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Stage 1
Number
Numbers and the number system
Calculation
Addition and subtraction
•
Understand addition as counting on and combining two sets; record related
addition sentences.
•
Understand subtraction as counting back and ‘take away’; record related
subtraction sentences.
•
Understand difference as ‘how many more to make?’
•
Add/subtract a single-digit number by counting on/back.
•
Relate counting on and back in tens to finding 10 more/less than a number (<
100).
•
Begin to use the +, – and = signs to record calculations in number sentences.
•
Understand that changing the order of addition does not change the total.
•
Add a pair of numbers by putting the larger number first and counting on.
•
Begin to add single- and two-digit numbers.
Multiplication and division
•
Double any single-digit number.
Geometry
Shapes and geometric reasoning
Position and movement
Measure
Money
•
Recognise all coins and work out how to pay an exact sum using smaller coins.
Length, mass and capacity
•
Compare lengths and weights by direct comparison, then by using uniform
non-standard units.
Time
Handling data
Organising, categorising and representing data
•
Answer a question by sorting and organising data or objects in a variety of ways,
e.g. – using block graphs and pictograms with practical resources; discussing
the results – in lists and tables with practical resources; discussing the
results – in Venn or Carroll diagrams giving different criteria for grouping
the same objects
Problem solving
Using techniques and skills in solving mathematical problems
•
Check the answer to an addition by adding the numbers in a different order.
•
Check the answer to a subtraction by adding the answer to the smaller number in
the question.
Stage 2
Number
Numbers and the number system
Calculation
Mental strategies
Addition and subtraction
Multiplication and division
Geometry
Shapes and geometric reasoning
Position and movement
Measure
Money
Length, mass and capacity
Time
Handling data
Organising, categorising and representing data
Problem solving
Using techniques and skills in solving mathematical problems
Stage 3
Number
Numbers and the number system
•
Understand and use fraction notation recognising that fractions are several
parts of one whole, e.g. 4
3
is three quarters and 3 2 is two thirds.
•
Recognise equivalence between 2 1 , 4 2 ,
8 4 and
10 5 using
diagrams.
•
Recognise simple mixed fractions, e.g. 1 2
1 and 2 4
1 .
•
Order simple or mixed fractions on a number line, e.g. using the knowledge that
2 1 comes
half way between 4
1
and 4
3 , and that 1 2 1 comes half way between 1 and 2.
•
Begin to relate finding fractions to division.
•
Find halves, thirds, quarters and tenths of shapes and numbers (whole number
answers).
Calculation
Mental strategies
•
Know addition and subtraction facts for all numbers to 20.
•
Know the following addition and subtraction facts: – multiples of 100 with a
total of 1000 – multiples of 5 with a total of 100
•
Know multiplication/division facts for 2×, 3×, 5×, and 10× tables.
•
Begin to know 4× table.
•
Recognise two- and three-digit multiples of 2, 5 and 10.
•
Work out quickly the doubles of numbers 1 to 20 and derive the related halves.
•
Work out quickly the doubles of multiples of 5 (< 100) and derive the related
halves.
•
Work out quickly the doubles of multiples of 50 to 500.
Addition and subtraction
•
Add and subtract 10 and multiples of 10 to and from two- and three-digit
numbers.
•
Add 100 and multiples of 100 to three-digit numbers.
•
Use the = sign to represent equality, e.g. 75 + 25 = 95 + 5.
•
Add several small numbers.
•
Find complements to 100, solving number equations such as 78 + = 100.
•
Add and subtract pairs of two-digit numbers.
•
Add three-digit and two-digit numbers using notes to support.
•
Re-order an addition to help with the calculation, e.g. 41 + 54, by adding 40
to 54, then 1.
•
Add/subtract single-digit numbers to/from three-digit numbers.
•
Find 20, 30, … 90, 100, 200, 300 more/less than three-digit numbers.
Multiplication and division
•
Understand the relationship between halving and doubling.
•
Understand the effect of multiplying two-digit numbers by 10.
•
Multiply single-digit numbers and divide two-digit numbers by 2, 3, 4, 5, 6, 9
and 10.
•
Multiply teens numbers by 3 and 5.
•
Begin to divide two-digit numbers just beyond 10× tables, e.g. 60 ÷ 5, 33 ÷ 3.
•
Understand that division can leave a remainder (initially as ‘some left over’).
•
Understand and apply the idea that multiplication is commutive.
•
Understand the relationship between multiplication and division and write
connected facts.
Geometry
Shapes and geometric reasoning
•
Identify, describe and draw regular and irregular 2D shapes including pentagons,
hexagons, octagons and semi-circles.
•
Classify 2D shapes according to the number of sides, vertices and right angles.
•
Identify, describe and make 3D shapes including pyramids and prisms;
investigate which nets will make a cube.
•
Classify 3D shapes according to the number and shape of faces, number of
vertices and edges.
•
Draw and complete 2D shapes with reflective symmetry and draw reflections of
shapes (mirror line along one side).
•
Relate 2D shapes and 3D solids to drawings of them.
•
Identify 2D and 3D shapes, lines of symmetry and right angles in the environment.
•
Identify right angles in 2D shapes.
Position and movement
•
Use the language of position, direction and movement, including clockwise and
anti-clockwise.
•
Find and describe the position of a square on a grid of squares where the rows
and columns are labelled.
•
Use a set square to draw right angles.
•
Compare angles with a right angle and recognise that a straight line is
equivalent to two right angles.
Measure
Money
•
Consolidate using money notation.
•
Use addition and subtraction facts with a
total of 100 to find change.
Length, mass and capacity
•
Choose and use appropriate units and
equipment to estimate, measure and record measurements.
•
Know the relationship between kilometres
and metres, metres and centimetres, kilograms and grams, litres and
millilitres.
•
Read to the nearest division or half division, use scales that are numbered or
partially numbered.
•
Use a ruler to draw and measure lines to the nearest centimetre.
• Solve word
problems involving measures.
Time
•
Suggest and use suitable units to measure time and know the relationships
between them (second, minute, hour, day, week, month, year).
•
Read the time on analogue and digital clocks, to the nearest 5 minutes on an
analogue clock and to the nearest minute on a digital clock.
•
Begin to calculate simple time intervals in hours and minutes.
•
Read a calendar and calculate time intervals in weeks or days.
Handling data
Organising, categorising and representing data
•
Answer a real-life question by collecting, organising and interpreting data,
e.g. investigating the population of mini-beasts in different environments.
•
Use tally charts, frequency tables, pictograms (symbol representing one or two
units) and bar charts (intervals labelled in ones or twos).
•
Use Venn or Carroll diagrams to sort data and objects using two criteria.
Problem solving
Using techniques and skills in solving mathematical problems
•
Choose appropriate mental strategies to carry out calculations.
•
Begin to understand everyday systems of measurement in length, weight, capacity
and time and use these to make measurements as appropriate.
•
Make sense of and solve word problems, single (all four operations) and
two-step (addition and subtraction), and begin to represent them, e.g. with
drawings or on a number line.
•
Check the results of adding two numbers using subtraction, and several numbers
by adding in a different order.
•
Check subtraction by adding the answer to the smaller number in the original
calculation.
•
Check multiplication by reversing the order, e.g. checking that 6 × 4 = 24 by
doing 4 × 6.
•
Check a division using multiplication, e.g. check 12 ÷ 4 = 3 by doing 4 × 3.
•
Recognise the relationships between different 2D shapes.
•
Identify the differences and similarities between different 3D shapes.
•
Estimate and approximate when calculating, and check working.
•
Make a sensible estimate for the answer to a calculation, e.g. using rounding.
• Consider
whether an answer is reasonable.
Using understanding and strategies in solving problems
•
Make up a number story to go with a calculation, including in the context of
money.
•
Explain a choice of calculation strategy and show how the answer was worked
out.
•
Explore and solve number problems and puzzles, e.g. logic problems.
•
Use ordered lists and tables to help to solve problems systematically.
•
Describe and continue patterns which count on or back in steps of 2, 3, 4, 5,
10, or 100.
•
Identify simple relationships between numbers, e.g. each number is 7three more
than the number before it.
•
Identify simple relationships between shapes, e.g. these shapes all have the
same number of lines of symmetry.
•
Investigate a simple general statement by finding examples which do or do not
satisfy it, e.g. when adding 10 to a number, the first digit remains the same.
•
Explain methods and reasoning orally, including initial thoughts about possible
answers to a problem.
Stage
4
Number
Numbers and the number system
•
Read and write numbers up to 10 000.
•
Count on and back in ones, tens, hundreds and thousands from four-digit
numbers.
•
Understand what each digit represents in a three- or four-digit number and
partition into thousands, hundreds, tens and units.
•
Use decimal notation and place value for tenths and hundredths in context, e.g.
order amounts of money; convert a sum of money such as $13.25 to cents, or a
length such as 125 cm to metres; round a sum of money to the nearest pound.
•
Understand decimal notation for tenths and hundredths in context, e.g. length.
•
Find multiples of 10, 100, 1000 more/less than numbers of up to four digits,
e.g. 3407 + 20 = 3427.
•
Multiply and divide three-digit numbers by 10 (whole number answers) and
understand the effect; begin to multiply numbers by 100 and perform related
divisions.
•
Recognise multiples of 5, 10 and 100 up to 1000.
•
Round three- and four-digit numbers to the nearest 10 or 100.
•
Position accurately numbers up to 1000 on an empty number line or line marked
off in multiples of 10 or 100.
•
Estimate where three- and four-digit numbers lie on empty 0–1000 or 0–10 000
lines.
•
Compare pairs of three-digit or four-digit numbers, using the > and < signs,
and find a number in between each pair.
•
Use negative numbers in context, e.g. temperature.
•
Recognise and extend number sequences formed by counting in steps of constant
size, extending beyond zero when counting back.
•
Recognise odd and even numbers.
•
Make general statements about the sums and differences of odd and even numbers.
•
Order and compare two or more fractions with the same denominator (halves,
quarters, thirds, fifths, eighths or tenths).
•
Recognise the equivalence between: 2
1 , 8
4 and 10
5 ; 4
1 and 8
2 ; 5
1 and 10
2 .
•
Use equivalence to help order fractions, e.g. 10
7 and 4
3 .
•
Understand the equivalence between one-place decimals and fractions in tenths.
•
Understand that 2 1 is
equivalent to 0.5 and also to 10 5 .
•
Recognise the equivalence between the decimal fraction and vulgar fraction
forms of halves, quarters, tenths and hundredths.
•
Recognise mixed numbers, e.g. 5 4 3 , and order these on a number line.
•
Relate finding fractions to division.
•
Find halves, quarters, thirds, fifths, eighths and tenths of shapes and numbers.
Calculation
Mental strategies
•
Derive quickly pairs of two-digit numbers with a total of 100, e.g. 72 + = 100.
•
Derive quickly pairs of multiples of 50 with a total of 1000, e.g. 850 + = 1000.
•
Identify simple fractions with a total of 1, e.g. 4 1 + = 1.
•
Know multiplication for 2×, 3×, 4×, 5×, 6×, 9× and 10× tables and derive
division facts.
•
Recognise and begin to know multiples of 2, 3, 4, 5 and 10, up to the tenth
multiple
•
Add three two-digit multiples of 10, e.g. 40 + 70 + 50.
•
Add and subtract near multiples of 10 or 100 to or from three-digit numbers,
e.g. 367 – 198 or 278 + 49.
•
Add any pair of two-digit numbers, choosing an appropriate strategy.
•
Subtract any pair of two-digit numbers, choosing an appropriate strategy.
•
Find a difference between near multiples of 100, e.g. 304 – 296.
•
Subtract a small number crossing 100, e.g. 304 – 8.
•
Multiply any pair of single-digit numbers together.
•
Use knowledge of commutativity to find the easier way to multiply.
•
Understand the effect of multiplying and dividing three-digit numbers by 10.
•
Derive quickly doubles of all whole numbers to 50, doubles of multiples of 10
to 500, doubles of multiples of 100 to 5000, and corresponding halves.
Addition and subtraction
•
Add pairs of three-digit numbers.
•
Subtract a two-digit number from a three-digit number.
•
Subtract pairs of three-digit numbers.
Multiplication and division
•
Double any two-digit number.
•
Multiply multiples of 10 to 90 by a single-digit number.
•
Multiply a two-digit number by a single-digit number.
•
Divide two-digit numbers by single digit-numbers (answers no greater than 20).
•
Decide whether to round up or down after division to give an answer to a
problem.
•
Understand that multiplication and division are the inverse function of each
other.
•
Begin to understand simple ideas of ratio and proportion, e.g. a picture is one
fifth the size of the real dog. It is 25 cm long in the picture, so it is 5 ×
25 cm long in real life.
Geometry
Shapes and geometric reasoning
•
Identify, describe, visualise, draw and make a wider range of 2D and tetrahedron;
use pinboards to create a range of polygons. Use spotty paper to record
results.
•
Classify polygons (including a range of quadrilaterals) using criteria such as
the number of right angles, whether or not they are regular and their
symmetrical properties.
•
Identify and sketch lines of symmetry in 2D shapes and patterns.
•
Visualise 3D objects from 2D nets and drawings and make nets of common solids.
•
Find examples of shapes and symmetry in the environment and in art.
Position and movement
•
Describe and identify the position of a square on a grid of squares where rows
and columns are numbered and/or lettered.
•
Know that angles are measured in degrees and that one whole turn is 360° or
four right angles; compare and order angles less than 180°.
•
Devise the directions to give to follow a given path.
Measure
Length, mass and capacity
•
Choose and use standard metric units and their abbreviations (km, m, cm, mm,
kg, g, l and ml ) when estimating, measuring and recording length, weight and
capacity.
•
Know and use the relationships between familiar units of length, mass and
capacity; know the meaning of ‘kilo’, ‘centi’ and ‘milli’.
•
Where appropriate, use decimal notation to record measurements,
•
Interpret intervals/divisions on partially numbered scales and record readings
accurately.
Time
•
Read and tell the time to nearest minute on 12-hour digital and analogue
clocks.
•
Use am, pm and 12-hour digital clock notation.
•
Read simple timetables and use a calendar.
•
Choose units of time to measure time intervals.
Area and perimeter
•
Draw rectangles, and measure and calculate their perimeters.
•
Understand that area is measured in square units, e.g. cm2.
•
Find the area of rectilinear shapes drawn on a square grid by counting squares.
Handling data
Organising, categorising and representing data
•
Answer a question by identifying what data to collect, organising, presenting
and interpreting data in tables, diagrams, tally charts, frequency tables,
pictograms (symbol representing 2, 5, 10 or 20 units) and bar charts (intervals
labelled in twos, fives, tens or
twenties).
•
Compare the impact of representations where scales have different intervals.
•
Use Venn diagrams or Carroll diagrams to sort data and objects using two or
three criteria.
Problem solving
Using techniques and skills in solving mathematical problems
•
Choose appropriate mental or written strategies to carry out calculations
involving addition or subtraction.
•
Understand everyday systems of measurement in length, weight, capacity and time
and use these to solve simple problems as appropriate.
•
Check the results of adding numbers by adding them in a different order or by
subtracting one number from the total.
•
Check subtraction by adding the answer to the smaller number in the original
calculation.
•
Check multiplication using a different technique, e.g. check 6 × 8 = 48 by
doing 6 × 4 and doubling.
•
Check the result of a division using multiplication, e.g. multiply 4 by 12 to
check 48 ÷ 4.
•
Recognise the relationships between 2D shapes and identify the differences and
similarities between 3D shapes.
•
Estimate and approximate when calculating, and check working.
Using understanding and strategies in solving problems
•
Make up a number story for a calculation, including in the context of measures.
•
Explain reasons for a choice of strategy when multiplying or dividing.
•
Choose strategies to find answers to addition or subtraction problems; explain
and show working.
•
Explore and solve number problems and puzzles, e.g. logic problems.
•
Use ordered lists and tables to help to solve problems systematically.
•
Describe and continue number sequences, e.g. 7, 4, 1, –2 ... identifying the relationship
between each number.
•
Identify simple relationships between shapes, e.g. these polygons are all
regular because ...
•
Investigate a simple general statement by finding examples which do or do not
satisfy it.
•
Explain methods and reasoning orally and in writing; make hypotheses and test
them out.
Stage
5
It
is important that learners become confident users of calculators.
They
need to recognise that the calculator is a tool of which they are in control
and to understand how it can help them to develop their mathematics. Learners
can be taught how to use a calculator effectively and to recognise how and when
it is appropriate to do so; by first deciding if mental and pencil-and-paper
methods are quicker or more reliable. Note that to use a calculator effectively
requires a secure knowledge of number, which has to be the prime aim.
Number
Numbers and the number system
•
Count on and back in steps of constant size, extending beyond zero.
•
Know what each digit represents in five- and six-digit numbers.
•
Partition any number up to one million into thousands, hundreds, tens and
units.
•
Use decimal notation for tenths and hundredths and understand what each digit
represents.
•
Multiply and divide any number from 1 to 10 000 by 10 or 100 and understand the
effect.
•
Round four-digit numbers to the nearest 10, 100 or 1000.
•
Round a number with one or two decimal places to the nearest whole number.
•
Order and compare numbers up to a million using the > and < signs.
•
Order and compare negative and positive numbers on a number line and
temperature scale.
•
Calculate a rise or fall in temperature.
•
Order numbers with one or two decimal places and compare using the > and
< signs.
•
Recognise and extend number sequences.
•
Recognise odd and even numbers and multiples of 5, 10, 25, 50 and 100 up to
1000.
•
Make general statements about sums, differences and multiples of odd and even
numbers.
•
Recognise equivalence between: 2 1 , 4 1 and
8 1 ;
3 1 and
6 1 and
10 1 .
•
Recognise equivalence between the decimal and fraction forms of halves, tenths
and hundredths and use this to help order fractions, e.g. 0.6 is more than 50%
and less than 10 7 .
•
Change an improper fraction to a mixed number, e.g. 4 7 to 1 4
3 ; order mixed numbers and place between
whole numbers on a number line.
•
Relate finding fractions to division and use to find simple fractions of quantities.
•
Understand percentage as the number of parts in every 100 and find simple
percentages of quantities.
•
Express halves, tenths and hundredths as percentages.
Numbers and the number system (continued)
•
Use fractions to describe and estimate a simple proportion, e.g. 5 1 of the beads are yellow.
•
Use ratio to solve problems, e.g. to adapt a recipe for 6 people to one for 3
or 12 people.
Calculation
Mental strategies
•
Know by heart pairs of one-place decimals with a total of 1, e.g. 0.8 + 0.2.
•
Derive quickly pairs of decimals with a total of 10, and with a total of 1.
•
Know multiplication and division facts for the 2× to 10× tables.
•
Know and apply tests of divisibility by 2, 5, 10 and 100.
•
Recognise multiples of 6, 7, 8 and 9 up to the 10th multiple.
•
Know squares of all numbers to 10 × 10.
•
Find factors of two-digit numbers.
•
Count on or back in thousands, hundreds, tens and ones to add or subtract.
•
Add or subtract near multiples of 10 or 100, e.g. 4387 – 299.
•
Use appropriate strategies to add or subtract pairs of two- and three-digit
numbers and number with one decimal place, using jottings where necessary.
•
Calculate differences between near multiples of 1000, e.g. 5026 – 4998, or near
multiples of 1, e.g. 3.2 – 2.6.
•
Multiply multiples of 10 to 90, and multiples of 100 to 900, by a single-digit
number.
•
Multiply by 19 or 21 by multiplying by 20 and adjusting.
•
Multiply by 25 by multiplying by 100 and dividing by 4.
•
Use factors to multiply, e.g. multiply by 3, then double to multiply by 6.
•
Double any number up to 100 and halve even numbers to 200 and use this to
double and halve numbers with one or two decimal places, e.g. double 3.4 and
half of 8.6.
•
Double multiples of 10 to 1000 and multiples of 100 to 10 000, e.g. double 360
or double 3600, and derive the corresponding halves.
Addition and subtraction
•
Find the total of more than three two- or three-digit numbers using a written
method.
•
Add or subtract any pair of three- and/or four-digit numbers, with the same
number of decimal places, including amounts of money.
Multiplication and division
•
Multiply or divide three-digit numbers by single-digit numbers.
•
Multiply two-digit numbers by two-digit numbers.
•
Multiply two-digit numbers with one decimal place by single-digit numbers, e.g.
3.6 × 7.
•
Divide three-digit numbers by single-digit numbers, including those with a
remainder (answers no greater than 30).
•
Start expressing remainders as a fraction of the divisor when dividing two-digit
numbers by single-digit numbers.
•
Decide whether to group (using multiplication facts and multiples of the
divisor) or to share (halving and quartering) to solve divisions.
•
Decide whether to round an answer up or down after division, depending on the
context.
•
Begin to use brackets to order operations and understand the relationship
between the four operations and how the laws of arithmetic apply to
multiplication.
Geometry
Shapes and geometric reasoning
•
Identify and describe properties of triangles and classify as isosceles, equilateral
or scalene.
•
Recognise reflective and rotational symmetry in regular polygons.
•
Create patterns with two lines of symmetry, e.g. on a pegboard or squared
paper.
•
Visualise 3D shapes from 2D drawings and nets, e.g. different nets of an open
or closed cube.
•
Recognise perpendicular and parallel lines in 2D shapes, drawings and the
environment.
•
Understand and use angle measure in degrees; measure angles to the nearest 5°;
identify, describe and estimate the size of angles and classify them as acute,
right or obtuse.
•
Calculate angles in a straight line.
Position and movement
•
Read and plot co-ordinates in the first quadrant.
•
Predict where a polygon will be after reflection where the mirror line is
parallel to one of the sides, including where the line is oblique.
•
Understand translation as movement along a straight line, identify where
polygons will be after a translation and give instructions for translating
shapes.
Measure
Length, mass and capacity
•
Read, choose, use and record standard units to estimate and measure length,
mass and capacity to a suitable degree of accuracy.
•
Convert larger to smaller metric units (decimals to one place), e.g. change 2.6
kg to 2600 g.
•
Order measurements in mixed units.
•
Round measurements to the nearest whole unit.
•
Interpret a reading that lies between two unnumbered divisions on a scale.
•
Compare readings on different scales.
•
Draw and measure lines to the nearest centimetre and millimetre.
Time
•
Recognise and use the units for time (seconds, minutes, hours, days, months and
years).
•
Tell and compare the time using digital and analogue clocks using the 24-hour
clock.
•
Read timetables using the 24-hour clock.
•
Calculate time intervals in seconds, minutes and hours using digital or
analogue formats.
•
Use a calendar to calculate time intervals in days and weeks (using knowledge
of days in calendar months).
•
Calculate time intervals in months or years.
Area and perimeter
•
Measure and calculate the perimeter of regular and irregular polygons.
•
Understand area measured in square centimetres (cm2).
•
Use the formula for the area of a rectangle to calculate the rectangle’s area.
Handling data
Organising, categorising and representing data
•
Answer a set of related questions by collecting, selecting and organising
relevant data; draw conclusions from their own and others’ data and identify
further questions to ask.
•
Draw and interpret frequency tables, pictograms and bar line charts, with the
vertical axis labelled for example in twos, fives, tens, twenties or hundreds.
Consider the effect of changing the scale on
the
vertical axis.
•
Construct simple line graphs, e.g. to show changes in temperature over time.
•
Understand where intermediate points have and do not have meaning, e.g.
comparing a line graph of temperature against time with a graph of class
attendance for each day of the week.
•
Find and interpret the mode of a set of data.
Probability
•
Describe the occurrence of familiar events using the language of chance or
likelihood.
Problem solving
Using techniques and skills in solving mathematical problems
•
Understand everyday systems of measurement in length, weight, capacity, temperature
and time and use these to perform simple calculations.
•
Solve single and multi-step word problems (all four operations); represent
them, e.g. with diagrams or a number line.
•
Check with a different order when adding several numbers or by using the inverse
when adding or subtracting a pair of numbers.
•
Use multiplication to check the result of a division, e.g. multiply 3.7 × 8 to
check 29.6 ÷ 8.
•
Recognise the relationships between different 2D and 3D shapes, e.g. a face of
a cube is a square.
•
Estimate and approximate when calculating, e.g. using rounding, and check
working.
•
Consider whether an answer is reasonable in the context of a problem.
Using understanding and strategies in solving problems
•
Understand everyday systems of measurement in length, weight, capacity,
temperature and time and use these to perform simple calculations.
•
Choose an appropriate strategy for a calculation and explain how they worked
out the answer.
•
Explore and solve number problems and puzzles, e.g. logic problems.
•
Deduce new information from existing information to solve problems.
•
Use ordered lists and tables to help to solve problems systematically.
•
Describe and continue number sequences, e.g. –30, –27, , , –18...;
identify the relationships between numbers.
•
Identify simple relationships between shapes, e.g. these triangles are all
isosceles because ...
•
Investigate a simple general statement by finding examples which do or do not
satisfy it, e.g. the sum of three consecutive whole numbers is always a
multiple of three.
•
Explain methods and justify reasoning orally and in writing; make hypotheses
and test them out.
•
Solve a larger problem by breaking it down into sub-problems or represent it
using diagrams.
Stage
6
As
in Stage 5, it is important that learners become confident users of
calculators.
They need to recognise that the calculator is a tool of
which
they are in control and to understand how it can help them to
develop
their mathematics. Learners can be taught how to use a
calculator
effectively and to recognise how and when it is appropriate
to
do so; by first deciding if mental and pencil-and-paper methods are
quicker
or more reliable. Note that to use a calculator effectively
requires
a secure knowledge of number, which has to be the prime
aim.
Number
Numbers and the number system
•
Count on and back in fractions and decimals, e.g. 3 1 s, 0.1s, and repeated steps of whole
numbers (and through zero).
•
Know what each digit represents in whole numbers up to a million.
•
Know what each digit represents in one- and two-place decimal numbers.
•
Multiply and divide any whole number from 1 to 10 000 by 10, 100 or 1000 and
explain the effect.
•
Multiply and divide decimals by 10 or 100 (answers up to two decimal places for
division).
•
Find factors of two-digit numbers.
•
Find some common multiples, e.g. for 4 and 5.
•
Round whole numbers to the nearest 10, 100 or 1000.
•
Round a number with two decimal places to the nearest tenth or to the nearest
whole number.
•
Make and justify estimates and approximations of large numbers.
•
Order and compare positive numbers to one million, and negative integers to an
appropriate level.
•
Use the >, < and = signs correctly.
•
Estimate where four-digit numbers lie on an empty 0 –10 000 line.
•
Order numbers with up to two decimal places (including different numbers of
places).
•
Recognise and extend number sequences.
•
Recognise and use decimals with up to three places in the context of measurement.
•
Recognise odd and even numbers and multiples of 5, 10, 25, 50 and 100 up to
1000.
•
Make general statements about sums, differences and multiples of odd and even
numbers.
•
Recognise prime numbers up to 20 and find all prime numbers less than 100.
•
Recognise the historical origins of our number system and begin to understand
how it developed.
•
Compare fractions with the same denominator and related denominators, e.g. 4 3 with 8 7 .
•
Recognise equivalence between fractions, e.g. between 100 1 s, 10 1 s
and 2 1 s.
•
Recognise and use the equivalence between decimal and fraction forms.
•
Order mixed numbers and place between whole numbers on a number line.
•
Change an improper fraction to a mixed number, e.g. 8 17 to 2 8
1 .
•
Reduce fractions to their simplest form, where this is 4 1 , 2
1 , 4
3 or a number of fifths or tenths.
•
Begin to convert a vulgar fraction to a decimal fraction using division.
•
Understand percentage as parts in every 100 and express 2 , 4 1 ,
3 1 ,
10
1 , 100
1 as percentages.
•
Find simple percentages of shapes and whole numbers.
•
Solve simple problems involving ratio and direct proportion.
Calculation
Mental strategies
•
Recall addition and subtraction facts for numbers to 20 and pairs of one-place
decimals with a total of 1, e.g. 0.4 + 0.6.
•
Derive quickly pairs of one-place decimals totalling 10, e.g. 7.8 and 2.2, and
two-place decimals totalling 1, e.g. 0.78 + 0.22.
•
Know and apply tests of divisibility by 2, 4, 5, 10, 25 and 100.
•
Use place value and number facts to add or subtract two-digit whole numbers and
to add or subtract three-digit multiples of 10 and pairs of decimals, e.g. 560
+ 270; 2.6 + 2.7; 0.78 + 0.23.
•
Add/subtract near multiples of one when adding numbers with one decimal place,
e.g. 5.6 + 2.9; 13.5 – 2.1.
•
Add/subtract a near multiple of 10, 100 or 1000, or a near whole unit of money,
and adjust, e.g. 3127 + 4998; 5678 – 1996.
•
Use place value and multiplication facts to multiply/divide mentally, e.g. 0.8
× 7; 4.8 ÷ 6.
•
Multiply pairs of multiples of 10, e.g. 30 × 40, or multiples of 10 and 100,
e.g. 600 × 40.
•
Double quickly any two-digit number, e.g. 78, 7.8, 0.78 and derive the corresponding
halves.
•
Divide two-digit numbers by single-digit numbers, including leaving a remainder.
Addition and subtraction
•
Add two- and three-digit numbers with the same or different numbers of
digits/decimal places.
•
Add or subtract numbers with the same and different numbers of decimal places,
including amounts of money.
•
Find the difference between a positive and negative integer, and between two
negative integers in a context such as temperature or on a number line.
Multiplication and division
•
Multiply pairs of multiples of 10, e.g. 30 × 40, or multiples of 10 and 100,
e.g. 600 × 40.
•
Multiply near multiples of 10 by multiplying by the multiple of 10 and adjusting.
•
Multiply by halving one number and doubling the other, e.g. calculate 35 × 16
with 70 × 8.
•
Use number facts to generate new multiplication facts, e.g. the 17× table from
10× + 7× tables.
•
Multiply two-, three- or four-digit numbers (including sums of money) by a
single-digit number and two- or three-digit numbers by two-digit numbers.
•
Divide three-digit numbers by single-digit numbers, including those leaving a
remainder and divide three-digit numbers by two-digit numbers (no remainder)
including sums of money.
•
Give an answer to division as a mixed number, and a decimal (with divisors of
2, 4, 5, 10 or 100).
•
Relate finding fractions to division and use them as operators to find fractions
including several tenths and hundredths of quantities.
•
Know and apply the arithmetic laws as they apply to multiplication (without
necessarily using the terms commutative, associative or distributive).
Geometry
Shapes and geometric reasoning
•
Classify different polygons and understand whether a 2D shape is a polygon or
not.
•
Visualise and describe the properties of 3D shapes, e.g. faces, edges and
vertices.
•
Identify and describe properties of quadrilaterals (including the parallelogram,
rhombus and trapezium), and classify using parallel sides, equal sides, equal
angles.
•
Recognise and make 2D representations of 3D shapes including nets.
•
Estimate, recognise and draw acute and obtuse angles and use a protractor to
measure to the nearest degree.
•
Check that the sum of the angles in a triangle is 180°, for example, by
measuring or paper folding; calculate angles in a triangle or around a point.
Position and movement
•
Read and plot co-ordinates in all four quadrants.
•
Predict where a polygon will be after one reflection, where the sides of the
shape are not parallel or perpendicular to the mirror line, after one
translation or after a rotation through 90° about one of its
vertices.
Measure
Length, mass and capacity
•
Select and use standard units of measure. Read and write to two or three
decimal places.
•
Convert between units of measurement (kg and g, l and ml, km, m, cm
and mm), using decimals to three places, e.g. recognising that 1.245 m is 1 m
24.5 cm.
•
Interpret readings on different scales, using a range of measuring instruments.
•
Draw and measure lines to the nearest centimetre and millimetre.
•
Know imperial units still in common use, e.g. the mile, and approximate metric
equivalents.
Time
•
Recognise and understand the units for measuring time (seconds,minutes, hours,
days, weeks, months, years, decades and centuries); convert one unit of time
into another.
•
Tell the time using digital and analogue clocks using the 24-hour clock.
•
Compare times on digital and analogue clocks, e.g. realise quarter to four is
later than 3:40.
•
Read and use timetables using the 24-hour clock.
•
Calculate time intervals using digital and analogue times.
•
Use a calendar to calculate time intervals in days, weeks or months.
•
Calculate time intervals in days, months or years.
•
Appreciate how the time is different in different time zones around the world.
Area and perimeter
•
Measure and calculate the perimeter and area of rectilinear shapes.
•
Estimate the area of an irregular shape by counting squares.
•
Calculate perimeter and area of simple compound shapes that can be split into
rectangles.
Handling data
Organising, categorising and representing data
•
Solve a problem by representing, extracting and interpreting data in tables,
graphs, charts and diagrams, e.g. line graphs for distance and time; a price
‘ready-reckoner’ for currency conversion; frequency tables and bar charts with
grouped discrete data.
•
Find the mode and range of a set of data from relevant situations, e.g.
scientific experiments.
•
Begin to find the median and mean of a set of data.
•
Explore how statistics are used in everyday life.
Probability
•
Use the language associated with probability to discuss events, to assess
likelihood and risk, including those with equally likely outcomes.
Problem solving
Using techniques and skills in solving mathematical problems
•
Choose appropriate and efficient mental or written strategies to carry out a
calculation involving addition, subtraction, multiplication or division.
•
Understand everyday systems of measurement in length, weight, capacity,
temperature and time and use these to perform simple calculations.
•
Check addition with a different order when adding a long list of numbers; check
when subtracting by using the inverse.
•
Recognise 2D and 3D shapes and their relationships, e.g. a cuboid has a
rectangular cross-section.
•
Estimate and approximate when calculating, e.g. use rounding, and check
working.
Using understanding and strategies in solving problems
•
Explain why they chose a particular method to perform a calculation and show
working.
•
Deduce new information from existing information and realise the effect that
one piece of information has on another.
•
Use logical reasoning to explore and solve number problems and mathematical
puzzles.
•
Use ordered lists or tables to help solve problems systematically.
•
Identify relationships between numbers and make generalized statements using
words, then symbols and letters, e.g. the second number is twice the first
number plus 5 (n,
2n +
5); all the numbers
are
multiples of 3 minus 1 (3n – 1); the sum of angles in a triangle is 180°.
•
Make sense of and solve word problems, single and multi-step (all four
operations), and represent them, e.g. with diagrams or on a number line; use
brackets to show the series of calculations necessary.
•
Solve simple word problems involving ratio and direct proportion.
•
Solve simple word problems involving percentages, e.g. find discounted prices.
•
Make, test and refine hypotheses, explain and justify methods, reasoning,
strategies, results or conclusions orally.
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