# Lesson Notes By Weeks and Term - Junior Secondary School 1

WHOLE NUMBERS

SUBJECT: MATHEMATICS

CLASS:  JSS 1

DATE:

TERM: 1st TERM

WEEK ONE

TOPIC: WHOLE NUMBERS

CONTENT

• Introduction
• System of Counting
• Counting in Millions
• Counting inBillions and Trillions

INTRODUCTION

• Counting

It is likely that mathematics began when people started to count and measure. Counting and measuring are part of everyday life.

Ancient people used fingers and toes to help them count or group numbers in different number bases. This led them to collect numbers in groups: sometimes 5s (fingers of one hand), sometimes 10s (both hands) and even in 20s (hands and feet). When people group numbers in 5s, we say they use a base five method. The most common bases used were five, ten and twenty. For example, a person with thirty two cows would say ‘I have six fives and two cows’ when counting in base ten. The most widely used base is base ten also called the denary system.

Other bases of counting: seven and sixty

7 days = 1 week

60 seconds = 1 minute

60 minutes = 1 hour

In English, ‘dozen’ means 12, ‘score’ means 20 and ‘gross’ means 144

System of Counting

• Tally System

Tally marks were probably the first numerals.

The ancient people employed tally marks to count large numbers. The tally marks were scratched on stones or sometimes cut on sticks but today we use tally marks to count or record large data, especially in statistics.

A tally mark of 5 is written by putting a line across a tally count of 4.

i.e          = 4   and          = 5

Example 1

Draw the tally marks for each of the following numbers:

1. 34     (b) 15

Solution

1. 34 =
2. 15 =

EVALUATION

1. During a dry season, it did not rain for 128 days. How many weeks and days is this?
2. What is the number represented by
3. Draw the tally marks for each of the following numbers: (a) 43   (b) 52

• Roman numerals

The Romans used capital letters of the alphabets to represent numbers. Many people believe that the Romans used the fingers to represent numbers as follows:

I for one finger, II for two fingers, III for three fingers, V for five fingers and X for the combination of two hands ( or two V’s) .

The Roman also used L for fifty, C for hundred, D for five hundred and M for one thousand as shown below.

 Hindu-Arabic Roman Numeral Hindu-Arabic Roman Numeral 1 I 20 XX 2 II 40 XL 3 III 50 L 4 IV 60 LX 5 V 90 XC 6 VI 100 C 7 VII 400 CD 8 VIII 500 D 9 IX 900 CM 10 X 1000 M

The Roman used the subtraction and addition method to obtain other numerals. For example

1. IV means V- I i.e.  5- 4 = 4
2. VI means V+ I, i.e. 5 + 1 = 6
3. IX means X- I, i.e. 10 – 1 = 9
4. XXIV means XX + IV = 20 + 4 = 24
5. CD means D- C = 500 – 100 = 400
6. MC means M + C = 1000 + 100 = 1100

Example 1

Change the following numbers to Roman numerals: (a) 2459       (b)  3282

Solution

1. 2459--- 2000 = MM

400 =  CD

50 =  L

9 = IX

2459 = MMCDLIX

1. 3282  = 3000    + 200   + 80    + 2

= MMM    CC   LXXX    II

i.e 3282 = MMMCCLXXXII

EVALUATION

1. Write the following Roman figures in natural ( or counting) numbers:
1. MMMCLIV     (b) MMCDLXXI   (c) MCMIX     (d) DCCCIV
1. Write the following natural numbers in Roman figures:
1. 2659     (b) 1009     (c) 3498     (d) 1584

• The Counting board

A counting board is a block of stone or wood ruled in columns. Loose counters, pebbles, stones or seeds in the columns show the value of the numbers in the columns.

Counters in the right-hand column (U) represent units, counters in the next column (T) represent tens, and so on.

 TH H T U âââ â ââ ââââ ââââ

2             7               5

The diagram below is a counting board showing the number 275.

• The Abacus

An abacus is a frame consisting of beads or disks that can be moved up or down (i.e. slide) on a series of wires or strings. Each wire has its own value. Both abacus and counting board work in the same way when carrying out calculations.

Example 1

M HTH   TH   H    T   U

An Abacus showing 2703

• Place Value of Numbers

Numbers of units, tens, hundreds,…….., are each represented by a single numeral.

(a).For a whole number:

- the units place is at the right-hand end of the number.

- the tens place is next to the units place on the left, and so on

For example: 5834 means ↓

5  thousands,  8 hundreds, 3 tens, and 4 units.

See the illustration below:

5           8           3            4

(b) for decimal fraction, we count the places to the right from the decimal point as tenths, hundredths, thousandths, etc.

See the illustration below:

↓         ↓         ↓         ↓          ↓

6         .          7          9         8

6    →   units

.     →   decimal

7    →    tenths

9    →    hundredths

8    →    thousandths

Example 1:

What is the place value of each of the following?

1. the 9 in 10269
2. the 2 in 2984

Solution:

1. the 9 in 10269 is = 9 units or nine units
2. the 2 in 2984  is = 2 thousands or two thousands

Example 2

What is the value of each of the following?

1. the 8 in  1.85
2. the 0 in 16.08

Solution:

1. the 8 in 1.85 is = 8 tenths or eight tenths
2. the 0 in 16.08 is =0 in tenths or zero tenths

Example 3

What is the value of each digit in 3 865 742

Solution

 3 8 6 5 7 4 2 M H. Th T.Th Th H T U Digit Value Word Form 3 3 000 000 Three million 8 800 000 Eight hundred thousand 6 60 000 Sixty thousand 5 5 000 Five thousand 7 700 Seven hundred 4 40 Forty 2 2 Two

EVALUATION

1 (a) The place value of 5 in 5763 is ……………

(b)What is the place value 1 in 5.691?

1. Give the value of each digit in 489 734
2. Write down the number shown in the following figures:

(a)

1. Essential Mathematics for JSS1 by AJS Oluwasanmi page 3-7
2. New General Mathematic for Jss1 by M. F. Macrae et al page 17-18.

Counting and Writing in millions, billions and trillions

The figures 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are called digits or units.

The table below gives the names and values of some large numbers.

 Name Value One thousand 1 000 Ten thousand 10 000 One hundred thousand 100 000 One million 1 000 000 Ten million 10 000 000 One hundred million 100 000 000 One billion 1 000 000 000 One trillion 1 000 000 000 000

Large numbers can be read easily by grouping the digits in threes starting from the right hand side as shown below.

Billion    Million   TH       H    T    U

25           800        074       4       3     0

The 1st gap separates hundreds from thousands and the second gap separates thousands from millions and the third gap separates million from billion.

Thus 25 800 074 430 reads twenty five billion, eight hundred million, seventy four thousand, eight hundred and ninety.

Example

Write the following in figures:

1. twelve billion, three hundred and nine million, ninety five thousand, six hundred and sixty three
2. six trillion, four hundred and thirty billion, one hundred and five million, two hundred and one thousand and fifty four
3. nine hundred and four billion, five hundred and forty million, three hundred and seventy thousand, seven hundred and fifty

Solution

1. You can work it out as follows:
 Twelve billion =   12 000 000 000 Three hundred and nine million =        309 000 000 Ninety five thousand =                 95 000 Six hundred and sixty three =                      663 Adding =   12 309 095 663
 Six Trillion =   6 000 000 000 000 Four hundred and thirty billion =      430 000 000 000 One hundred and five million =             105 000 000 Two hundred and one thousand =                    201 000 Fifty four =                             54 Adding =   6 430 105 201 054
 Nine hundred and four billion =   904 000 000 000 Five hundred and forty million =          540 000 000 Three hundred and seventy thousand =                 370 000 Seven hundred and fifty =                        750 Adding =   904 540 370 750

EVALUATION

1. Write the following in figures:
1. Ninety nine million, eighty thousand, nine hundred and forty one.
2. Fifteen trillion, six hundred and seventy one billion, three hundred and ninety one million, eighty eight thousand, five hundred and fifty five.
1. Write in figures, the number referred to in the statement: Last year a bank made a profit of ‘two hundred and twenty billion, five hundred and one thousand, four hundred and ninety three Naira ( N)

WEEKEND ASSIGNMENT

1. The value of 8 in 18214 is   (a) 8 units   (b) 8 tens  ( c) 8 hundreds  ( d) 8 thousands  (e) 8 ten thousands
2. The Roman numerals CXCIV represents the number (a) 194   (b) 186   (c ) 214   (d) 215  (e)  216.
3. What is the number represented by                   ? (a) 32  (b) 40  (c) 28  (d) 39
4. The value of 7 in 3.673 is (a) 7tenths    (b) 7 hundredths   ( c ) 7 units   ( d) 7 hundredth.
5. Three million and four in figures is (a) 300004  (b) 300040 (c) 30000004 (d) 3000004

THEORY

1. Change this Roman figure to natural numbers

(i)    MMCDLXXI        (ii)    MMMCLIV

1. Write the following in figures:

(a) fifteen trillion, six hundred and seventy one billion, three hundred and ninety one million, eighty eight thousand, five hundred and fifty five.

(b) three hundred and twenty-nine billion, five hundred and sixty two million, eight hundred and one thousand, four hundred and thirty three.