# Understanding Math Expressions

It’s difficult to get to grips with math problems if you don’t understand the expressions that are used, so let’s take a look at some of these.

FACTORS
Factors are numbers that divide exactly into other numbers without leaving a remainder. For example:
the factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24
the factors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42.

MULTIPLES
Multiples are numbers arrived at when one number is multiplied by another. For example:
the first ten multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50
the first ten multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, 63, 70.

Many students confuse factors and multiples. Remember that factors are smaller than or equal to the number, whereas multiples are bigger than or equal to the number.

PRIME NUMBERS
Prime numbers have only two factors: the number itself, and one. Since the number one has only one factor (which is of course one), it is not a prime number. Two is a prime number, as its factors are two and one. Two is, however, the only even number that is prime.

FRACTIONS
When dealing with fractions, every fraction has a numerator, which is the top number. The number at the bottom is called the denominator.

An improper fraction is one that has the numerator larger than the denominator; in other words, it is ‘top heavy’. This would usually be changed to a mixed number, which is a whole number followed by a fraction. Thus, fifteen quarters would be changed to three and three quarters.

A reciprocal of a fraction is its inverse in other words, it is turned upside-down. When dividing fractions, we actually multiply the first fraction by the reciprocal of the second.

PRODUCT
The product of two numbers is the answer arrived at when they are multiplied together.

QUOTIENT
The quotient is the answer arrived at when dividing one number by another. The number being divided is called the dividend, whilst the number it is being divided by is the divisor.

BIDMAS
BIDMAS is an acronym that helps you remember in what order operations must be carried out. This is what each letter stands for:
Brackets
Indices
Division
Multiplication
Subtraction

Thus, if you are solving a problem that contains brackets, whatever is inside the brackets must be worked out first. Division must be carried out before addition, and so on.

AVERAGES
When handling sets of data, you may be asked to find different types of average.

The mean average is calculated by adding up all the values in the set, and then dividing the total by the number of values. Thus the mean average of 3, 5 and 7 would be 5, as the total is 15, which is then divided by 3, giving the answer 5.

The median is the middle value which is found be arranging the values in order of size. For the set of data 3, 4, 6, 8, 9, 11, 13, the median would be 8.

The mode is the value that occurs most frequently in a set. Some students remember this with the acronym Most Often Digit Ever not perfect grammar, but it helps! For the set of data 12, 17, 15, 11, 17, 15, 13, 15, 16, 15, 19, the mode is 15.

The range is the difference between the highest and the lowest values in a set. For the set of data 30, 25, 21, 35, 29, 32, the range would be 35 minus 21, which is 14.

TRIANGLES
Triangles are three-sided figures of the following types:

equilateral three equal angles (each sixty degrees) and three sides of equal length;
isosceles two equal angles and two sides of equal length;
scalene no equal angles and no equal sides;
right-angled one angle measures ninety degrees. A triangle could be isosceles and right-angled, or scalene and right-angled.

A square has four equal sides, and its four angles are right angles.
A rectangle has two longer sides and two shorter sides, and its four angles are right angles.
A parallelogram has opposite sides parallel to each other.
A rhombus is a parallelogram whose four sides are all equal in measure.
A trapezium has one pair of parallel sides. The other two sides are not parallel to each other.
A kite has two adjacent short sides and two adjacent long sides. Its diagonals intersect at right angles.

POLYGONS
A polygon is a many-sided figure that can be regular (with equal sides) or irregular. Here are a few examples:

pentagon 5 sides
hexagon 6 sides
heptagon 7 sides
octagon 8 sides (like an octopus with 8 tentacles)
decagon 10 sides.

CIRCLES
The perimeter (distance round the outside edge) of a circle is the circumference.
The radius of a circle extends from the centre to any point on the circumference.
The diameter of a circle is a straight line that extends from a point on the circumference, through the centre of the circle, to another point on the circumference. It is therefore twice the length of the circumference.

3D SHAPES
The most common 3-dimensional shapes are:

cube, which has six faces, each of which is a square;
cuboid, which also has six faces, but all may be rectangles; there could alternatively be four rectangular faces and two square ones;
sphere, like a ball;
cylinder, like a tube with circular ends;
cone, circular at one end and tapering to a point, like an ice-cream cone;
pyramid, triangular sides with either a square base or a triangular one;
prism, like a wedge shape. Think of a bar of Toblerone that’s a triangular prism!