How do we help our children become mathematicians?

In preparing for the new statutory assessment arrangements in 2016, the government has identified the bullet points below as the ‘expected standard’ in mathematics by the end of:_

**Key Stage One**

**Number**

•Count in multiples of 2, 5 and 10, to 100, forwards and backwards

•Count forward in multiples of 3, to 18

•Count in steps of 10, forward and backwards (e.g. 97, 87…)

•Read & write numbers to at least 100 in numerals, and phonetically attempts to write numbers to 100 in words

•Use place value in whole numbers up to 100 to compare and order numbers, sometimes using < and > signs correctly

•Identify, represent and estimate within a structural environment (e.g. estimate 33 on a number line)

•Use place value and number facts to solve problems (e.g. 60 –XX = 20)

•Recall and use addition and subtraction facts

•Subtract two simple 2-digit numbers, which do not involve bridging ten (e.g. 36-24)

•Add three 1-digit numbers, where they use known addition or doubling facts

•Add and subtract numbers using concrete objects and pictorial representations, including:

oa 2-digit number and ones

oa 2-digit number and tens

oadding two 2-digit numbers

oadding three 1-digit numbers

•Use inverse operations to solve missing number problems for addition and subtraction

•Solve simple 2-step problems with addition and subtraction

•Recall and use multiplication and division facts for the x10 table using the appropriate signs

•Recognise odd and even numbers

•Solve simple problems involving multiplication and division

•Know that addition and multiplication of two small numbers can be done in any order (commutative) and subtraction of one number from another cannot

•Recognise and find half of a set of objects or a quantity and begin to find 1/3 or 1/4 of a small set of objects with support

•Recognise, find and name fractions ½, 1/3, ¼, 2/4, and ¾ of a shape

•Recognise the equivalence of two quarters and one half in practical contexts

__Measurement__

•Compare and order lengths, mass, volume/capacity

•Choose and use appropriate standard units to measure length/height in any direction (m/cm); mass (kg/g); temperature (°C); capacity (l/ml) to the nearest appropriate unit using rules, scales, thermometers and measuring vessels and begin to make good estimates.

•Recognise and use symbols for pounds (£) and pence (p); combine amounts to make a particular value and find different combinations of coins to equal the same amounts of money

•Recognise, tell and write the times: o’clock; half past and quarter past and are beginning to recognise quarter to the hour; draw hands on a clock face to show half past and o’clock times

•Solve simple problems in a practical context involving addition and subtraction of money using the same unit, including giving change

**Geometry**

•compare and sort common 2-D shapes (e.g. semi-circle, rectangle and regular polygons such as pentagon, hexagon and octagon) and everyday objects, identifying and describing their properties (e.g. the number of sides or vertices, and are beginning to recognise symmetry in a vertical line)

•compare and sort common 3-D shapes (e.g. cone, cylinder, triangular prism, pyramid) and everyday objects, identifying and describing their properties (e.g. flat / curved surfaces, and beginning to count number of faces and vertices correctly)

•identify 2-D shapes on the surface of 3-D shapes and images of them (e.g. a circle on a cylinder and a triangle on a pyramid)

•order and arrange combinations of mathematical objects in patterns (e.g. continue a repeating pattern)

•use mathematical vocabulary to describe position, direction (e.g. left and right) and movement including movement in a straight line and distinguishing between rotation as a turn and in terms of right angles for quarter and half turns

__Statistics__

•interpret simple pictograms (where the symbols show one to one correspondence), tally charts, block diagrams (where the scale is divided into ones, even if only labelled in multiples of two) and simple tables

•answer questions by counting the number of objects in each category and sorting the categories by quantity

•answer questions about totalling and begin to compare simple categorical data (e.g. when the pictures or blocks are adjacent)

**Solve problems, communicate and reason mathematically**

•solve problems by applying their mathematics in a range of contexts (including money and measures, geometry and statistics) using the content described above; use and interpret mathematical symbols and diagrams; and begin to communicate their reasoning; for example:

ouse place value and number facts to solve problems (e.g. 40 + XX = 70)

ouse inverse operations to solve missing number problems for addition and subtraction (e.g. There were some people on a bus, six get off leaving seventeen people on the bus. How many were on the bus to start with?)

osolve simple 2-step problems with addition and subtraction, which require some retrieval (e.g. There are 12 kittens in a basket, 6 jump out and only 2 jump back in. How many are in the basket now?)

osolve simple problems involving multiplication and division (e.g. Ahmed buys 3 packs of apples. There are 4 apples in each pack. How many apples does he buy?)

osolve problems with one or two computational steps using addition, subtraction, multiplication and division and a combination of these (e.g. Joe has 2 packs of 6 stickers; Mina gives him 2 more stickers. How many stickers does he have altogether?)

osolve simple problems in a practical context involving addition and subtraction of money of the same unit, including giving change (e.g. Identify three coins with a total value of 24p or find the two items which cost exactly £1 altogether from a list such as: 70p, 40p, 50p and 30p)

**Key Stage 2**

**Number**

•use place value in whole numbers up to 1 000 000 to compare and order numbers and are beginning to become confident with numbers up to 10 000 000

•round any whole number to the nearest power of ten

•use negative numbers in practical contexts such as temperature and calculate intervals across zero

•count forwards or backwards in steps of any whole number with one significant figure, e.g. 9, 20, 3000 to generate, describe and complete linear number sequences

•recognise and use multiples, factors, prime numbers less than 20 and square numbers up to 121 show evidence of using mental methods, including jottings where necessary to speed up the process, to add and subtract whole numbers with up to two significant figures (e.g. 95 + 36, 5700 − 2900)

•add and subtract whole numbers with more than four digits, using formal written methods where appropriate

•Use their understanding of place value to multiply and divide whole numbers and decimals with up to two decimal places by 10 or 100 (e.g. 1532 ÷100 = , XX ÷100 = 6.3)

•Multiply and divide whole numbers mentally drawing upon multiplication facts up to 12 ×12 and place value (e.g. 60 ×70) and begin to use these facts to work with larger numbers

•Multiply numbers with up to two digits by a two digit number using a formal written method and becoming more confident with multiplication with larger numbers; multiply and divide numbers with up to four digits by a single digit number using the formalwritten method and becoming more confident with two digit divisors

•Recognise and use equivalent fractions

•Recognise and use the equivalences between simple fractions, decimals and percentages and become more confident with calculating decimal fraction equivalents

•Find simple fractions and percentages of whole numbers and quantities

•Add and subtract fractions with the same denominator, using mixed numbers where appropriate for the context

•Add and subtract fractions with the same denominator and multiples of the same number and become more confident with more complex fraction calculations

•Add and subtract decimal numbers that have the same number of decimal places

•Multiply a one digit decimal number by a single digit number

•Use simple ratio to compare quantities

•Use simple formulae expressed in words (e.g. time needed to cook a chicken: allow 20 minutes plus 40 minutes per kilogram)

•Find possible values in missing number problems involving one or two unknowns (e.g. Ben thinks of two numbers: the sum of the two numbers is 10: multiplied together they make 24: What are Ben’s numbers?)

**Measurement**

•Read, write and convert time between analogue (including clock faces using Roman numerals) and digital 12 and 24 hour clocks, using am and pm where necessary

•Calculate the duration of an event using appropriate units of time (e.g. A film starts at 6:45pm and finishes at 8:05pm. How long did it last?)

•Convert between ‘adjacent’ metric units of measure for length, capacity and mass (e.g. 1.2 kg = 1200 g; how many 200 ml cups can be filled from a 2 litre bottle?; write 605 cm in metres)

•Find the perimeter of compound shapes when all side lengths are known or can be easily determined (e.g. a simple shape made from two identical rectangles joined together to make an L-shape with given dimensions of the rectangle)

•Estimate the area of irregular shapes by counting squares (including half squares and fractions of squares that join with others to make whole squares)

•Calculate and compare the area of squares and rectangles including using standard units, square centimetres (cm2) and square metres (m2) and estimate the area of irregular shapes by counting squares

__Geometry__

•Compare and classify 3D and 2D shapes based on their properties (e.g. for 2–D shapes: parallel sides, length of sides, type and size of angles, reflective symmetry, regular / irregular polygons; for 3–D shapes: faces, vertices and edges)

•Recognise, describe simple 3D shapes, including using nets and other 2D representations

•Complete simple shapes using given lengths, such as 7.5cm, (accurate to +/ −2 mm) and acute angles that are multiples of 5°(accurate to +/-2°)

•Know and use the facts that angles at a point sum to 360°, angles at a point on a straight line sum to 180°and angles in a triangle sum to 180°(e.g. calculate the base angles of an isosceles triangle where the other angle is 110°) and identify other multiples of 90°

•Identify, describe and represent the position of a shape following a reflection or translation

•Describe positions on a 2–D co-ordinate grid using axes with equal scales in the first quadrant (in the context of number or geometry) and use co-ordinates to complete a given rectangle; becoming more confident in all four quadrants

__Statistics__

•Complete, read and interpret information presented in tables and bar charts (e.g. find the difference between two bars showing temperatures, where one is 20°C and the other is 13°C, on a scale labelled in multiples of 5)

•Interpret line graphs (e.g. beginning to find the difference between two temperatures on a line graph, where one is 20°C and the other is 13°C, on a scale labelled in multiples of 5) and simple pie charts (e.g. a pie chart cut into eight pieces for favourite fruit using whole numbers for each section)

•Calculate the mean as an average for simple sets of discrete data (e.g. find the mean mass of three parcels weighing 5kg, 3kg and 10kg)

**Solving problems and reason mathematically**

•Develop their own strategies to solve problems by applying their mathematics to a variety of routine and non-routine problems, in a range of contexts (including money and measures, geometry and statistics) using the content described above

•Begin to reason mathematically making simple generalisations, using mathematical language and searching for solutions by trying out ideas of their own

•Use and interpret mathematical symbols and diagrams, and present information and results in a clear and organised way; for example:

oderive strategies to solve problems with two or three computational steps using addition, subtraction, multiplication and division and a combination of these (e.g. extract and add prices from a table and calculate change, or solve problems such as ‘Jason bought some bags of green apples (6 for 75p) and some bags of red apples (10 for 90p). He spent £4.20. How many bags of each type of apples did he buy?’)

osolve problems involving numbers with up to two decimal places (e.g. find the two numbers which sum to 10 from this list: 0.01, 0.11, 1.01, 9.09, 9.9, 9.99)

oselect appropriate strategies when calculating depending on the numbers involved

ouse rounding and estimation to check their answers and determine, in the context of the problem, appropriate levels of accuracy

oidentify simple patterns and relationships, and make simple generalisations. They can draw their own conclusions and explain their reasoning in simple contexts using mathematical language (e.g. an explanation to satisfy statements such as ‘If you add a two-digit number to a two-digit number you cannot get a four-digit number’