Form One Mathematics Notes Pdf

TOPIC 1: NUMBERS

we know that when we count we start 1,2 .... . But there are other numbers like 0, negative numbers and decimals. All these types of numbers are categorized in different groups like counting numbers, integers,real numbers, whole numbers and rational and irrational numbers according to their properties. all this have been covered in this chapter

Numbers are represented by symbols called numerals. For example, numeral for the number ten is 10. Numeral for the number hundred is 110 and so on.

The symbols which represent numbers are called digits. For example the number 521 has three (3) digits which are 5, 2 and 1. There are only tendigits which are used to represent any number. These digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

The Place Value in each Digit in Base Ten Numeration

Identify the place value in each digit in base ten numeration

When we write a number, for example 521, each digit has a different value called place value. The 1 on the right means 1 ones which can be written as 1 × 1, the next number which is 2 means 2 tens which can be written as 2 × 10 and the last number which is 5 means 5 hundreds which can be written as 5 × 100. Therefore the number 521 was found by adding the numbers 5 × 100 + 2 × 10 + 1× 1 = 521.

Note that when writing numbers in words, if there is zero between numbers we use word 'and'

Example 1

Write the following numbers in words:

7 008
99 827 213
59 000

Solution

7 008 = Seven thousand and eight.
99 827 213 = Ninety nine millions eight hundred twenty seven thousand two hundred thirteen.
59 000 = Fifty nine thousand.

Example 2

Write the numbers bellow in expanded form.

732.
1 205.

Solution

732 = 7 x 100 + 3 x 10 + 2 x 1
1 205 = 1 x 1000 + 2 x 100 + 0 x 10 + 5 x 1

Example 3

Write in numerals for each of the following:

9 x 100 + 8 x 10 + 0 x 1
Nine hundred fifty five thousand and five.

Solution

9 x 100 + 8 x 10 + 0 x 1 = 980
Nine hundred fifty five thousand and five = 955 005.

Example 4

For each of the following numbers write the place value of the digit in brackets.

89 705 361 (8)
57 341 (7)

Solution

8 is in the place value of ten millions.
7 is in the place value of thousands.

Numbers in Base Ten Numeration

Read numbers in base ten numeration

Base Ten Numeration is a system of writing numbers using ten symbols i.e. 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Base Ten Numeration is also called decimal system of Numeration.

Numbers in Base Ten Numeration up to One Billion

Write numbers in base ten numeration up to one billion

Consider the table below showing place values of numbers up to one Billion.

Billions	Hundred millions	Ten millions	Millions	Hundred Thousands	Ten Thousands	Thousands	Hundreds	Tens	Ones
									1
								1	0
							1	0	0
						1	0	0	0
					1	0	0	0	0
				1	0	0	0	0	0
			1	0	0	0	0	0	0
		1	0	0	0	0	0	0	0
	1	0	0	0	0	0	0	0	0
1	0	0	0	0	0	0	0	0	0

If you are given numerals for a number having more than three digits, you have to write it by grouping the digits into groups of three digits from right. For example 7892939 is written as 7 892 939.

When we are writing numbers in words we consider their place values. For example; if we are told to write 725 in words, we first need to know the place value of each digit. Starting from right side 5 is in the place value of ones, 2 is in the place value of tens and seven is in the place value of hundreds. Therefore our numeral will be read as seven hundred twenty five.

Numbers in Daily Life

Apply numbers in daily life

Numbers play an important role in our lives. Almost all the things we do involve numbers and Mathematics. Whether we like it or not, our life revolves in numbers since the day we were born. There are numerous numbers directly or indirectly connected to our lives.

The following are some uses of numbers in our daily life:

Calling a member of a family or a friend using mobile phone.
Calculating your daily budget for your food, transportation, and other expenses.
Cooking, or anything that involves the idea of proportion and percentage.
Weighing fruits, vegetables, meat, chicken, and others in market.
Using elevators to go places or floors in the building.
Looking at the price of discounted items in a shopping mall.
Looking for the number of people who liked your post on Facebook.
Switching the channels of your favorite TV shows.
Telling time you spent on work or school.
Computing the interest you gained on your business.

Integers

Identify integers

Consider a number line below

The numbers from 0 to the right are called positive numbers and the numbers from 0 to the left with minus (-) sign are called negative numbers. Therefore all numbers with positive (+) or negative (-) sign are called integers and they are denoted by Ζ. Numbers with positive sign are written without showing the positive sign. For example +1, +2, +3, … they are written simply as 1, 2, 3, … . But negative numbers must carry negative sign (-). Therefore integers are all positive and negative numbers including zero (0). Zero is neither positive nor negative number. It is neutral.

The numbers from zero to the right increases their values as the increase. While the numbers from zero to the left decrease their values as they increase. Consider a number line below.

If you take the numbers 2 and 3, 3 is to the right of 2, so 3 is greater than 2. We use the symbol '>' to show that the number is greater than i. e. 3 >2(three is greater than two). And since 2 is to the left of 3, we say that 2 is smaller than 3 i.e. 2<3. The symbol '<' is use to show that the number is less than.

Consider numbers to the left of 0. For example if you take -5 and -3. -5 is to the left of -3, therefore -5 is smaller than -3. -3 is to the right of -5, therefore -3 is greater than -5.

Generally, the number which is to the right of the other number is greater than the number which is to the left of it.

If two numbers are not equal to each to each other, we use the symbol '≠' to show that the two numbers are not equal. The not equal to '≠' is the opposite of is equal to '='.

Example 21

Represent the following integers Ζ on a number line

0 is greater than Ζ and Ζ is greater than -4
-2 is less than Ζ and Ζ is less than or equal to 1.

a. 0 is greater than Ζ means the integers to the left of zero and Ζ is greater than -4 means integers to the left of -4. These numbers are -1, -2 and -3. Consider number line below

b. -2 is less than Ζ means integers to the right of -2 and Ζ is less than or equal to 1 means integers to the left of 1 including 1. These integers are -1, 0 and 1. Consider the number line below

Example 22

Put the signs 'is greater than' (>), 'is less than' (<), 'is equal to' (=) to make a true statement.

Addition of Integers

Add integers

Example 23

Show a picture of 2 and 3 on a number line.

When drawing integers on a number line, the arrows for the positive numbers goes to the right while the arrows for the negative numbers goes to the left. Consider an illustration bellow.

The distance from 0 to 3 is the same as the distance from 0 to -3, only the directions of their arrows differ. The arrow for positive 3 goes to the right while the arrow for the negative 3 goes to the left.

Example 24

Solution

Subtraction of Integers

Subtract integers

Since subtraction is the opposite of addition, if for example you are given 5-4 is the same as 5 + (-4). So if we have to subtract 4 from 5 we can use a number line in the same way as we did in addition. Therefore 5-4 on a number line will be:

Take five steps from 0 to the right and then four steps to the left from 5. The result is 1.

Multiplication of Integers

Multiply integers

Example 25

2×6 is the same as add 2 six times i.e. 2×6 = 2 + 2 + 2 + 2 + 2 +2 = 12. On a number line will be:

Multiplication of a negative integer by a negative integer cannot be shown on a number line but the product of these two negative integers is a positive integer.

From the above examples we note that multiplication of two positive integers is a positive integer. And multiplication of a positive integer by a negative integer is a negative integer. In summary:

(+)×,(+) = (+)
(-)×,(-) = (+)
(+)×,(-) = (-)
(-)×,(+) = (-)

Division of Integers

Divide integers

Example 26

6÷3 is the same as saying that, which number when you multiply it by 3 you will get 6, that number is 2, so, 6÷3 = 2.

Therefore division is the opposite of multiplication. From our example 2×3 = 6 and 6÷3 = 2. Thus multiplication and division are opposite to each other.

Dividing two integers which are both positive the quotient (answer) is a positive integer. If they are both negative also the quotient is positive. If one of the integer is positive and the other is negative then the quotient is negative. In summary:

(+)÷(+) = (+)
(-)÷(-) = (+)
(+)÷(-) = (-)
(-)÷(+) = (-)

Mixed Operations on Integers

Perform mixed operations on integers

You may be given more than one operation on the same problem. Do multiplication and division first and then the rest of the signs. If there are brackets, we first open the brackets and then we do division followed by multiplication, addition and lastly subtraction. In short we call it BODMAS. The same as the one we did on operations on whole numbers.

Example 27

=3 + 6 -1 (first divide and multiply)

=8 (add and then subtract)

Example 28

=1 + 4 – 7 (do operations inside the brackets and divide first)

TOPIC 2: FRACTIONS

A fraction is a number which is expressed in the form of a/b where a - is the top number called numerator and b- is the bottom number called denominator.

A Fraction

Describe a fraction

A fraction is a number which is expressed in the form of a/b where a - is the top number called numerator and b- is the bottom number called denominator.

Consider the diagram below

The shaded part in the diagram above is 1 out of 8, hence mathematically it is written as 1/8

Example 1

(a) 3 out of 5 ( three-fifths) = 3/5

Example 2

(b) 7 0ut of 8 ( i.e seven-eighths) = 7/8

Example 3

5/12=(5 X 3)/(12 x 3) =15/36
3/8 =(3 x 2)/(8 X 2) = 6/16

Dividing the numerator and denominator by the same number (This method is used to simplify the fraction)

Difference between Proper, Improper Fractions and Mixed Numbers

Distinguish proper, improper fractions and mixed numbers

Proper fraction -is a fraction in which the numerator is less than denominator

Improper fraction -is a fraction whose numerator is greater than the denominator

Mixed fraction -is a fraction which consist of a whole number and a proper fraction

Example 6

(a) To convert mixed fractions into improper fractions, use the formula below

(b)To convert improper fractions into mixed fractions, divide the numerator by the denominator

Example 7

Convert the following mixed numbers into improper fractions

Addition of Fractions

Add fractions

Operations on fractions involves addition, subtraction, multiplication and division

Addition and subtraction of fractions is done by putting both fractions under the same denominator and then add or subtract
Multiplication of fractions is done by multiplying the numerator of the first fraction with the numerator of the second fraction, and the denominator of the first fraction with the denominator the second fraction.
For mixed fractions, convert them first into improper fractions and then multiply
Division of fractions is done by taking the first fraction and then multiply with the reciprocal of the second fraction
For mixed fractions, convert them first into improper fractions and then divide

Example 11

Find

Solution

Subtraction of Fractions

Subtract fractions

Example 12

Evaluate

Solution

Multiplication of Fractions

Multiply fractions

Example 13

Division of Fractions

Divide fractions

Example 14

Mixed Operations on Fractions

Perform mixed operations on fractions

Example 15

Example 16

Word Problems Involving Fractions

Solve word problems involving fractions

Example 17

Musa is years old. His father is 3¾times as old as he is. How old is his father?
1¾of a material are needed to make suit. How many suits can be made from

TOPIC 3: DECIMAL AND PERCENTAGE

The Concept of Decimals

Explain the concept of decimals

A decimal- is defined as a number which consist of two parts separated by a point.The parts are whole number part and fractional part

Example 1

Example 2

Conversion of Fractions to Terminating Decimals and Vice Versa

Convert fractions to terminating decimals and vice versa

The first place after the decimal point is called tenths.The second place after the decimal point is called hundredths e.t.c

Consider the decimal number 8.152

NOTE

To convert a fraction into decimal, divide the numerator by denominator
To convert a decimal into fraction, write the digits after the decimal point as tenths, or hundredths or thousandths depending on the number of decimal places.

Example 3

Convert the following fractions into decimals

Divide the numerator by denominator

Expressing a Quantity as a Percentage

Express a quantity as a percentage

The percentage of a quantity is found by converting the percentage to a fraction or decimal and then multiply it by the quantity.

Example 9

NOTE:The concept of percentage of a quantity can be used to solve the problems involving percentage increase and decrease as shown in the below examples:-

A Fractions into Percentage and Vice Versa

Convert a fraction into percentage and vice versa

To change a fraction or a decimal into a percentage, multiply it by 100%

Example 10

Convert the following fractions into percentages

A Decimal into Percentage and Vice Versa

Convert a decimal into percentage and vice versa

To change a percentage into a fraction or a decimal, divide it by 100%

Example 11

Convert the following percentages into decimals

Percentages in Daily Life

Apply percentages in daily life

Example 12

In an assignment, Regina scored 9 marks out of 12. Express this as a percentage

Solution

Example 13

A school has 400 students of which 250 are girls. What percentage of the students are not girls?

Solution

TOPIC 7: ALGEBRA

An algebraic expression – is a collection of numbers, variables, operators and grouping symbols.Variables - are letters used to represent one or more numbers

Symbols to form Algebraic Expressions

Use symbols to form algebraic expressions

The parts of an expression collected together are called terms

Example

x + 2x – are called like terms because they have the same variables
5x +9y – are called unlike terms because they have different variables

An algebraic expression can be evaluated by replacing or substituting the numbers in the variables

Example 1

Evaluate the expressions below, given that x = 2 and y = 3

Example 2

Evaluate the expressions below, given that m = 1 and n = - 2

An expression can also be made from word problems by using letters and numbers

Example 3

A rectangle is 5 cm long and w cm wide. What is its area?

Simplifying Algebraic Expressions

Simplify algebraic expressions

Algebraic expressions can be simplified by addition, subtraction, multiplication and division

Addition and subtraction of algebraic expression is done by adding or subtracting the coefficients of the like terms or letters

Coefficient of the letter – is the number multiplying the letter

Multiplication and division of algebraic expression is done on the coefficients of both like and unlike terms or letters

Example 4

Simplify the expressions below

Solution

An inequality – is a mathematical statement containing two expressions which are not equal. One expression may be less or greater than the other.The expressions are connected by the inequality symbols<,>,≤ or≥.Where< = less than,> = greater than,≤ = less or equal and ≥ = greater or equal.

Linear Inequalities with One Unknown

Solve linear inequalities in one unknown

An inequality can be solved by collecting like terms on one side.Addition and subtraction of the terms in the inequality does not change the direction of the inequality.Multiplication and division of the sides of the inequality by a positive number does not change the direction of the inequality.But multiplication and division of the sides of the inequality by a negative number changes the direction of the inequality

Example 11

Solve the following inequalities

Linear Inequalities from Practical Situations

Form linear inequalities from practical situations

To represent an inequality on a number line, the following are important to be considered:

The endpoint which is not included is marked with an empty circle
The endpoint which is included is marked with a solid circle

Example 12

Compound statement – is a statement made up of two or more inequalities

Example 13

Solve the following compound inequalities and represent the answer on the number line

Solution

TOPIC 8: NUMBERS

A Rational Number

Define a rational number

ARational Numberis a real number that can be written as a simple fraction (i.e. as aratio). Most numbers we use in everyday life are Rational Numbers.

Number	As a Fraction	Rational?
5	5/1	Yes
1.75	7/4	Yes
.001	1/1000	Yes
-0.1	-1/10	Yes
0.111...	1/9	Yes
√2(square root of 2)	?	NO !

The square root of 2 cannot be written as a simple fraction! And there are many more such numbers, and because they arenot rationalthey are calledIrrational.

The Basic Operations on Rational Numbers

Perform the basic operations on rational numbers

Real Numbers

Real Numbers

Define real numbers

he type of number we normally use, such as 1, 15.82, −0.1, 3/4, etc.Positive or negative, large or small, whole numbers or decimal numbers are all Real Numbers.

They are called "Real Numbers" because they are not Imaginary Numbers.

Absolute Value of Real Numbers

Find absolute value of real numbers

The absolute value of a number is the magnitude of the number without regard to its sign. For example, the absolute value of 𝑥 𝑜𝑟 𝑥 written as 𝑥 . The sign before 𝑥 is ignored. This is because the distance represented is the same whether positive or negative. For example, a student walking 5 steps forward or 5 steps backwards will be considered to have moved the same distance from where she originally was, regardless of the direction.