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MATHEMATICS STUDY

GUIDE

CSEC Mathematics cheat sheet

Curriculum Planning and Development Division

The booklet highlights some salient points for each topic in the CSEC Mathematics syllabus. At

least one basic illustration/example accompanies each salient point. The booklet is meant to be

used as a resource for “last minute” revision by students writing CSEC Mathematics.

Number Theory

Basic Rules

Points to Remember Illustration/ Example

To find the difference between two numbers when

one number is positive and one number is negative

the result will be “+” if the larger value is positive

or “–“ negative if the larger number is negative.

When multiplying, two positive numbers (+) × (+) = + e.g.

multiplied together give a positive product; and a (-) × (-) = + 8 × 5 = 40

negative number multiplied by another negative (+) × (-) = - -8 × -5 = 40

number gives a positive product. Also, a negative (-) × (+) = - 8 × -5 = -

number multiplied by a positive number gives a -8 × 5 = -

negative product

Number Theory

Positive and Negative Numbers

Points to Remember Illustration/ Example

The rules for division of directed numbers are (+) ÷ (+) = + e.g. 10 ÷ 5 = 2

similar to multiplication of directed numbers. (-) ÷ (-) = + -10 ÷ -5 = 2

Use manipulatives- counters (yellow and red) (+)^ ÷^ (-) =^ -^ 10 ÷^ -5^ =^ -

(-) ÷ (+) = - -10 ÷ 5 = -

There are different type of numbers:

Natural Numbers - The whole numbers from 1

upwards

Integers - The whole numbers, {1,2,3,...} negative

whole numbers {..., -3,-2,-1} and zero {0}.

Rational Numbers - The numbers you can make

by dividing one integer by another (but not

dividing by zero). In other words, fractions.

Irrational Number – Cannot be written as a ratio

of two numbers

Real Numbers - All Rational and Irrational

numbers. They can also be positive, negative or

zero.

Natural Numbers (N) : {1,2,3,...}

Integers (Z) : {..., -3, -2, -1, 0, 1, 2, 3, ...}

Rational Numbers (Q) :. 3/2 (=1.5), 8/4 (=2), 136/

Irrational Number : 𝜋, 3.142 (cannot be written as a

fraction)

Real Numbers (R): 1.5, -12.3, 99, √2, π

Number Theory

Decimals – Rounding

Points to Remember Illustration/ Example

Rounding up a decimal means increasing the 5.47 to the tenths place, it can be can be rounded up to

terminating digit by a value of 1 and drop off the 5.

digits to the right. 6.734 to the hundredths place, it can be rounded down to

Round down if the number to the right of our 6.

terminating decimal place is four or less (4,3,2,1,0)

Number Theory

Operations with Decimals

Points to Remember Illustration/ Example

Find the product of 3.77 x 2.8 =?

1. Line up the numbers on the right,
2. multiply each digit in the top number by

each digit in the bottom number (like

whole numbers),

3. add the products,
4. and mark off decimal places equal to the

sum of the decimal places in the numbers

being multiplied.

Find the product of 3.77 × 2.

3.77 (2 decimal places)

2.8 (1 decimal place)

10.556 (3 decimal places)

When dividing, if the divisor has a decimal in it,

make it a whole number by moving the decimal

point to the appropriate number of places to the

right. If the decimal point is shifted to the right in

the divisor, also do this for the dividend.

Fractions can always be written as decimals. (^) For example:

2 1 3

5 2 4

1 3 3

4 5 4

Number Theory

Computation – Fractions

Points to Remember Illustration/ Example

When the numerator stays the same, and the (^1 1 )

denominator increases, the value of the fraction 3

4

5

decreases 3 3 3 3

4

5

6 7

When the denominator stays the same, and the 7 8 9

numerator increases, the value of the fraction

2

2

2

increases.

Equivalent fractions are fractions that may look 1 2 3 4

different, but are equal to each other. 2

4

6 8

Equivalent fractions can be generated by 1 1 x 2 2

multiplying or dividing both the numerator and

2

2 x 2

4

denominator by the same number. 3 3 x 2 6

5

5 x 2

10

Fractions can be simplified when the numerator 6 3 x 2 3

and denominator have a common factor in them 10

5 x 2

5

Fractions with different denominators, can be

converted to a set of fractions that have the same

denominator

3 2 9 8

4

3

is the same as 12

12

Addition and subtraction of fractions are similar to

adding and subtracting whole numbers if the

9 8 1

12

12

12

fractions being added or subtracted have the same

denominator

When multiplying fractions, multiply the 5 2 10 5

numerators together and then multiply the 6

×

3

18

9

denominators together and simplify the results.

Number Theory

Prime Numbers

Points to Remember Illustration/ Example

A prime number is a number that has only two

factors: itself and 1e.g. 5 can only be divided

evenly by 1 or 5, so it is a prime number.

Numbers that are not prime numbers are referred

to as composite numbers

Number Theory

Computation of Decimals, Fractions and Percentages

Points to Remember Illustration/ Example

Percent means "per one hundred" 20 % = 20 per 100

To convert from percent to decimal, divide the 10

percent by 100 10 %^ =^100 =^ 0.

100

To convert from decimal to percent, multiply the

decimal by 100

0.10 as a percentage is 0.10 × 100 = 10%

0.675 is 0.675 × 100 = 67.5%

To convert from percentages to fractions, divide 12 12  4 3

the percent by 100 to get a fraction and then

100

100  4

25

simplify the fraction

To convert from fractions to percentages, convert 3

the fraction to a decimal by dividing the numerator 25 =^ 0.

by the denominator and then convert the decimal 0.12 as a percentage is 0.12 × 100 = 12%

to a percent by multiplying by 100.

Angles formed by a Transversal Crossing two Parallel Lines

Vertical Angles are the angles opposite each

other when two lines cross.

Vertically opposite angles are equal

a = d f = g

b = c e = h

The angles in matching corners are called

Corresponding Angles.

Corresponding Angles are equal

a = e c = g

b = f d = h

The pairs of angles on opposite sides of the

transversal but inside the two lines are called

Alternate Angles.

Alternate Angles are equal

d = e c = f

The pairs of angles on one side of the

transversal but inside the two lines are called

Consecutive Interior Angles.

Consecutive Interior Angles are supplementary

(add up to 180°)

d + f c + e

Illustration of all angles mentioned on a single

diagram. The transversal crosses two Parallel

Lines

Triangles

Pythagoras' Theorem

Points to Remember Illustration/ Example

Pythagoras' Theorem states that the square of the

hypotenuse is equal to the sum of the squares on

the other two sides

c

2

= a

2

• b

2

The Hypotenuse is c

Find c

c

2

= 5

2

• 12

2

c= (^) √ 169

= 13 units

Triangles

Similar Triangles & Congruent Triangles

Points to Remember Illustration/ Example

Definition: Triangles are similar if they have the

same shape, but can be different sizes.

(They are still similar even if one is rotated, or one

is a mirror image of the other).

There are three accepted methods of proving that

triangles are similar:

If two angles of one triangle are equal to two

angles of another triangle, the triangles are similar.

If angle A = angle D and angle B = angle E

Then ∆ABC is similar to ∆DEF

Show that the two triangles given beside are similar and

calculate the lengths of sides PQ and PR.

Solution:

∠ A = ∠ P and ∠ B = ∠ Q, ∠ C = ∠ R (because ∠C = 180

∠A - ∠B and ∠R = 180 - ∠P - ∠Q)

Therefore, the two triangles ΔABC and ΔPQR are

similar.

Curriculum Planning and Development Division - 11

Triangles

Similar Triangles & Congruent Triangles

Points to Remember Illustration/ Example

Therefore:

𝐷𝐸 𝐶𝐷

𝐴𝐵 𝐶𝐴

7 15

11 𝐶𝐴

7CA = 11 × 15

11 × 15

CA =

7

CA = 23.

x = CA – CD = 23.57 – 15 = 8.

Curriculum Planning and Development Division - 12

Mensuration

Areas & Perimeters

Points to Remember Illustration/ Example

The area of a shape is the total number of square

units that fill the shape.

Area of Square= a

2

Perimeter of Square= a+ a + a + a

a = length of side

Find the area and perimeter of a square that has a side-

length of 4 cm

Area of Square = a × a= a

2

= 4 × 4= 4

2

= 16 cm

2

Perimeter of Square = 4 + 4 + 4 + 4 = 16 cm

a represents the length; b represents the width

Area of Rectangle = a × b

Perimeter of Rectangle = a+ a + b + b = 2(a+b)

Find the area of a rectangle of length 5cm, width 3cm

Area of Rectangle = 5 cm x 3cm = 15 cm

2

Perimeter of Rectangle = 5 + 5 + 3 + 3 = 2(5+3) = 16cm

1

The area of a triangle is : 2

x b x h

b is the base

h is the height

1 1

Area = 2

x b x h = 2

x 20units x 12units = 120 units

2

Area of triangle using "Heron's Formula"- given

all three sides:

Step 1: Calculate "s" (half of the triangle’s perimeter):

Step 2: Then calculate the Area :

Example: What is the area and perimeter of a triangle

with sides 3cm, 4cm and 5cm respectively?

3  4  5 12

Step 1: s = 2

2

Step 2 : Area of triangle = (^) √6(6 − 3)(6 − 4)(6 − 5)

= (^) √ 6 (3)(2)(1) = 6cm

2

Perimeter of triangle = a + b + c

= 3 + 4 + 5 = 12cm

Curriculum Planning and Development Division - 14

Mensuration

Areas & Perimeters

Points to Remember Illustration/ Example

c d

Area of Trapezium = ½(a+b) × h

= ½(sum of parallel sides) × h

h = vertical height

Perimeter = a + b + c + d

Find the area of the trapezium

A = ½ (a+b) × h

= ½(10 + 8) × 4

=½ × (18) × 4

= 36 cm

2

Perimeter = a + b + c + d

= 26.4 cm

a

b

Area of Parallelogram = base × height

b = base

h= vertical height

Find the area of a parallelogram with a base of 12

centimeters and a height of 5 centimeters.

Area of parallelogram= b × h = 12cm × 5cm = 60 cm

2

Perimeter of parallelogram = a + b + a + b = 2 (a + b)

= 12cm + 7cm + 12cm + 7cm = 38cm

Curriculum Planning and Development Division - 15

Mensuration

Surface Area and Volumes

Points to Remember Illustration/ Example

Volume of Cylinder

= Area of Cross Section x Height = π r

2

h

Surface Area of Cylinder

= 2π r

2

• 2πrh = 2πr (r + h)

Find the volume and total surface area of a cylinder with a

base radius of 5 cm and a height of 7 cm.

22

Volume = π r

2

h = 7

× 5

2

× 7 =22 × 25 cm

3

= 550cm

3

Conversion to Litres : 1000 cm

3

= 1L

550

550 cm

3

= 1000

L= 0.55 L

Surface Area = 2π(5)(7) + 2π(5)

2

= 70π + 50π

= 120π cm

2

≈ 376.99 cm

2

• A prism is a three-dimensional shape which has the

same shape and size of cross-section along the entire

length i.e. a uniform cross-section

Prism- Since a cylinder is closely related to a prism, the

formulas for their surface areas are related

Volume of Prism = area of cross section × length = A l

Example: What is the volume of a prism whose ends

have an area of 25 m

2

and which is 12 m long

Answer: Volume = 25 m

2

× 12 m = 300 m

3

Volume of irregular prism = A h

Surface Area of irregular prism

= 2 A + (perimeter of base × h)

Curriculum Planning and Development Division - 17

Mensuration

Surface Area and Volumes

Points to Remember Illustration/ Example

Volume of cube = s

3

Surface Area of cube = s

2

• s

2

• s

2

• s

2

• s

2

• s

2

= 6 s

2

Find the volume and surface area of a cube with a side

of length 3 cm

Volume of cube = s × s × s = s

3

= 3 × 3 × 3 = 27 cm

3

Surface Area of cube = s

2

• s

2

• s

2

• s

2

• s

2

• s

2

= 6 s

2

2

= 6 × 9 = 54 cm

2

Volume of cuboid = length x breadth x height

= xyz

Surface area = xy + xz + yz + xy + xz + yz

= 2 xy + 2 xz + 2 yz

= 2( xy + xz + yz)

Find the volume and surface area of a cuboid with length

10cm, breadth 5cm and height 4cm.

Volume of cuboid = length × breadth × height

= 10 × 5 × 4

= 200cm

3

Surface Area of cuboid = 2 xy + 2 xz + 2 yz

= 220 cm

2

The Volume of a Pyramid

1

= 3 × [Base Area] × Height

Find the volume of a rectangular-based pyramid

whose base is 8 cm by 6 cm and height is 5 cm.

Solution:

1

V =

3

× [Base Area] × Height

Curriculum Planning and Development Division - 18

Mensuration

Surface Area and Volumes

Points to Remember Illustration/ Example

1

= 3 × [8 × 6] × 5

= 80 cm

3