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NRICH http://nrich.maths.org Problems Linked to the Primary National Curriculum for Mathematics in Years 3, 4, 5 and 6 NRICH tasks embrace the aims of the curriculum (problem solving, reasoning and fluency) as well as curriculum ‘content’ (further information). The stars indicate the level of confidence and competence needed to begin the activity. One star problems will be suitable for the whole class, two stars for the majority and three stars for those who like a serious challenge. The activity listings now include what type of activity they are: games are indicated by ‘G’, problems by ‘P’ and investigations by ‘I’. N.B. This is work in progress – we would really appreciate your comments. Please email [email protected] Year 3 Year 4 Year 5 Year 6 Strand 1 – Number Count from 0 in multiples of 4, 8, 50 Count in multiples of 6, 7, 9, 25 and Read, write, order and compare Read, write, order and compare numbers up and 100; find 10 or 100 more or less 1000 numbers to at least 1 000 000 and to 10 000 000 and determine the value of than a given number determine the value of each digit each digit NRICH: How Would We Count? * P Recognise the place value of each Find 1000 more or less than a given Count forwards or backwards in steps of Round any whole number to a required digit in a three-digit number (hundreds, number powers of 10 for any given number up to degree of accuracy tens, ones) 1 000 000 NRICH: Coded Hundred Square * P NRICH: Which Scripts? * P Compare and order numbers up to Count backwards through zero to Interpret negative numbers in context, Use negative numbers in context, and 1000 include negative numbers count forwards and backwards with calculate intervals across zero positive and negative whole numbers NRICH: First Connect Three * G P through zero NRICH: Tug Harder! * G NRICH: Swimming Pool* P NRICH: Sea Level * P I Identify, represent and estimate Recognise the place value of each digit Round any number up to Solve number and practical problems that numbers using different in a four-digit number (thousands, 1 000 000 to the nearest 10, 100, 1000, involve all of the above representations hundreds, tens and ones) 10 000 and 100 000 NRICH: Round the Four Dice * P I NRICH: Nice or Nasty * G Number andNRICH: Place Dicey Value Operations * G NRICH: The Deca Tree * P NRICH: Four-digit Targets * P Read and write numbers up to 1000 in Order and compare numbers beyond Solve number problems and practical numerals and in words 1000 problems that involve all of the above

January 2016 1 To make sure this document displays correctly in Word 2000, click on the Table menu, choose Table Properties and then select the option which has no text wrapping Solve number problems and practical Identify, represent and estimate Read Roman numerals to 1000 (M) and problems involving these ideas numbers using different representations recognise years written in Roman NRICH: Take Three Numbers * I numerals NRICH: Three Neighbours ** I NRICH: Square Subtraction *** I NRICH: Planning a School Trip * p NRICH: Magic Vs ** P NRICH: Number Differences * G P NRICH: Sitting Round the Party Tables * P NRICH: Number Match * G NRICH: A Mixed-up Clock * P NRICH: That Number Square! * I Round any number to the nearest 10, 100 or 1000 NRICH: Reasoned Rounding * G Solve number and practical problems that involve all of the above and with increasingly large positive numbers Read Roman numerals to 100 (I to C) and know that over time, the numeral system changed to include the concept of zero and place value Add and subtract numbers mentally, Add and subtract numbers with up to 4 Add and subtract whole numbers with including: digits using the formal written methods more than 4 digits, including using  a three-digit number and ones of columnar addition and subtraction formal written methods (columnar  a three-digit number and tens where appropriate addition and subtraction)  a three-digitAddition number and and Subtraction hundreds

Add and subtract numbers with up to Estimate and use inverse operations to Add and subtract numbers mentally with three digits, using formal written check answers to a calculation increasingly large numbers methods of columnar addition and subtraction Estimate the answer to a calculation Solve addition and subtraction two-step Use rounding to check answers to and use inverse operations to check problems in contexts, deciding which calculations and determine, in the

January 2016 2 To make sure this document displays correctly in Word 2000, click on the Table menu, choose Table Properties and then select the option which has no text wrapping answers operations and methods to use and why context of a problem, levels of accuracy NRICH: Money Bags ** P NRICH: Amy’s Dominoes ** P NRICH: Fifteen Cards * P I NRICH: Sealed Solution ** P NRICH: Roll These Dice ** I

Solve problems, including missing Solve addition and subtraction multi-step number problems, using number facts, problems in contexts, deciding which place value, and more complex operations and methods to use and why addition and subtraction NRICH: Twenty Divided Into Six ** P NRICH: Buying a Balloon * P NRICH: Reach 100 *** P NRICH: GOT IT ** G NRICH: Maze 100 ** P NRICH: Make 37 ** P NRICH: Six Ten Total ** P I NRICH: Consecutive Numbers ** P I NRICH: Six Numbered Cubes ** P NRICH: Super Shapes * P NRICH: Strike it Out * G NRICH: Dice in a Corner *** P I NRICH: Domino Square ** P NRICH: Dicey Addition * G NRICH: 4 Dom *** P NRICH: Finding Fifteen ** P

Recall and use multiplication and Recall multiplication and division facts Identify multiples and factors, including Multiply multi-digit numbers up to 4 digits by a division facts for the 3, 4 and 8 for multiplication tables up to 12x12 all factor pairs of a number, and two-digit whole number using the formal multiplication tables NRICH: Multiplication Square Jigsaw common factors of two numbers written method of long multiplication NRICH: Ordering Cards * G P * G P NRICH: Sweets in a Box * P I NRICH: Music to My Ears * P I NRICH: Shape Times Shape * P NRICH: Which Is Quicker? * P NRICH: Table Patterns Go Wild! ** I NRICH: Multiplication Squares * P I NRICH:Multiplication Let Us Divide! and * P Division NRICH: Flashing Lights * P NRICH: Carrying Cards * P NRICH: Abundant Numbers * I NRICH: Light the Lights Again * G P NRICH: Factor Track ** G P NRICH: Multiples Grid * I NRICH: Factors and Multiples Game * NRICH: Zios and Zepts * P G NRICH: Times Tables Shifts * G P NRICH: Pebbles ** I

January 2016 3 To make sure this document displays correctly in Word 2000, click on the Table menu, choose Table Properties and then select the option which has no text wrapping Write and calculate mathematical Use place value, known and derived Know and use the vocabulary of prime Divide numbers up to 4 digits by a two-digit statements for multiplication and facts to multiply and divide mentally, numbers, prime factors and composite whole number using the formal written method division using the multiplication tables including: multiplying by 0 and 1; (non-prime) numbers of long division, and interpret remainders as that they know, including for two-digit dividing by 1; multiplying together three NRICH: Two Primes Make One whole number remainders, fractions, or by numbers times one-digit numbers, numbers Square ** I rounding, as appropriate for the context using mental and progressing to formal written methods Solve problems, including missing Recognise and use factor pairs and Establish whether a number up to 100 is Divide numbers up to 4 digits by a two-digit number problems, involving commutativity in mental calculations prime and recall prime numbers up to 19 number using the formal written method of multiplication and division, including short division where appropriate, interpreting positive integer scaling problems and remainders according to the context correspondence problems in which n objects are connected to m objects NRICH: A Square of Numbers * G P NRICH: What do you Need? * P NRICH: This Pied Piper of Hamelin ** P NRICH: Follow the Numbers * P I NRICH: What's in the Box? * P NRICH: How Do You Do It? * P NRICH: Ip Dip * I NRICH: Journeys in Numberland * I

Multiply two-digit and three-digit Multiply numbers up to 4 digits by a one- Perform mental calculations, including with numbers by a one-digit number using or two-digit number using a formal mixed operations and large numbers formal written layout written method, including long NRICH: Exploring Number Patterns You multiplication for two-digit numbers Make ** P I NRICH: All the Digits ** P NRICH: Become Maths Detectives * P I NRICH: Trebling * P

Solve problems involving multiplying and Multiply and divide numbers mentally Identify common factors, common multiples adding, including using the distributive drawing upon known facts and prime numbers law to multiply two digit numbers by one NRICH: Mystery Matrix ** P I digit, integer scaling problems and NRICH: Factor Lines ** P I harder correspondence problems such NRICH: Factor-multiple Chains ** P as n objects are connected to m objects NRICH: The Moons of Vuvv * P NRICH: Round and Round the Circle ** P I NRICH: Counting Cogs ** P

January 2016 4 To make sure this document displays correctly in Word 2000, click on the Table menu, choose Table Properties and then select the option which has no text wrapping Divide numbers up to 4 digits by a one- Use their knowledge of the order of operations digit number using the formal written to carry out calculations involving the four method of short division and interpret operations remainders appropriately for the context NRICH: Division Rules * P I

Multiply and divide whole numbers and Solve addition and subtraction multi-step those involving decimals by 10, 100 and problems in contexts, deciding which 1000 operations and methods to use and why NRICH: Multiply Multiples 1 * P NRICH: Multiply Multiples 2 * P NRICH: Multiply Multiples 3 * P

Recognise and use square numbers and Solve problems involving addition, subtraction, cube numbers, and the notation for multiplication and division squared (²) and cubed (³) NRICH: Up and Down Staircases * P NRICH: One Wasn’t Square ** P NRICH: Cycling Squares ** P NRICH: Picture a Pyramid … ** P

Solve problems involving multiplication Use estimation to check answers to and division including using their calculations and determine, in the context of a knowledge of factors and multiples, problem, an appropriate degree of accuracy squares and cubes NRICH: Four Go * G NRICH: Curious Number *** P I NRICH: Division Rules * P I NRICH: Odd Squares * P NRICH: Cubes Within Cubes *** P Solve problems involving addition, subtraction, multiplication and division and a combination of these, including understanding the meaning of the equals sign NRICH: Make 100 ** P I NRICH: Multiply Multiples 1 * I NRICH: Multiply Multiples 2 * I NRICH: Multiply Multiples 3 * I

January 2016 5 To make sure this document displays correctly in Word 2000, click on the Table menu, choose Table Properties and then select the option which has no text wrapping NRICH: Highest and Lowest * P I NRICH: Four Goodness Sake *** P

Solve problems involving multiplication and division, including scaling by simple fractions and problems involving simple rates Count up and down in tenths; Recognise and show, using diagrams, Compare and order fractions whose Use common factors to simplify fractions; use recognise that tenths arise from families of common equivalent fractions denominators are all multiples of the common multiples to express fractions in the dividing an object into 10 equal parts NRICH: Fractional Wall * P same number same denomination and in dividing one-digit numbers or NRICH: Fractional Triangles * P quantities by 10 NRICH: Bryony’s Triangle * P

Recognise, find and write fractions of Count up and down in hundredths; Identify, name and write equivalent Compare and order fractions, including a discrete set of objects: unit fractions recognise that hundredths arise when fractions of a given fraction, represented fractions >1 and non-unit fractions with small dividing an object by one hundred and visually, including tenths and hundredths denominators dividing tenths by ten NRICH: Fraction Match * G Recognise and use fractions as Solve problems involving increasingly Recognise mixed numbers and improper Add and subtract fractions with different numbers: unit fractions and non-unit harder fractions to calculate quantities, fractions and convert from one form to denominators and mixed numbers, using the fractions with small denominators and fractions to divide quantities, the other and write mathematical concept of equivalent fractions including non-unit fractions where the statements > 1 as a mixed number (e.g. answer is a whole number 2/5 + 4/5 = 6/5 = 1 1/5) NRICH: Andy’s Marbles ** P NRICH: Balance of Halves * P NRICH: Fractions in a Box ** P- Fractions, Decimals, Percentages, Ratio and Proportion NRICH: Chocolate ** P I Recognise and show, using diagrams, Add and subtract fractions with the same Add and subtract fractions with the same Multiply simple pairs of proper fractions, equivalent fractions with small denominator denominator and denominators that are writing the answer in its simplest form [for denominators multiples of the same number example, ¼ x ½ = 1/8] NRICH: Matching Fractions * G Add and subtract fractions with the Recognise and write decimal Multiply proper fractions and mixed Divide proper fractions by whole numbers [for same denominator within one whole equivalents of any number of tenths or numbers by whole numbers, supported example, 1/3 ÷ 2 = 1/6] [for example, 5/7 + 1/7 = 6/7] hundredths by materials and diagrams

January 2016 6 To make sure this document displays correctly in Word 2000, click on the Table menu, choose Table Properties and then select the option which has no text wrapping Compare and order unit fractions, and Recognise and write decimal Read and write decimal numbers as Associate a fraction with division and calculate fractions with the same denominators equivalents to ¼; ½; ¾ fractions (e.g. 0.71 = 71/100) decimal fraction equivalents [for example, 0.375] for a simple fraction [for example, 3/8] Solve problems that involve all of the Find the effect of dividing a one- or two- Recognise and use thousandths and Identify the value of each digit in numbers above digit number by 10 and 100, identifying relate them to tenths, hundredths and given to three decimal places and multiply and the value of the digits in the answer as decimal equivalents divide numbers by 10, 100 and 1000 giving ones, tenths and hundredths answers up to three decimal places Round decimals with one decimal place Round decimals with two decimal places Multiply one-digit numbers with up to two to the nearest whole number to the nearest whole number and to one decimal places by whole numbers NRICH: Round the Dice Decimals 1 * decimal place P I NRICH: Round the Dice Decimals 2 *

Compare numbers with the same Read, write, order and compare Use written division methods in cases where number of decimal places up to two numbers with up to three decimal places the answer has up to two decimal places decimal places NRICH: Greater Than or Less Than? * I NRICH: Spiralling Decimals *** G

Solve simple measure and money Solve problems involving number up to Solve problems which require answers to be problems involving fractions and three decimal places rounded to specified degrees of accuracy decimals to two decimal places NRICH: Route Product ** P I NRICH: Forgot the Numbers ** I

Recognise the per cent symbol (%) and Recall and use equivalences between simple understand that per cent relates to fractions, decimals and percentages, including “number of parts per hundred”, and write in different contexts percentages as a fraction with NRICH: Doughnut Percents * P denominator 100, and as a decimal Solve problems which require knowing percentage and decimal equivalents of ½, ¼, 1/5, 2/5, 4/5 and those fractions with a denominator a multiple of 10 or 25 NRICH: Matching Fractions Decimals Percentages * G Solve problems involving the relative sizes of two quantities where missing values can be found by using integer multiplication and

January 2016 7 To make sure this document displays correctly in Word 2000, click on the Table menu, choose Table Properties and then select the option which has no text wrapping division facts NRICH: Orange Drink ** P NRICH: Pumpkin Pie Problem ** P NRICH: Jumping * P NRICH: Rectangle Tangle * P NRICH: Fraction Fascination *** P

Solve problems involving the calculation of percentages [for example, of measures, and such as 15% of 360] and the use of percentages for comparison NRICH: Would you Rather? * P

Solve problems involving similar shapes where the scale factor is known or can be found Solve problems involving unequal sharing and grouping using knowledge of fractions and multiples Use simple formulae Generate and describe linear number sequences Algebra NRICH: Domino Sets * P I NRICH: Break it Up! * P I NRICH: Button-up Some More ** I NRICH: Holes * P I Express missing number problems algebraically NRICH: Plenty of Pens * P NRICH: Two and Two *** P I Find pairs of numbers that satisfy an equation with two unknowns Enumerate possibilities of combinations of two variables Year 3 Year 4 Year 5 Year 6 Strand 2 - Measurement

January 2016 8 To make sure this document displays correctly in Word 2000, click on the Table menu, choose Table Properties and then select the option which has no text wrapping Measure, compare, add and subtract: Convert between different units of Convert between different units of metric Solve problems involving the calculation and lengths (m/cm/mm); mass (kg/g); measure [for example, kilometre to measure [for example, kilometre and conversion of units of measure, using decimal volume/capacity (l/ml) metre; hour to minute] metre; centimetre and metre; centimetre notation up to three decimal places where NRICH: Oh! Harry! ** P and millimetre; gram and kilogram; litre appropriate NRICH: Olympic Starters * I and millilitre] NRICH: Car Journey * I Measure the perimeter of simple 2-D Measure and calculate the perimeter of Understand and use approximate Use, read, write and convert between shapes a rectilinear figure (including squares) in equivalences between metric units and standard units, converting measurements of centimetres and metres common imperial units such as inches, length, mass, volume and time from a smaller pounds and pints unit of measure to a larger unit, and vice versa, using decimal notation up to three decimal places Add and subtract amounts of money to Find the area of rectilinear shapes by Measure and calculate the perimeter of Convert between miles and kilometres give change, using both £ and p in counting squares composite rectilinear shapes in practical contexts NRICH: Torn Shapes * P I centimetres and metres NRICH: How Much Did It Cost? ** P NRICH: Twice as Big? * P NRICH: Area and Perimeter * I NRICH: Through the Window * I Tell and write the time from an Estimate, compare and calculate Calculate and compare the area of Recognise that shapes with the same areas analogue clock, including using different measures, including money in rectangles (including squares), and can have different perimeters and vice versa Roman numerals from I to XII, and 12- pounds and pence including using standard units, square NRICH: Dicey Perimeter, Dicey Area * G hour and 24-hour clocks NRICH: Discuss and Choose * P centimetres (cm2) and square metres NRICH: What is the Time? * P (m2) and estimate the area of irregular NRICH: Two Clocks ** P shapes NRICH: Clocks * P NRICH: Numerically Equal ** P NRICH: The Time Is … ** P NRICH: Shaping It * I NRICH: How Many Times? * I NRICH: Cubes * P I NRICH: 5 on the Clock *** I NRICH: Fitted *** P NRICH: Brush Loads * P I NRICH: Making Boxes ** I NRICH: Ribbon Squares *** P Estimate and read time with increasing Read, write and convert time between Estimate volume [for example, using 1 Recognise when it is possible to use formulae accuracy to the nearest minute; record analogue and digital 12- and 24-hour cm³ blocks to build cuboids (including for area and volume of shapes and compare time in terms of clocks cubes)] and capacity [for example, using seconds, minutes and hours; use water] vocabulary such as o’clock, a.m./p.m., morning, afternoon, noon and midnight NRICH: Wonky Watches ** P NRICH: Watch the Clock *** P

January 2016 9 To make sure this document displays correctly in Word 2000, click on the Table menu, choose Table Properties and then select the option which has no text wrapping Know the number of seconds in a Solve problems involving converting Solve problems involving converting Calculate the area of parallelograms and minute and the number of days in from hours to minutes; minutes to between units of time triangles each month, year and leap year seconds; years to months; weeks to days Compare durations of events [for Use all four operations to solve Calculate, estimate and compare volume of example to calculate the time taken by problems involving measure [e.g. length, cubes and cuboids using standard units, particular events or tasks] mass, volume, money] using decimal including cubic centimetres (cm³) and cubic notation, including scaling metres (m³), and extending to other units [for example, mm³ and km³] NRICH: Next Size Up ** P Year 3 Year 4 Year 5 Year 6 Strand 3 – Geometry Draw 2-D shapes and make 3-D Compare and classify geometric shapes, Identify 3-D shapes, including cubes and Draw 2-D shapes using given dimensions and shapes using modelling materials; including quadrilaterals and triangles, other cuboids, from 2-D representations angles recognise 3-D shapes in different based on their properties and sizes NRICH: Making Spirals *** P orientations and describe them NRICH: Nine-pin Triangles *** I NRICH: Shape Draw * P NRICH: Building Blocks * P NRICH: Cut it Out *** P NRICH: Baravelle * P NRICH: The Third Dimension *** P I NRICH: Sorting Logic Blocks * G NRICH: Rolling ThatProperties Cube * P of ShapesNRICH: What Shape? * G P NRICH: Inky Cube *** P NRICH: Shapes on the Playground ** NRICH: Triple Cubes * I P NRICH: Sponge Sections ** P NRICH: A Puzzling Cube * P NRICH: Arranging Cubes * G NRICH: Board Block Challenge *** G NRICH: Square Corners ** P NRICH: Stick Images * G P NRICH: Overlapping Again ** P Recognise angles as a property of Identify acute and obtuse angles and Know angles are measured in degrees; Recognise, describe and build simple 3-D shape or a description of a turn compare and order angles up to two estimate and compare acute, obtuse shapes, including making nets right angles by size and reflex angles NRICH: Cut Nets ** P NRICH: Making Cuboids ** P I

January 2016 10 To make sure this document displays correctly in Word 2000, click on the Table menu, choose Table Properties and then select the option which has no text wrapping Identify right angles, recognise that Identify lines of symmetry in 2-D shapes Draw given angles, and measure them Compare and classify geometric shapes two right angles make a half-turn, presented in different orientations in degrees (°) based on their properties and sizes and find three make three quarters of a turn NRICH: Let Us Reflect * P NRICH: The Numbers Give the unknown angles in any triangles, and four a complete turn; identify NRICH: Stringy Quads ** P Design * I quadrilaterals, and regular polygons whether angles are greater than or NRICH: Counters in the Middle * G P NRICH: Six Places to Visit * P NRICH: Where Are They? * P less than a right angle NRICH: How Safe Are You? * P NRICH: Quadrilaterals *** P I NRICH: Square It * G NRICH: Olympic Turns *** P NRICH: Round a Hexagon * P Identify horizontal and vertical lines Complete a simple symmetric figure with Identify: Illustrate and name parts of circles, including and pairs of perpendicular and parallel respect to a specific line of symmetry  angles at a point and one whole radius, diameter and circumference and know lines NRICH: Symmetry Challenge *** I turn (total 360°) that the diameter is twice the radius NRICH: National Flags * P NRICH: School Fair Necklaces ** I  angles at a point on a straight line and ½ a turn (total 180°)  other multiples of 90° Use the properties of rectangles to Recognise angles where they meet at a point, deduce related facts and find missing are on a straight line, or are vertically lengths and angles opposite, and find missing angles

Distinguish between regular and irregular polygons based on reasoning about equal sides and angles NRICH: Egyptian Rope ** P I NRICH: Bracelets * I

Describe positions on a 2-D grid as Identify, describe and represent the Describe positions on the full coordinate grid coordinates in the first quadrant position of a shape following a reflection (all four quadrants) NRICH: Coordinate Challenge * P or translation, using the appropriate NRICH: Cops and Robbers * G NRICH: Eight Hidden Squares ** P language, and know that the shape has NRICH: Coordinate Tan ** P not changed NRICH: Ten Hidden Squares *** P NRICH: Transformations on a Pegboard * P NRICH: More Transformations on a Pegboard ** P I Position and Direction

Describe movements between positions Draw and translate simple shapes on the as translations of a given unit to the coordinate plane, and reflect them in the axes left/right and up/down

January 2016 11 To make sure this document displays correctly in Word 2000, click on the Table menu, choose Table Properties and then select the option which has no text wrapping Plot specified points and draw sides to complete a given polygon. NRICH: A Cartesian Puzzle * P

January 2016 12 To make sure this document displays correctly in Word 2000, click on the Table menu, choose Table Properties and then select the option which has no text wrapping Year 3 Year 4 Year 5 Year 6 Strand 4 - Statistics Interpret and present data using bar Interpret and present discrete and Solve comparison, sum and difference Interpret and construct pie charts and line charts, pictograms and tables continuous data using appropriate problems using information presented in graphs and use these to solve problems NRICH: How Big Are Classes 5, 6 graphical methods, including bar charts a line graph NRICH: Match the Matches ** P and 7? * P and time graphs NRICH: Our Sports * I NRICH: Class 5’s Names * P NRICH: Going for Gold * I NRICH: The Domesday Project * I NRICH: The Car That Passes * I NRICH: Now and Then ** P NRICH: Real Statistics *** P NRICH: If the World Were a Village * P NRICH: It's a Tie ** I Solve one-step and two-step Solve comparison, sum and difference Complete, read and interpret information Calculate and interpret the mean as an questions [for example, ‘How many problems using information presented in in tables, including timetables average more?’ and ‘How many fewer?’] using bar charts, pictograms, tables and other NRICH: Birdwatch * I information presented in scaled bar graphs charts and pictograms and tables NRICH: Venn Diagrams * P NRICH: The Olympic Flame: Are NRICH: More Carroll Diagrams * P You in the 95%? * P NRICH: Plants ** I

January 2016 13 To make sure this document displays correctly in Word 2000, click on the Table menu, choose Table Properties and then select the option which has no text wrapping