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Revision notes – Form 1 Maths Formula One A2 – 2011/12 (Mr. S. Azzopardi)

Chapter 1

How our numbers work

Read and write whole numbers in figures and words

Eg. (a) 23456 – twenty three thousand four hundred fifty six

(b) one million three thousand and fifty three – 1003053

Know how to write powers of ten. Trillions Billions Millions Hundred Thousands Ten Thousands Thousands Hundreds Tens Units

1012 109 106 105 104 103 102 101 100 = 1

Eg. Ten – 10 – 101, Hundred – 100 – 10

2, Thousand – 1000 – 10

3, … , Trillion – 1000000000000 – 10

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Multiply and divide whole numbers by 10, 100 or 1000

To multiply whole numbers by 10, 100 or 1000, just add zeros to the whole number and when dividing, the number of zeros of the

divisor corresponds to the number of digits of the remainder.

Multiply and divide large numbers that end in noughts ( zeros).

First multiply the digits of the numbers that are not zeros and then put the zeros on the right of the product.

Chapter 2

Position

Practice on Spreadsheet software (Excel Microsoft Office)

Chapter 3

Basic number

Add natural numbers up to 1000 and subtract numbers less than 1000.

Eg. 234 + 156 = 390 ; 789 – 268 = 521

Learn the multiplication facts up to 10 ×10. Multiply and divide natural numbers by a single-digit number.

Eg. Times 10 table; 14574 ÷ 3 = 4858

Multiply numbers by a two-digit number. – Eg.: 67 × 24 67

× 24

673 × 24 673

× 24

13460 673 × 20

+ 2692 673 × 4

16152

32 × 15

× 10 5

30 300 150

2 20 10

= 300 + 150 + 20 + 10 = 480 Ans

28 7 × 4

240 60 × 4

140 7 × 20

+ 1200 60 × 20

1608

Divide numbers by a two-digit number using repeated subtraction. – Eg.: 256 ÷ 16

16 ) 256

- 160 10 × 16

96

- 96 + 6 × 16

0 Answer 16 rem 0

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Chapter 4

Angles: An angle is formed by two rays that have a common end point. This common end point is referred to as the vertex and the rays

• A

are the sides of the angle. • • B C Angles are

represented by the symbol (∠). When naming angles the letter representing the vertex is either used

by itself (example “∠B”) or as the second point (example “∠ABC”)

Angles are measured in degrees, which is represented by the symbol ( ° ). The size of an angle is measured by using an instrument called a “protractor.” To use a protractor, place the center of the protractor at the vertex of the angle and the edge along a side of the angle. See below for an example of how a protractor is used to measure angles.

Center of the protractor side of the angle

Outer ring Inside ring

If the angle starts from the right side of the protractor (as in the example above), then you will use the numbers on the inside ring to get your measurement.

If the angle starts from the left side of the protractor, then you will use the numbers on the outer ring to get your measurement.

Distinguish between right, acute, obtuse and reflex angle.

• REVOLUTION – 1 COMPLETE TURN (360°)

• HALF A REVOLUTION (STRAIGHT LINE) – HALF A TURN (180°)

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• RIGHT ANGLE – QUARTER OF A REVOLUTION (90°)

• ACUTE – ANGLE GREATER THAN 0° AND SMALLER THAN 90°

• OBTUSE – ANGLE GREATER THAN 90° AND SMALLER THAN 180°

• REFLEX – ANGLE GREATER THAN 180° AND SMALLER THAN 360°

Distinguish between line and line segments.

A line is a continuous straight line which has no end point.

A line segment is part of a line that has two end points.

Identify parallel lines in geometric figures.

Lines in a plane can be parallel, perpendicular, or intersecting.

Parallel lines are lines that never intersect. The symbol “||” is used to l1

indicate parallel lines. Example: l1 || l2 means that l2

the line l1 is parallel to line l2.

l1

Perpendicular lines are lines that intersect at 90° angles. The

symbol “⊥” is used to represent perpendicular lines. Example: l2

l1 ⊥ l2 means that line l1 is perpendicular to line l2.

If the lines are not parallel or perpendicular, then they are just l1

two intersecting lines.

l2

Related Angles

Lines AB and CD are parallel to one another (hence the » on the lines).

a and d are vertically opposite angles. Vertically opposite angles are equal. (b and c, e and h, f and g are

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also vertically opposite).

g and c are corresponding angles. Corresponding angles are equal. (h and d, f and b, e and a are also

corresponding).

d and e are alternate angles. Alternate angles are equal. (c and f are also alternate). Alternate angles form a

'Z' shape and are sometimes called 'Z angles'.

a and b are adjacent angles or supplementary angles. Adjacent angles add up to 180 degrees. (d and c, c

and a, d and b, f and e, e and g, h and g, h and f are also adjacent).

d and f are interior angles. These add up to 180 degrees (e and c are also interior).

a, b, c and d are angles at a point that add up to 360 degrees (e, f, g and h are also angles at a point).

The interior angles of a triangle add up to 180°.

Similarly interior angles of a quadrilateral add up to 360°.

Summary of facts (very important)

Vertically opposite angles are equal

Angles on a straight line add up to 180° (adjacent angles)

Two angles on a straight line that add up to 180° are called Supplementary angles.

Two angles that add up to 90° are called Complementary angles.

Angles at a point add up to 360°

All the angles of an equilateral triangle are 60°

The base angles of an isosceles triangle are equal (upright sides equal)

The interior angles of a triangle add up to 180°

The interior angles of a quadrilateral add up to 360°

Corresponding angles are equal

Alternate angles are equal

Interior angles add up to 180°

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Chapter 5

Displaying data

Draw and interpret bar charts and pie charts.

Bar Chart

A bar chart is a chart where the height of bars represents the frequency. The data is 'discrete'

(discontinuous- unlike histograms where the data is continuous). The bars should be separated by

small gaps.

Pie Chart

A pie chart is a circle which is divided into a number of parts.

The pie chart above shows the TV viewing figures for the following TV programmes:

Eastenders, 15 million

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Casualty, 10 million

Peak Practice, 5 million

The Bill, 8 million

Total number of viewers for the four programmes is 38 million. To work out the angle that

'Eastenders' will have in the pie chart, we divide 15 by 38 and multiply by 360° (degrees). This is

142°. So 142° of the circle represents Eastenders. Similarly, 95° of the circle is Casualty, 47° is

Peak Practice and the remaining 76° is The Bill.

Chapter 6

Symmetry

Identify and draw the lines of symmetry of shapes.

A shape has line symmetry if when folded along a broken (dotted) line fits exactly over the other

half.

Determine the order of rotational symmetry in 2D.

The order of rotational symmetry is the number of times (more than 1) a shape fits onto itself when

it is rotated 360°.

A shape has no rotational symmetry if it has rotational symmetry of order 1.

Chapter 7

Decimals

The number 31.88 is in Decimal Notation. (more commonly referred to as Decimals)

A number written in Decimal Notation has three parts.

Whole Number Decimal Decimal

31 . 88

Part Point Part

Note:

(a) The Decimal part of a number represents a number less than one.

(b) The portion of a digit in a decimal determines the digits place value.

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4 Hundred

5 Tens

8 Ones

. Decimal

3 Tenths

0 Hundredths

2 Thousandths

7 Tenths Thousandths

1 Hundredths

Thousandths

9 Millionths

Note the relationship between fractions and numbers written in Decimal Notation.

Seven Tenths Seven Hundredths Seven Thousandths

7/10 = 0.7

7/100 = 0.07

7/1000 = 0.007

1 zero in 10 2 zeros in100 3 zeros in 1000

1 decimal place in .7 2 decimal place in .07 3 decimal place in .007

Scales

To read scales determine the value of a section.

Arrange numbers in ascending and descending order.(includes fractions and decimals)

To compare the sizes of fractions change all fractions to equivalent fractions with the same

denominator or change to decimals to two decimal places.

Change fractions to decimals.

To change a fraction to a decimal number, change the fraction to an equivalent fraction with

denominator 10, 100 or 1000; or else if it is not possible divide the numerator by the denominator

to two decimal places.

Addition and subtraction of decimals

To add decimals, write the numbers so that the decimal points are on a vertical line. Add as you would with whole numbers. Then write the decimal point in the sum directly below the decimal points in the addends. Add: 0.326 + 4.8 + 57.23

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Note that by placing the decimal points on a vertical line, digits of the same place value

were added.

Insert zeros so that it has the same number of decimal places

0.326

4.800

+57.230

62.356

Subtract: 31.642 - 8.759

Note that by placing the decimal points on a vertical line, digits of the same place value are

subtracted.

31.642

- 8.759

22.883

Chapter 8

Co-ordinates

Mark points in the first quadrant using an ordered pair of numbers.

When the axes are drawn in a coordinate plane, the plane is divided into 4 sections called

quadrants.

Mark points using an ordered pair of numbers in any quadrant.

Co-ordinates are a set of numbers which pin point a specific point on a graph. They are written in

the form ( x, y ), where x is the distance along the x-axis, the horizontal, and the y is the distance

along the y-axis, the vertical.

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Here are some examples:

A - ( 4 , 8 )

B - ( 6 , 2 )

C - ( 2 , -6 )

D - ( -8 , -8 )

E - ( -2 , 2 )

Co-ordinates are used for several things:

Drawing a shape by giving the co-ordinates of the vertices.

Drawing a graph by giving points along the graph.

Finding patterns between the x and y co-oridnates.

Used to show the center of transformations.

Chapter 9

Fractions

Understand the notion of a fraction.

A fraction is a part of a whole.

A Proper fraction is a fraction whose numerator is less than its denominator.

An improper fraction is a fraction whose numerator is greater than or equal to its denominator.

E.g.: 11

/7.

(To change an improper fraction to a mixed number, divide the numerator by the denominator)

A mixed number is a whole number together with a proper fraction e.g. 1 4/7

(To change a mixed number to an improper fraction, multiply the whole number by the

denominator and add the result with the numerator to get the new numerator. The denominator

remains the same)

Find a fraction of a quantity.

Eg. 9248

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Reduce fractions to their lowest terms.

Fractions, of course, can often be 'cancelled down' or simplified to make them simpler. For

example, 4/6 =

2/3. You can divide or multiply the top and bottom of any fraction by any number, as

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long as you do it to both the top and bottom. However, when there is more than one term on the top

and/or bottom, to cancel you must divide every term in the top and bottom by that number.

Multiply one fraction by another fraction.

Eg. 8

1

28

9

18

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Add and subtract 2 fractions with different denominators.

To add two fractions, the bottom (denominator) of the two fractions must be the same. 1/2 +

3/2 =

4/2

= 2; 1/10 +

3/10 +

5/10 =

9/10 . If the denominators are not the same, multiply the top and bottom of

one (or more) of the fractions by a number to make the denominators the same.

Example

5 + 2 = 5 + 4 = 9 = 3 = 1 1/2

6 3 6 6 6 2

The same is true when subtracting fractions.

To add or subtract mixed numbers, first change everything to improper fractions and then continue

as above.

Find the percentage of a quantity.

The value of a percentage of a given quantity is:

Percentage × given quantity

100

Use the ratio notation to compare two or more quantities.

A ratio is a quotient used to compare two or more quantities of the same units of measure.

Ratios compare two or more quantities which are in the same units. They are written with a colon

( : ) between each values. Ratios can be treated just like fractions and can be simplied.

Finding Ratios

Finding a ratio between a set of numbers means finding out numerically how the numbers compare

with each other.

For example:

Find the ratio between 50 euro cent and 1 euro .

Put these into the same units: 1 euro is the same as 100 euro cent.

Put them into the format of a ratio: 50 euro cent : 100 euro cent

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Simplify this to it's simplest form: 50 : 100 = 1 : 2

This shows that the 1 euro was twice as much as the 50 euro cent.

Using Ratios

A ratio shows the relation between each value. For a ratio a:b:c there are three different parts. Each

of these represents a part of a whole. This ratio means there are a+b+c parts. This means a value

which is to be split into the values with this ratio are in the parts a/(a+b+c),

b/(a+b+c) and

c/(a+b+c) which,

together, make a whole.

For example:

Split 100 euro in the ratio 3:2:5

This means there are 10 parts, found by the sum of the values in the ratio. Each of the 10 parts are

equal to each other. That means each part is 10 euro, found by 100 divided by 10. We can then find

how the 100 euro is split:

First part = ( 3 × 10 ) or ( 3/10 of 100 ) = 30 euro

Second part = ( 2 × 10 ) or ( 2/10 of 100 ) = 20 euro

Third part = ( 5 × 10 ) or ( 5/10 of 100 ) = 50 euro

This shows that 100 euro split by the ratio - 3:2:5

gives the values - 30 euro : 20 euro : 50 euro

Write ratios in their simplest form.

Ratios can be simplified like fractions. E.g. 12:36 = 1:3 (dividing both numbers by 12)

Chapter 10

Number patterns

Know the meaning of factor and multiple, even and odd.

A factor is a number that divides another number exactly. 2 is a factor of 12.

A multiple is a number contained in another number several number of times. 36 is a multiple of 3.

An even number is a number that is divisible by 2.

An odd number is not divisible by 2.

Find common factors and multiples of up to 2 numbers.

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The lowest common multiple (LCM) of two or more numbers is the smallest number into which

they evenly divide.

For example, the LCM of 2 and 3 is 6.

Also consider all multiples of: 2 – 2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,etc

3 – 3,6,9,12,15,18,21,24,27,30,33,36,etc

Note that 6 is the least common multiple of 2 and 3.

Understand and use squares and square roots.

Square numbers are numbers which can be obtained by multiplying another number by itself. E.g.

36 is a square number because it is 6 x 6.

Square roots are numbers that when squared give the value of the original numbers. E.g.: the

square root of 49 = 7 because 7² = 49.

Use the calculator to find squares and squares roots.

Use – ^ or x2 square – √ square root

Recognise prime numbers and write numbers as a product of their prime factors.

Prime numbers are numbers above 1 which cannot be divided by anything, other than 1 and itself,

to give a whole number. The first 8 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19.

We can use a factor tree to work out all the prime factors of a number.

E.g.: Write 60 in prime factors:

Answer: 22×3×5

Understand the terms cube and cube root.

Cube numbers are numbers which can be obtained by multiplying three identical numbers. E.g. 8 is

a cube number because it is 2 × 2 × 2.

Cube roots are numbers that when cubed give the value of the original numbers. E.g.: the cube root

of 27 = 3 because 33 = 27.

Chapter 11

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Everyday measures

Length

1 kilometre (km) = 1000 m 1 metre (m) = 100cm

1 centimetre (cm) = 10mm 1 m = 1000mm

km to m → amount x 1000 m to cm → amount x 100

cm to mm → amount x 10 m to mm → amount x 1000

m to km → amount † 1000 cm to m → amount † 100

mm to cm → amount † 10 mm to m → amount † 1000

Weight

1 tonne (t) = 1000 kg 1 kilogram (kg) = 1000g (grams) 1 gram (g) = 1000 milligrams (mg)

t to kg → amount x 1000 kg to g → amount x 1000 g to mg → amount x 1000

kg to t → amount † 1000 g to kg → amount † 1000 mg to g → amount † 1000

Note: Use the respective statement in order to:

(a) change from big to small units multiply for example m to cm multiply by 100 etc.

(b) change from small to big units divide for example cm to m divide by 100 etc.

Capacity – volume of liquids

1 litre = 1000 millilitres (ml) 1 litre = 100 centilitres (cl) 1 litre = 1000 cm³ 1 ml = 1 cm3

1000 litres = 1 m3

litres to ml → amount x 1000 litres to cl → amount x 100

litres to cm³ → amount x 1000 ml to cm3 → amount x 1 m

3 to litres→ amount x 1000

ml to litres → amount † 1000 cl to litres → amount † 100

cm³ to litres → amount † 1000 cm3 to ml → amount † 1 litres to m

3 → amount † 1000

Money: Lm1 = 100c (cents)

₤1 (Pound Sterling) = 100p (pence)

1$ (US dollar) = 100c (cents)

1€ (Euro) = 100c (euro cent)

Lm to c → amount x 100 ₤ to p → amount x 100 $ to c → amount x 100 € to c → amount x

100

c to Lm → amount † 100 p to ₤ → amount † 100 c to $ → amount † 100 c to € → amount †

100

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Time: 1 year = 365 days or 366 days (leap year every 4 years – divides exactly by 4)

1 year = 52 weeks

1 month = 30 or 31 days (February 28 days or 29 days – leap year)

1 week = 7 days

1 day = 24 hours

1 hour = 60 minutes

1 minute = 60 seconds

12-hour clock starts from 12.00pm and ends at 12.00 am (am is morning and pm is evening)

24-hour clock starts from 00.00 and ends at 23.59

In the case of pm, to change from 12-hour clock to 24-hour clock add 12 to the number of hours.

E.g. 8.20pm = (8+12) 20:20; 11.59pm = (11+12) = 23:59

6.23am = 06:23; 10.03am = 10:03

When you cannot subtract the minutes, you have to borrow 6 from the hours and not 10.

When you add, the minutes answer must be 59 or less.

Chapter 12

Flat shapes

A polygon is a shape made of straight lines. For e.g. triangle, quadrilateral, pentagon, etc.

A regular polygon is a shape made of straight lines of equal sides and of equal angles. For e.g.

equilateral triangle, square, regular pentagon, etc.

Number of Side Name of the Polygon 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon Properties of special quadrilaterals

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• SQUARE - ALL SIDES EQUAL, ALL ANGLES 90°, OPPOSITE SIDES PARALLEL,

DIAGONALS EQUAL IN LENGTH, DIAGONALS BISECT AT RIGHT ANGLES

• RECTANGLE - OPPOSITE SIDES EQUAL, OPPOSITE SIDES PARALLEL, ALL ANGLES 90°,

DIAGONALS EQUAL IN LENGTH, DIAGONALS BISECT EACH OTHER

• RHOMBUS - ALL SIDES EQUAL, OPPOSITE SIDES PARALLEL, OPPOSITE ANGLES EQUAL, DIAGONALS BISECT AT RIGHT ANGLES

• PARALLELOGRAM / RHOMBOID - OPPOSITE SIDES EQUAL, OPPOSITE ANGLES EQUAL, OPPOSITE SIDES PARALLEL, DIAGONALS BISECT EACH OTHER

• TRAPEZIUM - ONE PAIR OF OPPOSITE SIDES ARE PARALLEL

• KITE – TWO PAIRS OF EQUAL SIDES, ONE PAIR OF EQUAL OPPOSITE ANGLES

• ARROWHEAD – TWO PAIRS OF EQUAL SIDES, ONE PAIR OF EQUAL OPPOSITE ANGLES

Construction of Hexagon – (revise radius, diameter, circumference, arc)

To construct a hexagon of side 5cm, open the compasses 5cm. Draw a circle of radius 5cm. Place

the point of the compasses anywhere on the circumference of the circle and draw an arc on the

circumference. Repeat the procedure on the newly marked arcs. Finally with the ruler join the arcs

by straight lines until forming a regular hexagon (a 6-sided regular polygon).

Kite

Arrowhead

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Construction of a line.

To construct a line, draw a line, mark a point on one side of the line, use a ruler to adjust the size of

the compasses then put the point of the compasses on the point marked before and draw an arc

where it meets on the line.

Construction of a 60° angle (same as drawing an equilateral triangle)

Draw a line and mark a point on one side of the line. With compasses measure any length from

your ruler. With the point of your compasses on the point of the line draw a wide arc meeting with

the line. Then with the same length, with the point of the compasses where the arc met the line,

draw another arc that will intersect with the other end of the arc. Finally join the first point of the

line with the intersection of the two arcs.

Circle

The distance around a circle is called circumference.

The distance from the circumference passing through the center of the circle to the other side on

the circumference is called the diameter.

The distance of half the diameter is called the radius.

Triangles

In a triangle:

It has 3 sides and angles

The corners are called vertices (singular vertex)

Vertices named A, B and C

„Triangle ABC‟ or „ΔABC‟

Sides: „side AB‟ or „AB‟

Angle: „angle A‟ or

Types of triangles

A

B C A or A

radius

diameter

circumference

center

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An equilateral triangle has all sides equal and all angles 60°, an isosceles triangle has the base

angles equal and the upright sides equal. The scalene triangle has no sides and no angles equal.

The right angled triangle has a right angle.

Construction of triangles

1. Given one side and two angles: Construct ΔABC in which AB=7cm, angle A=30° and angle

B=40°.

Draw line AB

Use protractor to make angle A and angle B. Extend lines until they cross, this is point

C.

2. Given two sides and the angle between the two sides: Construct ΔPQR in which PQ=4.5cm,

PR=5.5cm and angle P=35°.

Draw line PQ

Use protractor and make angle P

Use compasses to measure PR on your ruler

Draw arc PR

Join R and Q

3. Given the length of the three sides: Construct ΔXYZ in which XY=5.5cm, XZ=3.5cm and

YZ=6.5cm.

Draw line XY

With compasses measure XZ from your ruler

With the point of your compasses at X draw a wide arc

With compasses measure YZ and then with point of your compasses at Y draw a large

arc to cut the first arc

Where the two arcs meet mark point Z

Join ZX and ZY

Chapter 13 (Use of calculator is encouraged)

Multiplying and dividing decimals (involving money)

Multiply and divide decimal numbert.

For multiplication use the same rule for multiplication of whole numbers. Also the decimal point

has to be placed according to the number of digits present in the number that are being multiplied.

To divide by decimals, change the divisor to a whole number by moving the point to the extreme

right. The decimal point of the dividend should be moved accordingly exactly the same number of

times as the divisor. Sometimes the quotient has to be corrected to a particular number of decimal

places.

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Use a calculator when multiplying or dividing numbers with more than two non-zero digits e.g.

43.76 × 2.36.

Multiply and divide decimal numbers by 10 and 100.

To multiply a decimal number by 10 or 100 move the point to the right according to the number of

zeros. For division move the point to the left according to the number of zeros.

Change fractions and percentages to decimals and vice-versa.

To change a decimal or fraction into a percentage, multiply by 100. So 0.3 = 0.3 × 100 = 30% .

To change a percentage to a decimal or fraction divide by 100.

Chapter 14

Number machines

Find input and output from a function machine.

A function machine is when we start with a number and finish with another number by means of a

rule. Now numbers are represented by letters.

For e.g.:

If x = 7 then y = 7 × 3 so y = 21

Similarly if y = 15 then x = 15 ÷ 3 so x = 5

Chapter 15

Scale

Apply ratio rules for scale plans.

Example

Simon made a scale model of a car on a scale of 1 to 12.5 . The height of the model car is 10cm.

(a) Work out the height of the real car.

The ratio of the lengths is 1 : 12.5 . So for every 1 unit of length the small car is, the real car is 12.5

units. So if the small car is 10 units long, the real car is 125 units long. If the small car is 10cm

high, the real car is 125cm high.

(b) The length of the real car is 500cm. Work out the length of the model car.

We know that model : real = 1 : 12.5 . However, the real car is 500cm, so 1 : 12.5 = x : 500 (the

ratios have to remain the same). x is the length of the model car. To work out the answer, we

convert the ratios into fractions:

2 ×3 6 x ×3 y

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1 = x

12.5 500

multiply both sides by 500:

500/12.5 = x

so x = 40cm

Example

Alex and Chloe divide €40 in the ratio 3 : 5. How much do they each get?

First, add up the two numbers in the ratio to get 8. Next divide the total amount by 8, i.e. divide

€40 by 8 to get €5. €5 is the amount of each 'unit' in the ratio. To find out how much Alix gets,

multiply €5 by 3 ('units') = €15. To find out how much Chloe gets, multiply €5 by 5 = €25.

Chapter 16

Averages

Compute the mean, mode and range.

The average of a set of numbers is the value which best represents it. There are three different

types of averages.

Mean

This is also known as the arithmetic mean. It is found by dividing the sum of the set of numbers

with the actual number of values.

For example: Find the mean of 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10.

Sum of values: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55

Number of values = 10

Mean of values = 55 / 10 = 5.5

Mode

The mode is the value which occurs most frequently in the set of values. The mode of the set of

values is also known as the modal value.

For example: Find the mode of 1, 2, 2, 3, 4, 4, 5, 5, 5, 5, 7, 8, 8 and 9.

Modal value = 5

Range - in statistics, the difference between the least and the greatest values in a set of

data.

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From the example above: Range = 9 – 1 = 8

Chapter 17

Formulae

Use letter symbols to represent unknown numbers.

The idea of equations

Form formulae or equations from story sum. Equations are made of numbers, letter terms with an

equals sign to separate two sides (a mathematical sentence containing an equal sign).

Evaluate simple formulae with positive inputs.

Substitution

Substitute letters with numbers and then simplify like terms as above.

Chapter 18

Negative numbers

Know the meaning of natural numbers and integers and represent them on a number line

Numbers can either be positive or negative. Often brackets are put around negative numbers to

make them easier to read, e.g. (-2). If a number is positive, the + is usually missed out before the

number. So 3 is really (+3). Adding and multiplying combinations of positive and negative

numbers can cause confusion and so care must be taken.

Example

2 – 3 + 5 = + 2 + 5 – 3 = 4 or + 4

– 3 – 4 = – 7

-8 0 8

1. On the number line numbers get larger to as we move from Left to Right.

2. Numbers that appear to the Right of a given number are Greater Than (>) the given number.

3. Numbers that appear to the Left of a given number are Less Than (<) the given number.

Integers are –4, -3, -2, -1, 0, 1, 2, 3, 4

Zero is not negative or positive. Zero is the origin on the number line.

Example:

1a. On the number line what number is 5 units to the right of negative (-2)

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

5 spaces

Solution: 3 is 5 units to the right of (-2)

Add/subtract directed numbers.

Addition and Subtraction

Two 'pluses' make a plus, two 'minuses' make a plus. A plus and a minus make a minus.

Example

3 + (-2)

A plus and a minus make a minus, so this is the same as 3 - 2 = 1

Example

(-2) + (-5) = -7

This is the same as (-2) - 5 = -7

Multiply/divide directed numbers

Multiplication and Division

If two positive numbers are multiplied together or divided, the answer is positive.

If two negative numbers are multiplied together or divided, the answer is positive.

If a positive and a negative number are multiplied or divided, the answer is negative.

Examples

(-2) ÷ (-4) = ½

(8) ÷ (-2) = (-4)

2 × (-3) = (-6)

(-5)2

= (-5) × (-5) = 25

Chapter 19

Equations

Solving equations (short cut)

To solve equations, first put letter terms on one side and numbers on the other. Secondly work out

the addition or subtraction of the whole numbers and of the letter terms. Finally divide both sides

by the number of the letter term so that on one side we have the letter and on the other the answer

of the equation.

Example 1:

The cost of a television with remote control is €649. This amount is €125 more than the cost

22

without remote control. Find the cost of the television without remote control.

Strategy

To find the cost of the television without remote control, write and solve an equation using C to

represent the cost of the television without remote control.

Solution

Step 1:

Let the cost of the TV without the remote = C

Step 2:

s

The cost of a

television with remote

control

Is

€125 more than TV without remote

Step 3:

The cost of a

television with remote

control

Is

€125 more than TV without remote

649 = 125 + C

Step 4: Solve the equation (normal method)

649

= C + 125

649 −125 = C + 125 −125

524 = C

The cost of the television without remote control is €524.

Example 2: Translate “three more than twice a number is seventeen” into an equation and

solve.

Step 1: Assign a variable to the unknown quantity.

Let the unknown number = n

23

Step 2: Find two verbal expressions for the same value.

Three more than twice a number

is

seventeen

Step 3: Write a mathematical expression for each verbal expression. Write the equals sign.

Three more than twice a number

is

seventeen

3 + 2n = 17

Step 4: Solve the resulting equation.

3 + 2n - 3 = 17 - 3 * subtract 3 from both sides of the equation

2n = 14

2 n =

14

* divide both sides of the equation by 2

2 2

n = 7

The number is seven.

Chapter 20

Solid shapes

Example of nets of cubes and cuboids:

A cube and a cuboid have 4 vertices, 12 edges and 6 faces.

Chapter 21

Graphs

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The equation of a line (the relation between the x and y co-ordinates) e.g. x = 1; x + y = 2; x – y = 3 etc.

For example the coordinates for the equation y = x + 3 are ...(-2, 1); (-1,2); (0,3); (1,4); (2,5); etc.

Because to get x, subtract 3 from y or to get y, add 3 to x :- which is the relation between x and y.

The following table illustrates the coordinates of the equation y = x + 3. Below we plot the graph from -5 to

5 for both x and y axis.

x -2 -1 0 1 2 3 4

y = x + 3 1 2 3 4 5 6 7

Chapter 22

Measuring

Find the perimeter of simple shapes by adding side lengths and or squares.

It is the total length of the line or lines which bound a figure. It is measured in mm, cm, m or km.

Understand and use units of area: mm2, cm

2 and m

2.

1 m2 = 10000 cm

2 (100cm × 100cm)

1 cm2 = 100mm

2 (10mm × 10mm)

1 m2 = 1000000mm

2 (1000mm × 1000mm)

m2 to cm

2 → amount x 100 x 100

cm2 to mm

2 → amount x 10 x 10

m2 to mm

2 → amount x 1000 x 1000

cm2 to m

2 → amount † 100 † 100

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mm2 to cm

2 → amount † 10 † 10

mm2 to m

2 → amount † 1000 † 1000

Find the area of a rectangle by counting squares / using formula.

The area of a rectangle is length × breadth

Derive the area of the triangle from the area of a rectangle.

The area of the triangle is half the area of a rectangle.

Find the area of a triangle.

Area of triangle – ½ × base× height

Find the area of composite shapes.

To work out the area of compound figures divide the shape into squares and rectangles and or triangles,

calculate their areas and finally add their results together to get the total are of the compound shape.

Understand and use units of volume: mm3 cm

3, and m

3.

Changing units of volume

1 m3 = 1000000cm

3 (100cm × 100cm × 100cm)

1 cm3 = 1000mm

3 (10mm × 10mm × 10mm)

1 m3 = 1000000000mm

3 (1000mm × 1000mm × 1000mm

m3 to cm

3 → amount x 100 x 100 x 100

cm3 to mm

3 → amount x 10 x 10 x 10

m3 to mm

3 → amount x 1000 x 1000 x 1000

cm3 to m

3 → amount † 100 † 100 † 100

mm3 to cm

3 → amount † 10 † 10 † 10

mm3 to m

3 → amount † 1000 † 1000 † 1000

Find the volume of a cuboid by counting cubes/using formula.

Volume of cuboid = l × b × h

Find the volume of compound shapes involving cubes, cuboids.

The volume of a cube and cuboid = l × b × h.

Chapter 23

Flow diagrams and number patterns

Examples:

Write down the missing numbers in the following sequences:

(a) 50, 45, 41, ____, 36, ____. Ans: 38, 35

26

(b) 35

1 , 3

5

4 , ____, 5, ____. Ans: 4

5

2, 5

5

3

This is a sequence of the patterns.

Ans (a)

(a) Draw the next pattern.

(b) How many white tiles are needed in pattern 8? Ans 2 × 8 = 16

(c) What is the total number of tiles in pattern 50? Ans 2 × 50 + 1 = 101

Chapter 24

Accuracy

Round numbers to the nearest 10, 100 or 1000.

Rounding or approximating a number is when giving a rough answer in tens, or in hundreds or in thousands

etc.

Example: Round 48 in tens. Answer is 50 since it is nearer to 50 than to 40 (2 units nearer to 50)

Round 1354 in hundreds. Answer is 1400 since it‟s nearer to 1400 than to 1300.

Round 23456 in thousands . Answer is 23000 since it is nearer to 23000 than to 24000

Round numbers to one, two decimal places.

Correcting to a given number of d.p.

Decimal Places is to give answers to a specific degree of accuracy. For example:

Change 12.3456789 to 2 d.p. (decimal places) Answer is 12.35 since the number on the right of the 4 is

bigger than 5.

Chapter 25

Probability

Describe events as certain, impossible, likely, unlikely ...

The probability of something which is certain to happen is 1.

The probability of something which is impossible to happen is 0.

The probability of something not happening is 1 minus the probability that it will happen.

Probability scale

0 ½ 1

Impossible Very Unlikely Unlikely Evens Likely Very Likely Certain

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Understand and work out the probability of an event.

Probability is the likelihood or chance of an event occurring.

Probability = the number of ways of achieving success

the total number of possible outcomes

For example,

1. The probability of flipping a coin and it being heads is ½, because there is 1 way of getting a head

and the total number of possible outcomes is 2 (a head or tail). We write P(heads) = ½ .

2. There are 6 beads in a bag, 3 are red, 2 are yellow and 1 is blue. What is the probability of picking a

yellow?

The probability is the number of yellows in the bag divided by the total number of balls, i.e. 2/6 = 1/3.

Chapter 26

Transformations

Rotations, Translation and Reflections.

Shapes can be put under transformations. This means a shape can be changed by a set of instructions.

The original shape is put under instructions which are given to you. You may be able to change a whole

shape or transfrom the shape vertex by vertex. Below are the main types of transformations:

Reflection

A reflection is made with the use of a mirror line. Each vertex

of the shape is reflected to the opposite side of the mirror line. The

new point is perpendicular to the mirror line and is equal in distance

from the mirror line as the original shape is from the mirror line.

Translation

The x means how many units you should move the shape along the

horizontal and the y means how many units it should be translated along the

verticle axis. A positive x means moving to the right while a negative x

means moving to the left. A positive y means moving up while a negative y

means moving down.

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For FORMS 1, shapes should only be reflected along the x or y axis while in translations, the words right or

left and up or down should be used. In the example above A is translated to B by being mapped 6 right and

4 up while C is the reflection of A along the y axis. Rotation is not covered in Form 1 (shape D). Remember

to describe fully a transformation for e.g. reflection along the y axis, translation 2 right 5 up, etc.

Chapter 27

Operations in algebra

To work out brackets of algebra, multiply the number attached to the brackets by all the numbers or letter

terms inside the brackets. When simplifying an expression such as 3 + 4 × 5 - 4(3 + 2), remember to work it

out in the following order: brackets, of (/indices), division, multiplication, addition, subtraction. (BIDMAS)

So do the thing in the brackets first, then any division, followed by multiplication and so on. The above is: 3

+ 20 - 4 × 5 = 3 + 20 - 20 = 3. You mustn't just work out the sum in the order that it is written down.

The Order of Operations Agreement:

Step 1: Do all operations in brackets

Step 2: Simplify any numerical expressions containing indices

Step 3: Do all multiplication or division as they occur from left to right.

Step 4: Do addition and subtraction as they occur from left to right.

Here are some examples:

Simplify: 2(4+1)-23 + 6 ÷ 2

2(4+1) -23 + 6 ÷ 2

Perform operations in brackets: = 2(5) - 23+ 6 ÷ 2

Simplify expressions with indices: = 2(5) – 8 + 6 ÷ 2

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Do multiplication or division L → R: = 10 – 8 + 6 ÷ 2

Do addition or subtraction L → R: = 2 + 3

= 5

Simplify: 18 ÷ (6+3) ∗ 9 - 42

18 ÷ (6+3) ∗ 9 - 42

= 18 ÷ 9 ∗ 9 - 42

= 18 ÷ 9 ∗ 9 – 16

= 2 ∗ 9 – 16

= 18 – 16

= 2

Simplify: 20 + 24(8-5) ÷ 22

= 20 + 24(3) ÷ 22

= 20 + 72 ÷ 4

= 20 + 18

= 38

Simplify: 5 ( x + 6 )

= 5(x) + 5(6)

= 5x + 30 OR

Simplify: 8x + 4y – 8x + y

= 8x – 8x + 4y + y use the commutative property to rearrange the order of the

= (8x – 8x) + (4y + y) use the associative property to group our like terms

= 0 + 5y combine like terms.

= 5y use the addition property of zero

x 6

5 5x 30