GCSE  Number

Foundation
only

Foundation and Higher

Higher only




INTEGERS, RECIPROCALS, FACTORS,
MULTIPLES AND PRIME NUMBERS
Specification

Learning objectives

Grade

Resources

recall
all positive integer complements to
100

Be
able to 'make 100' from an integer

G


recall
all multiplication facts to 10 x 10
and use them to derive quickly the
corresponding division facts

know
all times tables up to 10 x 10

G

P
P
W
W
W
division as reverse multiplication
W
30 division questions

Use
times tables to work out the answers to
simple division problems

F

understand
and use positive numbers and negative
integers, both as positions and
translations on a number line

Understand
positive and negative integers

G

W
blank number lines from 9 to 9
P
negative number scales and temperature
problems
W
temperature problems

order integers

Be
able to order a list of negative and
positive integers

F


add, subtract, multiply and
divide integers, and then any number
multiply and divide
by a negative number

Add
and subtract negative integers

F

W
4 in a line  adding a multiple of 10 to
a number
X
X
subtraction practice
W
F
F addition and subtraction
including negative numbers
W Four
rules 'Collect a letter'
W
F
calculating with negative numbers

Multiply
and divide negative integers

E

use brackets and the
hierarchy of operations

Understand
and use BIDMAS

E

W
BIDMAS Collect a letter
W
insert brackets to make the calculations
correct

understand
‘reciprocal’ as multiplicative
inverse, knowing that any nonzero
number multiplied by its reciprocal is
1 (and that zero has no reciprocal,
because division by zero is not
defined)

Find
the reciprocal of a number

C


know about prime
numbers and find the prime factor
decomposition of positive integers

Recognise
prime numbers

D

P shade the prime
numbers picture puzzle
F
Prime Factor Questions

Write
a number as a product of prime factors

C

use the concepts and
vocabulary of factor (divisor),
multiple and common factor, highest
common factor, least common multiple.

Find
the factors of a number

G

X
multiples on a hundred square
W
spider diagrams  fill in the factors
P
shade factors of 24 picture puzzle
P
Shade factors of 20 and 50 picture
puzzle
W
factor grid  shade in the factors
W
primes, factors and multiplies
W mixed
questions

Find
the LCM of two simple numbers

C

Find
the LCM of two or more numbers

B

Find
the HCF of two simple numbers

C

Find
the HCF of two or more numbers

B

ROUNDING AND APPROXIMATIONS
Specification

Learning objectives

Grade

Resources

round
to the nearest integer.

Round to the nearest integer

G

P
rounding to the nearest whole number
X
rounding to the nearest whole number or
nearest ten
W
4 in a line  nearest 10 or 100
W blockbusters  nearest
100 or 1000

use their previous
understanding of integers and place
value to deal with arbitrarily large
positive numbers and round them to a
given power of 10

Round numbers to given powers of 10.

F

round to a given
number of decimal places.
round
to one significant figure.
round to a given
number of significant figures.

Round numbers to a given number of
decimal places

F

P
examples of rounding to decimal places
and significant figures
X
rounding off  includes
decimal places and significant figures
W
rounding and accuracy

Round a number to one significant figure

D

Round
to a given number of significant figures

B

estimate
answers to problems involving decimals
derive unknown facts
from those they know

estimate
answers to problems involving decimals

G


Estimate
square roots

F


Estimate
answers to calculations involving
division

D


Estimate
answers to calculations involving
division by numbers less than one

C

W
estimating answers to complex
calculations

understand the calculator
display, knowing when to interpret the
display when it h as been rounded by a
calculator and not to round during the
intermediate steps of a calculation

Know
not to round a calculation until the
last step.

C


recognise upper and
lower bounds of a rounded value
use calculators to
calculate the upper and lower bounds
of calculations, particularly when
working with measurements.

Find
minimum and maximum values

C


Calculate the upper and lower
bound using a formula when the values have
a given degree of accuracy 
A 
WHOLE NUMBER AND
DECIMAL CALCULATIONS
Specification

Learning objectives

Grade

Resources

use
decimal notation and
order decimals

Use
decimal notation for money

G

W
decimals and money
W
understanding place value and ordering
decimals
W
ordering decimals
W
W
decimal pictures examples
P
Ordering decimals using decimal pictures

Write
down the place value of a digit, e.g.
what is the value of 4 in 0.24?

F

Order
decimals, e.g. which is bigger 0.24 or
0.3?

F

use
standard column procedures for
addition and subtraction of integers
and decimals

Add
and subtract whole numbers using
standard column procedures

G

W
3 addition methods
W
addition questions
W
3 subtraction methods
W
subtraction questions
W
999 problem
W
worded four rules problems

Add
and subtract decimals using standard
column procedures

E

add and subtract
mentally numbers with up to two
decimal places

Add
and subtract decimals using mental
procedures

E

W
adding and subtracting decimals with 1
dp
W
W
4 in a line activities  adding decimals

develop a range of
strategies for mental calculation



W
basic arithmetic

derive
unknown facts from those they know



W
W
use known facts for decimal calculations

use inverse
operations




use standard column
procedures for multiplication and
division of integers and decimals,
understanding where to position the
decimal point by considering what
happens if they multiply equivalent
fractions.
multiply and divide
numbers with no more than one decimal
digit, using the commutative,
associative, and distributive laws and
factorisation where possible, or place
value adjustments
solve a problem
involving division by a decimal (up to
2dp) by transforming it to a problem
involving division by an integer.

Multiply
and divide whole numbers using standard
column procedures

E

W
4 multiplication methods
W
3 division methods
W
2 by 1 multiplication grids
W
2 by 2 multiplication grids
W
3 by 1 multiplication grids
W
3 by 2 multiplication grids
W
2 by 1 Italian grids
X
2 by 1 & 2 by 2 Italian
questions
X
2 by 1 & 3 by 1 Italian
questions
X
X
3 by 2 Italian questions
W
4 in a line game  two digit by 1 digit
multiplication
W
worded four rules problems

Multiply
two decimals, such as 2.4 x 0.7

D

W
decimal multiplication (up to 3 decimal
places)
W
multiplying decimals using a grid method
W
Decimal fun multiplication
W
four rules calculations with decimals

Divide
a number by a decimal, such as 1 ÷ 0.2
and 2.8 ÷ 0.07

C

W
decimal division
X
decimal calculations derived from known
calculations

FRACTIONS
1
Specification

Learning objectives

Grade

Resources

recall
the fractiontodecimal conversion of
familiar simple fractions

Recall
fraction to decimal conversions for
simple fractions such as 1/4, 1/2, 3/4,
1/3 and 2/3

F

P
W
comparing decimals, fractions and
percentages
P
W
unscrambles (choose the biggest)

recognise that each
terminating decimal is a fraction
recognise that
recurring decimals are
exact
fractions, and that some exact
fractions are recurring decimals
perform short
division to convert a simple
fraction to a decimal

Convert
decimals to fractions

D

W
W
converting decimals to fractions

Convert
fractions to decimals

D

W
Collect a letter
X
Various Questions

convert
a recurring decimal to a fraction

Convert
recurring decimals to fractions

B


Convert
fractions to recurring decimals

B/A

distinguish between
fractions with denominators that
have only prime factors of 2 and 5
(which are represented by
terminating decimals), and other
fractions (which are represented by
recurring decimals)

know
that fractions with prime denominators
(other than 2 or 5) are recurring
decimals

B


FRACTIONS
2
Specification

Learning objectives

Grade

Resources

understand
equivalent fractions

Find
equivalent fractions

G


simplify a fraction
by cancelling all common factors

Simplify
fractions

F


order
fractions by rewriting them with a
common denominator

Arrange
fractions in order of size

F

W
Ordering fractions

calculate a given
fraction of a given quantity,
expressing the answer as a fraction

Work
out fractions of quantities.

E

P
finding fractions of pictures
P
fractions of amounts examples
P
fractions of amounts spider diagrams
P
fractions of amounts using division
W
4 in a line activity
W
W
shade fractions of diagrams
W
W
fractions of amounts
W
X
W
fractions of amounts
W
W
P
shade in the answers picture puzzle

express a given
number as a fraction of another

Find
one number as a fraction of another

E


add and subtract
fractions by writing them with a
common denominator

Do
calculations with simple fractions
involving addition

E

W
4 in a line activity  adding two
fractions

Do
calculations with simple fractions
involving subtraction

D

use efficient
methods to calculate with fractions,
including cancelling common factors
before carrying out the calculation,
recognising that, in many cases, only
a fraction can express the exact
answer

Do
calculations with mixed numbers

C

W
H
higher arithmetic
W
mixture of fraction questions

Be
able to cancel down a calculation in
order to work out the answer.

C

multiply and divide
a fraction by an integer, by a unit
fraction and by a general fraction

Do
calculations with simple fractions
involving multiplication

E


Do
calculations with simple fractions
involving division

C

understand and
use unit fractions as multiplicative
inverses




SQUARES, CUBES, POWERS AND
ROOTS
Specification

Learning objectives

Grade

Resources

use the terms
square, positive and
negative square root , cube and cube root

Calculate
squares and square roots (with and
without the use of a calculator)

F

W
4 in a line activity  square, cube,
factor and multiples
W
square numbers from pictures
W
square numbers and cube numbers
W
cube numbers from pictures
P
P
shade the square numbers picture puzzle
P
shade the square and cube numbers
picture puzzles

Calculate
cubes and cube roots (with and without
the use of a calculator)

E

use
the terms square, positive and negative
square root, cube and cube root

D

use calculators
effectively and efficiently:
know how to
enter complex calculations
use function
keys for reciprocals, squares and
powers.

Use
function keys on a calculator for powers
and roots

E

W
using a calculator  notes on functions
followed by questions

recall integer squares from
2x2 to 11x11 (15x15)
and the corresponding square
roots.

recall
integer squares from 2x2 to 15x15 and
the corresponding square roots.

D

P squares and cubes

recall the cubes of
2, 3, 4, 5 and 10

recall
the cubes of 2, 3, 4, 5 and 10

D

use index notation for
squares, cubes and powers of 10
use index laws
to simplify and calculate the value
of numerical expressions involving
multiplication and division of
integer, fractional and
negative powers

Use
index notation and index laws for
positive and negative powers

C

W
introducing powers of 10 including
negative powers
W
some notes on index laws
W
powers and indices including fractional
and negative powers and use of index
laws
W
index laws and standard form
W
index laws
W
indices and index laws

Use
index notation and index laws for
fractional powers where the power
is a unit fraction

A

Use
index notation and index laws for
fractional powers where the power
is not a unit fraction

A*

understand that the
inverse operation of raising a
positive number to power n is
raising the result of this operation
to power 1/n
recall the facts
that n^0= 1 and n^1 = 1/n for
positive integers n, the
corresponding rule for negative
numbers, n^1/2 and n^1/3 for any
positive number n

Interpret
fractional and negative powers

A*

W
blank table for investigating indices
W
notes on indices  fractional and
negative powers
W
indices including fractional and
negative powers
W
mixed questions on indices and decimals
W
negative and fractional indices

use surds in exact
calculations without a calculator.

Simplify
surds

A*

W
recurring decimals, surds and irrational
numbers

Carry out calculations with surds
without a calculator 
A* 
rationalise a
denominator

Rationalise
a denominator.

A

P
explaining the concept of irrational
numbers

POWERS OF 10 AND
STANDARD INDEX FORM
Specification

Learning objectives

Grade

Resources

multiply
or divide any number by powers of 10,
and any positive number by a number
between 0 and 1

Know
how to multiply a number by a power of
10 such as 10, 100, 100

F

W
multiplying by powers of 10
W
multiplying and dividing by powers of 10

Know
how to divide a number by a power of 10
such as 10, 100, 100

F

Know
how to multiply a number by a power of
10 such as 0.1, or 0.01

D

Understand the effect of
multiplying a number by a number between 0
and 1 
D 
W
multiplying by numbers between 0 and 1
W
dividing by numbers between 0 and 1

Understand the effect of dividing
a number by a number between 0 and 1

D 
express standard
index form both in conventional
notation and on a calculator display
enter a range of
calculations onto a calculator,
including those involving standard
index form.
use standard
index form display and how to enter
into a calculator numbers in
standard index form
calculate with
standard index form

Use
standard index form both with and
without a calculator

C

W
W
standard form
W
index laws and standard form
W
standard form questions
P
millionaire cards including standard
index form

convert between
ordinary and standard index form
representations, converting to
standard index form to make sensible
estimates for calculations involving
multiplication and/or division

Convert
between ordinary and standard index form
notation

B

W
F
F
converting between ordinary and standard
index form
W
questions in standard index form

PERCENTAGES
Specification

Learning objectives

Grade

Resources

understand
that ‘percentage’ means ‘number of
parts per 100’ and use this to compare
proportions

Understand
that percentage means 'out of 100'

F


interpret
percentage as the operator ‘so many
hundredths of’
convert simple
fractions of a whole or decimals to
percentages of the whole and vice
versa

Change
a percentage to a fraction and vice
versa

F

W
match cards
W
Fractions, decimals and percentages
conversion

Change
a percentage to a decimal and vice versa

F


Compare
percentages, fractions and decimals

E


use
percentage in reallife situations
solve
simple percentage problems, including
percentage increase and decrease
understand the
multiplicative nature of
percentages as operators
represent repeated
proportional change using a
multiplier raised to a power

Work
out a percentage of a given quantity
without a calculator

E

P
P
blank percentage spider diagram
P
blank percentage spider diagram
(reversed for printing on OHT)
P
W
using 10% and 50% to work out other
percenatges
P
find the reward  simple percentages
P
blanks for working out mental pecentages
W
Collect a letter  includes percentages
of amounts and four rules questions
P
P
spider diagrams for explaining mental
percentages
P
blank spider diagrams for students
W
X
W
W working
out percentage mentally
W
W
W
4 in a line activities  4 in a line
P
millionaire cards
W
percentages unscramble puzzle
W
W
P
shade in the answers picture puzzle
W
collect a joke

Work out a percentage of a given
quantity with a calculator 
E 
W
W
W
calculator percentages
W
worded percentage questions

Calculate
simple interest

E


Increase
or decrease a given quantity by a
percentage

D

W
percentage increase and decrease
W
percentages including increase and
decrease

Work
out percentage increase or decrease

C


express
one quantity as a percentage of another

D


Understand
how to use successive percentages
(compound)

B

W compound percentages 
Solve
problems to do with compound percentage
such as compound interest.

B


calculate an
original amount when given the
transformed amount after a
percentage change
(reverse percentages
problems)
use calculators for
reverse percentage calculations by
doing an appropriate division.

Work
out reverse percentage problems

B


use calculators to
explore exponential growth and
decay, using a multiplier and the
power key.

Work
out compound percentage problems using a
multiplier and the power key

A


RATIO AND PROPORTION
Specification

Learning objectives

Grade

Resources

use ratio notation,
including reduction to its simplest
form and its various links to
fraction notation

Solve problems to do
with 'best value'

E

P
W
W
W
W
Best value

Solve simple ratio and
proportion problems such as finding the
ratio of teachers to students in a school

D 
W
identify ratio from pictures and
simplify

divide a quantity in
a given ratio
solve word problems
about ratio and proportion,
including using informal strategies
and the unitary method of solution
represent repeated
proportional change using a multiplier
raised to a power

Solve
more complex ratio and proportion
problems such as sharing out money
between two groups in the ratio of their
numbers

C

W
sharing £240 into given ratios
W
using fractions to share into a ratio
W mixed
ration questions
W
understanding map scales
W
W
map problems (questions and maps)

Solve
ratio and proportion problems using the
unitary method

C

Calculate
proportional changes using a multiplier

B

calculate an unknown
quantity from quantities that vary in
direct or inverse proportion

Solve
direct and inverse proportion problems

A


set up and use
equations to solve word and other
problems involving direct proportion
or inverse proportion and relate
algebraic solutions to graphical
representation of the equations

Interpret
the graphs of direct and inverse
proportion relationships

A

P
explaining direct and inverse proportion
W
questions on direct and inverse
proportion

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