: (i) Express as fractions

(a) 5 % = 5 = 1

100 20

(b) 15% = 15 = 3

100 20

(c) 33 ¹/₃ % = 100 % = 100
¸
100

3 3 1

 = 100 x 1 = 100 = 1

3 100 300 3

(ii) Fractions as percentages

1. 4 = 4 x 100 % = 400 % = 80 %

   5 5 5

1. 2 = 2 x 100 % = 200 % = 66 ²/₃ %

   3 3 3

Activity

New MK pg 72-74

Understanding mtc pg 113

Remarks _____________________________________________________________________

LESSON23

Subtopic: Decimals as percentages.

Content: – Express decimals as percentages

 – Change percentages to decimal

Examples: (i) Convert 0.6 to percentage

0.6 = 6

10

6 x 100% = 6 x 100 % = 600 % = 60%

10 10 10

(ii) What is 2.8 as a percentage?

2.8 = 28

10

28 x 100 % = 28 x 100 % = 28%

10 10 1

(iii) Express 0.014 as percentage

0.014 = 14

1000

14 x 100 % = 1400 % = 1.4 %

1000 1000

(iv) Change 2.5% to decimal

2.5 = 25 % = 25
¸
100 = 25 x 1

100 100 1 100 100