whole numbers

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```					                                 Whole Numbers

You should be able to

   Read and write whole numbers expressed in figures and words
   Order whole numbers
   Recognise the place value of each digit in a number
   Use mental methods to carry out addition and subtraction
   Carry out accurately non-calculator methods for addition and subtraction
   Know the multiplication tables up to 10 x 10
   Carry out multiplication by a number less than 10
   Multiply whole numbers by 10, 100, 1000 etc
   Multiply whole numbers by 20, 30, 40 etc
   Carry out division by a number less than 10
   Divide whole numbers by 10, 100, 1000 etc
   Divide whole numbers by 20, 30, 40 etc
   Carry out long multiplication
   Carry out long division
   Know the order of operations on a calculator
Decimals

Without using a calculator you should be able to

   Use decimal notation for money and other measures
   Multiply and divide decimals by powers of 10
   Multiply decimals by other decimals
   Divide decimals by other decimals
   Change decimals to fractions
   Carry out a variety of calculations involving decimals
   Know that:
When a number is multiplied by a number between 0 and 1 the result will be
smaller than the original number
When a number is divided by a number between 0 and 1 the result will be
bigger than the original number
Approximation and Estimation

   A number can be rounded to an approximate number

   How to approximate using decimal places
When rounding a number to one, two or more decimal places
1.     write the number to one more decimal place than asked for
2.     look at the last decimal place and
   if the figure is 5 or more round up
   if the figure less than 5 leave the previous number alone

   How to approximate using significant figures
When rounding a number to one, two or more significant figures
1.    Start from the most significant figure and count the
required number of figures
2.    Look at the next figure on the right of this and
   if the figure is 5 or more round up
   if the figure is less than 5 leave the previous number alone
3.    Add noughts, as necessary, to locate the decimal point and
preserve the place value

   When answering a problem remember to include any units and state the
degree of approximation used

   Choose a suitable degree of accuracy

   Use approximation to estimate that the actual answer to a calculation is
of the right magnitude (size).
Negative Numbers

You should be able to

   Use negative numbers in context such as temperatures, bank accounts
   Realise where negative numbers come on a number line
   Put numbers in order (including negative numbers)
   Add, subtract, multiply and divide negative numbers.
Working with Number

 Multiples of a number are found by multiplying the number by 1, 2, 3, 4, etc.
 Factors of a number are found by listing all the products that give that
number
 A Prime Number is a number with only two factors, 1 and the number itself.
 The prime factors of a number are those factors of the number which are
themselves prime
 Product of prime factors Powers can be used to help write any number as
the product of prime factors
 The least common multiple of two numbers is the smallest number that is a
multiple of them both
 The highest common factor of two numbers is the largest number that is a
factor of both of them
 Numbers raised to the power of 2 are squared. Whole numbers squared are
called square numbers. The opposite of squaring a number is called finding
the square root.
 The rules of indices
Multiplying powers with the same base – add the index numbers
Dividing powers with the same base – subtract the index numbers
Raising a power to a power – multiply the powers
Raising any power to the power of 0 always = 1
 Numbers raised to the power of 3 are cubed. Whole numbers cubed are
called cubed numbers. The opposite of cubing a number is called finding the
cubed root.
 An expression such as 3 x 3 x 3 x 3 x 3 can be written in a shorthand way as
35 this is read as “3 to the power of 5”. The number 3 is the base of the
expression and the 5 is the power or index.
 The square root of a number can be positive or negative. The square root of
4 can be +2 or -2.
 Reciprocals
The reciprocal of a number is the value obtained when the number is divided
into 1. A number times its reciprocal equals 1.
 A Surd is the root of a rational number that is not rational. A surd is an
irrational number
Standard form

 Standard Index Form
 Numbers in Standard Form use the powers of 10 to write very large
and very small numbers in shorthand form.
Fractions

 The top number of a fraction is called the numerator the bottom is called
the denominator
 To write equivalent fractions the numerator and denominator of a fraction
are multiplied (or divided) by the same number
 In its simplest form the numerator and denominator of a fraction have no
common factor
 2½ is an example of a mixed number. It is a mixture of whole numbers and
fractions
 5/2 is an improper (or top-heavy) fraction
 Fractions must have the same denominator before adding or subtracting
 Mixed numbers must be changed to improper fractions before multiplying or
dividing
 All fractions can be written as decimals. Some decimals have recurring
digits. These are shown by a single dot above a single recurring digit or a dot
above the first and the last digit of a set of recurring digits.
Percentages

   “Per cent” means “out of 100”
   To change a decimal or a fraction to a percentage – multiply by 100
   To change a percentage to a decimal or a fraction – divide by 100
   Percentage increase = actual increase x 100
Initial value
   Percentage decrease = actual decrease x 100
Initial value
   Hourly pay is paid at basic rate for a fixed number of hours
Overtime pay is usually paid at a higher rate such as time and a half, which
means each hour’s work is worth 1.5 times the basic rate.
   Tax is only paid on income earned in excess of the tax allowances. This is
called taxable income.
   Value added tax or VAT is a tax on some goods and services and is added to
the bill
   Gas, electricity and telephone bills are paid quarterly. The bill consists of a
standing charge plus a charge for the amount used
   When considering a best buy compare quantities by using the same units. For
example find which product gives more grams per penny
   Money invested in a savings account at a bank or a building society earns
interest which is usually paid once a year.
With simple interest the interest is paid out each year and not added to
With compound interest the interest earned each year is added to your
account and earns interest the following year.
Time

 Time can be given using either the 12 hour clock or the 24 hour clock
When using the 12 hour clock times before midday are given as am
after midday are given as pm
 Timetables are usually given using the 24 hour clock.
Ratio

 The ratio 3:2 is read 3 to 2

 A ratio is used only to compare quantities
A ratio does not give information about the exact values of quantities being
compared

 In its simplest form a ratio contains whole numbers which have no common
factors. All quantities in a ratio must have the same units before the ratio
can be simplified.

 When two different quantities are always in the same ratio the two
quantities are in direct proportion
Speed and other compound measures

 Speed is a compound measure because it involves two other measures
 Speed is a measure of how fast something is travelling. It involves measures
distance and time.

Speed = distance
Time

 The formula linking speed, distance and time can be re-arranged and
remembered as:
(average) speed = (total) distance ÷ (total) time
(total) distance = (average) speed x (total) time
(total) time = (total) distance ÷ (average) speed

 Two other commonly used compound measures are density and population
density

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