The Groundwork Maths programme

Pupils make a ten from ten ones and build every number from 11 to 29 as some tens and some ones, saying what each digit is worth. This is the first step away from seeing a teen number as a pile of single ones.

Pupils build and read any number to 50 on a place value grid, matching the grid to bundles of tens and ones. The grid gives each digit a column, so its place shows its value.

Pupils build and read any number to 99 with Base 10 equipment, knowing that ten ones-cubes make one tens-rod. Moving to Base 10 keeps the tens-and-ones structure while letting pupils work with larger numbers more quickly.

Pupils partition a two-digit number into tens and ones and rebuild it, write the partition as an equation, and find a missing part. They also partition in non-standard ways, such as seeing 68 as 67 and 1, which prepares the ground for calculation.

Pupils place and read two-digit numbers on a marked number line, and work out which two tens a number sits between. This adds a sense of where a number sits to their understanding of what it is made of.

Pupils estimate where a number sits on a line with few or no marks, using the midpoint and both ends. Working from the midpoint lets them place a number sensibly even when the intervals are not all marked.

Pupils compare two two-digit numbers using the symbols for less than, greater than and equal to, and order a small set. They learn to compare the tens first and then the ones, rather than judging by the first digit they notice.

Pupils build and read three-digit numbers with Base 10, exchanging ten tens for one hundred, and say what each digit is worth. They meet the hundreds column as the next step in the same tens-and-ones structure, where ten of one unit make one of the next.

Pupils represent three-digit numbers with place value counters, match across Base 10, counters and the written number, and find 1, 10 or 100 more or less. Place value counters hold their value by position rather than size, so a pupil reads a hundreds counter by where it sits, not how big it is.

Pupils partition three-digit numbers and rebuild them, both in the standard way and in non-standard ways such as seeing 342 as 200 and 140 and 2. They recognise parts given in any order and find a missing part, which prepares the ground for calculation.

Pupils place and read three-digit numbers on marked lines, work out which two hundreds or tens a number sits between, and name the previous and next multiples of 100 and 10. They count fluently across hundreds and tens boundaries.

Pupils estimate where a three-digit number sits on a line with few or no marks, using halfway and the endpoints to reason proportionally. They take approximate readings from measures and statistics scales, connecting estimation to numbers met in the world.

Pupils compare two three-digit numbers using the symbols for less than, greater than and equal to, and order a set. They compare the hundreds first, then the tens, then the ones, working through the places in turn.

Pupils divide 100 into 2, 4, 5 and 10 equal parts and count in the steps that result, including past 100, then read and mark jugs, dials, rulers and bar charts. The key move is working out what one step on the scale is worth before reading it.

Pupils build and read four-digit numbers with Base 10, exchanging ten hundreds for one thousand, and say what each digit is worth. The thousands column is the next step in the same structure, where ten of one unit make one of the next.

Pupils represent four-digit numbers with place value counters, match across Base 10, counters and the written number, and find 1, 10, 100 or 1,000 more or less. The counters hold their value by position rather than size, so a pupil reads a thousands counter by where it sits.

Pupils partition four-digit numbers and rebuild them, both in the standard way and in non-standard ways such as seeing 4,274 as 3,000 and 1,200 and 70 and 4. They recognise parts given in any order and find a missing part.

Pupils place and read four-digit numbers on marked lines, work out which two thousands or hundreds a number sits between, and name the previous and next multiples of 1,000 and 100. They count fluently across thousands and hundreds boundaries.

Pupils estimate where a four-digit number sits on a line with few or no marks, using halfway and the endpoints to reason proportionally. They take approximate readings from measures and statistics scales, connecting estimation to numbers met in the world.

Pupils round a four-digit number to the nearest 10, 100 and 1,000, first on a number line and then from the digits. They handle the halfway case and cope when rounding rolls over into the next place.

Pupils divide a scale into 2, 4, 5 and 10 equal parts and count in the steps that result, then read and mark jugs, dials, rulers and bar charts. The key move is working out what one step on the scale is worth before reading it.

Pupils understand what a tenth is, and build, write and count in tenths, including past one whole. They see that the digit after the point follows the same ten-for-one rule as the whole numbers before it.

Pupils understand what a hundredth is, and build, write and count in hundredths, including past one tenth. They meet the zero place-holder, so three hundredths is written as 0.03 rather than 0.3.

Pupils build and read decimals on a grid and with counters, and say what each digit is worth. Moving off concrete pieces onto counters means the value comes from the place, not the size of the piece.

Pupils partition a decimal into parts and rebuild it, in standard and non-standard ways, and find a missing part. Recognising parts given in any order keeps the focus on value rather than the order the digits appear.

Pupils place, read and estimate any tenth or hundredth on a number line, both on marked lines and by estimating on unmarked ones. This gives them a feel for where a decimal sits, not only how it is written.

Pupils round a decimal to the nearest whole and the nearest tenth, first on a number line and then from the digits, and know which digit decides. This builds directly on the rounding of whole numbers met earlier.

Pupils divide one whole into 2, 4, 5 or 10 equal parts, work out what one step is worth, and read and mark scales, including past one. The key move is the same as in earlier phases, finding the value of one step before reading the scale.

Pupils compare and order decimals, including decimals of different lengths and equivalent decimals, find missing digits, and place decimals into sections of a line. Reasoning from place value is what stops the common error of thinking 0.45 is greater than 0.5.

Pupils see that a number can be split into parts and put back together, so seven is made of three and four, or five and two. This part-whole idea is the foundation that number bonds and all later calculation rest on.

Pupils recall the pairs of numbers that make each total within 10, so they know without counting that three and four make seven. Quick recall here is what later lets them bridge through 10 and add within 100.

Pupils secure the pairs that make 10 until they come to mind instantly, so six calls up four straight away. Bonds to 10 are used so often in later calculation that they need to be the most secure of all.

Pupils count forwards and backwards in multiples of 2, 5 and 10, building the rhythm of counting that leads into the multiplication tables. Counting back as well as forwards prepares the way for division as well as multiplication.

Pupils build the 3 multiplication facts and the matching division facts through skip counting and arrays, and recognise the products as multiples of three. Fact families link each multiplication to its two divisions, so one array gives several facts.

Pupils build the 4 facts and their division facts, seeing fours as doubled twos. Reaching a new table from one they already know makes the facts easier to secure and shows how tables connect.

Pupils build doubling as a skill in its own right, doubling small numbers using bonds and larger numbers by partitioning into tens and ones. Doubling then becomes a strategy for deriving facts, since doubling twos gives fours, and doubling again gives eights.

Pupils build the 8 facts and their division facts, using doubling of the 4 facts alongside skip counting in eights. Having more than one way to reach a fact makes recall more secure.

Pupils build the 6 facts and their division facts, deriving sixes by doubling the threes they already know. Reaching a new table from a familiar one keeps the number of genuinely new facts small.

Pupils build the 9 facts and their division facts using the tens-minus-one-group structure, so nine sevens is ten sevens with one seven taken away. This gives a reliable way to reach a nine fact that is not yet secure.

Pupils build the 7 facts and their division facts through arrays, skip counting and fact families. The structural work and links to known facts matter most here.

Pupils build the 11 and 12 facts and their division facts, taking elevens as tens plus one group and twelves as doubled sixes. With these secured, pupils have the whole set of tables to twelve times twelve.

Pupils apply place value to known multiplication and division facts, replacing ones with tens or hundreds so that three fours are twelve gives three forties are a hundred and twenty. They use the fact they know and then read the unit, keeping the same array structure with larger counters.

Pupils apply the same place-value thinking in the other direction, scaling known facts by a tenth or a hundredth, so three fours are twelve gives three times four tenths is one and two tenths. The fact stays the same; only the value of the unit changes.

Pupils add across 10 by splitting the calculation into two jumps, first making 10, then adding what is left, using the ten frame to make both jumps visible. Eight plus five becomes eight plus two to reach 10, then three more, so the calculation rests on bonds they already know.

Pupils subtract across 10 by jumping back to 10 first and then beyond it, using the ten frame to show the split. This mirrors adding through 10, so subtraction is built on the same picture rather than learnt separately.

Pupils recognise the difference structure of subtraction, answering questions of the form how many more, where two amounts are compared rather than one taken away. Seeing that the gap between two numbers is also subtraction widens what the operation means.

Pupils add and subtract only ones to and from a two-digit number where the tens stay the same, so the calculation relies on a one-digit fact within the ones. This keeps the focus on applying a known fact in a larger number.

Pupils add and subtract ones where the tens digit changes, bridging a multiple of ten. This brings the bridging-through-10 method from Phase 1 into two-digit numbers.

Pupils add and subtract whole tens to and from a two-digit number, seeing that four tens add three tens works just like four add three. Counting in tens makes the tens digit behave like the ones they already know.

Pupils add and subtract two two-digit numbers where no boundary is crossed, dealing with the tens and the ones separately. Partitioning each number into tens and ones keeps the calculation manageable.

Pupils add two two-digit numbers where the ones cross a multiple of ten, combining adding tens with bridging through 10. This draws together the two skills built earlier in the phase.

Pupils subtract two-digit numbers where the calculation crosses a multiple of ten, applying bridging back through 10 within larger numbers. Seeing it alongside addition keeps the inverse relationship in view.

Pupils calculate the complement of a two-digit number to 100 in two steps, first to the next ten, then to 100. This builds on bonds to 10 and complements within a ten, now scaled up to whole tens.

Pupils understand the inverse relationship between addition and subtraction, and how both connect to the part-whole structure, alongside the commutative property of addition. Knowing that a subtraction can be checked by an addition gives them a way to reason about a calculation, not just carry it out.

Pupils add numbers using columns, first combining each column on the place value grid with Base 10, then recording in column layout and working right to left from the ones. Exchange is met one column at a time, so when a column reaches ten, ten of one unit are exchanged for one of the next, such as ten ones for one ten.

Pupils subtract using columns, building the larger number on the place value grid and working right to left from the ones. Where a column cannot be subtracted, one of a unit is exchanged for ten of the unit to its right, such as one ten for ten ones, the same move as in addition run the other way.

Pupils extend column addition and subtraction to four-digit numbers, including calculations that need exchange across more than one column and across a zero. The method is unchanged; what grows is the size of the numbers it is applied to.

Pupils use a given calculation to derive related ones, adjusting by compensation, applying the inverse, and using place value to scale. Knowing that a small change to one number can be balanced by a change to another lets them reach a new answer from one they already have, without starting again.

Pupils solve problems with two unknowns, using bar models and working systematically through the possible values. They reason about how many solutions a problem has, recognising when there is one, when there are several, and when there are infinitely many.

Pupils recognise when groups have the same number in each, build equal groups, and find the total by skip counting in 2s, 5s and 10s. Describing a set as so many groups of so many gives them the language multiplication is built on.

Pupils organise equal groups into an array of rows and columns, and write a multiplication to match. The array turns a calculation into something they can see and skip count along, and it is the picture they return to right through multiplication and division.

Pupils use the array to see that three rows of five and five rows of three have the same total, so a multiplication can be read either way. This halves how many facts feel new, and lets them switch to a count they find easier.

Pupils make and draw equal groups from a total and count how many groups there are, linking this to a missing-factor multiplication such as something times five is fifteen before meeting the division symbol for the same thinking. Working out how many fives make fifteen connects division straight back to the facts they know.

Pupils share a total equally between a known number of groups and write the division to match, then see that fourteen divided by two can mean shared between two or how many twos are in fourteen. Grouping and sharing are two interpretations of the same division, and both give the same answer.

Pupils multiply and divide whole numbers by 10 and 100, understanding this as making a number ten or a hundred times the size rather than as adding zeros. They apply a known fact and then read off the unit, so three fours are twelve tells them three forties are a hundred and twenty.

Pupils meet multiplication as comparison, the idea of one quantity being a number of times as many as another, and represent it on a bar model. This is a different structure from equal groups, and pupils use division to find the smaller quantity or the number of times as many.

Pupils manipulate multiplication and division equations and apply the commutative property, so a missing number can be found by rearranging the equation. Seeing that a multiplication and its two related divisions describe the same relationship gives them three facts from one.

Pupils apply the distributive property, breaking a multiplication into easier parts and adding them, so seven sevens becomes five sevens and two sevens. This is the reasoning that later underpins written multiplication.

Pupils divide two-digit numbers by one-digit numbers where there is a remainder, and interpret the remainder according to the context, deciding whether to round up, round down, or keep it. The calculation is the same skill they have built; what is new is reasoning about what is left over.

Pupils find factors and multiples of whole numbers, including common factors and common multiples, and express a number as a product of two or three factors. Knowing the factors of a number underpins simplifying fractions and choosing efficient ways to calculate.

Pupils multiply a whole number of up to four digits by a one-digit number using short multiplication. The method builds directly on the distributive property they met earlier, so the written steps record reasoning they already understand.

Pupils divide a number of up to four digits by a one-digit number using short division. They connect each step to grouping within each place value, so the method is a record of dividing rather than a sequence to memorise.

Pupils use short division where the answer has a remainder, and interpret it according to the context, deciding whether to round up, round down, or express it as a fraction. This joins the written method to the reasoning about remainders built earlier.

Pupils see that a fraction comes from dividing a whole into equal parts, and that the parts must be equal for the fraction to mean anything. This is the idea that all of fractions rests on.

Pupils interpret and write proper fractions to represent one or several equal parts of a whole, using unit fractions as the basis for understanding non-unit fractions. They connect the notation, the words and the picture so a fraction is more than a symbol.

Pupils find unit fractions of quantities using known division facts, then extend to non-unit fractions. Finding a quarter of 12 becomes a question of dividing into equal groups, which links fractions to what they already know about division.

Pupils reason about the location of any fraction within 1 in the linear number system, placing fractions on a number line as numbers in their own right. This sits fractions alongside whole numbers rather than treating them as something separate.

Pupils add and subtract fractions with the same denominator within 1, seeing a non-unit fraction as repeated unit fractions, so three-eighths is three one-eighths. Adding then means counting in eighths, just as they count in ones.

Pupils reason about the location of mixed numbers in the linear number system, placing numbers like two and a third on a number line. Seeing them in order with whole numbers shows that a mixed number is just a number with a whole part and a fraction part.

Pupils convert mixed numbers to improper fractions and the other way round, seeing that the two forms are different ways of writing the same amount. Picturing seven-quarters as one whole and three-quarters makes the conversion something they reason out rather than a rule to recall.

Pupils add and subtract improper and mixed fractions with the same denominator, including calculations that cross a whole. This works the same way as bridging 10 in whole-number addition, so a familiar idea carries them through.

Pupils find equivalent fractions and understand that they have the same value and the same position in the linear number system, so a half and two-quarters are the same point on a number line. Seeing equivalence on the line keeps it grounded in size rather than a trick with numerators and denominators.

Pupils find non-unit fractions of quantities, building on finding unit fractions earlier. Three-quarters of 20 becomes a quarter found first, then three of them, which links back to what they already know.

Pupils reason about fractions of quantities in less routine situations, such as working back from a part to the whole. This moves them from carrying out a method to thinking flexibly about what a fraction of an amount means.

Pupils recall decimal equivalents for common fractions such as a half, a quarter and three-quarters, and for multiples of these. Knowing these by heart connects fractions to the decimal place value they have already met and speeds up later work.

Pupils recognise when a fraction can be simplified and use common factors to simplify it, so they see that six-eighths and three-quarters are the same amount. Simplifying first often makes a later comparison far easier.

Pupils express fractions in a common denomination and use this to compare fractions that are close in value. Rewriting both fractions over the same denominator means they are comparing parts of the same size.

Pupils compare fractions with different denominators, including fractions greater than 1, using reasoning, and choose between reasoning and common denomination as a strategy. Knowing when a quick piece of reasoning will do and when to fall back on a common denominator is the mark of a confident fraction-user.

Pupils recognise common 2D and 3D shapes presented in different orientations, so a triangle is still a triangle when it is turned or upside down. They name shapes confidently wherever they meet them, not only the usual way up.

Pupils compose 2D and 3D shapes from smaller shapes to match an example, and see how the same shapes come apart again. Fitting two triangles together to make a square shows that shapes, like numbers, are made of parts.

Pupils use precise language (sides, vertices, edges and faces) to describe 2D and 3D shapes, and compare them by reasoning about what is the same and what is different. This moves them from rough comparison by appearance to comparing shapes by their actual properties.

Pupils recognise a right angle as a property of a shape and as a description of a turn, so it is both a square corner and a quarter turn. They find right angles in 2D shapes presented in different orientations, not only when a shape sits straight.

Pupils draw polygons accurately by joining marked points, building shapes themselves rather than only naming ready-made ones. Making a shape this way fixes their attention on its sides and corners.

Pupils identify parallel sides, the lines that never meet, and perpendicular sides, the lines that meet at a right angle. They use this to describe and compare polygons more precisely.

Pupils read and plot points in the first quadrant, and draw polygons from a set of given coordinates. This gives them a precise way to say exactly where a shape sits.

Pupils translate shapes within the first quadrant, describing how far a shape moves across and up or down. The shape keeps its size and form; only its position changes.

Pupils identify regular polygons, such as equilateral triangles and squares, as those whose sides and angles are all equal, and distinguish them from irregular ones. They reason from the properties rather than from how a shape looks.

Pupils find the perimeter of regular and irregular polygons by working out the total distance around the edge. For regular shapes they can use the equal side lengths to work more efficiently.

Pupils identify lines of symmetry in 2D shapes presented in different orientations, and reflect a shape accurately in a line of symmetry. They also complete a symmetric figure or pattern from one half.

Pupils compare angles by size and estimate them against familiar benchmarks such as a right angle or a half turn. This builds a secure sense of how big an angle is before any measuring tool comes in.

Pupils measure angles accurately in degrees with a protractor, and draw angles of a given size. Estimating first and then measuring to check keeps their answers sensible and guards against the usual protractor errors.

Pupils understand area as the amount of surface a shape covers, and compare the areas of different shapes using standard units. Counting squares gives them a concrete picture of what area means before any formula.

Pupils calculate the area of rectangles, including squares, by multiplying the side lengths, and connect this to the rows-and-columns structure of an array. They see why multiplying works rather than only applying a rule.