Understanding Numbers and Place Value in Mathematics Education
Explore cardinal, ordinal, and nominal numbers with a focus on place value understanding. Learn to read, represent, and interpret numbers up to one million using various representations. Develop skills in rounding, estimation, and problem-solving involving measurements and bounds.
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Statistics Geometry Number Algebra Patterns and sequences Collecting data Measurement - Time The number system - Place value Representing data by grouping and classifying The number system - Reading and representing numbers Forming Measurement - Units The number system - Comparing, estimating and rounding Manipulating Representing data in graphs and charts Measurements - Estimating The number system - Estimating and checking Solving Interpreting and exploring data Shape and Space - 2-D and 3-D shapes Shape and Space - Symmetry and transformations Modelling Interpreting and evaluating data The number system - Counting The number system - Ordering and sequencing Graphical Methods Probability Shape and Space - Area and perimeter Relationships - Fractions, decimals and percentages Shape and Space - Scale and ratio Shape and Space - Triangles Relationships Calculations Relationships Multiplicative reasoning Shape and Space - Volume Mathematics and Numeracy strands developed from Descriptions of Learning Relationships - Times tables, multiples and factors Position Angles Financial Literacy
WM1: Number The Number System - Place value Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I have experienced and explored numbers, including cardinal, ordinal and nominal numbers, in number-rich indoor and outdoor environments. I can understand that the value of a number can be determined by the position of the digits. I can use a range of representations to develop and secure my understanding that the value of a digit is related to its position. I can read, record and interpret numbers, using figures and words up to at least one million. I can use standard index form to represent large and small numbers, performing calculations in context. I can use appropriate rounding methods, including significant figures, to estimate values. I can use my knowledge that measurements are not always accurate, and are subject to tolerance and margins of error, to solve problems involving upper and lower bounds. Knowledge awareness of number in the environment moving, touching items to show awareness of 'counting' - not necessarily accurately saying numbers aloud count forwards to 10 and back (then 10-20) name numbers primary, ordinal and nominal numbers concepts of:- number conservation, cardinality, subitizing Knowledge use the terms tens and ones/units compare and order numbers within 10 (then 20) use comparison language to compare numbers within 10 (then 20), for example; more than, less than, equal to names of different digits; hundreds, tens, units (or ones) know one ten = ten ones ordinal numbers to compare values order and compare numbers to 50 represent 50 using a variety of concrete materials group two-digit numbers to 50 in tens and ones use the term partitioning understand that groups of 10 also make 100's understand zero as a place holder Knowledge difference between digits / numerals / numbers counting, then counting in different steps exchange when bridging 10, then 100 base 10 and modelling with base 10 equipment names of the values in base 10 partitioning and use of place value cards read numbers to at least 1 million write numbers to at least 1 million in words and digits exchange units for tens and units, tens for hundreds, hundreds for thousands, thousands for tens of thousands ... to 1 million (with base 10 equipment, then place value counters, then pictorial representations understand the role of '0' as a place holder understand how to 'read' a number and which digits have the most significant value when ordering Knowledge powers of ten rules of indices addition, subtraction, multiplication and division of whole numbers and decimals understand what significant figures are and their relationship to estimation / accuracy understand that rounding to significant figures is a form of estimation understand Standard Form and why/where it is used identify the limitation of decimal form identify numbers written in standard Form (a x 10n - where 0 < a < 10) understand that negative indices indicate values between 1 and 0 Knowledge understand rounding / estimation use of measures and compound measure (see WM3: Geometry) conversion between standard metric and imperial units use of formulae to calculate perimeter, area and volume use of Pythagoras / Trigonometry to calculate length and/or size of angle understand the meaning of tolerance within mathematics understand margins of error understand the meaning of upper and lower bounds understand direct and inverse proportionality and its effect on upper/lower bounds when working with compound measures Skills recognise that 2 comes after 1 recognise 0-3; 0-5; 0-10 explore and recognise numbers in the environment subitising make accurate amounts represent a number - count 5 blocks and show the number 5 relate ordinal numbers to stories and organise e.g. simple characters or events count in board / yard / dice / card games make marks attributing meaning to them - then numbers Correctly write numbers to 10, then 20 Skills show how numbers 11 to 19 can be formed by combining a ten and ones, and can be partitioned into a ten and ones explore different ways of making numbers to 20, then 50, then 100, then to 1000 (concrete and pictorial) compare numbers to 20, then 50, then 100, then to 1000 using the language more than , less than and equal to compare two or more sets of objects using more than , less than and equal to, then use > , < and = accurately order numbers to 20/50/100/1000 (use concrete and pictorial representations and moving to abstract) in ascending and descending order find 1 more and 1 less to numbers to 20/50/100/1000. find 10 more/less, then 100 more and less compare and order numbers beyond 10 to 20, then 50, then 100, then 1000 represent numbers in different ways; first 20, then 50, then 100, then 1000 estimate and write numbers to 20, then 100, 1000 on a number line Skills partition and identify the value of any digit in a given number up to 1000, then beyond correctly record a number when a pictorial or concrete representation is given write 12,567 = 10,000 + 2,000 + 500 + 60 + 7 explain which value will change if 12,567 becomes 12,067 solve problems in this format to make a given total build a given number with manipulatives choose the correct number when given different types of pictorial representations arrange given numbers in ascending and descending order - including those which include the same digits in a different order e.g. 12,356 and 12,563 make adjustments to the correct value when adding or subtracting / increasing or decreasing a given value or counting in different steps Skills round numbers to one, two and three significant figures use significant figures to estimate calculations convert numbers from decimal to standard form and vice versa (large numbers and those less than 1) round numbers (up to three significant figures) when choosing to represent or calculate in standard form addition and subtraction of numbers in standard form (with and without calculator) multiplication and division of numbers in standard form (with and without calculator) investigate more calculations in standard form eg. (5.3 x 105)2 Skills calculate upper and lower bounds of a given unit of measurement apply upper and lower bounds to a correctly chosen formula ie. area, perimeter, area, speed, density etc use upper and lower bounds within addition and subtraction problems eg. whether a desk will always fit in a given space use upper and lower bounds within multiplication and division problems use algebra to represent a generalisation of upper and lower bounds Vocabulary 1st, 2nd, 3rd, 3rd ... nominal, primary, ordinal, arithmetic, subitising, cardinality, conservation Vocabulary number, digit, position, place value, column, value, place, zero, hundreds, tens, ones/units, number line, compare, contrast, more than, less than, greater than, fewer than, between Vocabulary unit, ten, hundred, thousand, ten thousand, hundred thousand, million numeral , base, place value, equal to, increasing, decreasing, value, column Vocabulary standard form, Decimal form, Significant figures Vocabulary upper bound, lower bound, tolerance, margin of error, variance
WM1: Number The Number System - Reading and representing numbers Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I can notice, recognise and write numbers in a range of media, through a multisensory approach, from 0 to 10 and beyond. I can read, write and interpret larger numbers, up to at least 1000, using digits and words. I can use a range of representations to extend my understanding of the number system to include negative values, decimals and fractions. I can accurately place integers, decimals and fractional quantities on a number line. I can apply my understanding of number value to round and approximate appropriately. Knowledge awareness of number in the environment recognise that 2 comes after 1 identify and recognise 0-3; 0-5; 0-10 recognise numbers 0-10 initially, then to 20 if readiness shown recognise numbers of significance to them in the environment understand 0 = no objects = zero numbers can be represented in a number of ways number songs number story books Knowledge experience of reading and writing numerals to 10, (then 20) know that a two digit number is made up of T and Ones/Units know that a three digit number is made up of H,T and Ones/Units Knowledge use of number lines for rounding and estimating of whole numbers counting forwards and backwards in different steps awareness of negative numbers in real life understand that numbers continue below 0 understand 0 what is an integer what is a fraction including improper fraction and mixed numbers what is a decimal what is estimation and when/why it is used - understand the estimation is used as a form of approximation IT IS ESSENTIAL TO REFER TO AND TEACH IN CONJUNCTION WITH THE PLACE VALUE STRAND WHEN DEVELOPING THESE SKILLS Skills Skills Skills make marks attributing meaning to them in a variety of media practise over writing numbers in a variety of media try to write numbers up to 10 in a variety of media write numbers from 0-10 initially (then to 20) write number words to 10 match numbers with number / number frames / objects / pictures develop arithmetic by doing 'number of the day and representing numbers in different ways everyday awareness of numbers, role play, stories, games etc application of incidental mathematics (e.g. register, lunch numbers, choosing book of the week) begin to read and write numbers in words read and write numbers to 20 in words and numerals explore the meaning of teen' read and write two-digit numbers to 50, then 100 in words and numerals read and write three-digit numbers to 1000 in words and numerals (consider phonics / spelling when words are used) count numbers to 20, then 50, then 100, then 1000 accurately read a given number - understanding the conventions of place value (see place value strand) accurately write a given number in digits and words - understanding the conventions of place value (see place value strand) show a concrete / pictorial representation of a given number count in different steps count to and from different points count forwards and backwards count in steps that are NOT multiples use a number line to count accurately below and above zero place fractions and decimals on number line with segments and approximate when no segments are provided (include improper fractions, mixed numbers, negative decimals and fractions) use position on number line to investigate comparisons of fractions / decimals use position on number line to investigate comparisons of negative numbers round numbers to the nearest one, two and three decimal places as well as nearest integer use estimation as a form of verifying accuracy of answers Vocabulary number, arithmetic, number names Vocabulary digit, numeral, number, place value, place, position, thousands, hundreds, tens, ones/units, compare, contrast, larger, smaller, teen, between Vocabulary rounding, estimation, approximation, verify, integer, decimal, fraction, improper, mixed number, negative, minus, numerator, denominator, decimal point, fractional amounts e.g. half, quarter, third, fifth, sixth etc, decimal place values i.e. tenth, hundredth,
WM1: Number The Number System - Comparing, estimating and rounding Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I can use mathematical language to describe quantities, and to make estimates and comparisons such as more than , less than and equal to . I have engaged in practical tasks to estimate and round numbers to the nearest 10 and 100. Knowledge subitising experience of counting groups of objects understand more / less / more than / less than understand equal / not equal, more, less understand one more than, one less than understand 'most' 'least develop mental arithmetic e.g. being able to mentally recall one more than or one less than name numbers Knowledge read and write numerals to 10 (then 20) use the terms tens and ones/units. compare and order numbers within 10 then (20) use comparison language to compare numbers within 10 e.g.more than, less than, equal to being able to estimate numbers using visual representations e.g. base 10 blocks accurately subitise place value of numbers to 1000 order and compare numbers to 100 count in different steps forwards and backwards e.g. 1s, 10s, 100s Work with multiples such as 2,5, 10 halve two and three-digit numbers understand and use the term 'estimate' correctly estimate groups of objects- link to previous experiences of subitising use of bead strings, then number lines how to draw own number line with a wide variety of scales work with a range of empty and scaffolded number lines rounding rules Skills find or create a specific amount use counting to balance (equal), more, less - including with scales group objects equally, not equal, more / less - use manipulatives count several objects in groups then identify groups which have more or less count objects within groups; mark in order to group pictures or group objects into specific sized groups subitise to say how many are in a group / set - perceptual up to 4/5 and conceptual up to 5/7/10/20 compare sets using various manipulatives compare mixed objects e.g. 5 golf balls and 4 tennis balls - which one has the most? compare - match the specific number needed e.g. 4 teddy bears at a tea party, 4 plates compare - set up a tea party table and use the correct number needed organise groups of objects (1 block, 2 blocks, 3 blocks ....) Or backwards (5 blocks, 4 blocks, 3 blocks ...) or is there one missing? Which one? play board games and throw 2 dice and choose the best one to make a move within the game. solve a simple problem - encourage children to correct and explain e.g. which is the longest - blocks from the table or blocks from the floor? estimate groups of objects or pictures e.g group which is largest? must first be able to subitise before estimating games including an estimation jar Skills to 100: round to the nearest 10 to 100 in practical tasks (bead strings, base 10, place value counters) estimate and write numbers to a 100 on a numberline. compare numbers using < and > symbols (algebra link) identify and complete missing numbers on number lines with different scales and midpoints to 1000: round to the nearest 10 and 100 to 1000 in practical tasks estimate and write numbers to a 1000 on a numberline. compare numbers using < and > symbols (algebra link) identify and complete missing numbers on number lines with different scales and midpoints estimate a given number on a blank number when the start and end are labelled estimate a number on a number line using demarcations between 2 points count in steps of different values- forwards and backwards find the midpoint compare and order two and three-digit numbers continue to use apparatus to represent values of a different numbers to support rounding Vocabulary how many, how many more, how many less than, equal to, equal, the same, more, less, about, largest, smallest Vocabulary round, estimate, compare, value, order, scales, midpoint
WM1: Number The Number System - Estimating and checking Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I am beginning to estimate and check the accuracy of my answers, using inverse operations when appropriate. Knowledge accurate touch counting know number names read and write numbers to 10 (then 20) double and halve using a range of practical and written methods addition and subtraction using written and mental methods multiplication and division using grouping, sharing, array methods and mental/written methods know and understand the term inverse know the relationships between; halving and doubling, subtraction and addition, multiplication and division vocabulary linked to estimation and approximation Skills use estimation to check answers use approximation to check answers estimate and round numbers in calculations (see above) estimate and approximate to check answers in different contexts and ways including concrete and pictorial representations check and prove halving using doubling. check and prove subtraction using addition check and prove addition using subtraction check and prove division/sharing/grouping practically, using multiplication and times tables knowledge check and prove multiplication using division/sharing/grouping practically Vocabulary guess, estimate, round, approximate, add, subtract, total, inverse, multiply, divide/group/share
WM1: Number The Number System - Counting Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I can use my experience of the counting sequence of numbers and of one-to-one correspondence to count sets reliably. I can count objects that I can touch, and ones that I cannot. What knowledge is needed? awareness of 'counting' - not necessarily accurately informal 'counting' of objects during play say numbers aloud (not necessarily accurately) develop 'number talk' in informal play situations read counting books, sing counting songs/rhymes etc informal counting opportunities linked to snack time, register and daily routines etc know that the last number recited is the total of the group know number names know to put objects in a line to avoid miscounting / counting the same object multiple times know that anything can be counted e.g. hops, skips, claps etc not just items you can see Skills accurately count everyday manipulatives and objects (up to 5 to start then 10) act counting songs and rhymes with objects pupils join in - counting with an adult - in stories and books - saying the numbers together sing counting songs and rhymes role-play with objects for stories and rhymes accurately count up to five objects, using one number name for each object recognise and represent numbers using pictorial representations and on a number track / line, tens frame / five frame place objects on a number track to ensure accuracy match the appropriate number of objects with the corresponding number match dot cards, manipulatives and numerical cards x-ray vision games e.g. count 5 - hide some, how many are hiding? hidden box games - count 3 items for children to see then hide their eyes and count 2 others (listen to them fall into the tin), how many? board games e.g. snakes and ladders; card games pupils follow verbal instructions - 'bring me 5 ...', 'show me 3 ...' etc. make a specific number in games - target number, and create the correct number on each biscuit for example count claps, hops etc (pupils could combine hearing a clap with moving counter to keep track of sounds / movements) pupils correct an adult who has counted objects 'wrongly Vocabulary count, number, manipulatives, number frames, dot cards, dice, number names
WM1: Number The Number System - Ordering and sequencing Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I have experienced the counting sequence of numbers in different ways, reciting forwards and backwards, and starting at different points. I can order and sequence numbers, including odd and even numbers, and I can count on and back in step sizes of any whole number and simple unit fractions. Knowledge informal 'counting' of objects during play chanting / reciting numbers aloud choral counting names of numbers recite numbers forwards and backwards in various steps how to count past a boundary e.g. 9,10,11 Knowledge count forwards and backwards in different steps to 10 (then 20) count forwards and backwards in steps of smaller whole numbers originally; for example steps of 2, 5 and 10 count forwards and backwards in different sized steps (beyond 20) place value to 1000 odd and even numbers (originally to 100, then 1000) count to and across 100 forwards and backwards to and from any given number count forwards and backwards in steps of larger numbers from a given number, such as; 20, 50, 100 place value of numbers to 1000 and order and compare numbers to 1000 make connections between counting in steps of different values and multiples of numbers Skills join an adult to count out loud during stories and rhymes - telling the numbers to others sing songs with numbers and join familiar numbers recite / sing 1-3 e.g. clap clap 1,2,3 sing songs and rhymes in number order e.g. one and two and three bananas ; fingers dancing; 5 crocodiles swimming in the river . recite numbers to 10 children skip a particular number, practise counting on or back to that number e.g. omit 7, practise counting 1 to 7 count from 10 to 0 count up to 20 count in 2's to 10 (20 if readiness shown) count from a small specific number to another fixed number e.g. 3 -10; 9-15; 8-18 count forward and backwards starting at different numbers Skills continue to use concrete and pictorial representations to order and sequence numbers order and sequence odd and even numbers practically to 20, then 50, then 100, then 1000. count forwards in steps of whole numbers from a given number count backwards in steps of whole numbers from a given number count in steps of uniform size solve and complete number patterns and sequences of different sizes (see WM2: Algebra) count forwards and backwards in step sizes of simple unit fraction (see WM2: Algebra) explore unit fractions read, write, order and compare unit fractions count forwards and backwards in units of fractions, such as; 1/2, 1/4 count forwards and backwards in units of fractions, such as; 1/3, 1/5 Vocabulary forwards, backwards, count, number, count in 2's Vocabulary sequences, odd, even, steps of, forwards and backwards (in terms of counting), generate, unit fractions
WM1: Number The Number System - Fractions, decimals and percentages A Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I have experienced fractions in practical situations, using a variety of representations. I can demonstrate my understanding that a fraction can be used as an operator or to represent division. I can understand the inverse relation between the denominator of a fraction and its value. I have derived and can apply the rules of indices, using integer exponents I have explored the relationship between powers, roots and fractional indices and can use it to solve problems. I am beginning to understand that unit fractions represent equal parts of a whole and are a way of describing quantities and relationships. Knowledge equipartition a whole shape, such as a circle or rectangle a whole can be divided into equal parts (any shape / item) experience of arrays, practical grouping and sharing explain how fraction notation can be used to describe an equal part of a whole one equal part of a whole is called a unit fraction and each unit fraction has a name one half is one of two equal parts and one quarter is one of four equal parts fractions can be represented in different ways; fraction circles, bar models, Cuisenaire rods, paper folding, geoboards, fraction blocks, number lines fractions of amounts can be found in different ways; counters, objects, number frames, shapes link fractions with division Knowledge multiplication and division skills (see Multiplication and Division strand PS2) experience of arrays, practical grouping and sharing one equal part of a whole is called a unit fraction and each unit fraction has a name one half is one of two equal parts and one quarter is one of four equal parts understand what is meant by 'equivalent fractions different fractions may be equivalent; understand that fractions represent division understand that the value decreases as the denominator increases and vice versa understand the proportional relationship between a particular fraction and the whole e.g. 1/3 of a number is 20 what is the whole number Knowledge MUST BE TAUGHT IN CONJUNCTION WITH ALGEBRA understand the rules of indices (multiplication, division, power of power, power of 0, negative - integer exponents only) understand what reciprocal is know when to apply rules of indices eg. 25 x 26 = 211 however rules can not be applied to 35 x 26 Knowledge MUST BE TAUGHT IN CONJUNCTION WITH ALGEBRA understand the rules of indices including fractional exponents understand the equivalence of fractional powers and roots eg.x = x, x3/2 = ( 2)3 Skills can apply the rules of indices to various calculations using numbers or algebraic expressions see WM1: Number, Calculations Skills can apply the rules of indices to various calculations using numbers or algebraic expressions Skills Skills count forwards and backwards in quarters and halves count and place and quarters on a number line colour shapes showing and 2/4 and compare visual representations draw and shade shapes in relation to a given fraction label simple fractions with written fraction notation make simple unit fractions using a range of representations identify examples and non examples of fractions compare simple common fractions using models and representations link to WM2: Algebra -use vocabulary to compare quantities and values such as greater than, less than, equal to. Then use <> = explore representations of simple fractions explore fractions of shapes, objects, quantities, money and measure find fractional quantities linked to known multiplication facts with use of concrete and pictorial representations investigate fractions of whole numbers by dividing into equal groups name and record fractions accurately investigate finding a fraction of a whole number using manipulatives - begin with unit fractions develop understanding using a range of manipulatives and representations link representations with multiplication and division facts investigate different types of questions finding fractions of whole number and the original whole number when given the fraction find a fraction of a given quanitity e.g.such as of 8 = 4 how can we work out of 8? Or 1/2 of ?= apply above skill in different contexts develop understanding of non-unit fractions calculate the total value where questions include visual representations e.g. 1/3 of a number is 20 what is the whole number / 20% of a number is 10 what is the whole number? Vocabulary names of fractions; halves and quarters, unit fraction, representations, represent, denominator, numerator, equal parts, equivalence, equivalent, fraction names; half, quarter, third, quantity, share, division, divide, multiply, whole Vocabulary names of fractions; halves and quarters, equivalent, equivalences, equal parts, whole, numerator, denominator, unit fraction, divide Vocabulary index, power, simplify, express, prove, reciprocal Vocabulary indices, roots, fractional exponents
WM1: Number The Number System - Fractions, decimals and percentages B Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I have explored equivalent fractions and understand equivalent fraction relationships I can use my knowledge of equivalence to compare the size of simple fractions, decimals and percentages and I can convert between representations. I can use my knowledge of the equivalence of fractions, decimals and percentages to understand that numbers or proportions may be represented in different ways. I can recognise the difference between rational and irrational numbers, and I have derived rules and applied them to simplify and decompose surds. I can extend my knowledge of the equivalence of fractions, decimals and percentages to understand that recurring decimals may be represented in different ways. I can demonstrate my understanding that non-integer quantities can be represented using fractions (including fractions greater than 1), decimals and percentages. Knowledge counting in multiples strategies to double and halve amounts understand the relationship between unit fractions e.g. 1/10 is half of 1/5 understand that equivalent fractions have the same value but different representations understand that simplifying a fraction means writing a fraction with the smallest possible denominator Knowledge understand that you can compare the values of fractions with different denominators e.g. number line, diagram, calculations understand that a percentage is a fraction with the denominator 100 understand that decimals, fractions and percentages can represent equal quantities know and recall the equivalence of fractions, decimals and basic percentages eg. 10%, 20%, 25%, 50%, 75%, 100% understand that percentages can be more than 100% understand the equivalence of simple fractions, decimals and percentages, e.g. find 25% of 60cm and know that this is equivalent to of 60cm. understand what is meant by integer and non-integer understand that non-integer quantities can be represented as decimals, fractions (including >1) and percentages develop conceptual understanding of what happens when fractions are added or subtracted Knowledge understand the equivalence of fractions, decimals and percentages can help better understand and compare numerical information presented in different formats express percentage change as a decimal or fractional multiplier represent fractions, decimals and percentages in a range of different visual representations e.g. bar models, number lines, percentage / pie charts understand that one representation may be more accurate in a calculation e.g. 0.1111... x 81 = 1/9 x 81 order integers, fractions, decimals and percentages by considering equivalent representations understand and use place value in decimals understand that some fractions correspond exactly to a certain decimal/percentage understand that fractions can represent both terminating and recurring decimals understand the proportional relationship between percentages and how these can be used to perform more complex calculations eg. 10% is 30, what is 40% understand that ratio is another representation of a fraction where the parts represent the whole eg. 3:4 represents 3/7 and 4/7 respectively Knowledge recognise the difference between rational and irrational numbers e.g.know that irrational numbers cannot be expressed as a fraction p/q whereas rational numbers can know the notation to represent recurring decimals understand that recurring decimals can be expressed as fractions know examples of common irrational numbers understand that a surd is a representation of number left in its root form to avoid expressing it as a truncated/rounded decimal use surds in exact calculations without a calculator use multiples of in exact calculations without a calculator Understand how to simply surds by rationalising the denominator Decompose surds by reducing to roots of prime numbers Skills recognise and label simple fractions with written fraction notation read and write fractions using fraction notation find unit fractions of shapes, objects and quantities find equivalent fractions; and 2/4 with use of concrete and pictorial representations divide a shape into equal pieces to represent a given fraction e.g. and 2/4 create a fraction representation with equal parts and the correct number of repetitions of a unit fraction compare fractions and identify equivalences with use of concrete and pictorial representations e.g. fraction wall investigate number patterns with equivalences write fractions in their simplest form Skills (i) test equivalence between / comparing fractions (including mixed numbers and decimal fractions) with different denominators (use symbols < / = /> / ) convert between fractions, decimals and percentages using multiples / common factors e.g. 16/40 = 8/20 = 40/100 = 40% = 0.4 calculate the percentage of a given value with and without a calculator (10%, 20%, 25%, 75%, 100%) use equivalence between fractions, decimals and percentages to choose the most efficient calculation eg. use 1/4 of 60 to calculate 25% use a percentage to represent the numerical value of a statement e.g. John scores 12 out of 20 in his test what percentage is this? represent a fraction out of 100 as a percentage represent a value greater than one as a mixed number and an improper fraction using manipulatives to model then using number facts place fractions, decimals and percentages on a number line (including mixed f/d/p) identify errors in number line placements identify equivalence or non-equivalence add and subtract fractions using appropriate manipulatives or representations Skills express a recurring decimal as a fraction identify and justify which fractions will be recurring and which will be terminal by looking at the denominator solve problems which include different number types e.g. Is 2 + 3 irrational? provide examples of rational / irrational numbers for given problems e.g. give an example of an irrational number whose square is rational decompose surds to their simplest form eg. 50 = 5 2 by recognising appropriate square numbers Skills convert fractions to decimals (inc recurring) using non- calculator methods use simple percentages and fractions to calculate other proportions represent a ratio as a fraction and vice versa write fractions / ratios in their simplest form calculate the percentage increase / decrease of a given value use multipliers to calculate percentage values and percentage increase / decrease Vocabulary unit fraction, denominator, numerator, equal parts, equivalent, fraction names;, quantity, share, divide, multiply, simplify Vocabulary percentage, mixed number, improper fraction, integer, non-integer Vocabulary ratio, multiplier, percentage increase / decrease, original value, recurring decimal, terminating decimal Vocabulary surd, irrational, rational, decompose, recurring, terminal, decompose, rationalising
WM1: Number Relationships Calculations Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I have explored forming a quantity in different ways, using combinations of objects or quantities. I have explored additive relationships, using a range of representations. I can add and subtract whole numbers, using a variety of written and mental methods. I can verify calculations and statements about number by inverse reasoning and approximation methods. I can fluently and accurately apply the four arithmetic operations in the correct order with integers, decimals and fractions, consolidating my understanding of reciprocals when dividing fractions. I can use the four arithmetic operations confidently, efficiently and accurately with integers and decimals, and I can combine these using distributive, associative and commutative laws where appropriate. I can communicate how sets change when objects are added to and taken away from them. Knowledge 1 to 1 counting Order numbers understand the value of numbers understand all gone understand more than match quantities to numerals. understand one more, one less. know the final number is the total. altogether. Knowledge accurate counting read and write numbers to 10 sequence and order numbers place value inverse operations concept of equal and not equal counting on / back Knowledge (i) understand the relationship between addition and subtraction / multiplication and division understand how to estimate efficiently and accurately (also see WM1: NUMBER, Comparing, estimating and rounding) determine whether an answer is reasonable using understanding of place value and context Knowledge know the conventional order for performing calculations involving brackets, four rules and powers, roots and reciprocals know that addition and subtraction, multiplication and division, and powers and roots, are inverse operations and use this to simplify and check calculations use non-calculator methods with the four arithmetic operations with positive and negative whole numbers use the four arithmetic operations with simple fractions (proper and improper), including mixed numbers and negative fractions convert between fractions and decimals as appropriate for calculations e.g. 0.8 x 45 is 4/5 x 45 calculate with integer powers and roots estimate or check, without a calculator, the result of a calculation by using suitable approximations know why dividing by a fraction is equivalent to multiplying by its reciprocal calculate positive integer powers and exact roots Skills (i) Skills check answers using inverse operations check answers using alternative methods approximate answers using when reasoning and problem solving choose plausible answers from a given set and explain reasons please note that strategies and methods used should be in line with the school calculation policy appropriate use of manipulatives to model mathematical operations (e.g. build it, draw it, say it, solve it) add single digit numbers with manipulatives and representations subtract single digit numbers with manipulatives then representations add single and two digit numbers with manipulatives then representations subtract single digit and two digit numbers with manipulatives and representations add and subtract zero link addition and subtraction and check using inverse operations add multiple single digit numbers investigate number bonds to 10, then 20 and beyond investigate bridging ten solve practical problems introduce and use appropriate symbols to represent and solve number problems use manipulatives then representations addition and subtraction within 20 using a range of methods addition and subtraction using a number line develop mental strategies Skills recite numbers backwards and forwards start counting at different points. use one to one correspondence to count sets reliably to 10 play games to identify totals investigate cardinality investigate conservation with tens frames and counters make sets of different objects linked with role play and stories, add or take away items describe what has happened when a total changes compare quantities Identify groups with more than / less than find one more / one less develop language skills ensure use of a range of representations and contexts investigate what if..? questions Knowledge (ii) secure understanding of place value understand how to model a question with manipulatives / representations understand what happens when bridging 10,100 etc understand exchanging and regrouping consistent use of accurate mathematical language understand distributive, associative and commutative laws understand the order of operations Skills use a calculator accurately and efficiently know a range of non-calculator methods for the four arithmetical operations and select as appropriate use a number line and number facts as appropriate for jottings be able to change between different representations of numbers e.g. decimals to fractions and vice versa Skills (ii) please note that strategies and methods used should be in line with the school calculation policy fluency in application of four operations build on strategies introduced in PS2 choose an appropriate method based on the context and values presented explain reason for choice of method explain why a particular method is more appropriate than another e.g. what method is best for 1001-999? Why? discuss misconceptions missing digit questions Vocabulary how many, more than, less than, altogether Vocabulary add, subtract, altogether, minus, take away, increase, decrease, digit, number, zero, total Vocabulary strategy, inverse, distributive, associative, commutative, bridge, exchange, regroup Vocabulary order of operations, inverse, non-calculator, proper, improper, integer, decimal, powers, roots, reciprocal, jottings, representations
WM1: Number Relationships Multiplicative reasoning Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I have experienced grouping and sharing with objects and quantities, and I can group or share small quantities into equal-sized groups. I have explored and can use my understanding of multiplicative relationships to multiply and divide whole numbers, using a range of representations, including sharing, grouping and arrays. I have extended my understanding of multiplicative reasoning to include the concept and application of ratio, proportion and scale. I have used proportional reasoning to compare two quantities, using direct or inverse proportion, and I can solve problems involving repeated and inverse proportional reasoning. Knowledge experience 'counting' of objects during play understand items can be grouped groups Knowledge what is meant by groups of the term multiply understand multiplication as adding the same number repeatedly (manipulatives) splitting a group of items into equal groups is division (grouping) what is meant by array understand multiplication and division as inverse operations = x symbols Knowledge knowledge of multiplication facts knowledge of place value understand the different ways to represent a remainder knowledge that fractions have an equivalent decimal understand that more than two items can be compared using ratios understand that values can be shared unequally understand that the order is relevant when expressing ratio Knowledge solve ratio and proportion problems in mathematical and non-mathematical contexts solve simple problems involving quantities in direct proportion including algebraic proportion. solve problems in direct proportion where y x (use of notation) formulate equations and solve problems involving a quantity in direct proportion to a power or root of another quantity. solve word problems involving quantities in inverse proportion including those to powers and roots. solve problems step-by-step involving multipliers over a given interval e.g. time intervals with compound interest, depreciation, growth, decay etc. Skills count items to create groups of a given size make groups of equal sizes compare groups identify similarities and difference link with stories and rhymes link with role play activities share items unequally and discuss why this isn t fair adjust given sets to ensure they are equal in quantity identify where did it go wrong? in unequal groups Skills identify relationships from an image investigate relationships with visual representations. E.g. if have 1 green sweets for every 2 red sweets.. solve problems by scaling quantities (including finding the original value) using models and representations investigate relationships with manipulatives e.g. Cuisenaire rods or representations e.g. double number line express the relationship of items A to items B e.g. when scaling ingredients in a recipe share unequally (image then numbers) use : notation to express ratio find a missing ratio using an appropriate representation e.g. bar model simplify ratios investigate fraction and ratio comparisons express a ratio as a fraction and a fraction as a ratio Skills demonstrate multiplication with manipulatives by creating groups of items multiplication problems with manipulatives and pictorial representations division problems with manipulatives and pictorial representations demonstrate the link between multiplication and division e.g. Numicon plates model repeated addition of manipulatives - discuss groups of model division with manipulatives (no remainders, then remainders) represent multiplication and division using arrays tell the story of an array create number sentences using arrays, rearrange the array to investigate patterns use an array to find all the possibilities of number sentences and record these systematically compare two arrays to discus the same/ different use appropriate multiplication and division strategies to solve problems create number sentences using correct symbols missing number questions missing symbol questions investigate and match number sentences investigate and match missing symbol / number sentences e.g. 4x2= 16?2 word problems choose an appropriate strategy Skills effective calculator skills or spreadsheet skills recognise and use symbol for direct and inverse proportion link proportion to appropriate graphs e.g. linear recognise proportion as rates of change Be able to substitute into given formulae for repeated proportional change e.g. AER manipulation of formulae for repeated proportional change Vocabulary share, same as, more than, less than, altogether, halving , halves, equal, unequal, group Vocabulary doubling, halving, lots of, multiply, divide, array, groups of, multiple of, times, dividing by, grouping, sharing, split, left over, row, column, remainder Vocabulary product, remainder, fraction, ratio, proportion, scale, numerator, denominator, relationship Vocabulary proportional, ratio, direct proportion, inverse proportion, multipliers, symbol
WM1: Number Relationships - Times tables, multiples and factors Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I can use my understanding of multiplication to recall some multiplication facts and tables starting with tables 2, 3, 4, 5 and 10 and I can use the term multiples . I can fluently recall multiplication facts up to at least 10 x 10 and use these to derive related facts. I have experienced and explored simple multiplicative relationships that allow me to discuss the properties of number, including factors, multiples, prime and square numbers. Knowledge understand how to group manipulatives into equal groups recognise patterns when counting (e.g. using a number square) understand equal and unequal groups recognise what the multiplication symbol (x) means, - 'groups of understand that 'multiple' is the product of two numbers understand multiplication as repeated addition Knowledge understand the term 'multiple begin to understand inverse operations recognise factor pairs identify that a product is the result of factor x factor use fact families and associated fact understand the links between times tables (e.g, 3x, 6x, 9x and 2x ,4x, 8x) use knowledge of place value to understand x100, x1000 know that a prime number has only two factors know a square number is a product of multiplying a number by itself knowledge of square numbers, square roots, prime numbers knowledge of indices Skills identify odd and even numbers count regularly to rehearse number patterns count aloud in 2s, 5s and 10s (forwards and backwards) make equal groups (concrete, pictorial and abstract) use arrays to represent groups use arrays to investigate multiples solve missing number problems investigate number patterns using puzzles identify and prove multiples and non-multiples predict number patterns, then prove (e.g. will be in this number sequence? Why/not?) Skills count in multiples of 10, 100 and 1000 calculate square numbers (use concrete and pictorial representations) use appropriate written methods to multiply larger numbers create and extend fact families solve varied questions types including missing numbers, sequences, predictions, what if ? solve missing digit questions to apply skills with inverse operations identify the error in where did it go wrong questions develop skills using counting stick activities identify factors and multiples identify prime numbers, square numbers, square roots Vocabulary groups of, multiples, equal, odd, even, patterns, arrays, repeated addition, times, lots of, count on, count back, sequence, predict Vocabulary lots of, groups of, multiply, multiplication, factor, product, array, row, column, double, halve, share equally, inverse
WM1: Number Financial Literacy Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I have used money, and the language of money, in play and real-life situations and I can understand that I need to exchange money for items. I can understand the equivalence and value of coins and notes to make appropriate transactions in role play. I can demonstrate an understanding of income and expenditure, and I can apply calculations to explore profit and loss. I can apply percentages and ratio to solve problems including simple and compound interest, appreciation and depreciation, calculating budgets, foreign currencies, and basic taxation on goods and services. I have developed my understanding of finance in personal, local and global contexts. i) I have further developed my understanding of finance to include annual equivalent rate (AER) and annual percentage rate (APR) so that I can evaluate and compare financial products. I can calculate income tax and understand the implications of taxation including using the Welsh rates of income tax and other taxes devolved to Wales ii) Knowledge awareness of numbers Understand items can be exchanged. sort and recognise difference distinguish between colour, shape and size experience shops understand that goods can be exchanged for an amount understand that goods have different values recognise that there are different types of coins with different names begin to understand the value of coins Knowledge secure understanding of place value to 1000 read and write numbers to 1000 working understanding of a decimal point knowledge of the pound sign ( ) and p sign for pence. understand that goods have different values when exchanging coins for items understand that totals can be made in different ways understand that when exchanging money for goods you may get change Knowledge secure understanding of decimal place value. secure understanding of addition and subtraction (including decimals) read a record monetary value accurately (understand decimal notation) understand what the term 'budgeting' means develop understanding of terms 'profit' and 'loss Knowledge percentage calculations non calc and calc methods including use of decimal multipliers basic indices definitions multiply / divide decimals conversion graphs e.g. to convert currencies, calculate paying/earning interest on investments, savings, loans understand when to multiply or divide to convert currencies. understand of simple interest and the idea of interest compounding over time, including using multipliers and powers understanding that investments can increase or decrease in value (appreciate, depreciate) explore different types of savings accounts, loans, investments including comparing interest rates to choose the best option. knowledge of a variety of foreign currencies. understand the concept of an exchange rate, including the fact that it varies and that bought currency and sold currencies may have different exchange rates understand that VAT is tax paid on many goods and services. understand how to calculate VAT using calculator and non calculator methods understand the idea of personal budgeting using given conditions Knowledge i substitution use of decimal multipliers knowledge of compound interest understand splitting the year into monthly/quarterly time periods understand that banks use AER/APR to compare annual rates of savings or loans. understand how to substitute into the given AER formula. definitions of nominal interest rate, per annum, savings accounts, loans, mortgages, investments Skills i identify when AER is used versus compound interest identify nominal interest rates and compounding periods per annum from worded questions. substitute into the given formula to calculate AER. Skills Skills plan and calculate a simple budget calculate profit and loss from a simple set of accounts record accurately using decimal notation in context of money begin to be able to read and say amounts of objects, not always accurately exchange objects for one another pay for an items given amount using the correct amount. use objects/money in a role play scenario using appropriate vocabulary use 1p, 2p, 5p and 10p coins to pay for Items make simple amounts Skills Knowledge ii awareness of different types of incomes including salaries, pensions, inheritance and taxes including income tax, land allowance tax, VAT, road tax, inheritance tax etc. knowledge of the concept of tax free personal allowance and basic, higher and additional tax bands used to calculate tax. definitions of gross income, net income, taxable income calculate percentages non calculator and calculator methods substitution into formulae read and say amounts of money confidently record money amounts in pounds and pence use combinations of coins and notes to make different amounts and to represent the same amount recognise and use and p accurately Skills calculate simple percentages and multiply by time periods to work out simple interest. calculate compound interest by repeated percentage change and by original quantity x multiplier to the power of n where n is the number of compounded percentage changes. reverse calculations to find the original quantity, the interest rate or the number of years. Skills ii substitution, order of operations. sharing gross income into the relevant bands percentage calculations Vocabulary money coin penny, pence, pound price cost buy sell spend, spent pay change dear, costs more, cheap, costs less, cheaper, costs the same as, how much ? how many ? total Vocabulary bought, sold, note. more/most expensive and less/least expensive, amount, value, worth. Vocabulary discount, currency, profit, loss. percentage Vocabulary simple interest, compound interest, multiplier, power, per annum, quantity, appreciation, depreciation. currency, exchange rates, coin and note denominations, value added tax (VAT), budget. Vocabulary annual equivalence rate, annual percentage rate, simple, compound interest, nominal interest rate, per annum, mortgage, investments, loans tax bands, income, pension, inheritance, gross income, net income, personal allowance, basic rate, higher rate, additional rate, taxable income
WM2: Algebra Patterns and sequences Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I am beginning to recognise, copy, extend and generalise patterns and sequences around me I have explored patterns of numbers and shape. I can recognise, copy and generate sequences of numbers and visual patterns. I can explore and create patterns of numbers and shapes. I can explain numerical sequences and spatial patterns in words and by generalising them. I can explore, generate, identify and represent both numerical and spatial linear sequences, including finding and using a general term. I can explore, generate, identify and represent both numerical and spatial patterns, using linear and non-linear sequences. Knowledge become aware of patterns through use of stories and discussions notice and talk about patterns and sequences in the environment- e.g. seasons, sequence of daily routine show awareness of different types of patterns outdoors, clapping, syllables in words, nursery rhymes, repetition recognise a simple colour pattern (e.g. red, blue, red, blue) - use both real life resources and mathematics manipulatives recognise of a simple shape pattern (e.g. triangle, square, triangle, square - at this point pupils may not necessarily be naming shapes accurately) explore and discuss patterns around them. recognise patterns in other forms, explore and develop language Knowledge WM2: Algebra PS1 is interwoven throughout develop in conjunction with W1: Number PS1 and PS2 - to identify and continue patterns- e.g. odds and evens, multiples, counting in steps, forward, backwards, place value recognition of shapes, properties and names a pattern is a series or sequence that repeats colours, shapes, actions, anything that repeats number patterns are sequences of numbers that are ordered based upon a rule use of language to describe patterns appropriate to the context of the pattern Knowledge ability to verbally describe patterns and changes from one term to another - what is the same, what has changed, what do you notice? develop understanding alongside number skills at an appropriate level to ensure access to identifying the pattern / sequence e.g.- multiples, addition, subtractions, differences, counting on and back, in steps understand that sequences can be ascending, descending or a combination Knowledge understand what nth term' means generate a sequence by spotting a pattern or using a term-to-term rule given algebraically. find a position-to-term rule for simple arithmetic sequences, algebraically or in words. generate a sequence from a formula for the nth term. find a formula for the nth term of an arithmetic sequence. recognise sequences of triangular, square and cube numbers, and simple arithmetic progressions. recognise Fibonacci and simple geometric progressions. Knowledge substitute a value for x into an equation verbalise changes from one pattern to the next using language of sequences look at patterns that don t have a constant difference but have a constant difference of differences - link to quadratic equations Generate complex sequences given the nth term rule e.g. geometrical Understand subscript notation for position- to-term and term-to-term rules e.g. xn= n + 2 xn+1= 2xn- 3 recognise Fibonacci and simple quadratic sequences (e.g. n2+3), and simple geometric progressions. recognise when sequences are linear or non- linear Skills move from recognising patterns to creating their own recognise and explain different types of sequences (numerical and spatial) which include one consistent step (e.g. add 2) recognise and explain sequences (numerical and spatial) which include inconsistent steps (e.g. add 3, add 4, add 5) recognise and explain you can have more than one rule within a pattern ( - e.g. with blue and orange counters, increase orange by 1 each time and increase blue by 2 each time) find the missing number in a sequence research various numerical patterns within 100 square complete patterns what is the next term? identify the rule between term and term use accurate vocabulary create their own sequence from a given rule make generalisations explain sequences and patterns nth term linear Skills Skills extend PS1 skills to recognise and copy a more complex colour pattern (e.g. red, red, green) or shapes (ABC / AAB) generate a pattern following a given criteria e.g.using multilink blocks; Numicon, shapes etc identify an error in a given pattern / sequence fill in a missing object / item showing an understanding of the pattern / sequence use a number square to colour patterns of numbers (this could include multiples and other number patterns) count e.g. in steps of 2s, 3s, 4s, 5s, 10s; multiples; odd and evens; forward and backwards whole numbers and simple unit fractions (e.g counting in steps of ) - ensure that number skills are appropriate to this task complete missing 'values' within a given sequence generate sequences starting at (and ending at different points, include counting back) use appropriate language to begin identifying and explaining the features of sequences and patterns, including rules/criteria play and sort toys free play with musical instruments modelling play notice and talk about what is the same and what is different in play activities or when sorting items identify which item doesn't belong in a set talk about an AB pattern copy an AB pattern continue an AB pattern spot an error in an AB pattern copy a simple colour pattern/shape pattern using resources (e.g. real life resources and mathematics manipulatives ) extend a simple colour/shape pattern make their own AB pattern make patterns in a variety of ways; e.g. horizontal/vertical/circular make non-patterns, explaining why they are not patterns begin to predict what will come next, through shared stories, pictures and rhymes that have a pattern element describe the pattern and begin to make generalisations Skills generate equations using pictorial representations generate nth term using pictorial representations and sequences of numbers, involving linear and quadratic predict and find next terms in a pattern involving non linear patterns- verbalise and form equation use inverse operation -looking at the nth term and working backwards learn to find the nth term in a quadratic equation by halving the difference of the differences and adjusting) Recognise square and cube numbers Skills explain the rule in a given sequence find the missing term in a sequence continue a sequence (working forwards and backwards) find the nth term rule from a sequence of numbers be able to use the nth term rule to generate a sequence of numbers (substitution is important here rather than just guessing or using inefficient methods) generate a sequence from a function machine Vocabulary first, then, next, now and then, before, after, patterns, non-pattern, ordinal numbers Vocabulary odd, even, multiples, repeating, increasing, decreasing, larger, smaller, equal, more than, less than Vocabulary sequence, position, term, ascending and descending, inverse Vocabulary nth term, sequence, generate , term-to- term, arithmetic, triangular, Fibonacci, geometric Vocabulary difference of differences, linear, quadratic, term-to-term, arithmetic, triangular, Fibonacci, geometric
WM2: Algebra Forming Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I am beginning to demonstrate, using objects, an understanding of the concepts of equal and not equal . i) I can use the equals sign to indicate that both sides of a number sentence have the same value and I can use inequality signs when comparing quantities to indicate more than and less than . I can use commutativity, distributivity and associativity to explore equality and inequality of expressions. I can demonstrate my understanding of the concept of a variable, using algebraic notation to form linear expressions, equations and inequalities. I can interpret algebraic expressions because I understand the way symbols are used to represent operations, multiples and powers. I can explore the concepts of equality and identity, connecting geometric, algebraic and graphical representations. ii) I have explored commutativity with addition and multiplication and I can recognise when two different numerical expressions describe the same situation but are written in different ways Knowledge say numbers play' at counting talk about same / different accuracy when counting understand heavy and light understand more and less understand one more than and one less than understand same and different understand balance (scales) understand concept of grouping understand counting of objects within groups understand groups with more or less concepts of counting, cardinality and conservation Knowledge (i) know the purpose of the equals signs use of addition, subtracting signs then multiplication and division signs (when appropriate) understand place value show links between representations understand equal groups (objects, pictures) understand that 5 blocks = 5 understand the use of = > and < Knowledge know that the commutative, distributive and associative laws hold for all real numbers. recognise and understand that commutativity means a + b = b + a and a x b = b x a and be able to use in algebraic and numerical calculations recognise and understand that distributivity means a x (b + c) = a x b + a x c and be able to use in algebraic and numerical calculations recognise and understand that associativity means (a + b) + c = a + (b + c) and be able to use in algebraic and numerical calculations know that the commutative law does not work for subtraction or division know that the distributive law does not work for division know that the associative law does not work for subtraction or division Knowledge understand the difference between equations, expressions and formulae. know the meaning of the word term understand the basic conventions of algebra know that a variable represents an aspect of a quantity e.g. weight, dimension, cost, etc. know the misconceptions of using letters be able to simplify linear expressions using the conventions of algebra be able to create expressions, equations and inequalities in context simplify expressions including expansion of a single bracket, including a(b + c) + d(e + f), and double brackets give solutions for inequalities < > , recognising that there are an infinite number of solutions formation and simplification of expressions and equations involving sums, differences, products and powers. formation and manipulation of linear equations and linear inequalities Knowledge understand meaning and vocabulary of equality and identity use algebraic conventions to show two expressions are equal and/or identical understand and use different representations to show algebraic equations e.g. y = mx + c represent a straight line be able to examine features of linear and quadratic functions, read an intercept from a graph, and recognise positive and negative gradients sketch graphs of quadratic functions, identifying the turning point by completing the square. recognise and sketch graphs of exponential functions in the form y = kx for positive k formation and manipulation of simple linear inequalities. identifying the equation of lines parallel or perpendicular to a given line. constructing and using equations that describe direct and inverse proportion Skills (i) use equals signs with objects and numbers to show equal value use numbers to create number sentences use > more than and less than e.g. list numbers that are > 4 or list numbers that are < 6 Skills count: one-to-one correspondence spot same and different in pictures identify more and less in pictures/with objects use scales with objects to balance and solve a simple problem regarding equal/not equal examples and non-examples match pictorial representations number sentences with the same totals read and discuss number stories pupils begin investigating the laws of commutativity with toys, manipulatives (Cuisenaire / Numicon)and everyday objects questions with manipulatives, if I can see....how many can you see? Knowledge (ii) know the purpose of the equals signs use of addition, and multiplication signs addition A+B=B+A then investigate concrete examples of this multiplication AxB=BxA then investigate concrete examples of this Skills use the three laws to highlight equality and inequality in calculations e.g. (9 5) 3 9 (5 3) use of the equals sign partitioning of numbers in multiple ways substitution of numerical values into algebraic expressions use place value to determine inequalities with numbers e.g. 9.563 <9.57 Skills (ii) find the missing number in an equation (concrete examples) model and investigate fact families and inverse operation questions explore and build representations of problems match statements of multiplication and addition match appropriate number sentences to word 'stories Skills solve equations to find unknown and then using to find dimensions of a shape link particular terms in algebraic equations to properties of a graph. recognise symbol(s) for direct and inverse proportion Skills understand and use the different symbols for operations e.g.squared , x/2 means divide by 2 understand that 2x means 2 times x or x + x use the order of operations on algebraic expressions, equations and inequalities Vocabulary equal, not equal, same, different, more, less, heavy, light, balance, group Vocabulary inequality, more than, less than, equal, expression, equation addition, multiplication, lots of, sets of, groups of, arrays, columns, rows, multiplier, multiplicand, total, product, commutativity Vocabulary term, rule, unknown, variable, formula, one- step equation, two-step equation, substitution, pairs of unknowns Vocabulary Equations, expressions, inequalities, term, variable, conventions, expansion, simplify, inequalities, manipulation Vocabulary Equality, identity, geometric, algebraic, graphical, conventions, intercept, exponential, parallel, perpendicular, direct, inverse, proportion
WM2: Algebra Manipulating Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I can manipulate algebraic expressions fluently by simplifying, expanding, substituting and factorising by extracting a common factor. I can manipulate algebraic expressions fluently by expanding double brackets, factorising quadratic expressions and simplifying algebraic fractions. Knowledge know and state the meaning of the vocabulary simplifying, expanding, substituting and factorising simplify algebraic expressions by multiplying a single term over a bracket. take out common factors e.g numerical, algebraic simplify algebraic products and quotients. simplify algebraic expressions by collecting like terms. substitute positive and negative numbers into simple expressions and formulae to find the value of the subject. changing the subject of a formula when the subject appears in one term. substitute positive numbers into simple expressions and formulae to find the value of the subject. Knowledge expand products of two binomials. expand products of more than two binomials. factorise quadratic expressions of the form x2+ bx + c factorise quadratic expressions of the form ax2+ bx + c (where a 0 or 1) simplify and manipulate algebraic fractions. complete the square on a quadratic expression simplify algebraic products and quotients using the laws of indices. changing the subject of a formula when the subject appears in one or more term. substitute positive or negative numbers into more complex formulae, including powers, roots and algebraic fractions. Skills recognise factors of numbers calculate with directed numbers inverse operations Skills recognise factors of numbers and common algebraic terms calculate with directed numbers inverse operations Vocabulary, expanding, substituting, factorising, simplifying, factors, common, quotients, collecting, change the subject Vocabulary Manipulate, expressions, expanding, brackets, factorising, simplifying, complete the square, products, quotients, binomials
WM2: Algebra Solving Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I can find missing numbers when number bonds and multiplication facts are not complete. I can demonstrate an understanding of the idea of input, application of a rule (including inverse operations) and output, using a function machine or other appropriate methods, and I have applied this idea to solve problems. I can explore and use efficient methods of solving equations and inequalities in the first degree, and I can apply this knowledge to rearrange formulae where the subject appears in one term. I can explore and use efficient methods of solving simultaneous, quadratic and trigonometric equations, and I can apply this knowledge to rearrange formulae where the subject appears in more than one term. ALTHOUGH THERE IS NO CONTENT FOR PS1, THE KNOWLEDGE, SKILLS AND VOCABULARY INDICATED BELOW SHOULD BE TAUGHT DURING THIS STAGE AND DEVELOPED AS PREREQUISITE SKILLS FOR PS2 Knowledge place value understand position of numbers on a number line understand '=' sign accurate use of symbols: '= + - x commutative and inverse laws - concrete activities to exemplify this Knowledge use function machines with four operations to generate output following a given rule use simple algebraic expressions / equations to generate output link function machine input and output to writing coordinates use function machines with inverse operations express function machine calculations in an algebraic format understand writing a rule as a function machine Knowledge what makes an equation or inequality rather than an expression set up and solve linear equations in mathematical and non-mathematical contexts, including those with the unknown on one or both sides of the equation solve linear inequalities in one variable, expressing solutions on a number line using appropriate notation. changing the subject of a formula when the subject appears in one term to solve for a given variable formulate simple formulae in one term from real world contexts Interpret, where appropriate, simple expressions as functions with inputs and outputs. Knowledge understand and apply the quadratic formula. rearrange and solve quadratic equations by factorising, completing the square or using the quadratic formula. set up and solve two linear simultaneous equations in two variables algebraically. set up and solve two simultaneous equations in two variables algebraically (including where one is a quadratic) solution of a range of cubic equations by trial and improvement methods, justifying the accuracy of the solution. know that the coordinates of the points of intersection of a curve and a straight line are the solutions to the simultaneous equations for the line and curve. Solve quadratic equations with coefficient of x2equal to 1 by factorising. solve quadratic inequalities in one variable. use sine and cosine rules to solve and find missing values in mathematical and non- mathematical contexts. changing the subject of a formula when the subject appears in one or more term to solve for a given variable know algebraic conventions for trigonometric equations Knowledge comparison - more/less/ the same (equal) composition of number - the idea of 'howmuchness' (integer value) number bonds / number stories to 5 and then 10, include use of concrete manipulatives and representations) know that all numbers, quantities can be composed of smaller numbers or parts understand the part/whole relationship be able to see smaller numbers within a number (conceptual subitising, seeing groups and combining to make a total) Skills concrete activities to exemplify commutative and inverse laws recall of number bonds recall of multiplication facts halve and double of numbers using appropriate strategies partition counting in steps / skip counting - link to multiplication/division recognition and identification of pattern in sequence. fluency in four operation tasks complete missing number tasks match number sentences which yield the same answer identify errors in completed number sentences formulate and solve problems involving e.g. halving and doubling tell the story of the problem, selecting which is the most appropriate mathematics to use select appropriate manipulatives to model, then solve the problem explain the relationships using concrete, visual and abstract representations the justify the solution and the thinking Skills calculate with four operations use of calculator know inverse operations order of operations Skills Skills effective use of a scientific calculator using a number line to show solutions of an inequality being able to apply inverse operations rearrange equations / formula by considering the equals sign as a "balance" partition numbers into two or more numbers and putting them back together to develop understanding of aggregation as a structure of addition and partitioning as a structure of subtraction as the inverse operation explain part/whole relationship to solve hidden numbers or empty box questions make a reasonable guess at a hidden number in a known number of things, e.g. Five people go into a tent, then two come out. How many are left in the tent? concrete objects You can see...how many can I see? work towards creating, then reading a number sentence - orally at first to exemplify a concrete representation Skills use a range of methods to solve equations in mathematical and non-mathematical contexts effective use of a scientific calculator Vocabulary how many, more many more, match, the same as, partition, altogether Vocabulary equals, same as, increasing, decreasing, more than, less than, inverse, commutative, sequence Vocabulary input, output, rule, function, relationship, inverse Vocabulary inequality, solving, equation, expression, collecting like terms, rearrange, formulae, subject, hypotenuse, adjacent, opposite, linear Vocabulary simultaneous, factorise, quadratic, trigonometric, sin, cos, tan, symbols such as , completing the square, intersection, trial and improvement,
WM2: Algebra Modelling Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I can model problems, using expressions and equations involving symbols or words to represent unknown values, adopting the conventions of algebra. I can use inverse operations to find unknown values in simple equations. I can use equations and inequalities in the first degree to represent and model real-life situations and solve problems, using a range of representations. I can use equations and inequalities, and relevant graphs, to represent and model real-life situations and solve problems, including those which describe proportion and exponentiation. Knowledge count forwards and back interpret negative numbers solve problems involving addition, subtraction, multiplication and division place value commutative law the inverse Knowledge use simple formulae including cross-curricular express missing number problems algebraically represent/generate a worded problem using symbols, starting with expressions and moving on to equations and graphs generate inequalities linked to problems- representing through graphs and a number line understand linear equations in the context of a problem e.g.gradient and rate, c and a fixed charge use linear equations and inequalities to model optimum problems (linear programming)e.g. Cost of materials versus profits Knowledge understand linear equations in the context of a problem e.g.gradient and rate, c and a fixed charge using direct and indirect proportion to solve problems in mathematical and non-mathematical contexts Use quadratic graphs to model the 2D movement of an object e.g. use dynamic geometry package to investigate coefficients of quadratic to model the situation use equations that represent exponential growth or decay as a formula. use equations that represent financial models e.g. AER, APR use linear equations and inequalities to model optimum problems (linear programming)e.g.Cost of materials versus profits Skills assign values missing value questions practical activities to investigate relationships (e.g. Cuisenaire rods) use a range of manipulatives to model conceptual understanding and represent equations and expressions spot errors in completed questions Skills Skills represent problems using algebra- e.g. generate simultaneous equations/inequalities for cost of tickets for adult/child draw / sketch graphs from equations by plotting points- reading the graph to find relevant information- e.g. a linear graph to show the cost of a party ( price per person plus the cost of the room) the gradient representing the cost per person and the starting point (intercept) as the cost of the room.( link to physics SUVAT equations) plot and read quadratics linked to problems - what does the gradient mean? What does the minimum/maximum point refer to in real life? read and interpret graphs e.g. gradients and intercepts draw conclusions from relevant data collect like terms, rearranging, solving simple equations shade regions more than/less than on a graph to satisfy given criteria Interpret the gradient and y-intercept in a given context Vocabulary expression, equation, unknown values, algebra, inverse Vocabulary region, inequality, solving, equation, expression, collecting like terms, rearrange, formulae, subject, hypotenuse, adjacent, opposite, linear Vocabulary proportion, exponential, equations, inequalities, intercept, simultaneous, maximum, minimum, AER/APR
WM2: Algebra Graphical methods Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I can explore linear equations graphically and I can demonstrate an understanding of the effect on the line when the constant or coefficient of x is changed. i) I can investigate a variety of non-linear graphs, including quadratic, cubic and reciprocal, to develop an understanding of the effect of the coefficients and constants on the shape of the graph. I can determine or approximate the rate of change at a point on a graph and I can investigate the area under a graph, understanding what these represent in real-life contexts ii) Knowledge co-ordinates- reading, plotting points, meaning of gradient understand the meaning and significance of the gradient- how to find it in terms of proportional changes, linking x and y coordinates explore how the graph changes with change of gradient (m) and constant (c) e.g. dynamic changes Understand interception of line with x- and y-axes Understand interception of two lines and link to the solution of equations Understanding and recognising equations of parallel lines understand negative gradients - how this is reflected in an equation. Knowledge (i) recognition of different graphs - link graph to equation sketch ( quadratic, cubic, exponential) use different representations of a quadratic to obtain graphical information and to understand the effect of changing the coefficients of the quadratic interpreting and applying the transformation of functions in the context of their graphical representation, including y = f(x + a), y = f(kx), y = kf(x) and y = f(x) + a, applied to y = f(x). Skills (i) sketch a quadratic find output of an equation using different x values (substituting different values of x into an equation) sketch graphs using key information (intercepts, algebraic representation) interpret graphs Skills how to sketch and interpret a linear graph using gradient and intercept. Rearranging linear equation into the form y = mx + c from ax + by = d Understanding equations of horizontal and vertical lines Knowledge (ii) use the trapezium rule to find the area under a graph link acceleration to rate of change and what this looks like on a graph understand the relationship between gradient and ratio. interpret straight line gradients as rates of change e.g. gradient of a distance-time graph as a velocity, and as a compound measure. calculate or estimate gradients of graphs and interpret in contexts such as distance-time graphs and velocity-time graphs. apply the concept of (instantaneous) rate of change (e.g. gradients of tangents) in numerical, algebraic and graphical contexts. calculate or estimate areas under graphs, and interpret in contexts e.g. distance-time graphs and velocity- time graphs, and financial contexts Skills (ii) calculate gradients interpret the story of the graph analyse graphical data and draw conclusions e.g. maximum, minimum identify anomalies Vocabulary positive, negative gradient, co-efficient, constant, linear, straight line, constant, reciprocal Vocabulary exponential, cubic, maximum, minimum, acceleration, distance, time, rate of change, represent, quadratic, coefficients, transformation, trapezium rule, gradients of tangents
WM3: Geometry Measurement - Time Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I can understand and apply the language of time in relation to my daily life. I am beginning to tell the time using a variety of devices. I have explored and used different ways of showing the passing of time, including calendars, timelines, simple timetables and schedules. I can read analogue and digital clocks accurately and I can make interpretations and perform calculations involving time. I can represent and use compound measures, using standard units, and I can demonstrate an understanding of the relationship between a formula representing a measurement and the units used. I can represent and use compound measures, using standard units, and I can demonstrate an understanding of the relationship between a formula representing a measurement and the units used. Knowledge identify elements of daily routine use terms such as lunchtime, home time etc understand the terms 'before' and 'after name days of the week anticipate the elements of daily routine respond to the terms 'before' and 'after sequence days of the week Knowledge know that there are a variety of devices for telling the time including analogue and digital clocks understand a week is seven days, a year is twelve months etc understand a month is broken into days and the number of days in each month know that there four seasons understand how to use time measuring devices to record a start and finish time number skills counting accurately Knowledge understand halves and quarters understand positional language count accurately in fives understand symmetry (link with to / past) understand that digital time does not use a decimal system i.e. :59 is followed by :00 read an analogue clock - o clock > half past > quarter past > increments of five minutes > nearest minute (link with fractions, symmetry and number skills) understand the links with the number system and time e.g. counting in multiples of five, number bonds to 12, then 24 understand the function of the hands know how many seconds in. a minute, minutes in an hour, hours in a day read aloud a digital time correctly know how to count on to the next hour understand that different time zones exist Knowledge convert units of time sec/min/ hr/ days etc understand and use quantities (such as length, time, mass) and recognise / convert between the units in the SI system understand and using compound measures including speed. understand and use compound measures such as m/s, km/h, mph and mpg. understand that compound measures represent a rate. interpret time as measured in fractions and decimals Knowledge understand and use compound measures including density and population density. using compound measures such as kg/m3, g/cm3, population per km2. use and convert compound units in algebraic contexts. distinguish between formulae for length, area and volume by considering dimensions. Skills Skills talk about today, yesterday and tomorrow. recite/sing/chant the days of the week, months and seasons of the year in meaningful contexts, e.g. when changing the class calendar develop a sense of how long tasks and everyday events take sequence a daily routine sequence a story discuss weekdays and weekends discuss special occasions and holidays Skills time conversions, estimating and rounding in context convert between quantities and apply knowledge in context to solve problems use appropriate times when representing and analysing data e.g. DST graphs understand and use appropriate formulae involving time tell the story of the graph (link with statistics) use and interpret a calculator for time calculations measure and record time in non- mathematical contexts Skills understand and order the days of the week, the months and seasons of the year in meaningful contexts to know the day and date practical tasks with calendars use a visual timetable to discuss daily routines use a calendar to identify special occasions count accurately in fives number bonds to 12 positional language such as clockwise / anticlockwise convert between quantities and apply knowledge in context to solve problems use appropriate times when representing and analysing data e.g.DST graphs understand and use appropriate formulae involving time ) use and interpret a calculator for time calculations Skills label analogue clocks say / write the time when displayed on an analogue clock label digital clocks say / write the time when displayed on a digital clock match analogue and digital times represent time on 24 hr clock solve authentic / real life time problems calculate time taken for different events order times from quickest - slowest and v.v. understanding that a winning time is lowest number not highest Vocabulary morning / afternoon / evening / night, days of the week, dinner time / play time / home time, today / yesterday / tomorrow, time, birthday, holiday, bedtime, before / after, next /last, now / soon / early /late, quick / quicker / quickest / quickly, slow /slower /slowest /slowly, old /older /oldest ,new /newer /newest, takes longer / takes lest time, hour / o'clock, clock / watch /hands Vocabulary seasons: spring, summer, autumn , winter, month, year, weekend, midnight, fast /faster /fastest , half past , how long ago? how long will it be to.....? how long will it take to .? how often?, always, never, often, sometimes, usually, once, twice, months of the year, fortnight, minute, second, quarter to / quarter past, digital /analogue, timer, century, calendar, date, earliest /latest Vocabulary leap year, millennium ,date of birth, noon, timetable, arrive, depart, 24-hour clock, 12-hour clock, Greenwich Mean Time, British Summer Time, international date line Vocabulary gradient, graph, distance, speed, time, acceleration, variable, average, velocity, compound, rate, interpret Vocabulary gradient, graph, distance, speed, time, acceleration, variable, average, velocity, compound, rate, interpret, density,
WM3: Geometry Measurement - Units Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 I can represent and use compound measures, using standard units, and I can demonstrate an understanding of the relationship between a formula representing a measurement and the units used. Progression Step 5 I have used a variety of objects to measure. I am beginning to understand the need to repeat the same physical unit without any gaps when measuring. I have explored measuring, using counting, measuring equipment and calculating, and I can choose the most appropriate method to measure. I can convert between standard units, including applying my understanding of place value to convert between metric units. I can use a variety of measuring devices from different starting points. Knowledge the concept and language of larger, smaller, shorter, longer, lighter, heavier, full, empty etc for direct comparisons recognise a gap when measuring (link with jigsaws and block play) how to order items according to given criteria (e.g. length) Knowledge there can be no gaps when measuring using objects accurate vocabulary the difference between long, tall and big units must be equal Knowledge basic metric unit conversions approximate weights of familiar items sound place value skills e.g. vlaueof digits and the ability to multiply and divide by 10,100,1000 Knowledge convert and apply metric units in the SI system convert and use imperial units recognise and calculate common compound measures e.g. Speed or rate (mph, kmph) rates of pay (per week, per month, annual salary), density, population density etc unit pricing plot, read, interpret measures from graphs recognise and apply compound measures from formulae (speed = distance/time; density = mass/volume etc) link gradients to rates Skills compare size and weight physically then verbally compare and describe the size and weight of familiar objects compare and describe the length and capacity of familiar objects measure length using a repeated object leaving no gaps estimate and predict prior to measuring - ensure that this is done practically with a reference point recognise the relationship between the size of the object and the number. of units identify equipment for measuring Skills Skills Skills in all cases estimate and answer prior to measuring choose and use appropriate measuring equipment to measure lengths (then mass, capacity and temperature) measure, arrange ,order and compare lengths and heights using appropriate language (repeat with weights and capacity) record measurements with increasing accuracy e.g. cm, m, cm and m etc choose appropriate units of measurement for different objects use <>= to compare measurements solve problems using a range of measures (link with number and calculation skills) choose a sensible unit, equipment and answer for measuring lengths, weights and capacities including metric measurements, mixed measurements and decimal notation (include matching answers) solve problems in the context of lengths, weights and measures including totals, differences, fractions and decimals and mIxed metric units (include estimations) use <>= to compare measurements including decimals and mIxed metric units choose and apply appropriate units read rates from graphs plot measures on graphs and interpreting rates interpolate rates from a graph distinguish between formula for different measures / rates calculate gradients and interpret as a rate between two quantities use proportion in rate problems e.g. travelled 48 miles in 24min what is the speed in miles per hour? solve rate problems in a range of contexts (mathematical and real-life e.g. best price per unit) convert metric rates into other metric rates e.g. cm/sec into kmph. Vocabulary longer shorter, the same, lighter, heavier, full, empty, how many? Vocabulary millimetre, centimetre, metre, kilometre, gram, kilogram, millilitre, litre Vocabulary round, estimate, fraction, decimal, metric, convert Vocabulary compound measures, rates, density, formulae, gradient,
WM3: Geometry Measurement - Estimating Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I can make estimates and comparisons with measures, such as shorter than , heavier than . I can estimate and measure, using non- standard units, before progressing onto standard units. I can estimate and measure length, capacity, mass, temperature and time, using appropriate standard units. SEE - MEASUREMENT UNITS SEE - MEASUREMENT UNITS SEE - MEASUREMENT UNITS Vocabulary Vocabulary Vocabulary
WM3: Geometry Shape and Space - 2D and 3-D shapes Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I have explored, compared, and used the general language of shapes through investigative play. I have explored two-dimensional and three- dimensional shapes and their properties in a range of contexts. (also see 3-D shapes) I can explore and consolidate my understanding of the properties of two-dimensional shapes to include the number of sides and symmetry. I can explore and demonstrate an understanding of the effect of scale when comparing measurements of similar shapes in all three dimensions. I can explore vertices, edges and faces of three-dimensional shapes and I can use these characteristics to describe a three-dimensional shape. Knowledge begin to match shapes and names point to a given shape from a selection or in a picture begin to give reasons for answers begin to use accurate language as modelled by adults develop spatial awareness and spatial reasoning skills Knowledge recognise and name common 2-D shapes [e.g. rectangles (including squares), circles and triangles] recognise and name common 3-D shapes [e.g. cuboids (including cubes), pyramids and spheres] basic properties (beginning with informal language) difference between 2-D and 3-D shapes understand tessellation (based on properties) also see WM3: Geometry Shape and space - Symmetry and Transformations Knowledge (i) recognise and classify regular and irregular 2-D shapes based on properties understand what symmetry is identify and explain examples and non- examples understand the terms irregular and regular polygon also see WM3: Geometry Shape and space - Symmetry and Transformations see WM3: Geometry Shape and space - Symmetry and Transformations Knowledge know and recognise vocabulary for scale, three dimensions, similar, length, area, volume. understand scaling in three dimensions (build on effect of scaling from 2-D) know and apply scale volume measure (cubed from a length measure) know difference between congruence and similarity understand of ratio and proportion (direct) understand transformations of shapes to pair equivalent measures Skills (i) identify symmetry in shapes (include different orientations of shapes) pattern making including rotations explore and identify lines of symmetry sort, classify, order the number of sides on regular and irregular 2-D shapes compare and classify a range of 2-D shapes based on properties and size draw a range of 2-D shapes accurately (link with measures and angles as appropriate Skills Skills ensure indoor and outdoor learning opportunities read books and sing songs about shapes play with malleable materials squash, twist and create shapes build models using appropriate shapes. Ask - why did you use those shapes? Why is this a better shape than that? (ensure opportunities for large scale construction) construction tasks (build, add to, improve, change, copy etc) play with a range of shapes stack to build towers and nest shapes to identify when it fits sand wand water play filling, emptying, transferring with a range of containers complete and discuss jigsaw puzzles feely bag to identify shapes through properties pattern blocks to investigate patterns and shapes (focus on orientation and turning shapes to fit) find the missing piece / shape / block identify which one doesn t belong / odd one out / where did it go wrong? investigate properties through play ensure indoor and outdoor learning opportunities read books about shapes investigate and identify properties of 2-D and 3-D shapes small world play pattern making pattern blocks puzzles investigate, reason and problem solve construction investigate, reason and problem solve increasingly complex jigsaws identify shapes through use of examples and non- examples compare and sort shapes based on properties (link with statistics skills) identify 2-D properties of faces on 3-D shapes make 3-D shapes (Lego, clay etc) recognise 3-D shapes in different orientations and describe them using accurate vocabulary draw 2-D shapes (including in different orientations and describe them using accurate vocabulary) barrier games to develop understanding of properties and language always / sometimes / never, true / false, which one doesn t belong, same but different, shapes in the environment Skills calculate cubes and cubes roots of numbers calculate scale factors in a range of contexts for forward and inverse calculations identify and distinguish shapes that are similar or not apply similarity to calculate unknown lengths in similar shapes calculate measures of volume and capacity and apply scaling to them Knowledge (ii) know correct terminology understand what is meant by 2-D and 3-D names of common 3-D shapes recognise 3-D shapes from a 2-D representation on paper know that a 3-D shape can be constructed from 2-D shapes Skills (ii) identify and describe the properties of 3-D shapes, including the number of edges, vertices and faces use accurate terminology identify 3-D shapes based on 2-D representations identify and build 3-D shapes from nets paper folding and nets activities identify nets and non-nets investigate links with 2-D and 3-D properties Vocabulary same, different, build, make, add, change, piece, shape, block, fit, fill, full, empty, move, twist, squash, turn, curvy, flat, straight, roll Vocabulary side, edge, vertices, vertex, corner, straight, face, curved, shape names, turn, rotate, opposite, symmetry, line, angle, tesselate, roll, slide Vocabulary 2-D and 3-D shape names, regular, irregular, parallel, perpendicular, net, angle, enclosed, dimension, geometry, polygon, quadrilateral, equilateral, scalene, right-angle, isosceles Vocabulary scale factor, 3-D, similar / similarity, congruent, proportion, ratio, cubed, cubed root
WM3: Geometry Shape and Space - Symmetry and transformations Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I can explore and consolidate my understanding of the properties of two-dimensional shapes to include the number of sides and symmetry. (NB this skill is also in strand 2-D shapes) I can use a variety of approaches to investigate, predict and demonstrate the effect of transformations on two-dimensional shapes. I can apply my understanding of the effect of transformations on the properties of shapes in order to explain why they are similar, congruent or neither. I have explored reflective symmetry in a range of contexts and I can discuss it as a property of shapes and images. see WM3: Geometry 2-D and 3-D shapes Knowledge see WM3: Geometry 2-D and 3-D shapes what is symmetry? basic properties of 2-D shapes and associated vocabulary find half of a shape by cutting or folding Knowledge see WM3: Geometry 2-D and 3-D shapes what is and what isn t symmetry what is meant by reflectional symmetry understand the terms vertical and horizontal basic properties of shapes and associated vocabulary Knowledge accuracy of language - sketch , draw, polygons, regular, irregular, reflective, rotational, horizontal, vertical, diagonal how to accurately draw polygons properties and names of an increasing range of geometric of shapes understand the difference between translations and transformations identify co-ordinates read and plot points in 4 quadrants Knowledge congruent shapes are exactly the same shape and size but could have been reflected, rotated or translated similar shapes have been enlarged by a certain scale factor similar shapes have equal angles conditions for congruent triangles ASA, SAS, SSS similar triangles AAA recognise and use the mathematical symbols for equal sides, angles Skills identify lines of reflectional symmetry in 2-D shapes recognise when there are no lines of symmetry identify how many lines of symmetry a 2-D shape has draw lines of symmetry on a given shape identify reflectional symmetry when a shape is rotated / shown in a different orientation complete patterns and pictures with a horizontal line of symmetry complete patterns and pictures with a diagonal line of symmetry reflect a pattern / image with a horizontal line of symmetry reflect a pattern / image with a diagonal line of symmetry reflect a shape or pattern about a given mirror line identify an error in a reflectional pattern and explain why draw the reflection of a shape in a co-ordinate grid identify co-ordinates on a grid to show reflection / original position when shown a reflection plot and complete a reflection using co-ordinates investigate and identify rotational symmetry sketch, then accurately draw regular polygons that have reflective and rotational symmetry sketch, then accurately draw regular polygons and show the lines of symmetry describe, identify and classify an increasing range of regular and irregular polygons according to rotational and reflective symmetry prove / disprove statements regarding rotational, reflective symmetry (examples / non-examples, always, sometimes, never etc) Skills Skills Skills identify whether a shape has a vertical line of symmetry use a vertical line of symmetry to create a mirror image make a mirror image of pictures, patterns and shapes (practical opportunities) translate a given shape on a on a co-ordinate grid following instructions describe the translation find missing co-ordinates of a translation recognise a reflection translation recognise a reflection transformation plot vertices of given shapes, translate those shapes on a co-ordinate grid identify the new co-ordinates following translation describe a range of transformations apply a range of transformations to a given shape recognise multiple steps in a transformation apply multiple steps in a transformation find and use scale factors to classify shapes as similar, congruent or neither see WM3: Geometry 2-D and 3-D shapes Vocabulary shape names, mirror, reflect , symmetry, vertical Vocabulary increasing range of shape names, vertical, horizontal, diagonal, reflective, rotational, co-ordinates, polygons Vocabulary sketch, draw, translation, transformation, quadrant Vocabulary similar, congruent, enlargement, scale factor, vertices
WM3: Geometry Shape and Space - Area and perimeter Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I can use efficient methods for finding the perimeter and area of two- dimensional shapes, understanding how basic formulae are derived. I can explore and calculate the areas and perimeters of simple and compound two-dimensional shapes, including circles, and I have demonstrated an understanding of pi ( ) as the ratio of the circumference of a circle to its diameter. I can apply my understanding of area to be able to calculate the surface area of simple prisms. I can use my knowledge of measurement to calculate the perimeter, area (or surface area) and volume of compound two-dimensional and three-dimensional shapes. (NB this skill is also in strand Volume) ALTHOUGH THERE ARE NO STATEMENTS AT THIS STEP PUPILS CAN EXPLORE AND INVESTIGATE MEASUREMENTS, DISTANCES AND MISSING VALUES IN CALCUALTIONS IN PREPARATION FOR PERIMETER TASKS Knowledge perimeter is the distance around the edge of a 2-D shape perimeter is measured in units of length and can be calculated by adding together the lengths of the sides of a 2-D shape. perimeter of a regular polygon can be calculated by multiplying the length of one side by the total number of sides area is the measurement of the surface of a flat item area is measured in square units, e.g. (cm2) and (m2). area of a rectangle can be calculated by multiplication know that when a 2-D shape is decomposed and the parts rearranged that the area remains the same (prior to finding area of compound shapes) shapes with the same area can have different perimeters shapes with the same perimeters can have different areas Knowledge shape names and properties definition of perimeter definition of area accuracy in measuring basic formulae for calculating area and perimeter units of measurement of area and perimeter which formulae to use and when a parallelogram can be decomposed and rearranged to form a rectangular parallelogram two congruent triangles can be composed to for a parallelogram Knowledge distinguish between area / surface area and volume draw nets of 3D shapes know and use terminology of faces, edges and vertices recall of formulae for volume of 3D shapes (sphere, pyramid) calculate with powers and roots know that angles are conserved in similar shapes Skills calculate perimeter, area / surface area and volume of common 2D and 3D shapes split 2D / 3D compound (composite) shapes into known 2D / 3D shapes to calculate separate area / surface area and volumes to obtain overall area and volume measure with appropriate units a range of 3D shapes and calculate surface area and volume select appropriate lengths and units for use in surface area / volume formulae use volume calculations with inverse methods to calculate unknown lengths apply a general formula for volume of a pyramid (1/3 x base area x height) by identifying an appropriate base and measures leave area and volume calculations in terms of e.g. 3 Skills Skills build on perimeter work in PS3 by finding perimeters of more complex compound shapes and missing value questions Incorporate into word problems linking area and perimeter changes calculate the area of a parallelogram (then solve problems) calculate the area of a triangle solve a range of problems using area problem solving using a range of perimeter questions (range of shapes, units. Include missing values and word problems) calculate the area of trapezium area of compound shapes using previous knowledge calculate the surface area of cuboids calculate the area of a circle using calculate the area of compound shapes including circles and semi- circles calculate the surface area of a prism by finding the sum of the area of its faces calculate the surface area of a cylinder recognise when a formula may be used to find the surface area of a prism investigate perimeter by walking around the outside identify the error in perimeter statements e.g. missing side, inaccurate calculations, counting the same side twice etc. measure the perimeter of rectangles, then regular polygons (count squares around the outside prior to measuring in cm) find the perimeter of polygons using given measurements explore, identify and explain different. shapes with the same perimeter calculate the perimeter of more complex shapes using given side lengths (include missing side lengths too) solve perimeter problems begin to calculate area of rectangles (then equilateral triangles and regular polygons) using formulae investigate area of a range of shapes on grids. by counting square (go on to include half squares) begin to record answers in cm2 (make the link with counting squares) solve missing side questions when given the area of a rectangle compare areas and find differences investigate possible dimensions for a shape with a given area begin to use formula for areas of rectangles find areas of shapes made of a combination of rectangles and squares investigations increasing area / perimeter. E.g. make a shape where A=P, increase P, keep A the same, increase P, decrease A etc Vocabulary area, perimeter, side, length, calculate, formula, cm2 , shape names Vocabulary compound, prism, cylinder, parallelogram, trapezium, surface area, Vocabulary composite, vertices, faces, edges, volume, formulae, inverse, powers, roots
WM3: Geometry Shape and Space - Scale and ratio Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I can use my knowledge of scale and ratio to calculate the lengths and areas of fractions of shapes, including arcs and segments of circles. I can explore and demonstrate an understanding of the effect of scale when comparing measurements of similar shapes in all three dimensions. (see Shape and space 2 - 3D shapes) Knowledge understand that an arc is a fraction of the circumference of a circle understand that a sector is a fraction of the area of a circle understand that the area of a segment is the area of the (sector triangle) understand that scale factors represent the change in length scale factors are squared to represent the change in area and cubed to represent the change in volume Skills use the formula for circumference of a full circle to find the length of an arc use the formula for area of a full circle to find the area of a sector identify radius, diameter. find fractions of 360 in a circle rearrange the formulae for arcs, sectors to find lengths of radius or diameter or the size of angles solve problems involving arcs, sectors, segments, circles substitution of formulae for circles, arcs, sectors change the subject of a formula including order of operations and square roots to find radius find scale factors from given lengths use length scale factors to find area and volume scale factors (square, cube) use area, volume scale factors to find length scale factors (square root, cube root) use scale factors for length, area, volume to find missing lengths, areas, volume solve problems involving scale factors and similar shapes use ratios or decimals to represent scale factors. Vocabulary arc (major and minor), sector (major and minor), segment (major and minor), chord, radius, diameter, circumference, tangent, use of theta to label unknown angle similar, congruent, scale factor, square, cube, root, length, area, volume
WM3: Geometry Shape and Space - Triangles Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I can use efficient methods for finding the perimeter and area of two-dimensional shapes, understanding how basic formulae are derived. (NB this skill is also in strand Perimeter and Area) I can apply my understanding of area to demonstrate and use the relationship between right-angled triangles and squares in the context of Pythagoras theorem. I can explore trigonometric ratios in right-angled triangles and I can use my knowledge of them to solve problems involving lengths, angles and area of any triangle. see WM3: Geometry 2-D and 3-D shapes see WM3: Geometry Area and perimeter Knowledge identify the hypotenuse in a right-angled triangle with any orientation reason whether to add or subtract the squares of the sides knowledge and understanding of a few proofs of Pythagoras' theorem additional, if possible - the history of Pythagoras and the link to square root of 2 Knowledge label hypotenuse, opposite, adjacent sides from given angle in right angled triangles choose the correct ratio using SOHCAHTOA or from knowledge of the formulae for sin, cos, tan understand that the reverse operation of sin, cos, tan is inv/arcsin, cos, tan and how to find these using calculators label a non right angled triangle using lower case a, b, c etc for sides and capital A. B. C etc for angles know that the opposite angles and sides generally use the same letter in lower case and capital form know the formulae for the sine rule and cosine rule use given information in the question to identify which formula to use. know the formula for the area of a triangle and how this is used for any triangle including right angled triangles Skills Skills identify hypotenuse to apply Pythagoras' theorem. use a range of triangle orientations to apply Pythagoras' theorem. identify the application of Pythagoras' theorem in real-life contexts. apply Pythagoras' theorem to compound shapes for calculating lengths. use Pythagoras' theorem to show a triangle is right-angled appropriate units applied to Pythagorean problems extension: using Pythagoras' theorem with different shapes e.g. semicircles, equilateral triangles etc. substitute into correct formula for sin, cos, tan for right angled triangles rearrange these formulae to find the top or bottom of the fraction or find the angle using the inverse trig ratios. use the calculator accurately to find lengths and angles Vocabulary Pythagoras, theorem, hypotenuse, squares, square roots, area Vocabulary hypotenuse, adjacent, opposite, sine, cosine, tangent, theta
WM3: Geometry Shape and Space - Volume Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I can relate a three-dimensional shape to its two- dimensional nets. I can derive and apply the formulae for the volume of simple prisms. I can use my knowledge of measurement to calculate the perimeter, area (or surface area) and volume of compound two-dimensional and three-dimensional shapes. (NB this skill is also in strand Perimeter and Area) Knowledge see WM3: Geometry 2-D and 3-D shapes know that the same 3D shape can be composed from different 2D nets understand volume is measured in cm3 Knowledge concept of volume through counting cubes work out the volume of a simple shape made of cubes (practical) understand that a volume of an object is a measure of the 3-dimensional space within the object, whereas Surface Area measures the flat 2-dimensional space on the outside of the object to know that volume is measured in cm3 understand what a prism is (and isn t) recognise cross sections of prisms as polygons or composite shapes Knowledge be able to distinguish between area / surface area and volume be able to draw nets of 3D shapes know and use terminology of faces, edges and vertices recall of formulae for volume of 3D shapes (sphere, pyramid) calculate with powers and roots know that angles are conserved in similar shapes Skills Skills Skills use knowledge of shape properties to identify, sketch and draw nets and 3D shapes identify nets and non-nets recognise the nets of common 3D shapes recognise common 3D shapes from their nets recognise common 3D shapes from isometric drawings draw 3D shapes on isometric paper introduce the concept of volume through counting cubes work out the volume of a simple shape made of cubes (practical) calculate how many cubes can fit in a box when given the dimensions or volume of the box calculate the volume of cubes and cuboids by using the rule that Volume = Length Width Height. calculate the volume of prisms find the area of cross sections of prisms that are polygons or composite shapes identify a missing dimension when given the volume calculate the volume of a cylinder express the volume of a cylinder in terms of calculate perimeter, area / surface area and volume of common 2D and 3D shapes split 2D / 3D compound (composite) shapes into known 2D / 3D shapes to calculate separate area / surface area and volumes to obtain overall area and volume measure with appropriate units a range of 3D shapes and calculate surface area and volume select appropriate lengths and units for use in surface area / volume formulae use volume calculations with inverse methods to calculate unknown lengths apply a general formula for volume of a pyramid (1/3 x base area x height) by identifying an appropriate base and measures eave area and volume calculations in terms of e.g. 3 Vocabulary nets, isometric Vocabulary prism, oblique, right, perpendicular, bisect, constant, regular, irregular, cross-section Vocabulary composite, vertices, faces, edges, volume, formulae, inverse, powers, roots
WM3: Geometry Position Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I have explored movements and directions and I am beginning to use mathematical language to describe position. I can describe and quantify the position of objects in relation to other objects. I have developed an understanding of the ways in which co-ordinates are used to solve problems involving position, length and shape. I can locate and describe the locus of points defined by a range of different criteria. Knowledge development of vocabulary begin to appreciate that we see different things from different viewpoints Knowledge describe position, direction and movement understand left and right understand that position affects our viewpoint e.g. for opposite partners left is right and vice versa Knowledge using co-ordinates in the first (and four) quadrants properties of regular polygons simple map reading positional language measures skills Knowledge apply ruler and compass constructions to construct figures and identify the loci of points, to include real-world problems construct the locus of a point which moves such that it satisfies certain conditions understand the word equidistant in relation to loci criteria e.g.equidistant from two fixed points or lines Skill Skill Skill outdoor play physical activities to develop awareness of movement and position develop spatial awareness tangram games and puzzles talk about position in play activities e.g. jigsaws, riding a bike, running around and changing direction, playing with shapes and making patterns, using a beebot sit opposite a partner, If you can see , what can I see? treasure hunts giving and following clues following / giving instructions obstacle courses links with stories and books e.g. Rosie s Walk, We re Going on a Bear Hunt describe a journey using a simple map create a simple map to reflect a journey design an area e.g. school garden link with physical, creative, ICT and geography tasks for real life context develop the skills introduced in PS1 by using more complex language and instructions e.g. introduce measurements of distance e.g. number of steps to take or direction such as left / right / clockwise / anticlockwise or size of rotation e.g. half turn to the left etc investigate routes and paths focus on distance, position and direction create, follow, describe obstacle courses predict and re-create familiar routes, focusing on places of importance, landmarks, distance, orientation (link with stories) make a pattern / picture using given shapes in the appropriate orientation creative work with shapes in different orientations use increasingly accurate mathematical language to describe position of objects in relation to each other barrier games follow instructions to create the same picture. / pattern true / false, always/sometimes/ never, where did it go wrong? questions develop navigational skills in a practical context build on mapping skills in PS1 and PS2 create maps and models based on stories or thematic contexts investigate distances, direction and navigation skills begin to look at the concept of scale look at books that focus on position, direction, maps, journeys and directions use a grid to give directions from one position to another move polygons on a grid, note the original position and the new position see WM4: Geometry, Symmetry and transformation which includes work on translations use a square grid to translate polygons describe the translation follow instructions to translate a polygon on a grid use a translated polygon to draw the location of the original shape use co-ordinates to plot points and draw polygons identify missing co-ordinates in a given polygon identify a polygon using given co-ordinates true / false, always/sometimes/ never, where did it go wrong? questions barrier games to describe location giving clues / recreate a picture or pattern using co-ordinates and polygons Skills construct 2-D shapes from given information accurate use of ruler, pair of compasses and protractor to do constructions bisecting a given line bisect a given angle construct the perpendicular from a point to a line construct angles of 30 , 45 , 60 and 90 Vocabulary In, on, under, down, up, across, forwards, backwards, through, across, over Vocabulary in front of, behind, between, to the side, left, right, top, middle, bottom, on top of, above, around, inside, outside whole turn, half turn, quarter turn, three-quarter turn, direction, clockwise, anticlockwise, nearer, closer, further Vocabulary quadrant, co-ordinate, polygon, translate, plot, vertices Vocabulary loci, locus, constructions, equidistant, bisect, conditions, perpendicular
WM3: Geometry Angles Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I have explored movements and directions and I am beginning to use mathematical language to describe position. I have explored the concept of rotation and I am beginning to use simple fractions of a complete rotation to describe turns. I can demonstrate my understanding of angle as a measure of rotation and I can recognise, name and describe types of angles. I can use angle and shape facts to deduce further features and relationships of triangles and quadrilaterals. I can use logical arguments and my knowledge of polygons, intersecting lines, angle and the circle theorems to deduce and calculate the size of angles and length of lines. I can explore and calculate angles formed by parallel lines and by a transversal. I have applied my understanding of angles to model and solve problems involving bearings. see WM3: Geometry Position Knowledge describe position, direction and movement, including whole, half, quarter and three-quarter turns. understand what is a rotation understand what is a half understand what is a quarter understand left and right understand clockwise and anti-clockwise understand that direction doesn t matter in a half turn begin to understand an angle as a measurement of rotation Knowledge accurate use of language (see PS2) a right angle is a square angle a rectangle has four right angles four right angles make a full turn, two right angles make a half turn angle names and properties right angle, acute angle, obtuse angle, reflex angle know that angles are measured in basic number and calculation skills (e.g. to find totals and differences to 90, 180, 270, 360) Knowledge know the basic properties (angles and lengths) of 2D shapes e.g. polygons including triangles, quadrilaterals, etc. be able to provide geometrical reasons to justify these properties use angle and measure facts to find lengths and angles in rectilinear figures (and in simple proofs) Know and use: vertically opposite angles are equal, alternate angles on parallel lines are equal, corresponding angles on parallel lines are equal, supplementary angles add to 180 use angle facts in mathematical and non- mathematical contexts e.g. bearings Knowledge see PS4 skills understand and use the conditions for congruent triangles in formal proofs understand and apply the relationship between lengths, areas and volumes of similar shapes use vectors in geometric arguments and proofs Know and apply circle theorems understand and construct geometrical proofs using circle theorems and angle facts know the basic properties of 2D shapes including angle, length and symmetry properties understand how to calculate interior and exterior angles of a given polygon Skills outdoor learning to make turns and follow / give instructions make a quarter turn, half turn, three quarter turn in a given direction link clockwise with the direction of movement on an analogue clock begin to recognise quarter turn as a right angle move around a circle in a clockwise / anti-clockwise direction prior to work on Time . true / false, always/sometimes/ never, where did it go wrong? questions identify right angles in the environment identify greater than a right angle , equal to a right angle , smaller than a right angle. binary classification right angle / not right angle Skills create an angle measurer with paper and split pins rotate two lines around a point to make different sized angles reinforce understanding of right angles and identify them in the environment identify right angles / not right angles in regular polygons investigate and draw other polygons identify right angles/ no right angles physical activities to make angles of different sizes compare angles and order visually sort and classify angles match angles and their names estimate size of angles giving reasons Skills Skills use the sum of the interior angles of a triangle is 180 and be aware of a proof of this fact use the sum of the exterior angles of a polygon is 360 and be aware of a proof of this fact know and use the sum of the angles at a point is 360 know and use that the sum of the angles at a point on a line is 180 Use angle facts and lengths to draw the bearing of one object from another use SSS, SAS, ASA and RHS conditions for congruency apply opposite angles of a cyclic quadrilateral are supplementary know that for a point P on the circumference, the angle between the tangent and a chord through P equals the angle subtended by the chord in the opposite segment Know that for a point P on the circumference, the radius or diameter through P is perpendicular to the tangent at P. a radius or diameter bisects a chord if and only if it is perpendicular to the chord two angles in the same segment are equal. the angle on the circumference subtended by a diameter is a right angle. the angle subtended by an arc at the centre is twice the angle at the circumference Vocabulary in front of, behind, between, to the side, left, right, top, middle, bottom, on top of, above, around, inside, outside, whole turn, half turn, quarter turn, three- quarter turn, direction, clockwise, anticlockwise, nearer, closer, further Vocabulary right angle, acute angle, obtuse angle, reflex angle, rotation, degrees Vocabulary transversal, parallel, bearings, rectilinear, vertically opposite, alternate, corresponding, interior, exterior, north Vocabulary logical, reasoning, polygons, congruent, congruency, similar, proofs, theorems, segment, chord, cyclic, subtended, interior, exterior
WM4: Statistics Collecting data Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I can investigate, collect and record data found in my environment. I can collect and organise data to ask and answer questions in relevant situations. I can collect different types of data to answer a variety of questions that have been posed, demonstrating an understanding of the importance of collecting relevant data. I can choose a sensible hypothesis to investigate. I have explored the relationship between the type of data I have collected (including qualitative and quantitative) and how this can be manipulated and represented. I can explore different sampling methods, including systematic and stratified sampling, understanding the need to select appropriate sampling methods when collecting data. Knowledge why it is useful to collect information? understand same / different understand properties and appropriate descriptive language understand 'data' as information Knowledge know notation for tally charts begin to consider suitable questions which can be answered through simple data collection understand he purpose of a questionnaire identify the difference between open and closed questions within a questionnaire Knowledge know that data comes in different types (e.g., discrete, continuous, quantitative, qualitative) understand what is a census Knowledge to understand that a hypothesis is an assumption or idea which can be tested by collecting data to independently construct relevant processes for collecting and analysing data. to understand the difference between data types. to understand the use of scale to gather data relating to opinions (e.g. 1 - 4: Strongly disagree - Strongly agree) Knowledge to understand the reasons for and examples of using sampling. to understand examples of sampling methods and their impact (e.g., stratified, systematic, random, bias) Skills collecting objects to meet set criteria collecting information through asking questions collecting data using a story book link with other AoLES / topic where relevant - however, ensure the correct mathematical skills are taught record their information in their own way - do not lead too much at this stage, ask pupils to explain their work Skills populate a table that's provided complete a provided questionnaire create a simple questionnaire using closed questions only to collect the appropriate information to answer their given question Skills Skills pose relevant questions to collect data (e.g., data collection sheets, class census) ,and apply relevant statistical processes. identify advantages and disadvantages of different questioning techniques when designing questionnaires Skills critically evaluate the suitability of a designed survey (e.g., location, bias etc) to be able to choose the most appropriate sampling method for the given situation and justify their choice Vocabulary collect, count, the same, different, how many, most, least, how many more, fewer Vocabulary count, tally, same, different, more than, less than, data, total, questionnaire, open questions, closed questions Vocabulary data, primary data, secondary data, census, survey, questionnaire, population, interpret, multiple choice, discrete, continuous Vocabulary qualitative, quantitative, hypothesis, bias, validity, scale, discrete, continuous Vocabulary sample, stratified sample, systemic sample, random sample, sample size, population, sampling, generalisation,
WM4: Statistics Representing data by grouping and classifying Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I can group sets into categories and I am beginning to communicate the rule(s) I have used. I can sort and classify using more than one criterion, including the use of Venn diagrams and Carroll diagrams. Knowledge to understand why it is useful to collect and sort information to identify the properties of an object / group of objects to be able to communicate the properties of an object and use descriptive language to be able to match objects according to a given criteria e.g. colour, object, shape etc to identify similarities and differences between objects - one to one correspondence to understand that objects have many properties and can be classified in different ways Knowledge know that a Carroll diagram works on binary choices and that all data must fit onto the diagram to understand the use of a Venn diagram to distribute a set of elements including the meanings of each section of the diagram and where to place data that does not fit the given criteria (e.g., the Universal set) Skills Skills sort objects using similar attributes sort and count objects use knowledge of similarities and difference to identify rules of classification for pre-sorted groups identify the odd one out in a set add an extra object to the correct group using the rule sort objects e.g. books using one attribute, but realise they can be sorted in a different ways using different attributes guess the rule (same objects different ways of sorting) distribute sets of data according to the Carroll diagram/Venn diagram criteria use the intersection of a Venn accurately place data that doesn't meet the criteria in the Universal set place data into one of two sets in a Carroll diagram extend and place data in to one of four sets in a Carroll diagram look at a pre-prepared diagram and identify the reasons for the sets (label) label sets correctly on diagrams this skill can be linked with work on shape / number to sort and classify data appropriately e.g. multiples of 3 and multiples of 4 Vocabulary some, all, the same, different, sort ,match, similar , different , group , set , together Vocabulary count , tally , sort , data, group, set , list , chart , Carroll, Venn
WM4: Statistics Representing data in graphs and charts Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I am beginning to represent and interpret data, using a range of methods I am beginning to record and represent data in a variety of ways, including the use of tally charts, frequency tables and block graphs, when appropriate axes and scales are provided. I can represent information by creating a variety of appropriate charts of increasing complexity, including tally charts, frequency tables, bar graphs and line graphs. I can make informed choices about how to organise and represent data, using a wide range of graphs and charts, including pie charts, frequency diagrams and frequency polygons. I can extend my methods for representing data, including cumulative frequency, box and whisker, and histograms, to interpret measures of central tendency and measures of spread. Knowledge also see WM4 Statistics, Interpreting and exploring data understand same / different develop understanding of properties and descriptive language accurate counting skills Knowledge know how to construct a tally chart know how to 'read a given chart (e.g., identify the axes) understand that data can be recorded by use of a consistent scale (link with counting in multiples) understand that a pictogram requires a key (link with basic fractions) how to count accurately in different steps (link with counting in multiples) the importance of accuracy in construction when modelling Knowledge understand which type of graph is appropriate to display given data sets, (e.g.,vertical line graph, line graph, dual bar graph) know how to construct an appropriate chart (e.g., scale, when and why gaps are needed between bars, units, consistent width of bars , where to label the axes) know that labels and titles are needed when presenting a graph know that a key is needed for a pictogram (link with basic fractions) understand the notion of misleading presentations using bar graphs/line graphs e.g. ( graphs on inconsistent axes) Knowledge know and construct more complex charts (e.g., composite bar graphs, frequency polygon, frequency diagram, pie chart) to understand the difference between discrete and continuous data know when to group data know how (and why) to construct a frequency polygon know how and why the midpoint is used to represent a value for grouped data Knowledge know how to construct and justify the use of complex charts (e.g., cumulative frequency, box and whisker, histogram) understand that the area of the bar within the histogram represents the frequency (axis shows frequency density) Skill make groups and sets using given criteria make whole class graphs (e.g., human, objects, pictograms) make a simple graph / chart / pictogram to represent given data look at pre-prepared data -what questions would you like to ask? what do you notice / wonder? what can you tell me? identify their own 'data' in the whole class graph understand the 'data' as a set of information e.g. find their favourite colour in the class pictogram then find somebody else's favourite colour Skills construct a range of graphs using given data (e.g., 3D blocks to graph, adding data to a partially constructed bar chart, pictograms) construct a range of tables (e.g., collect data into given format) accurately transfer data from their table into a graph and check that there are no transfer errors construct their own bar (bar line) graphs with a given scale - this may be data they have collected or data they have been given Skills Skills Skills draw an appropriate chart (with accuracy) to display given data use digital skills - when appropriate, e.g. Excel and J2E group data to create appropriate class width in order to construct a frequency table transfer a written narrative into a suitable table / chart complete a pre-made frequency table (link with previous work on tally charts) choose an appropriate graph to display their data choose appropriate scales for the data justify choice of scale used to present data correctly labelling graphs and charts accurately construct frequency polygons accurately construct frequency diagrams accurately construct pie charts choose appropriate scales for the data transfer information from frequency table to cumulative frequency graph / box and whisker transfer information from a cumulative frequency graph / box and whisker into the frequency table complete partially presented graphs/charts accurately construct cumulative frequency graph accurately construct box and whisker accurately construct histogram use cumulative frequency graph to identify median, upper quartile, lower quartile, inter- quartile range Vocabulary the same, different, how many, most, least, how many more, fewer Vocabulary count , tally , sort , vote , questionnaire , data, scale, graph, block graph, bar line graph, bar graph, pictogram, key, represent , group , set , list, chart Vocabulary bar chart , line graph , grouped data chart , conversion graph tally chart , table , frequency table , timetable , Carroll diagram , Venn diagram , diagram , label , title , axis, axes , most popular , most common , least popular , least common Vocabulary scale, axes, frequency polygon, pie chart, frequency diagram, line graph, midpoint Vocabulary cumulative frequency, box and whisker, histogram, median, interquartile range, upper quartile, lower quartile, frequency density, class width.
WM4: Statistics Interpreting and exploring data Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I am beginning to represent and interpret data, using a range of methods. I am beginning to interpret and analyse simple graphs, charts and data. I can use different scales to extract and interpret information from a range of diagrams, tables and graphs, including pie charts with simple fractions and proportions. I can recognise any trends that are seen. I can explore trends and anomalies in data sets, investigating correlation between two variables. I can critically analyse statistics, considering how data is represented, its reliability, and whether and how the data has been manipulated to tell a particular story. I can make informed decisions based on statistical evidence, identifying bias and anomalies. I can use data to draw conclusions about hypotheses and I have communicated my findings clearly. I can critique my own methods and findings. Knowledge also see WM4 Statistics, Representing data in graphs and charts understand same / different develop understanding of properties and descriptive language accurate counting skills Knowledge read, compare and create a range of simple charts (e.g., tables, tally charts, bar charts, pictogram, tables, Carroll diagrams and Venn diagrams) read and show data in tables, understand that a fraction of a symbol can be used in a pictogram e.g., half a cat (link with basic fractions and counting in multiples) Knowledge understand that charts can be used to identity trends use charts to help draw conclusions, identify anomalies and make comparisons between data read more complex charts (e.g., pie chart, line graph and 'double' line) count accurately in different steps work accurately with fractions Knowledge (i) understand how to interpret and identify correlation for a scatter diagram understand the difference between correlation and causation understand that a line of best fit can be drawn to represent the trend of a data set understand the effect of outliers when drawing a line of best fit Knowledge understand the reliability of statistics consider the method of representation understand the term correlation understand that correlation does not equal causality understand 'outliers' and how they affect the data set Skill Skills Skills make groups and sets using given criteria make whole class graphs (e.g., human, objects, pictograms) make a simple graph / chart / pictogram to represent given data look at pre-prepared data -what questions would you like to ask? what do you notice / wonder? what can you tell me? identify their own 'data' in the whole class graph understand the 'data' as a set of information e.g. find their favourite colour in the class pictogram then find somebody else's favourite colour solve problems involving simple data in tables, pictograms and bar charts read and interpret the data from a given chart find most / least popular / common find the total / difference order results create and answer questions based on the data answer what if style questions answer 'comprehension' style questions about the data tell the story of the data make statements and draw conclusions about data presented as a chart or table identify the trends and suggest reasons as to what may cause them tell the story of a graph (use a line graph and explain what it might show) draw conclusions from their class census (*see representing data) answer questions on data presented in a pie chart - begin with interpreting halves, quarters and thirds (familiar fractional quantities) interpret data, make comparisons and draw conclusions using a double line graph solve 'comprehension' style problems using a variety of graphs / charts / diagrams answer questions on data in a frequency table Begin to recognise anomalies and give reasons Skills (i) Skill accurately construct a scatter diagram draw a line of best fit use of line of best fit to estimate values identify anomalies and outliers identify whether correlation exists then, if so, describe and interpret correlation identify, describing and interpreting outliers describe the relationship between two variables understand the relationship between variables Knowledge (ii) understand the different types of average understand that a hypothesis is an assumption or idea which can be tested by collecting data communicate findings clearly and accurately in charts and graphs link findings back to hypotheses Skills (ii) summarise findings and link back to hypotheses justify methods of collecting and representing data justify conclusion by referring back to the data reflect and critique methods Vocabulary the same, different, how many, most, least, how many more, fewer Vocabulary bar chart , bar line graph , tally chart , table , frequency table, timetable, Carroll diagram, Venn diagram, diagram , label , title , axis, axes , most popular , most common , least popular , least common Vocabulary bar chart , line graph , grouped data chart , conversion graph , pie chart, tally chart , table , frequency table , timetable , Carroll diagram , Venn diagram , diagram , label , title , axis, axes , most popular , most common , least popular , least common Vocabulary positive, negative, correlation, trend, anomaly, estimate, scatter graph, variable Vocabulary correlation, causation, outlier, data, dependent variable, independent variable, context hypothesis, summarise, justify, critique, conclusion, reflect
WM4: Statistics Interpreting and evaluating data Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I can explain my findings and I am beginning to evaluate how well my method worked. I can find and use the mean of a simple set of data to explain how the statistics do, or do not, support an argument. I can recognise how anomalies affect the mean. I can understand that different averages can be used to compare data, including grouped data, recognisingthe advantages and disadvantages of each average Knowledge how to read a bar graph / tally chart / pictogram also see Statistics Interpreting data Knowledge mean is only one type of average *care should be taken to avoid introducing misconceptions with median and mode) understand why finding the mean is useful how to calculate the mean Knowledge understand that there are different types of averages understand how to find the mode (from a list and a table) understand how to find the median (from a list and a table) understand the advantages and disadvantages and best use of each average (e.g. mode - can be used for non-numerical data )use of contextual and real- life examples use of the different averages on one data set to highlight which average represents the data set best use of different averages to compare different data sets Skills calculate the mode form a list and a table calculate the median from a list and a table justify the choice of best average - explaining the advantages and disadvantages compare differences Skills Skills NOTE THAT THIS PS STATEMENT DOES NOT REFER TO MEDIAN / MODE find the mean of small data sets make comparisons begin by representing findings with blocks in a 3D block graph - to find the mean the rearrange their blocks so all columns are of equal height (2 coloured counters could also be used) use the mean to make predictions find the mean of a given data set find the mean of a data set they have collected explore how adding an extra value to a data set will change the mean find the missing value in a data set when the mean is given investigate how an anomaly / outlier in the data can affect the mean e.g. having full marks in 3 tests then scoring 0 in the fourth explain what they have found from their data draw simple conclusions e.g. what is the favourite subject of the class link their conclusion back to their original question / statement consider whether the method they chose to represent their data was the most appropriate - would they choose a different method next time? be aware of an anomalies / outliers in the results Vocabulary bar chart , bar line graph , tally chart , table , frequency table, timetable, Carroll diagram, Venn diagram, diagram , label , title , axis, axes , most popular , most common , least popular , least common Vocabulary mean , average , total, range, maximum/ minimum value , classify, outcome , statistics , distribution , divide, Vocabulary mode, median, mean, reliable, appropriate, represent, outlier, anomaly
WM4: Statistics Probability Progression Step 1 Progression Step 2 Progression Step 3 Progression Step 4 Progression Step 5 I can explore outcomes and chance, using appropriate language, and I am beginning to use numerical values to represent probability. I can systematically explore all the possible mutually exclusive outcomes of successive and combined events. I can use modelling to solve problems involving probabilities of mutually exclusive, independent and dependent events. I have explored the relationship between relative frequency and theoretical probability, and I can make judgements on the outcomes of experimental data. I can use probabilistic arguments, drawing on theory, information, research and experimentation, to make informed decisions. Knowledge understand that some events are likely and some are impossible understand that context can affect the likelihood of events happening / not happening understand we can order these on a scale from 0-1 understand we can represent probability as a fraction, decimal and percentage Knowledge understand and explain mutually exclusive events state whether two events are mutually exclusive or not understand that the total of all possible outcomes must be one - as one of those events must happen apply ideas of randomness and fairness in simple experiments. Know when the addition of probabilities for mutually exclusive and multiplication of probabilities for independent events can be applied. Knowledge know when and how to use the multiplication rule of probabilities for dependent events in context understand when and how to estimate conditional probabilities understand that relative frequencies approach the theoretical probability as the number of trials increases estimate probabilities based on experimental evidence identifying all the outcomes of a combination of two experiments, e.g. tree diagrams, sample space diagrams, Venn diagrams or other diagramatic representations of compound events Skills use appropriate vocabulary to describe the probability of an event (e.g., certain, impossible, even chance, likely and unlikely) use contextual information to decide the likelihood of an event occurring understand how these contextual factors may affect the outcome if they changed order events from certain to impossible estimate, calculate and list the probability of an event occurring carry out probability experiments list the outcomes of these experiments (e.g. spinners, coins, counters, number cards etc) Skills Skills identify and give examples of mutually exclusive events calculate probabilities of mutually exclusive events conduct experiments and find all possible outcomes find all possible outcomes in a given situation find missing probabilities when a total is given find the probability of an event not occurring compare an estimated probability from experimental results with a theoretical probability raphical representation of relative frequency against the number of trials record experimental outcomes using appropriate diagrams use graph skills to record and interpret results apply knowledge of fractions to calculating probabilities Identify if events are dependent , independent or mutually exclusive Vocabulary Probability, possible, impossible, likely, unlikely, even chance, certain, fair, random, frequency, probability scale, outcome, fraction, decimal, percentage, Vocabulary independent, mutually exclusive, exhaustive , sum, sample space, successive, combined Vocabulary Relative frequency, theoretical probability, dependent, experimental
