FORM ONE MATHEMATICS STUDY NOTES TOPIC 4-6.

TOPIC 4: UNITS

A unit – is defined as a symbol or sign which is assigned to a number to describe a kind of measurement made

Units of Length

Convertion of One Unit of Length to Another

Convert one unit of length to another

The conversion of one unit to another is done by considering the arrangement below

Example, from the above we get

Computations on Metric Units of Length

Perform computations on metric units of length

Example 1

Convert

Solution

Unit of Mass

Convertion of One Unit of Mass to Another

Convert one unit of mass to another

The conversion of one unit to another is done by considering the arrangement below

Example, from the above

The conversion of tonneto other units is done converting it to kilogramfirst and then fromkilogramto the required unit.

1 tonne = 1000kg

Computation on Metric Units of Mass

Perform computation on metric units of mass

Example 2

Solution

Units of Time

The units of time are of two types, smaller and larger units of time.Smaller units of time includes seconds, minutes, hours and days.Larger units of time includes week, month, year, decade, century,millennium.

Convertion of One Unit of Time to another

Convert one unit of time to another

The conversion of units of time to another can be done by considering the arrangement below

Example, from the circle above

Also

Example 3

Convert

Solution

Convertion of Unit Time of 12 Hour Clock to 24 Hour Clock and Vice Versa

Read and convert unit time of 12 hour clock to 24 hour clock and vice versa

The hours can exist in two systems: 12- hour clock and 24 - hour clock.

A 12- hour clock has 12 hours between midnight and midday(a.m)and 12 hours between midday and midnight(p.m).

A 21 - hour <!--[endif]-->has 24 hours in a day.Times in the morning are the same in both systems.For times in the afternoon, convert by adding or subtracting 12 hours

Example 4

Convert the following times from the 12 - hour clock to 24 - hour clock.

Example 5

Convert the following times from the 24 - hour clock to 12 - hour clock.

Units of Capacity

Standard Unit of Measuring Capacity

State the standard unit of measuring capacity

Capacity is related to the volume.

Definitions:

• Capacity-is defined as the ability hold or contain something
• The S.I unit of capacity is litre.
• Volume –is defined as the amount of space occupied by a substance
• The S.I unit of volume is cubic metres (m³)

Capacity is related to the volume as follows:

1 litre = 1000cm³ = 0.001m³ = 1dm³

Also 1 ml = 1 cm³

Other units related to litre are kiloliter (kl), hectoliter (hl), decalitre (dal), litre (l), deciliter (dl), centiliter (cl) and mililitre (ml).

The conversion of one unit to another is done by considering the arrangement below

Example, from the above

The Litre in Daily Life

Use the litre in daily life

Example 6

Convert the following units into

Solution

TOPIC 5: APPROXIMATIONS
Measurements can be rounded to a certain number of significant figures.Approximation- is a process of rounding numbers to a certain degree of accuracy.A number can be rounded to a certain required place value such as to the nearest ten, hundred, and thousand

Rounding Off Numbers

Rounding off Whole Numbers to Given Place Values

Round off whole numbers to given place values

STEPS

• When rounding a number, stand at the digit of the required place value, then look at the next digit to the right; if it is or more, round up (i.e, increase the digit of the required place value by 1) and if it is or less, do not change the digit of the required place value
• Replace all the remaining digits to the right of the required place value with the zeros

Example 1

The population of Tanzania in a census of 2002 was 42,850,671. Round this to the nearest

1. million
2. ten million

Solution

1. The million digit is 2, since the next digit to the right is greater than 5, then we can increase 2 by 1 and put the remaining digits to the right of 2 zeros. There fore;42,850,671 ≈ 43,000,000
2. The ten million digit is 4, since the next digit to the right is less than 5, then we do not change 4 but we put the remaining digits to the right of 4 zeros. There fore;42,850,671 ≈ 40,000,000

Decimals to a Given Number of Decimal Place

Round off decimals to a given number of decimal place

STEPS

• When rounding a decimal, stand at the digit of the required decimal place, then look at the next digit to the right; if it is or more, round up (i.e increase the digit of the required decimal by 1) and if it is or less, do not change the digit of the required decimal place
• Replace all the remaining digits to the right of the required decimal place with the zeros

NOTE

• The first digit after the decimal point is the first decimal place, the second digit after the decimal point is the second decimal place e.t.c
• Example 0.568 is the first decimal place and is the second decimal place

Example 2

Round 0.24736 to the nearest

1. 1 decimal place
2. 2 decimal places
3. 3 decimal places

Solution

1. 0.24736≈ 0.02 (1 d.p)
2. 0.24736≈ 0.025 (2 d.p)
3. 0.24736≈ 0.0247 (1 d.p)

Significant Figures

Significant figures of a number - are the significant digits, counted from left of the number.The first significant figure must be non-zero; following significant figures may take any value.

A Number to a Given Number of Significant Figures

Write a number to a given number of significant figures

Is the number of significant digits including 0 if it is between the first and the last

Examples

1. 13 – has two significant figures
2. 709.43 – has five significant figures
3. 0.0004001 – has four significant figures

STEPS

• When rounding a number to a certain significant figure, stand at the digit of the required significant figure, then look at the next digit to the right; if it is or more, round up (i.e increase the digit of the required significant figure by 1) and if it is or less, do not change the digit of the required significant figure
• Replace all the remaining digits to the right of the required significant figure with the zeros

Example 3

Given the number 45.274 round to

1. 1 first significant figure
2. 2 significant figure
3. significant figure

Solution

1. 50
2. 45
3. 45.3

Example 4

Round 146 400 to

1. 2 first significant figure
2. 4 significant figure
3. 3 significant figure

Solution

1. 150 000
2. 146 400
3. 146 000

Approximations in Calculations

The Knowledge of Rounding Off of Numbers in Computations Involving Large Numbers and Small Numbers

Use the knowledge of rounding off of numbers in computations involving large numbers and small numbers

Approximation can be used in operation to check whether a calculation is correct or not, i.e in addition, subtraction, multiplication and divisione.g 446 x 45 = 20 070

The above calculation can be checked quickly whether it is correct or not by rounding each number to 1 significant figure and then multiply i.e 400 x 50 = 20 000

Therefore, the approximation of 20 000 is close to 20 0070 is correct.Before carrying an operation, each number in a calculation is rounded to 1 significant figure

Example 5

Find the approximate value of

Example 6

A school trip of 32 people went to a tour, which costs a transport fee of 580/- each people. What was the approximate total transport cost?

Solution

32 × 580 ≈30×600 = 18 000

The approximate transport cost was 18 000/- 

TOPIC 6: GEOMETRY

Points and Lines

The Concept of a Point

Explain the concept of a point

A point – is a smallest geometric figure which gives a position of object in a plane

A line segment – is a straight line joining two points in a plane

The Concept of a Point to Draw a Line

Extend the concept of a point to draw a line

A line segment – is a straight line joining two points in a plane

A line passing through two points e.g A and B and extends without end (i.e infinitely) in both directions is denoted by

The Difference Between a Line, a Line Segment and a Ray

Distinguish between a line, a line segment and a ray

A ray - is a line starting from a point, say A and pass through a point, say B and extends without end in one direction.It is denoted by

Angles and Lines

Angles

Draw angles

An angle – is a measure of an amount of turn. For instance, a complete turn has an angle of 360º

Measuring Angles of Different Size Using a Protractor

Measure angles of different size using a protractor

There are several types of angles including:- acute, right, complementary, obtuse, supplementary and reflex angle

Example 1

1. Two angles are supplementary. One angle is three times the other. What are the angles?
2. Two angles are complementary. One angle is 40º greater than the other. What are the angles?

Solution

Drawing Angles Using a Protractor

Draw angles using a protractor

The angles formed by crossing lines includes vertically opposite angles, alternate angle and corresponding angles

Vertically opposite angles

The angles on the opposite sides of the crossing lines are equal

Alternate angles

Consider a line segment crossing two parallel line segments. This line is called a transversal

<!--[endif]-->The angles within the parallel line segments on the opposite sides of the transversal are equal

They are also called Z - angles

<!--[endif]-->Corresponding angles

The angles on the same side of the transversal and on the same side of the parallel lines are equal.They are called corresponding angles and sometimes called F - angles

There are also three other pairs of corresponding angles in the diagram above.When showing that two angles are equal you must give reason whether they are vertically opposite, or alternate or corresponding angles.

Constructions

Construction of a Perpendicular Bisector to a Line Segment

Construct a perpendicular bisector to a line segment

Perpendicular Bisector to a Line Segment is shown below

Contruction of an Angle of 60° Using a Pair of Compasses

Construct an angle of 60° using a pair of compasses

Angle of 60°

Bisection of a Given Angle

Bisect a given angle

Activity 1

Bisect a given angle

Copying a Given Angle by Construction

Copy a given angle by construction

Activity 2

Copy a given angle by construction

Parallel Lines

Construct parallel lines

Parallel lines can be shown as below:

Different Types of Angles Formed by Parallel Lines and a Transversal

Identify different types of angles formed by parallel lines and a transversal

Different types of angles are shown below.

Polygons And Regions

A Polygon and a Region

Describe a polygon and a region

Apolygonis a plane figure whose sides are three or more coplanar segments that intersect only at their endpoints. Consecutive sides cannot be collinear and no more than two sides can meet at any one vertex.

Apolygonal regionis defined as a polygon and its interior.

Different Types of Triangles

Construct different types of triangles

A triangle – is a polygon with three sides.The sides connect the points called vertices

A right – angled triangle – has one angle equal to 90º

An isosceles triangle – has two equal sides and two equal angles

An equilateral triangle – has three equal sides and all angles equal

NOTE:A triangle with all sides different and all angles different is called scalene triangle

A triangle with vertices A, B and C is denoted as

A triangle has two kinds of angles

1. Interior angles
2. Exterior angles

Interior angle – is an angle inside the triangle.The sum of interior angles of a triangle is

Example, consider the triangle below

Exterior angle - is an angle outside the triangle.Consider the triangle below

Example 2

Find the angles x and y in the diagrams below

Different Quadrilaterals

Construct different quadrilaterals

A quadrilateral – is a polygon with four sides.Examples of quadrilaterals are a square, a rectangle, a rhombus, a parallelogram, a kite and a trapezium

A square – has equal sides and all angles are 90º

A rectangle – has two pairs of opposite sides equal and all angles are 90º

A rhombus – has all sides equal.Opposite angles are also equal

A parallelogram – has two pairs of opposite sides equal.Opposite angles are also equal

A kite – has two pairs of adjacent sides equal.One pair of opposite angles are also equal

A trapezium – has one pair of opposite sides pair

Any quadrilateral is made up of two triangles. Consider the below quadrilateral.

Sum of angles of quadrilateral = 2 ×180º = 360º

Example 3

Find the angles x and y in the diagrams below

Solution

Circles

A Circle

Draw a circle

To make a circle: Draw a curve that is "radius" away from a central point.

And so:All points are the same distance from the center.

You can draw it yourself:Put a pin in a board, put a loop of string around it, and insert a pencil into the loop. Keep the string stretched and draw the circle!

Different Parts of a Circle

Describe different parts of a circle

Theradiusof the circle is a straight line drawn from the center to theboundaryline or the circumference. The plural of the word radius isradii.

Thediameteris the line crossing the circle and passing through the center. It is the twice of the length of the radius.

Thecircumferenceof a circle is theboundaryline or theperimeterof the circle.

Anarcis a part of thecircumferencebetween two points or acontinuouspiece of a circle. The shorter arc betweenandis called theminor arc. The longer arc betweenandis called themajor arc.

Thechordis a straight line joining two points on thecircumferencepoints of a circle. The diameter is a special kind of the chord passing through the center.

Asemi-circleis an arc which is half of thecircumference.

Atangentis a straight line which touches the circle. It does not cut thecircumference. The point at which it touches, is called thepoint of contact.

 FORM ONE MATHEMATICS OTHER TOPICS
FORM ONE MATHEMATICS TOPIC 1-3.
FORM ONE MATHEMATICS STUDY NOTES TOPIC 4-6.
FORM ONE MATHEMATICS STUDY NOTES TOPIC 7-9.
FORM ONE MATHEMATICS STUDY NOTES TOPIC 10-11.

O'LEVEL MATHEMATICS NOTES 
FORM ONE MATHEMATICS STUDY NOTES

FORM TWO MATHEMATICS STUDY NOTES

FORM THREE MATHEMATICS STUDY NOTES

FORM FOUR MATHEMATICS STUDY NOTES