Mathematics
Units
Compound units, two quantitative forms of measurement need to be combined (m/s)
Number
Definitions:
= equal to < less than (≤ less than or equal to)
≠ not equal to > greater than (≥ greater than or equal to)
Multiple, lies in the times table of that number
Common multiple, multiple of both numbers
Lowest common multiple (LCM), smallest number that divides all the numbers in question.
Factor/divisor, divide into that number exactly (with no remainder)
Common factor, number that will divide exactly into all the numbers in the question.
Highest common factor (HCF), largest factor that a set of numbers have in common
Prime numbers, numbers that have exactly two factors (1 and themselves)
Prime factorization, writing a number as a product of prime numbers ex 12 = 2 x 2 x 3
Prime numbers that are less than 30:
2, 3, 5, 7, 11, 13, 17, 19, 23 and 29
Ex
180 = 2 x 2 x 3 x 3 x 5 so, LCM = 2 x 2 x 3 x 3 x 5 x 7 = 1260
420 = 2 x 2 x 3 x 5 x 7 HCF = 2 x 2 x 3 x 5 = 60
6 4 2
Ex given 129600 = 2 x 3 x 5
So, 129600 = 23 x 32 x 5 x 23 x 32 x 5
So, √ 129600 = 23 x 32 x 5 = 360
Order of operations (left to right):
1. Brackets/parentheses
2. Indices/exponents
3. Division and Multiplication
4. Addition and Subtraction
If there are m ways of doing one task and for each of these, there are n ways of doing
another task, then the total number of ways the two tasks can be done in order is m x n.
Squares, multiplying the number by itself
Square roots, positive number that when multiplied by itself gives the original number.
Cube numbers, multiplying a number by itself, then multiplying by this number again.
Cube roots, multiplied by itself and multiplied by itself again gives the original number.
Standard form, form a x 10n with 1 ≤ a < 10 and n is a positive or negative integer.
Terminating decimal, decimal which has a finite number of digits.
Recurring decimal, decimal which has repeating digits or a repeating pattern of digits with an
infinite number of digits.
Recurring decimal fraction
(pictures)
Fraction recurring factors,
denominator to 9, 99 or 999
etc. same amount of numbers
in the numerator
Fractions lowest terms, by cancelling by prime numbers
Improper fraction, numerator is greater than the denominator
, Surd, expression that includes square roots or cube roots etc., where the square root (or
cube root etc.) cannot be simplified to an integer or fraction, but can be simplified by
splitting Ex √ 12=√ 4 x 3=2 √ 3
Rationalising the denominator, write the expression without surds in the denominator.
Single surd then, multiply numerator and denominator by the surd
Form x + √ y then multiply numerator and denominator x−√ y
Lower bound, smallest number which would round up to x.
Upper bound, smallest number which would round up to a number bigger than x.
Ex 3.84
Lower bound would be 3.835
Upper bound would be 3.845
Truncate, cut off decimals without rounding
Ratio and proportions
Shapes A and B are mathematically similar and the ratio of corresponding lengths on shape A
to those on shape B is x: y then:
Areas x2: y2
Volumes is x3: y3
Growth and decay, qxn with for growth x > 1 and for decay 0 < x < 1
Compound interest, A = P (1 + (r/100)) n
Iterative processes, set of instructions are used again and again, usually over time ex interest
Algebra
Identity, true for all values of the variable two sides are the
same
Equality, only true for certain values of a variable x = …
Equation of a straight line: y = mx + c (m, gradient and c,
intercept with y-axis)
Parallel lines, same gradient
Perpendicular lines, product of their gradients is −1.
Quadratic function/parabola: f(x) = ax2 + bx + c
Roots, intercept with x-axis
Turning point, lowest or highest point
Completing the square, writing x2 + kx as a difference of two squares
Ex
Simple cubic functions: y = ax3
Reciprocal function: y = 1/x (x ≠ 0)
Exponential function: y = kx
Trigonometric functions: sin(x), cos(x) and tan(x)
Sequence, list of terms together with a rule for generating them.
Quadratic sequence: an2 + bn + c
Term-to-term rule, how to move from one term in the sequence to the next term in
the sequence
- ex tn – 2 with t1 = 7 7, 5, 3, 1 etc.
Units
Compound units, two quantitative forms of measurement need to be combined (m/s)
Number
Definitions:
= equal to < less than (≤ less than or equal to)
≠ not equal to > greater than (≥ greater than or equal to)
Multiple, lies in the times table of that number
Common multiple, multiple of both numbers
Lowest common multiple (LCM), smallest number that divides all the numbers in question.
Factor/divisor, divide into that number exactly (with no remainder)
Common factor, number that will divide exactly into all the numbers in the question.
Highest common factor (HCF), largest factor that a set of numbers have in common
Prime numbers, numbers that have exactly two factors (1 and themselves)
Prime factorization, writing a number as a product of prime numbers ex 12 = 2 x 2 x 3
Prime numbers that are less than 30:
2, 3, 5, 7, 11, 13, 17, 19, 23 and 29
Ex
180 = 2 x 2 x 3 x 3 x 5 so, LCM = 2 x 2 x 3 x 3 x 5 x 7 = 1260
420 = 2 x 2 x 3 x 5 x 7 HCF = 2 x 2 x 3 x 5 = 60
6 4 2
Ex given 129600 = 2 x 3 x 5
So, 129600 = 23 x 32 x 5 x 23 x 32 x 5
So, √ 129600 = 23 x 32 x 5 = 360
Order of operations (left to right):
1. Brackets/parentheses
2. Indices/exponents
3. Division and Multiplication
4. Addition and Subtraction
If there are m ways of doing one task and for each of these, there are n ways of doing
another task, then the total number of ways the two tasks can be done in order is m x n.
Squares, multiplying the number by itself
Square roots, positive number that when multiplied by itself gives the original number.
Cube numbers, multiplying a number by itself, then multiplying by this number again.
Cube roots, multiplied by itself and multiplied by itself again gives the original number.
Standard form, form a x 10n with 1 ≤ a < 10 and n is a positive or negative integer.
Terminating decimal, decimal which has a finite number of digits.
Recurring decimal, decimal which has repeating digits or a repeating pattern of digits with an
infinite number of digits.
Recurring decimal fraction
(pictures)
Fraction recurring factors,
denominator to 9, 99 or 999
etc. same amount of numbers
in the numerator
Fractions lowest terms, by cancelling by prime numbers
Improper fraction, numerator is greater than the denominator
, Surd, expression that includes square roots or cube roots etc., where the square root (or
cube root etc.) cannot be simplified to an integer or fraction, but can be simplified by
splitting Ex √ 12=√ 4 x 3=2 √ 3
Rationalising the denominator, write the expression without surds in the denominator.
Single surd then, multiply numerator and denominator by the surd
Form x + √ y then multiply numerator and denominator x−√ y
Lower bound, smallest number which would round up to x.
Upper bound, smallest number which would round up to a number bigger than x.
Ex 3.84
Lower bound would be 3.835
Upper bound would be 3.845
Truncate, cut off decimals without rounding
Ratio and proportions
Shapes A and B are mathematically similar and the ratio of corresponding lengths on shape A
to those on shape B is x: y then:
Areas x2: y2
Volumes is x3: y3
Growth and decay, qxn with for growth x > 1 and for decay 0 < x < 1
Compound interest, A = P (1 + (r/100)) n
Iterative processes, set of instructions are used again and again, usually over time ex interest
Algebra
Identity, true for all values of the variable two sides are the
same
Equality, only true for certain values of a variable x = …
Equation of a straight line: y = mx + c (m, gradient and c,
intercept with y-axis)
Parallel lines, same gradient
Perpendicular lines, product of their gradients is −1.
Quadratic function/parabola: f(x) = ax2 + bx + c
Roots, intercept with x-axis
Turning point, lowest or highest point
Completing the square, writing x2 + kx as a difference of two squares
Ex
Simple cubic functions: y = ax3
Reciprocal function: y = 1/x (x ≠ 0)
Exponential function: y = kx
Trigonometric functions: sin(x), cos(x) and tan(x)
Sequence, list of terms together with a rule for generating them.
Quadratic sequence: an2 + bn + c
Term-to-term rule, how to move from one term in the sequence to the next term in
the sequence
- ex tn – 2 with t1 = 7 7, 5, 3, 1 etc.
