Numerical Concepts

TEACHER’S GUIDE

MATHEMATICS

TOPIC: NUMERICAL CONCEPTS

SUB-TOPIC: NATURAL AND WHOLE NUMBERS

CLASS: S1

BRIEF DESCRIPTION OF UNIT:

Natural/Counting numbers and whole numbers are very common in our daily life.

We use them to count, add, subtract, multiply and divide. This enables us to interpret information about our riches, population etc. Many other types of numbers are a subset of the whole numbers.

TIME REQUIRED: Minimum: 40 minutes Maximum: 80 minutes

MAIN CONTENT AND CONCEPTS TO EMPHASISE:

Lesson 1 covers:

a) Roman Numerals

b) Sets of Natural numbers and Whole numbers

c) Addition, Subtraction, Multiplication and Division of Whole numbers

By the end of this lesson learners should be able to:

(i) Write down the first 1000 Roman numerals

(ii) Write down sets of Natural numbers and Whole numbers

(iii) Add, subtract, multiply and divide Whole numbers

MATERIALS REQUIRED:

1. Charts showing: a) Roman Numerals b) Egyptian Numerals c) Babylonian and any other numerals

2. A cube/dice/simple box

3. Chalkboard ruler

SUB-TOPIC: Even and Odd numbers, Factors and Multiples

BRIEF DESCRIPTION OF UNIT:

Odd and even numbers are positive integers. Each even number is exactly divisible by two whereas the odd numbers always give a remainder 1 when divided by 2. A set of factors of a number is finite but a set of multiples of any number is infinite.

TIME REQUIRED: Minimum: 40 minutes Maximum: 80 minutes

MAIN CONTENT AND CONCEPTS TO EMPHASISE:

By the end of this lesson learners should be able to:

i) Tell the difference between an odd and an even number,

ii) Write down a set of odd/even numbers less than 100,

iii) Find a) factors of numbers,

b) Multiples of numbers,

iv) Find the LCM and HCF of two numbers by listing multiples and factors respectively.

TEACHING SEQUENCE

EVEN AND ODD NUMBERS

Write down the set of the first fourteen Natural numbers.

Answer:{1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14}

Which of these numbers are divisible by 2?

Answer: 2, 4, 6, 8, 10, 12, and 14

Definition: A number, which is exactly divisible by 2, is called an even number.

Take 8/2=4, no remainder

7/2 =3 remainder 1.

Definition: A number, which is not exactly divisible by 2, is called an odd number.

A set of even numbers is written as,

Even numbers ={2, 4, 6, 8, 10, 12, 14, 16…}

A set of odd numbers is written as,

Odd numbers ={1, 3, 5, 7, 9, 11, 13…}

Note that;

1. All even numbers will have a last digit which is exactly divisible by two e.g. 0, 2, 4, 6 and 8. such as 90,398.

2. All odd numbers will have a last digit, which leaves a remainder when divided by two, for example, 1, 3, 5, 7 and 9. e.g. 248,749.

3. Any given natural number will either be an odd number or an even number.

4. Odd numbers are a subset of Natural numbers.

5. Even numbers are also a subset of the Natural numbers.

6. Natural numbers are a subset of Whole numbers since 0 is not a Natural number.

Hence, even numbers and odd numbers are a subset of Whole numbers.

7. Natural numbers are a subset of integers.

Question: Of the even and odd numbers which is a subset of Integers?

Answer: Both are subsets of Integers.

ACTIVITY 1

Write the following numbers where they belong: 49, 38, 117, 360, 984, 41, 43, 5479.

Even Numbers
Odd Numbers

FACTORS

a) Work out i) 18 2 ii) 18  3

Answers: i) 9 ii) 6

b) Which other numbers can divide 18 and leave no remainder?

Answers: 1,6,9,18

So 1, 2, 3, 6, 9 and 18 are exact divisors of 18 and are called factors of 18.

Definition: Factors of a number are whole numbers that divide exactly into the number.

The factors include 1 and the number itself.

c) Write down the factors of (i) 42 (ii) 75

Answer: (i) 1, 2, 3, 6, 7, 14, 21, 42 (ii) 1, 3, 5, 15, 25, 75

HIGHEST COMMON FACTOR

Consider 36 and 45.

Factors of 36 are {…,…,…,…,…,…,…,…,…,…,…,…,…,…,…,…,…}

Factors of 45 are {…………………………………………….}

Factors that are common to 36 and 45 are {………………………………………………….}

Answer: 1,3,9

The greatest/highest of these is ………….. And is called the Highest Common Factor (HCF) or the Greatest Common Divisor (GCD)

Definition: The Highest Common Factor (HCF) of two numbers is the highest factor common to both of them.

Example: Find the HCF of a) 4 and 12 b) 8 and 21.

SOLUTION:

a) The set of factors of 4 is {1, 2, 4}

The set of factors of 12 is {1, 2, 3, 4, 6, 12}

The highest number that appears in both sets is 4, Therefore, the HCF is 4.

NB. The HCF here is one of the numbers.

b)The set of factors of 8 is {1, 2, 4, 8}

The set of factors of 21 is {1, 3, 7, 21}

The highest number that appears in both sets is 1, the HCF is 1.

Definition: When the HCF of two (or more) numbers is 1, the numbers are said to be co-prime.

MULTIPLES

a) Multiply a) 5 x 1 b) 5 x 2 c) 5 x 3 5 x 8

Answer: a) 5 b) 10 c) 15 d) 40

b) What do the answers in (a) have in common?

Expected answers: 5 can divide all.

They are multiples of 5

Definition:

Multiples of a number are the products of multiplying the number by a positive whole number.

If we multiply two numbers together, the product is a multiple of each of the two numbers we multiplied together.

For example, 7 x 8 = 56

56 is a multiple of 7, and 56 is a multiple of 8.

c) The first seven multiples of 4 are {………………………………………}

Answer:4, 8, 12, 16, 20, 24, 28}

d) The set K is of multiples of 3 between 10 and 40.

 K = {…………………………………………………..}

Answer: 12, 15, 18, 21, 24, 27, 30, 33, 36, 39.

e) Each packet contains 12 pencils. If Tom has 9 packets, how many pencils does he have?

Answer: 12 x 9 =108. He has 108 pencils.

Lowest Common Multiple

Consider 4 and 5

The first fifteen multiples of 4 are {……………………………………………..}

Answer: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60.

The first twelve multiples of 5 are {…………………………………………….}

Answer: 5,10,15,20,25,30,35,40,45,50,55,60.

Multiples that are common to both sets are {----, ----,---- }

Answer: 20,40,60

The lowest of these is 20, so the Lowest Common Multiple of 4 and 5 is 20.

Definition: The Lowest Common Multiple of two numbers is the lowest number that is a multiple of both of them.

ACTIVITY 2

Write down at least 20 multiples of a) 3 b) 4 c) 5

Identify the LCM of 3,4 and 5.

Answer: 60

ADVICE

You need to emphasise that the set of factors of a number is finite while that of multiples is infinite. There are only a few numbers that can divide a given number but there is no end to the numbers that can be multiplied with it.

The LCM of two co-prime numbers needs special attention; it is still a product of these numbers. The largest factor of a number is the number its self and the lowest is 1. Note that these factors can be paired up as products of factors to give the required number; the first goes with the last (e.g. 1, 18) the second with the second last (e.g. 2, 9) etc. if only one factor remains unpaired in the middle we know that it is squared to obtain the answer. Such factors show us square numbers.

SUB-TOPIC: INTEGERS

BRIEF DESCRIPTION OF UNIT:

Integers are classified into two major types. Those on the right of zero are the positive integers, while those on the left are the negative integers. Note that zero is also an integer.

Like whole numbers, integers also can be added, subtracted, multiplied and divided. Integers are always represented on a number line. Addition and subtraction of integers can be done using a number line.

TIME REQUIRED: Minimum: 40 minutes Maximum: 80 minutes

MAIN CONTENT AND CONCEPTS TO EMPHASISE:

Main content and concepts to be emphasised in lesson 3

By the end of this lesson learners should be able to:

i) Represent negative and positive Integers on a Number line,

ii) Tell some practical application of integers e.g. –50m (50m below sea level),

iii) Add integers.

iv) Subtract integers

TEACHING SEQUENCE

INTEGERS

CASE 1

Consider a piece of land demarcated into plots as shown below.

The space reserved for the construction of a bank is clearly labeled.

Q

BANK

P

Plot Q is the third from the bank plot.

Plot P is also the third from the bank plot but both plots are on opposite sides to the bank.

a)How would you direct some one to go to plot Q but not plot P?

Answer: Q is the third plot on the left of the bank plot.

CASE 2

An eagle was 80 meters above a lake and saw a fish 20m below the lake surface.

a) What was the distance between the eagle and the fish?

b) The eagle descended through 60m from its original height.

What is its new height above the lake surface?

Answer: 80-60=20m

c) Is the eagle and the fish now in the same position?

Answer: No, there are not.

d) What similarity is there in the new position of the eagle and the position of the fish?

Answer: They are both 20m from the lake surface.

e) How different is the position of the eagle from that of the fish?

Answer: The eagle is 20m above the lake surface while the fish is 20m below the lake surface.

The lake surface is acting as our reference point (all measurements of height are based on it)

We are now adopting two signs, positive (+) to show height above the lake and negative (-) to show the height below the lake.

f) What was the original position of the eagle?

Answer:+80m.

g) What was the original position of the fish?

Answer:-20m

-20 is read as negative twenty, +20 is read as positive twenty

Each of the Natural numbers 1, 2, 3, 4, 5…has an opposite number or an additive inverse.

The additive inverse of +7 is –7, the additive inverse of –11 is +11.

Numbers 1, 2, 3, 4, 5…are called Positive Integers.

Numbers –1, -2, -3, -4, -5…are called Negative Integers.

Positive integers, negative integers and zero together form the set of integers.

Representing Integers on a Number line

Zero is used as a reference point.

-5 -4 -3 -2 -1 0 1 2 3 4 5

The arrows on both ends of the number line mean that integers continue in both directions without a limit.

The set of negative integers is {…-5, -4, -3, -2, -1}

The set of positive integers is {1, 2, 3, 4, 5, 6…}

The set of all the integers is {…-5, 4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6…}

The dot show that the two sets go on forever.

-3 is the same distance on the left of 0 as +3 is on the right. Thus, -3 is said to be the additive inverse of +3.

Order on the Number line

On the number line the larger number is to the right of a smaller one.

Consider the fish, which were initially 20m below the lake surface, if it came up to appoint 7m below the lake surface, -20m is lower than -7m. Then consider the maximum and minimum themes –7 is on the right of –20 on the number line.

ACTIVITY 1

Arrange the numbers below in order starting from the smallest.

(9, -3, -6, 7, 4, -1, 11, -2, -14, -5, 0)

Addition of Integers

Adding two positive integers is the same as adding whole numbers. (+5) + (+4) = (+9)

Adding two negative integers is the same as adding, except we attach a negative sign to the sum.

Consider a fish 2m below the lake surface, it further goes down by 3m, then it is 2 +3 = 5m below the surface.

So (-2) + (-3) = -5.

Addition using a number line

Add (+5) + (+4)

The lines always start from zero.

-2 -1 0 1 2 3 4 5 6 7 8 9 10

Answer: +9

5 steps to the right, followed by 4 steps to the right.

Also for (-2)+(-3)

2 steps to the left, followed by three steps to the left.

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

Answer: -5

EXAMPLE:

Use a number line to add.

a) (-4) + (+3) b) (-4) + (+6) c) (-5) + (5) d) (+4) + (-4)

SOLUTION

Case 1: When the magnitude of the negative integer is greater than that of the negative integer, the sum will be a negative integer.

-5 -4 -3 -2 -1 0 1 2 3

Answer –1

Case 2: When the magnitude of the positive integer is greater than that of negative integer, the sum will be a positive integer.

b)

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

Answer +2

Case 3: When the magnitudes are the same, the integer is the additive inverse of the other; the resulting sum is zero integer.

c)

-5 -4 -3 -2 -2 0 1 2 3 4

Therefore (-5) + (+5) = 0

d)

-3 -2 -1 0 1 2 3 4 5

So (+4) + (-4) = 0

Observation

In (c) and (d) the answer is zero.

Conclusion: When we add an integer and its additive inverse the

answer is zero

Subtraction of Integers

To subtract a positive number from any given number on the number-line we move those many steps from that number to the left. To subtract a negative number from any given number, move from that number to the right.

Example:

Use the number line to subtract

a) 3-(+4) b) 3 - (-2) c) –2-(+3) d) -2 - (-3)

SOLUTION:

a)

-4 -3 -2 -1 0 1 2 3 4

3 – (+4) = -1

b)

-3 -2 -1 0 1 2 3 4 5

3 – (-2) = +5

c)

-5 -4 -3 -2 -1 0 1 2 3 4

So –2 – (+3) = -5

d)

-3 -2 -1 0 1 2 3

So (-2) – (-3) = + 1

NOTE:

a) Subtracting (–3) has the same effect as adding its inverse (+3)

b) Subtracting (+4) has the same effect as adding its inverse (-4)

In summary, to subtract any integer, we add its inverse.

Subtraction is obtained by changing the sign of the term to be subtracted and then adding.

For example, a) (-8) - (-2) = -8+2 = -6

b) (-6) - (+3) = -6+-3 = -6 -3 = -9

STUDENTS’ ACTIVITY:

Work out. a) (-6)-(-2) b) (-13)- (+7) c) (+6) - (10)

Answer: a) –4 b) –20 c) –4

STUDENTS’ ACTIVITIES

ACTIVITY 1

ROMAN NUMERALS

1. Define a numeral?

Answer: A numeral is a symbol for a number.

2. Fill in the missing Roman numerals in the table below

Roman Numeral	I						VII
Hindu-Arabic numeral	1	2	3	4	5	6	7	8	9	10	50	100	500	1000

Answers: II, III, IV , V, VI,….,VIII,IX,L,C.D and M respectively.

A combination of these Roman numerals was used to write other figures.

For example,

II	For 2
III	For 3
IIII	For 4	Which was shortened to appear as IV
IV	Means	“One” before “five”
VI	For 6	NB IV is not the same as VI, the order matters
IX	For 9	“One” before “ten”
XIX	For 19	“Ten” and “nine”
XL	For 40	“Ten” before “fifty”
LXX	For 70	“Fifty” and “twenty”

c) Fill in the correct Hindu-Arabic figure below:

Roman Figure	Hindu-Arabic Figure
a) XC
b) CM
c) CD
d) DC
e) DCC
f) DCCCX
g) CL
h) LX
i) MDC
j) MCD
k) CMIX

ANSWERS a) 90, b) 900, c)400, d)600, e)700, f)810, g)150, h)60, i)1600, j)1400, k)909.

WORKED EXAMPLES

Example 1:

What is the number MMCCCXXI in Hindu-Arabic?

SOLUTION:

MM is two thousand

CCC is three hundred

XX is twenty

I is one

The number is 2321.

Example 2:

Write i) 657 ii) 2904 in Roman Numerals.

SOLUTION:

i) Six hundred is DC

Fifty is L

Seven is VII

So 657 is DCLVII

ii) 2904

Two thousand is MM

Nine hundred is CM

Four is IV

So 2904 is MMCMIV.

Fill in the missing gaps in the table below:

Hindu-Arabic numeral	Roman Numeral
a) 657	DCLVII
b) 660
c) 670
d) 680
e) 690
f) 700
g)	MMCD
h)	MCDXLIX
i)	CCCXC
j)	MMMCMXC

ANSWERS: b) DCLX c) DCLXX d) DCLXXX e) DCXC f)DCC g) 2400 h) 1449 i) 390 j) 3990

ACTIVITY 2

NATURAL NUMBERS

Fill in the most likely answer/numeral to questions in the table below:

	Question	Answer / Numeral
A	How many heads does a cow have?
B	How many legs does a person have normally?
C	How many letters are in the name TOM?
D	How many sides does a square have?
E	Sarah started walking at 2:0pm and ended at 7:0pm. How many hours did she use?
F	How many faces does a cube have?
G	How many days are in a week?
H	How many vertices has a cube?
I	How many sides have a nonagon?

Answers .a) 1 b) 2 c) 3 d)4 e) 5 f) 6 g)7 h) 8 i) 9

1. Comment on the pattern made by the numbers in the table above.

Answer: These are numbers used in counting arranged in ascending order.

Mathematical knowledge to be learnt:

i) These numbers are therefore called COUNTING NUMBERS OR NATURAL NUMBERS.

ii) The set of Natural numbers is represented as

Natural Numbers={1,2,3,4,5,6,7,8,9,10,11,………………………….}

2. What is the meaning of the dots “………” after 11?

Answer: ……….means “and so on”

3. Write down the next five natural numbers after 11 arranged in ascending order.

Answer:12,13,14,15,16,..

Such an ordered set where numbers are written in a definite order is called a SEQUENCE OR AN ORDERED SET.

4. What is the difference between the sets{4,1,3,2…..} and {1,2,3,4,……..}?

Answer: The first set has no definite order while the second set has a definite order/is an ordered set.

5.Which of the two sets is a sequence?

Answer: The second set.

SUB-TOPIC: Multiplication and Division of Integers

CLASS: S1

BRIEF DESCRIPTION OF UNIT:

Integer multiplication and division need application of usual multiplication and division of whole numbers. In this topic you will meet simple rules governing the two operations mainly based on the signs of the integers involved.

TIME REQUIRED: Minimum: 40 minutes Maximum: 80 minutes

MAIN CONTENT AND CONCEPTS TO EMPHASISE:

Lesson 4 covers:

Multiplication and Division of Integers

By the end of this lesson learners should be able to:

i) Multiply integers,

ii) Divide integers

LESSON PROGRESSION

MULTIPLICATION OF INTEGERS

Consider 3 x (+2)

This is the same as (+2) + (+2) + (+2)

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

So 3 x (+2) = +6

Now 3 x (-2) is (-2)+(-2)+(-2)

-7 -6 -4 -2 0

3x-2 = -6

Consider the pattern of the answers to the multiplications below:

4x3 =12

4x2 =8

4x1 =4

4x0 =0

4x-1 = ---

4x-2 = ----

What happens to the product as we decrease the multiplier by 1?

Answer: The product decreases by 4.

Fill in the answers to the last two multiplications.

Answers: -4, -8

A second situation is shown below:

4 x -3 = -12

3 x -3 = -9

2 x -3 = -6

1 x -3 = -3

0 x -3 = 0

-1 x -3 = ----

-2 x -3 = -----

What happens to the succeeding product as the multiplier is decreased?

Answer: The product increases by 3

Fill in the answers to the last two multiplications.

Answer: +3, +6

What happens to the succeeding product as the multiplier is decreased?

Answer: The product increases by 3

Fill in the answers to the last two multiplications.

Answer: +3, +6

CONCLUSION

1. A positive integer x a positive integer =a positive integer

(+6) x (+2) = +12

2. A positive integer x a negative integer = a negative integer

(+6) x (-2) = -12 and (-2) x (+6) = -12

3. A negative integer x a negative integer = a positive integer

(-2) x (-6) = +12

Work out: a) (+5) x (-4) b) (-3) x (-7) c) (-4) x (-8) d) (-5) x (+7)

Answers: a) –20 b) +21 c) +32 d) -35

Division of Integers

Dividing is the inverse of multiplication.

The rules of division follow from those of multiplication.

(+4) x (+3) = +12

This is the same as (+4) = +12/+3 or +3 =+12/+4

= A positive integer

i.e. A positive integer

A positive integer

+7 x –3 = -21

This is the same as +7 = -21/-3

i.e.

A

= a positive integer

negative integer

a negative integer

Or –3 = -21/(+7)

A

= a negative integer

negative integer

a positive integer

(-9) x (-5) = +45

This is the same as -9 = +45/-5 or -5 = +45/-9

= a negative integer

A positive integer

a negative integer

Work out: a) (+10)  (-2) b) (-90)  (+3) c) (-39)  (-1) d) (-96)  (-8)

Answers: a) –5 b) –30 c)+39 d) +12

Application of Integers in daily life

Learners should give their answers first.

1. We can differentiate two equal distances above and below a reference point

e.g. 50m above sea-level is (+50m)

50m below sea-level is (-50m)

2.We can quote temperature below 0C, say if water solidified to form ice at 0C, ice can be cooled further by 5C and its temperature will not be+5C but –5C.

3. Customers can buy items on credit and represent their debts as negative.

If Peter has 50 dollars and the bill of items on his shopping list is 55 dollars, he would need 5 more dollars. Since he is a regular customer of Jane, he is allowed to pay later.

Debt =50-55 = -5 dollars.

4. Integers are used in books of accounts and in banks to show overdrawn accounts.

Take John on whose account is deducted Sh.100.000/= every 29^th of a month to save fees for his children. On a given 29^th he had Sh,95,000/= on his account, the bank effected the deduction as follows:

95,000-100,000 = -5,000/=

A balance of Sh –5,000/= is not the same as that of Sh 5,000/=

5…

WORKED EXAMPLES

1. A thermometer shows a temperature of –13C at 8:00am.If the temperature is rising at a rate of 3C per hour, what temperature will the thermometer show at 5:pm on the same day?

SOLUTION:

Time interval in hours:5pm –8am

=(12+5)-8

=17-8

=9 hours

Rate of temperature rise = 3C per hour

Rise in temperature = 3C per hour x 9 hours

=27C

Since it is a temperature rise, we add i.e (+27).

Initial temperature = -13C

Thermometer reading at 5:pm =-13+27

=14C

The temperature will be 14C.

2. In a quiz a correct answer scores 3 marks and an incorrect answer scores –2 marks. John guesses all the answers .He gets 9 correct and 6 wrong answers. Work out his total marks.

SOLUTION:

Marks from correct answers:3 x 9 = 27

Marks from wrong answers: -2 x 6 = -12

Total marks = 27 + (-12)

=27 - 12

=15

John’s total mark is 15.

SUB-TOPIC: PRIME NUMBERS

CLASS: S1

BRIEF DESCRIPTION OF UNIT

Prime numbers are a subset of whole numbers. A prime number has only two factors: 1 and itself. The smallest prime number is 2.There is no established pattern that can be used to generate the next prime number.

TIME REQUIRED: Minimum: 40 minutes Maximum: 80 minutes

MAIN CONTENT AND CONCEPTS TO EMPHASISE

Lesson 6 covers:

Prime numbers

Composite numbers

By the end of this lesson learners should be able to:

i) Define a prime number,

ii) Write a set of prime numbers less than 100.

iii) Decompose a composite number into prime factors.

iv) Find the LCM and HCF using prime factors

STUDENTS’ ACTIVITIES

PRIME NUMBERS

Have a prepared worksheet of 10 by 10 squares to be used on the Sieve of Eratosthenes

ACTIVITY 1

1. Fill in the gaps in the table below.

	Number	Factors	Number of factors
a)	10
b)	9
c)	7
d)	5
e)	3
f)	2

Answer: a){1,2,5,10} 4 factors

b){{1,3,9} 3 factors

c){1,7} 2 factors

d){1,5} 2 factors

e){1,3} 2 factors

f){1,2} 2 factors

b) Comment on your answers in c, d, e and f.

Answer: All have two factors only.

The second factor other than 1 is the number itself.

Definition:

A prime number is a number greater than 1 which has only two factors: itself and 1.

In other words a prime number is not divisible by any number other than 1 and itself.

a) What is the smallest prime number?

Answer: 2

b) Is the smallest prime number an odd or even number?

Answer: It is an even number.

c) Write down the first six prime numbers.

Answer :{2, 3, 5, 7, 11, 13}

d) Is there any other prime number that is an even number?

Answer: No.

NOTE: The smallest prime number (2) is the only one that is an even number, the rest of the prime numbers are odd numbers.

e)The set of prime numbers is Prime numbers = {2,3,5,7,11,13,17,19,…………………..}

Write down more members of the set in (e) up to 97.

ACTIVITY 2 Sieve of Eratosthenes

In this activity you will generate prime numbers less than 100.

1	2	3	4	5	6	7	8	9	10
11	12	13	14	15	16	17	18	19	20
21	22	23	24	25	26	27	28	29	30
31	32	33	34	35	36	37	38	39	40
41	42	43	44	45	46	47	48	49	50
51	52	53	54	55	56	57	58	59	60
61	62	63	64	65	66	67	68	69	70
71	72	73	74	75	76	77	78	79	80
81	82	83	84	85	86	87	88	89	90
91	92	93	94	95	96	97	98	99	100

Procedure

a) Cross out 1.

b) Put a circle round 2, then cross out all the multiples of two.

c) Find the next number which is not crossed out and put a circle round it and cross out all its multiples.

d)Repeat procedure ( c ) until all the numbers are either circled or crossed out.

e) List the set of circled numbers that you are left with.

This is the set of prime numbers below 100.

Compare your answer with that in Activity 1 part (e).

This method of finding the prime numbers is called the Sieve of Eratosthenes

f) What is a sieve?

Answer: [A sieve is an utensil consisting of a wire mesh or a gauze on a frame, used for separating solids or coarse matter (which do not pass through) from liquid or fine matter.]

NB. Eratosthenes used this method to get prime numbers less than 100 from the first 100 Natural numbers, thus the name sieve of Eratosthenes.

PRIME FACTORS

Definition: A factor of a number which is a prime number is called its prime factor.

Factors of 36 are {………………………………………………………………}

Answer:1, 2, 3, 4, 6, 9, 12, 18, 36}

Of these 2 and 3 are prime numbers. These are called prime factors.

PRIME FACTORISATION

A number can be written as a product of its prime factors and this is known as prime factorization or prime decomposition of a number.

[Decomposition means breaking something down into smaller parts]

COMPOSITE NUMBERS

These are numbers that can be written as a product of prime numbers

WORKED EXAMPLES

Express (i) 48 (ii) 210 as the product of its prime factors.

SOLUTION

(i) 2 48 Divide by 2, the smallest prime factor.

2 24 The result is even, so divide by 2 again

2 12

2 6

3 3

So 48 = 2 x 2 x 2 x 2 x 3

(ii)

.

2 210 The result is even, so divide by 2 again

3 105

5 35

7 7

1

So 210 = 2 x 3 x 5 x 7

Quick Activity: Find the prime factors of a) 56 b)63 c)252 d)3610

Using the prime factors to calculate the Highest Common Factor

Example: Find the HCF of 720 and 84.

SOLUTION:

(i)First write each number in prime factor form:

720 = 2₁ x 2₂ x 2₃ x 2₄ x 3₁ x 3₂ x 5

84. =2₁ x 2₂ x 3₁ x 7

[The small figure below another like 2₁ is called a subscript, read as “two subscript one”]

The subscript is just used to identify the so many similar figures like 2, 3 etc..

(ii)Pick out the common factors: those that appear in both numbers.

These are 2₁, 2₂, 3₁

(iii) The product of the common factors is the HCF

HCF of 720 and 84 is 2 x 2 x 3 = 12

Question: Find the HCF of I) 36 and 48 ii) 12, 18 and 30

Answer: i) 12 ii) 6

Using the prime factors to calculate the Lowest Common Multiple (LCM)

Example: Find the LCM of i) 6 and 8 ii) 12 and 15

SOLUTION:

i)First write the numbers prime factor form. 6= 2₁ x 3₁

8 = 2₁ x 2₂ x 2_3: (Note that 21 appears twice but is written once)

The product of the above factors (with no subscripts) is the LCM.

The LCM of 6and 8 = 2 x 2 x 2 x 3

= 24

ii) Prime factors of 12 =2₁ x 2₂ x 3₁

15 = 3₁ x 5₁

Common factors are 2₁, 2₂, 3₁, 5_1. Members of the union set of factors.

So the LCM of12 and 15 = 2 x 2 x 3 x 5 = 60

Questions:

1. Find the LCM of i) 36 and 16 ii) 12, 18 and 30

Answer: i) 144 ii) 180

2. Decompose 160 into prime numbers.

Answer:2 x 2 x 3 x 3 x 5

6. If you continued writing down the Natural Numbers N={1,2,3,4,5,6,………………..}, what is the largest natural number possible?

Answer:

1.There is no largest natural number

2.What ever number you may think of, there is always a number one more than it.

3.There is an infinite number of Natural numbers.

7.What is the meaning of the term/word infinite?

Answer: endless, without limits, -------------,-------------------------,---------------------

ACTIVITY 3

WHOLE NUMBERS

Question: How many fingers does a person have normally?

Answer: Ten (10) fingers.

Mathematical knowledge to be learnt:

Ten (10) is written using two numerals 1 and 0 (zero).

The origin of Zero.

Zero was added to numerals by Hindu mathematicians using a symbol 0 for ‘sunya’ which means empty/nothing in the Hindu language.

This made a new set, which is called the set of WHOLE NUMBERS.

We can write, {Whole Numbers}={0,1,2,3,4,5,6,7,8,9,10,11,…….}

NOTE:0 is a Whole number but is NOT considered to be a Counting/Natural number.

ADDITION OF WHOLE NUMBERS

Mental Work for learners:

a) 3+4= ---- b) 5+3= ----- c) 9+4= ----- d) 9+7= ---------e)8+5= -------

f) 12+5= ---- g) 11+8= ------

ADDITION OF 28+26.

a) Use a pencil to put stars (*) in the rows A, B…. to F.

Each star represents a thumb pin

Sarah has 28 pins while George has 26 pins.

Each top row is to be filled first before filling the next until the pins for each person are over.

b)”Fix” the 28 pins for Sarah.

c)”Fix” the 26 pins for George

SARAH HAS 28 THUMB PINS
Row A	*
Row B
Row C
	GEORGE HAS 26 THUMB PINS
Row D
Row E
Row F

d) How many pins are in row A when it is full?

e) How many full rows are there in total for Sarah and George?

f) Get pins from row F to fill up row C.

i) how many did you take to fill up row C?

ii) How many remained in row F?

g) What is the total number of full rows now?

Answers: d) 10 e) 4 f (i) 2 f (ii) 4 g) 5

The normal addition of 28+26 is carried as follows:

28

+26 8+6=14=10+4

54 10=1ten which is added to 2+2 making 5 tens.

More Examples:

Add i) 341+123 ii) 238+17

Remember to arrange your numbers vertically.

1. 341 ii) 238

+123 + 17

464 255

The result obtained after adding up numbers is called the SUM.

Add a) 238+171+51 b) 3,435+239+10,049

Answer a) 460 b) 13.723

SUBTRACTION OF WHOLE NUMBERS

We can also use columns when finding the difference between whole numbers.

Examples:

1.a) 853-341 b) 1008-634

SOLUTION:

1. 853 b) 1008

-341 -524

512 484

The result obtained after subtracting two numbers is called the DIFFERENCE.

STUDENT TEST

a) 864-391 b) 792-609 c) 3982-2078 d) 5064-1097

Answer: a) 473 b) 183 c) 1904 d) 3967

MULTIPLICATION OF WHOLE NUMBERS

Consider 10 + 10 +10 +10.This is ten written four times, which is 10 x 4=40.

Similarly,10x7=70

10x9=90

Write down the answer for:

a) 360 x 10 = ---

b) 3600 x 10 = ----

c) 3000 x 8 = -----

Answer: a) 3600 b) 36,000 c) 24,000

The result obtained after multiplying numbers is called the PRODUCT.

NB. The number of zeroes in the product is the sum of zeroes in the figures multiplied.

Example:

Work out: i) 25x13x 4 ii)5 x 19 x 2 iii)5 x 14 x 8

SOLUTION:

Rearranging can be useful especially for numbers whose products are multiples of 10.

i)25x13x4

=25x4x13

=100x13

=1300

ii)5x19x2

=5x2x19

=10x19

=190

iii)5x14x8

=5x8x14

=40x14 but 4x14=56

=560

Example:

Multiply a) 83x6 b)57x8 c)43x26

SOLUTION:

a)

83 b) 57 c) 43

x 6 x 8 x 26

18 6x3

480 6x80 56 8x7 258

498 400 8x50 860

456 1118

DIVISION OF WHOLE NUMBERS

Example: i) Divide 325 by 13 ii) 1565 by 11

25 ii)

1. 325 1565

- 26 - 11

65 46

- 65 44

0. 25

- 22

3

The number by which we divide, 13 in(i) and 11in (ii) is the divisor(d)

The number we divide, 325 in(i) and 1565 in(ii) is the dividend(D).

The number which tells you how many times the divisor is contained in the dividend is called quotient (Q)

When the dividend is not exactly divisible by the divisor, a number is left over and this is called the remainder(R)

In general,

Dividend=Divisor x Quotient + Remainder

(i) 325 = 13 x 25 + 0

(ii) 1565 = 11 x142 + 3

REFERENCES:

School Mathematics for East Africa (SMEA) Book 1 Ch.7

Secondary Mathematics for Uganda (SMU) Book 1 Ch.1

Fountain School Mathematics for Uganda (FSMU) Book 1 Ch2

Secondary School Mathematics (SSM) Book 1 Ch 2

MODEL QUESTIONS AND MARKING GUIDE

1. Jesca harvested 6,960 kg of beans. She filled the beans in bags of 80 kg each.

i) How many bags did she fill?

ii) If she sold each bag for Sh.56,000/=, how much money did she get?

SOLUTION	MARK	COMMENT
Total harvest =6960 kg, Cost per bag =56,000/= Weight of each bag =80 kg i)Number of bags =Total harvest in kg Weight of each bag =6960 80 =87 She filled 87 bags ii)Money received = Number of bags x Cost per bag =87 x56,000 =4,872,000/= She got Sh 4,872,000/=	B1 M1 A1 M1 A1 5 MARKS	Correct formula Substitution Correct answer Only Substitution Correct answer

2.The sum of four consecutive odd numbers is 40.What is the product of the four numbers?

SOLUTION	MARK	COMMENT
Let the first odd number =n Second number =n +2 Third number =n+4 Four number =n+6 Sum of the numbers =n +(n+2)+(n+4)+(n+6) = 40 =4n + 12 = 40 4n =40 – 12 4n = 28 n = 7 (first odd number) Second number =7 + 2 =9 Third number =7 + 4 = 11 Fourth number = 7 + 6 = 13 Product of the numbers =7 x 9 x 11 x 13 =9009	B1 M1 A1 M1 A1 5 MARKS	Identifying the difference of two from each odd number Expression for the sum Correct answer Correct n and the next three numbers Correct answer

Juma and Catherine take their cars for service to the same service center. Juma’s car is taken for service every 30 days while Catherine’s car is taken every 40 days. Once they took their cars for service on the same day. After how many days will they meet for the second time at the service center?

SOLUTION	MARK	COMMENT
They will always meet when the number of days is a common multiple of 30 and 40. Second meeting will be there when the number of days is the LCM of 30 and 40. Prime factors of 30={21x31x51} Prime factors of 40={21x22x23x51} Union of prime factors ={21,22,23,31,51} LCM =2 x 2 x 2 x3 x 5 =120 They will next meet after 120 days.	B1 M1 A1 M1 A1 5 MARKS	Identifying the use of LCM Prime factorization Correct answer Multiplication of elements of Union set Answer

Numerical Concepts

Timed Group Activity (10 Minutes)

Materials required: 5sheets of paper, pair of scissors /razor blade, a box

In a group of 5 or 10 learners:

1. Cut 100 pieces of paper about 4 cm x 4 cm.
2. Label them from 1 to 100.
3. Fold them such that the labeled figure is not seen.
4. Divide the 100 by the number of members in your group such that each one will receive an equal number of folded papers.
5. Let each one pick randomly the equal number of papers as found in (4) above.
6. Each of you should now unfold the papers and record all the numbers on the papers she/he obtained.
7. Arrange your names in alphabetical order, the one coming first should start reading out her/his numbers one by one as she records it in the set(s) to which it belongs in the table below.

Prime Number
Even Number
Odd Number

8.Let each member add the numbers s/he recorded to get the sum of all his numbers.

9.Add the sums of each individual to get the group sum that you should immediately give to your teacher to record to rank you according to who completes first, second third…

The answer from all groups MUST be the same. Inaccuracy of one member will lead to the group’s failure to arrive at the correct answer. Do your best to contribute to the success of your group by getting the required sum in the least time possible.

A word to all groups in the class

(i) If you have got the required sum, thank you for working as a team.

(ii) If your answer was different, let the members check together every member’s addition to find why the group failed to get it right.

(iii) This is not to find the person(s) to blame but to identify the source of inaccuracy. Thank for your effort.

(iv) Remember to keep our class clean. All pieces of paper must be put in the dustbin.