The Dyscovery Programme

The Intervention is intended to be structured and cumulative. It is important that each learner starts at thebeginning and works their way through. One cannot see what parts of the brain are working when alearner is undertaking an activity; many people with dyscalculia can undertake complex calculations, butcannot understand the underlying concepts. It would be easy to assume that since the learner cancomplete such complex tasks that there can’t possibly be any underlying difficulty. Whilst I appreciate thatan in depth assessment of specific areas of strength and weakness is important (e.g. Emerson and Babtie), Istrongly believe that it is important that we cover aspect of number sense that could well be hidden bywhat is observable.There is no set time for each session, although an hour is the assumed maximum. Learners can repeatsessions as necessary until mastery is achieved; over-learning is a helpful strategy for those with SpecificLearning Difficulties and it is important to ensure that all concepts introduced are fully understood beforemoving on to the next topic. Conversely, learners may cover a more than one session within their hour ofclass time, although this is unlikely; extension activities are provided for more able learners.Lessons are based around various frameworks. Each session will include elements of the Concrete-Representational-Abstract, Scaffolding-and-Fading, Learner Powers and Social Construction frameworks.These frameworks map to the various elements of the whole-brain learning approach. In turn, theframeworks, ‘whole-brain approach’ and individual activities will be charted to the cognitive, neurological,behavioural and environmental aspects of dyscalculia. There are various overlaps within and betweenframeworks and activities, which will become apparent through this intervention rationale and the sessionnarratives. The Dyscovery Programme |Eleanor Machin 2

The Dyscovery Programme |Eleanor Machin 3

Session Topic Overview and Rationale Resources Curriculum Links1 Subitisation The main aim of this session is to start improving the skill of subitisation (i.e. the ability to Dot enumeration Cards N1/E1.1 recognise the number of objects in a group without counting). This is commonly a weak N1/E1.3 area for learners with dyscalculia. It is useful to be able to automatically recognise small Dice, playing cards, groups of objects without having to count them individually. dominoes Drill and rote practice of counting is not helpful for learners with dyscalculia. Emphasis will Counters/beads remain on the use of number words, seeing numbers as clusters of ‘ones’, the utilisation of Sticky dots fingers and more laborious methods of doing calculations. This session aims to introduce number patterns to enable learners to ‘see’ numbers as quantities and sizes rather than a list of number words. Learners will also begin to recognize how larger numbers can be ‘split’ into groups of smaller numbers.2 Creating The session begins with practise on the skill of subitization. Research has shown that ‘over- Array cards and N1/E1.1, 1.3 and Numbers learning’ and repeated practice will generally improve performance in individuals with counters 1.4 dyscalculia. Cuisenaire rods The aim of the next part of the session is to transpose arrays into base 10. Learners are Diene’s Blocks & base essentially ‘building’ numbers using base 10 materials. It is important that learners with 10 template dyscalculia can conceptualise the Base 10 system in order to move away from using their fingers, number words and counting ones. Using the base 10 materials means that the concepts is transposed into a format that they can see and touch. This session is an introduction to Base 10 materials and will be built on in session 5. Finally, learners will construct their own usable number line.

Session Topic Overview and Rationale Resources Curriculum Links3 Calculations This session looks at completing simple addition and subtraction using the familiar arrays Learners’ own number N1/E1.1, 1.2,4 with the and the number line created in the last session. line. 1.4, 1.5, 1.7 Number Line Learners will identify how arrays and the number line link together in simple calculations 1, 5 and 10 rods lengths and, hence, build on knowledge gained in Session 1 (creating larger numbers using smaller Numeral cards ones) and prepare them for the next session on number bonds and partitioning. Dot pattern cards Random array cards Learners will develop and practice the visualisation and use of a mental number line. It is Dry wipe markers hoped that this should, implicitly, build a foundation for the concept of rounding and, hence, aid estimation. Number bonds Using dot patterns and random arrays, learners practise splitting numbers, up to at least Dot enumeration test N1/E1.1, 1.4, and Partitioning 10, into various component parts. They discover, by creating number triads, how number sheets 1.5, 1.7 partitioning relates to addition and subtraction through 10. They will start to develop a ‘counting-on’ instead of ‘counting-all’ strategy, using partitioning to help speed up the Number triad wipe-off N1/E2.1, 2.5 process. templates Learners will be able to recognise number bonds and create their own rules of working out Numeral and function number bonds. These activities should also provide the basis for learning about reverse cards calculations for checking. Cuisenairre rods Students have further opportunity to work with concrete materials, but this session Counters encourages them to move through the process to the abstract by way of using visualisation and Arabic numerals. Envelopes The Dyscovery Programme |Eleanor Machin 5

Session Topic Overview and Rationale Resources Curriculum Links5 N1/E2.1 Base 10 and This session will introduce the concept of base 10 and the idea that the position of digits Diene’s blocks N1/E2.2 Place Value within the place value chart effects their value. Dyscalculic learners often struggle to make Laminated Place Value sense of the number structure above 100 and find calculations involving borrowing and mats (A3 and A4) carrying confusing. Dry wipe markers Base 10 number strips This session aims to equip learners with knowing-why digits can have different values and Laminated digit cards knowing-how base 10 units fit together. Learners will be confident in ‘translating’ the spoken numbers (number words) with concrete representations and written numerals.6 Calculations This session will cover simple calculations using place value; addition with carrying, Base 10 materials N1/E2.3 using Place subtraction with borrowing and multiplying and dividing by 10, 100 and 1000. ‘Trading’ worksheets N1/E2.4 Value Place value mats N1/E2.7 The session aims to make explicit the reasons behind the processes and clear up common misconceptions (i.e. take the smaller number from the bigger number, multiplying by 10 is adding a zero etc). Place value slider Exercises using concrete manipulatives will make it clear what is actually happening when Numeral cards and digit you carry a digit during addition, or borrow a digit during subtraction. This deeper strips. understanding will help learners remember the process and identify when something in a calculation has gone wrong. The Dyscovery Programme |Eleanor Machin 6

Session Topic Overview and Rationale Resources Curriculum Links7 N1/E2.5 Multiplication This session will use the repeated adding to introduce the concept of multiplication. Using Array patterns N1/E2.7 as Repeat their ability to subitise up to 5, they will be able to count a group of objects without Laminated Resource (N1/E3.6) Addition ‘counting-all’. Sheets (7a, b, c and d) Learners will discover that numbers used in multiplication are reversible (i.e. 2 x 3 is the Dry-wipe markers same as 3 x 2) so they can choose their preferred times table when completing a sum. They will develop confidence in their 2, 3, 4 and 5 times tables. Counters In addition to the idea of ‘chunking’ numbers, learners will also start to think about breaking up numbers, which forms the basis of division. Some knowledge of the relationship between multiplication and division may begin to be formed.8 Multiplication: Following on from the introduction to multiplication, this session looks into patterns Multiplication Square N1/E3.5 Hints, Tips and created by the multiplication tables. ‘Bridge Method’ hand- N1/L1.6 Tricks outs (N1/L1.5) Learners will be introduced to the multiplication square, and will use this and other methods to find patterns in the numbers. Learners will discover how to use their ‘Bridge Method’ knowledge of the ‘easier’ times tables to help them with the more difficult ones. They will laminated A3 work- also practice various methods of recalling times tables, including the ‘fingers’ and the mats ‘bridge’ method of the 7s, 8s and 9s. Learners will identify which methods work best for them as individuals and why. The Dyscovery Programme |Eleanor Machin 7

Session Topic Overview and Rationale Resources Curriculum Links9 Resource Sheet 9a N1/E3.6 Division: Sharing Revisiting, and following on from, session 7, learners are introduced to the concept of Multiplication Square and Inverse division both as repeated equal grouping and inverse/reversed multiplication. Multiplication Learners will be able to discuss and identify mathematical vocabulary related to division. Learners will be guided to discover how to use their multiplication square to complete straight-forward division sums, and will identify that multiplication and division sums can be placed into number triads in the same way as addition and subtraction sums in partitioning.10 The Continuous This session revisits and consolidates learning on the number line. It also moves on from 30cm ruler, tape N2/E3.1 Number Line: the Diene’s/Cuisenairre version of the number line, which incorporates only whole measure, thermometer, Parts of numbers, to the continuous number line, which involves starting to look at parts of number line, measuring N2/E3.2 Numbers numbers. strips, resource sheet, (N2/L1.1) Base 10 materials. Students should begin to develop a concept of fractions and possibly decimals in the context of simple measures. The Dyscovery Programme |Eleanor Machin 8

Overview and RationaleThe main aim of this session is to start improving the skill of subitisation (i.e. the ability to recognise thenumber of objects in a group without counting). This is commonly a weak area for learners withdyscalculia. It is useful to be able to automatically recognise small groups of objects without having tocount them individually.Drill and rote practice of counting is not helpful for learners with dyscalculia; emphasis will remain on theuse of number words, seeing numbers as clusters of ‘ones’, the utilisation of fingers and more laboriousmethods of doing calculations. This session aims to introduce number patterns to enable learners to ‘see’numbers as quantities and sizes rather than a list of number words. Learners will also begin to recognizehow larger numbers can be ‘split’ into groups of smaller numbers.ResourcesDot enumeration CardsDice, playing cards, dominoesCounters/beadsSticky dotsCurriculum LinksN1/E1.1N1/E1.3Part 1Start the session by challenging learners to represent a small number without using the word or thewritten numeral. Draw different suggestions on the board. Make a point of exaggerating some of the sizesof the images you draw. Ask, ‘does the size matter? Is it still x if one group is bigger than the other?’ (SeeFig 1) Fig. 1 Representing ‘six’

Part 2This episode starts with a brief discussion about where we see numbers represented as arrays of dots orimages. The answers we are looking for are dice, playing cards and dominoes (or any other similarresponse). Look at the patterns on the items. Which are the same and which are different? Why are they ina pattern? What would happen if the dots were painted on randomly?You could give an example here of how 8 is represented on a playing card and different ways of how itcould be represented by two die or a domino…Part 3Learners should recreate the dot patterns with counters. For numbers 7-10, learners should work in pairsto decide if there are any easier ways of representing the numbers. Why is this easier? Some learners maynotice that, for example, an 8 may be better represented as two arrays of 4 (see Fig 2). Fig. 2 Some examples of dot patterns for larger numbers The Dyscovery Programme |Eleanor Machin 10

Part 4This episode incorporates an element of formative self-assessment. Individually, learners should identifyhow far they can subitise. They should then get into pairs and investigate ways of creating larger numbers(up to 10) using only the numbers that they can currently recognise automatically, for example, if a learnercould recognise 4 but not 5, then in order to make 7, they might opt to use a 4 and a 3 rather than a 2 anda 5. After investigating various ‘designs’ learners can create their own number patterns using circles of cardand sticky dots. These are then the learners’ own number pattern cards to keep and use in the future.Part 5The final episode in this session is all about using visualisation. Learners are given random piles ofcounters, usually up to 10. Consider the ability of the learner; for learners who are struggling you maydecide to give them six or seven counters. For learners who are finding the activities easy you may decideto give more than 10. Encourage the learner to firstly estimate how many counters they have got. Nowarrange the counters into the recognised number pattern. Did they estimate correctly? If not, how closewere they? Repeat this two or three more times.Now give out random sets again, but this time, encourage the learners to close their eyes and seethemselves rearranging the counters in their ‘mind’s eye’, rather than physically moving the counters.Practice this with various numbers in random arrangements. If the learner still needs to practice physicallymoving the counters then this should be allowed for as long as necessary.Now repeat the first activity in this episode, this time encouraging the learners to use the visualisationtechnique when estimating. Did they estimate correctly? If not, was their estimate better than last time? The Dyscovery Programme |Eleanor Machin 11

Overview and RationaleThe session begins with practise on the skill of subitization. Research has shown that ‘over-learning’ andrepeated practice will generally improve performance in individuals with dyscalculia. The aim of the nextpart of the session is to transpose arrays into base 10. Learners are essentially ‘building’ numbers usingbase 10 materials. It is important that learners with dyscalculia can conceptualise the Base 10 system inorder to move away from using their fingers, number words and counting ones. Using the base 10materials means that the concepts is transposed into a format that they can see and touch. This session isan introduction to Base 10 materials and will be built on in session 5.Finally, learners will construct their own usable number line.ResourcesArray cards and countersCuisenaire rodsDiene’s Blocks & base 10 templatePreliminaryHave ready one of each dot enumeration card – one set of patterns and one set of random arrangements.Shuffle the cards, but make sure you keep a note of their order; you need to note both the number andwhether they are pattern or random. Give each learner one dot enumeration test sheet. Hold each card upfor 3 seconds, then allow a couple more seconds for the learner to record their answers. Your learnersshould circle the number they think is represented on the card.Let the learners mark their own work. We would expect to find that smaller numbers and numbers that arein patterns will be answered more accurately.Ask the learners:Did you use any strategies from last session? What strategies might you have used? The Dyscovery Programme |Eleanor Machin 12

Fig.1 Completed dot enumeration test sheetPart 1Learners should work in pairs or small groups. Introduce the Cuisenairre rods; give out a full set each.Instruct learners to give each rod a number value. Compare answers between groups – if answers differ,ask learners to use the materials available (let the learners use their initiative – if necessary, direct themtowards the Base 10 rods and blocks, Base 10 template etc.) to check their answer and construct anargument as to why their answer is the correct one. They are encouraged to deduce that the Cuisenairrerods can be ‘measured’, and therefore have a direct relationship with, the Base 10 materials (see Fig 2). Fig 2 – Comparing the Cuisenairre staircase to various Base 10 blocksEach pair/group shares their methods and reasoning with the rest of the class. The Dyscovery Programme |Eleanor Machin 13

Part 2This part of the session is adapted from the Let’s Think Through Maths activity, ‘How Big is the Number?’The aim is to create a number line that can be used for calculations involving simple addition andsubtraction.Each learner receives a 1, a 5, and a 10 rod and a copy of a blank number line. You will need enlarged(scale) versions of the 1, 5 and 10 rods. Draw an empty number line on the board to enable modelling. Askthe learners how we can mark the number 10 on the number line. What is the easiest way? What if weonly had ‘5’ and ‘1’ rods? Or if we only had ‘5’s? Only ‘1’s?At this point, the learners should be starting to develop their own number line, modelling the tutor. Whilstyou are modelling the creation of the number line, make the 10 marks big and bold, the 5 marksmoderately thick, and the 1 marks relatively narrow (see Fig 3). Ask the group why you have done this.Why is it useful? What numerals are worth writing on? What are the most important numerals? Fig. 3 An example of a number lineLearners should now build their number line to 25 either in pairs or independently. The Dyscovery Programme |Eleanor Machin 14

Part 3The final part of the session requires the learners to compare all the different representations of numbersthey have come across in the last two sessions. Learners should attempt this task independently, but tutorsshould use their judgement as to whether struggling learners may need pairing up with a more ablelearner. Give each learner a number between 7 and 25 (according to their ability) to create in as manydifferent ways as possible.Learners should be encouraged to use all the materials at their disposal to create their given number (seeFig 4). Give them a time limit to complete the task (suggest just a couple of minutes). Their representationsshould be displayed on their table. Ask the learners to go around the class and view everyone else’sdisplay, then return to their own and improve it if necessary.ExtensionYou could take photos of the displays and use them to create a pictoral number line. The Dyscovery Programme |Eleanor Machin 15

Overview and RationaleThis session looks at completing simple addition and subtraction using the familiar arrays and the numberline created in the last session.Learners will identify how arrays and the number line link together in simple calculations and, hence, buildon knowledge gained in Session 1 (creating larger numbers using smaller ones) and prepare them for thenext session on number bonds and partitioning.Learners will develop and practice the visualisation and use of a mental number line. It is hoped that thisshould, implicitly, build a foundation for the concept of rounding and, hence, aid estimation.ResourcesLearners’ own number line.1, 5 and 10 rods lengthsNumeral cardsDot pattern cardsRandom array cardsDry wipe markersCurriculum LinksN1/E1.1, 1.2, 1.4, 1.5, 1.7 The Dyscovery Programme |Eleanor Machin 16

PreliminaryArray and numeral Snap!This is a quick game for pairs of learners and is based on the classic game of Snap!Learners are given shuffled cards including random arrays, dot patterns and numerals. Give the learnerstheir instructions (most will already know the concept of the game). A ‘Snap!’ is when the quantity is thesame, regardless of whether it is a numeral, a random array or a pattern that has been placed on the pile(See fig. 1) Fig 1. Two examples of array and numeral SnapPart 1The first part of the session is adapted from the Let’s think through Maths activity ‘Groups in a Crowd’ andrequires learners to physically add two amounts using familiar concrete manipulatives.Recap the work completed in Parts 4 and 5 of Session 1. Ask learners to find some examples of largerarrays that they created from smaller ones. Sketch some examples onto the board and show how thelarger number has been created by adding two smaller ones (see Fig. 2) Fig 2. Some examples of variations for the number 7. The Dyscovery Programme |Eleanor Machin 17

Now, model ‘7+6’ using a choice of arrays. Firstly, sketch various ‘7’ patterns. Emphasise the groupings bylassoing the numbers (see Fig 3). Gather ideas from the class about how we can now add any ‘6’ pattern tothe ‘7’. Invite learners up to draw their ideas on the board. ___________________________________ Fig 3 – Representing 7+6, from LTTM! 5&6, p27Learners should work in pairs to complete sums given according to their ability. Higher ability pairs canchallenge each other to complete sums they create themselves.Part 2The second part of the session follows on from session two and is based on the LTTM activity ‘How Big isthe Number?’Make sure the learners have their number lines from last week. Hand out the 1, 5 and 10 lengthsCuisenairre rods and draw an enlarged version of the number line on the board, as you did in Session 2(you will need at least eight 1s, four 5s and two 10s per pair).Model the procedure for adding 7+8, asking the learners to follow using their number line and Cuisenairrerods. First put 7 on the board, starting from zero and placing on a 5, a 1, then a 1. Next build up another 8(5+1+1+1). Now show how the lengths can be rearranged for ease of reading (see fig 4). (Fig. 4 Modelling 7+8, LTTM! 6-9, p44)Ask the learners, ‘could we swap anything else to make it any easier?’ You are looking for the answer ‘a 10and a 5’. The Dyscovery Programme |Eleanor Machin 18

Learners then work in pairs to complete 16+6 and 19+5. Allow 5 to 10 minutes for this task, depending onthe ability of your learners. As an extension activity for pairs who complete the task quickly, each memberof the pair could challenge each other to complete other ‘harder’ sums. After 5 to 10 minutes answersshould be shared. You may find some different processes and constructions. Encourage learners to valueand explore any differences that occur.Part 3Whole group discussion:How has this session related to previous sessions?Further extension and reflectionAsk learners:How far can the number line be extended in their mind? And what sums could they complete with their‘imaginary’ number line?Can you draw halves onto your number line?Could you change the numbers on the number line to make it reach 100 or more (i.e. smaller scale)? The Dyscovery Programme |Eleanor Machin 19

Overview and RationaleUsing dot patterns and random arrays, learners practise splitting numbers, up to at least 10, into variouscomponent parts. They discover, by creating number triads, how number partitioning relates to additionand subtraction through 10. They will start to develop a ‘counting-on’ instead of ‘counting-all’ strategy,using partitioning to help speed up the process.Learners will be able to recognise number bonds and create their own rules of working out number bonds.These activities should also provide the basis for learning about reverse calculations for checking.Students have further opportunity to work with concrete materials, but this session encourages them tomove through the process to the abstract by way of using visualisation and Arabic numerals.ResourcesDot enumeration test sheetsNumber triad wipe-off templatesNumeral and function cardsCuisenairre rodsCountersEnvelopesCurriculum LinksN1/E1.1, 1.4, 1.5, 1.7N1/E2.1, 2.5PreliminaryGive each learner the same three dot enumeration sheets as they had in Session 1. Allow the learners 20seconds to complete as many as possible and make a note of their results.Ask the learners:Did you use any strategies from last session? What strategies might you have used?Have you improved since last time?Are you ready to move on to bigger numbers? The Dyscovery Programme |Eleanor Machin 20

Part 1This part combines an investigation and some consolidation of prior learning.Using knowledge from session 1, learners find as many ways as possible of splitting numbers 7, 8 and 9.Learners to work in pairs to create number triads, first using arrays, then using numerals (printed on card)(see Fig 1). (Fig 1: Splitting the number 7, firstly using arrays, then moving on to number triads)Part 2Learners use numeral cards and + and = function cards to create as many number bonds as possible.Investigate – is there any pattern or rule we can use to identify ALL number bonds? How do we know thatwe’ve got them all? Learners should be encouraged to notice that each time the number on the left of thesum goes up by one, the number on the right of the sum goes down by one. They could use Cuisenairrerods to help them find a pattern if they are struggling using numerals only (see fig 2a and 2b). The Dyscovery Programme |Eleanor Machin 21

(Fig 2a – the start of the investigation into bonds of 7 using Cuisenairre rods) (Fig 2b – A completed pattern which identifies all bonds to 7)Part 3This part of the session is adapted from the LTTM! Activity, ‘Hidden Treasure’. Students will need theirnumber line and you should have your enlarged number line on the board. You will start off by modellingthe process of adding a number to a hidden collection, through 10, using partitioning.Place 7 counters in an envelope, labelled with the number ‘7’, and put another 5 on the table. Pose thequestion of how we can add, as fast as we can, as easily as possible, without getting the 7 counters out ofthe envelope and counting them all? Get ideas from the class. Don’t accept simply ‘7 add 5 is 12’ – drawout ideas of how it is done. Encourage ideas which utilise partitioning and number bonds to 10. Now getthe counters out and model the arrangements as shown in figure 2. Invite learners up to draw arepresentation of these techniques on the number line on the board. Fig 2 – 7 + 5 using two different methods (Shayer, Adhami and Robertson, 1994, p35) The Dyscovery Programme |Eleanor Machin 22

Learners should now work in pairs to complete their own addition sum in the same way. Give them sumsaccording to their ability or allow them to challenge each other. Allow around 5 minutes before asking theclass to share their ideas.Now to do a similar activity with subtraction. Put 12 counters in an envelope, labelled with the number‘12’. Take out 4 and put them on the table. Ask again, what is the quickest and easiest way of working outhow many are left in the envelope without getting the rest of the counters out? Give the learners a fewminutes and, again, draw out ideas. Encourage or model imagining 12 as various groupings (see fig 3a) andthen focus on the ‘taking away’ action, eg counting back in ones, taking 2 at a time or taking 4 from a 6 etc.(fig 3b) Fig 3a – representations of 12 Fig 3b – taking 4 away from 12Part 4Learners should use their knowledge of number bonds (from Part 2) and their learning in Part 3 toinvestigate subtraction’s relationship to addition. Give out the resources that you gave in Part 2, swappingthe + cards for – cards. Ask the learners to create as many subtraction number bonds as possible and find arule or pattern that links the addition number bonds to these new subtraction number bonds. Encouragelearners to verbalise their thinking and record their thinking using diagrams.As a class, agree on how to describe the rule in words – you are aiming for the definition of reversecalculation, which could be denoted by various sketches (see Fig 3) Fig. 3 – Diagrams could be used to denote partitioning and reverse calculation The Dyscovery Programme |Eleanor Machin 23

Overview and RationaleThis session will introduce the concept of base 10 and the idea that the position of digits within the placevalue chart effects their value. Dyscalculic learners often struggle to make sense of the number structureabove 100 and find calculations involving borrowing and carrying confusing. This session aims to equiplearners with knowing-why digits can have different values and knowing-how base 10 units fit together.Learners will be confident in ‘translating’ the spoken numbers (number words) with concreterepresentations and written numerals.Curriculum LinksN1/E2.1 and E2.2ResourcesDiene’s blocksLaminated Place Value mats (A3 and A4)Dry wipe markersBase 10 number stripsPreliminaryArray and numeral Snap! Play for 5 minutes, as in session 3. Ask the learners, did they perform better thanlast time? Have they used any techniques? Are they any more confident about responding quickly? The Dyscovery Programme |Eleanor Machin 24

Part 1Start with a discussion: What representations of numbers are used today (Arabic numerals, Romannumerals, Base 10 number system)? If students need a prompt, refer them back to the Diene’s blocks theyare already familiar with. Write a mutually agreed definition of our number system on the board.Introduce the A3 laminated place value mat. Give one per group of 4 (max). Ask learners to position eachDiene’s block in the correct position. Demonstrate how to create a simple number using the blocks (forexample, 23 using two ‘ten’ rods and 3 ‘ones’ blocks, see fig. 1). Learners now challenge each other, withintheir groups, to build various other numbers, starting with two digits and moving up to four digits for thosewho are ready.Fig 1Part 2When you are confident that the learners have mastered creating the numbers with Diene’s blocks, moveon to using the digit strips. The learners are moving from the concrete to the representational phase.Discuss how each pack of strips map to the Diene’s blocks. Demonstrate how the simple number created inepisode 1 is made using the strips.Learners are now challenged, in pairs or small groups, to recreate the numbers they made in Episode 1using the strips. (See Fig. 2)Fig 2 – Using the number strips to create 1, 586 The Dyscovery Programme |Eleanor Machin 25

Part 3This episode represents another step through the concrete-representational-abstract process. We nowremove the strips and give out laminated digit cards. Again, learners challenge each other to create variousnumbers using the digit cards.Can any learner create the number without the use of the place value mat?Part 4The final episode’s aim is for learners to be able to write numbers, up to 4 digits, independently. Learnershave now moved to the final ‘abstract’ stage.Some learners may be ready to write down numbers without the chart, some may need help in usingvisualisation techniques to ‘see’ the chart in their minds eye, others may need to copy from digit cards tobegin with. Ensure learners are comfortable using whichever technique suits.Hand out three digit cards to each learner. Ask them to make the largest number possible and write itdown without the chart (if they need to use the chart to begin with and then transpose their answer toblank paper, then this should be allowed). What is the smallest number that can be made with the samedigits? Now do the same with 4 cards.NoteAt some point during the session, you may come across a number where one of the values is ‘zero’. Keepyour eye out for this. If it occurs naturally in the process you can pause the session to discuss the conceptof zero as a placeholder (see Fig 3a and 3b). Ask the learners ‘what does the zero mean?’ and ‘what wouldhappen if we just left it out’. If this doesn’t occur during Parts 1, 2 or 3, ensure you cover this in Part 4.Fig 3a - Using blocks to represent 1,064 The Dyscovery Programme |Eleanor Machin 26

Fig 3b – Recreating 1,064 using number strips The Dyscovery Programme |Eleanor Machin 27

Overview and RationaleThis session will cover simple calculations using place value; addition with carrying, subtraction withborrowing and multiplying and dividing by 10, 100 and 1000.The session aims to make explicit the reasons behind the processes and clear up common misconceptions(i.e. take the smaller number from the bigger number, multiplying by 10 is adding a zero etc).Exercises using concrete manipulatives will make it clear what is actually happening when you carry a digitduring addition, or borrow a digit during subtraction. This deeper understanding should help learnersremember the process and identify when something in a calculation has gone wrong.ResourcesBase 10 imagesBase 10 materials‘Trading’ worksheetsPlace value matsPlace value sliderNumeral cards and digit strips.Blu-tackCurriculum LinksN1/E2.3, 2.4 and 2.7PreliminaryGive each learner three dot enumeration sheets according to their performance in session 3. Allow thelearners 20 seconds to complete as many as possible and make a note of their results.Ask the learners:What strategies did/might you have used?Have you improved since last time?Are you ready to move on to bigger numbers? The Dyscovery Programme |Eleanor Machin 28

Part 1Part 1 is a re-cap of the last session. Learners work in pairs and challenge each other to make various multi-digit numbers using Base 10 materials, digit strips and numeral cards. Alternatively, one of the pair couldcreate a number and challenge their partner to identify it. More able pairs should be encouraged to becreative and use Cuisenairre rods and arrays (See Fig 1)Fig 1Part 2Introduce the concept of trading units for rods and rods for flats. This can take some time to get used to,but is a vital part of understanding carrying and borrowing in calculations. You may need to assure thelearners that it is time well spent! Demonstrate how rods can be traded for 10 units and vice versa (see Fig2). Give out rods and flats to groups of up to 4. Learners should practice trading in the same way beforeworking in pairs on the trading worksheets. The Dyscovery Programme |Eleanor Machin 29

Part 3Part 3 of the session involves addition with carrying. Model a multi-digit addition sum that doesn’t requireany carrying, for example 123 + 456 (encourage participation from the group but don’t put anyone on thespot). Now ‘model’ a sum that doesn’t work (see Fig 3). Gather ideas from the group as to why this iswrong. Give credit to those who identify that it is wrong and those who may give the correct answer, butensure the emphasis is on why.Fig 3 – Example of an addition sum that ‘doesn’t work’Now model the process of doing the sum using the Base 10 materials and trading technique. If you have asmall enough group you could gather them around a table at the front and use the Base 10 materials,otherwise, use the Base 10 images on the board.Group learners into pairs or threes in matched ability groups. Give one sum to each group to work through,matching the difficulty of the sum to their ability.Part 4Model a subtraction sum that ‘doesn’t work’ (See fig. 4). Again, gather ideas from the group as to why thisis wrong.Fig 4 – A subtraction sum that ‘doesn’t work’As in Part 3, model the correct method of completing the sum using trading.Group learners into pairs or threes in matched ability groups. Give one sum to each group to work through,matching the difficulty of the sum to their ability. The Dyscovery Programme |Eleanor Machin 30

Part 5Ask learners how they multiply a number by 10. Some may not know, most would probably say ‘add azero’, some may say ‘move the decimal point one space to the right’, some may say ‘move the numbersone space to the left’. You may also get some very bizarre answers. Note the answers on the board. Whichstatements are sometimes, always or never true? Give learners just two or three minutes to consider thisin pairs or small groups.Now give out calculators, dry-wipe markers and a place-value slider per group. Give another five to tenminutes for learners to investigate different sums, multiplying and dividing by 10 and 100. Allow them timeto figure out the place value slider for themselves, but be prepared to model an example if needed.After the allotted time, bring the class together to share their ideas. Write a mutually agreed method onthe board. The main areas of understanding that you are looking for are that:  the number of places you move is the same as the number of zeros in the multiplier  the digits move along the columns  digits move to the left when you times and to the right when you divide  you can add as many zeros as you like to the beginning of a number that appears before the decimal point or the end of a number after the decimal point without changing its value, but not in the middle.Extension ActivitiesLearners who find the activity in Part 3 easy can be challenged with adding three digits together. Make apoint of ensuring the sums require the learners to carry more than just one ‘ten’ or ‘hundred’, for example,389+257+685.For learners who find Part 4 easy, give them a sum that involves a ‘zero-as-place-holder’ in the largernumber, for example 1305-686.Place more able learners into groups to investigate how to take away numbers from ‘100’ or ‘1000’(borrowing from two or more columns to the left). Allow five to ten minutes for learners to experimentwith materials. After this time, learners should share their ideas/answers with the class. What reasoningdid they use? How do they know they are right? The Dyscovery Programme |Eleanor Machin 31

Overview and RationaleThis session will use the repeated adding to introduce the concept of multiplication. Using their ability tosubitise up to 5, they will be able to count a group of objects without ‘counting-all’.Learners will discover that numbers used in multiplication are reversible (i.e. 2 x 3 is the same as 3 x 2) sothey can choose their preferred times table when completing a sum. They will develop confidence in their2, 3, 4 and 5 times tables.In addition to the idea of ‘chunking’ numbers, learners will also start to think about breaking up numbers,which forms the basis of division. Some knowledge of the relationship between multiplication and divisionmay begin to be formed.Curriculum LinksN1/E2.5, E2.7 and E3.6ResourcesArray patternsLaminated Resource Sheets (7a, b, c and d)Dry-wipe markersCountersPart 1Discuss what learners understand about multiplication. You may find that some learners have someunusual misconceptions. Ensure that any misconceptions are cleared up.Learners should work in pairs to think of as many words as possible that could mean ‘multiply’ or arerelated to multiplication. Allow two or three minutes then ask pairs to share their answers with the class.Write ideas on the board. If any pertinent terms are missing, ensure they are added in at the end;preferably by drawing them out of the learners by giving hints and clues.Ensure the term ‘multiple’ is covered – this will be an important piece of vocabulary throughout thesession. The Dyscovery Programme |Eleanor Machin 32

Part 2Remind the learners of the array patterns for the numbers 8, 9 and 10. Invite some up to sketch some ofthe patterns on the board. Pose the question, ‘How can we best visualise these numbers in ways that showthe 2, 3, 4 and 5 times tables?’ Model an example of how 6 could be represented as two groups of 3 orthree groups of 2 (see Fig 1). Ensure you emphasise ‘multiples’ here, for example, 10 is a multiple of both 2and 5, 9 is a multiple of 3 but not 2 and so on. Fig 1 – Arrays showing multiplesPart 3Each learner should use their ‘least favourite’ multiplication table between 2 and 5. Put the learners intomatched pairs if possible. Instruct the learners to build repeated patterns of their chosen number up tofive times and then practice counting up using only the group numbers, and not counting all., i.e. 4, 8, 12…and not, 1,2,3,4,5,6,7,8,….If learners are struggling with this, they could draw their patterns out, numbering and highlighting the lastnumber in each group (order irrelevance could be revisited here); they are only allowed to count thehighlighted number (see Fig 2) The Dyscovery Programme |Eleanor Machin 33

Part 4This part of the session is adapted from the Let’s Think Through Maths activity ‘Groups in a Crowd’.Give out one Resource Sheet 7a and one dry-wipe marker per group of up to 4. Ask learners to find asmany ways as they can of grouping the objects so that they can count them without counting all. ResourceSheet 7a shows 20 items, so objects could be grouped in 2s, 4s or 5s. Model how these results would bewritten down as ‘sums’ (see fig 3).Learners should now work in pairs with Resource Sheets 7b, c and d, and record the corresponding sums.Allow 5-10 minutes for this activity, depending on the ability of the group. For a more able group you mayask learners to group in 6s and identify remainders.The whole class should now share their answers. Take a couple of minutes to identify all possible ways ofgroupings and all the sums identified. Ask the learners, which groupings were easiest? Why? Did you useany of your subitisation skills to help you?Part 5In the final part of the session, learners should be split into four groups and each group is given a task: 1) Create an array pattern of 10 that shows how it can be made out of 5s 2) Create an array pattern of 10 that shows how it can be made out of 2s 3) Group the items on Resource Sheet 7a into 4s 4) Group the items on Resource Sheet 7a into 5sInvite one or two learners from each group to show their representation to the class and write thecorresponding sum on the board (it may help to keep Groups 1 and 2’s results on one side of the boardsand Groups 3 and 4’s results on the other – see Fig 4). What do they notice?Learners should realise that the sums are ‘reversible’, that is, it doesn’t matter which way around themultipliers are, the answer will be the same.ExtensionUsing the rule identified in Part 4, learners should be challenged to prove whether this rule is ‘sometimes’or ‘always’ true. The Dyscovery Programme |Eleanor Machin 34

Overview and RationaleFollowing on from the introduction to multiplication, this session looks into patterns created by themultiplication tables. They will be introduced to the multiplication square, and will use this and othermethods to find patterns in the numbers.Learners will discover how to use their knowledge of the ‘easier’ times tables to help them with the moredifficult ones. They will also practice various methods of recalling times tables, including the ‘fingers’ andthe ‘bridge’ method of the 7s, 8s and 9s. Learners will identify which methods work best for them asindividuals and why.ResourcesMultiplication Square (blank)CountersBridge Method A3 MatBridge MethodCurriculum LinksN1/E3.5, L1.5 and L1.6PreliminaryArray and multiplier Snap! This is a slightly different take on the game they are used to. Make sure only touse the array patterns on this occasion, and add ‘Multiple of 2’, ‘Multiple of 3’ and ‘Multiple of 5’ to themix. The Dyscovery Programme |Eleanor Machin 35

Part 1Carrying on with the notion of ‘patterns’ within multiplication tables, this part of the session requireslearners to identify patterns in the 9x table. Write the 9x table on the board as shown in Fig 1. Learnersshould work in pairs for five to ten minutes identifying as many patterns as possible. After the allowed time(stop when it appears that all ideas are exhausted) ask the pairs to share their ideas with the class. Note allideas on the board. You are looking to obtain rules such as: - Digits in the answers add up to 9 - The digit in the tens column is always one less than the multiplier - As the digit in one column goes up, the other column goes down.(Fig 1 – Setting out the 9x table for investigation)If any of the patterns have been missed by the class, ensure these are covered before moving on.Learners should experiment with these rules and check if they are always or only sometimes true.Part 2Learners now work in pairs or small groups to investigate the 3x and 6x tables. Prompt the learners to setout their tables as you did in Part 1, and use the ideas they gained in that section. Allow approximately 10minutes for this activity before asking pairs/groups to share their ideas.For full class discussion, ask ‘why are these patterns useful?’ The Dyscovery Programme |Eleanor Machin 36

Part 3Model the ‘fingers’ method of the 9 times table. Learners should copy your example physically. Ifnecessary, learners could draw out a ‘tens and units’ chart on paper – the ‘bent under’ finger should beplaced on the line separating the columns (see Fig 2). This can help the learners identify which fingersrepresent the tens and which represent the units.When the learners seem comfortable with this method, ask why this method works. What patterns doesthis follow? If the learners understand why it works, they are more likely to remember how to do it.(Fig 2 – showing 4x9=36 using the fingers method and a tens and units chart)Part 4Give out the ‘bridge method’ hand-outs (one each) and the ‘Bridge Method’ A3 mats (one per group).Model the method. If necessary, labels can be put on learners’ fingers so they can quickly identify whichfinger represents which number. You may need to spend some time on this as it can be quite a complexmethod – especially if the ‘lower’ times tables are still not automatic. It is worth persevering; however, itmay be preferable to revisit this method at a later date than have the learners become confused andfrustrated. The Dyscovery Programme |Eleanor Machin 37

Part 5In this final section, learners will complete their own multiplication square. Give out the partiallycompleted squares. Ask learners what they notice about the patterns of the numbers already there. Theyshould notice the diagonal symmetry (although they are unlikely to use this term!) that occurs.Learners should use their preferred methods of working out their times tables to complete the square asquickly as they can.They should keep these safe as they will need to use them in the next session.Extension ActivityAny learners who find any part particularly easy should be challenged to find a method for the 7 timestable. This is a near impossible challenge, but learners may identify the use of days in a week or a fortnight,or suggest that the ‘other’ times table is used (i.e. reverse the sum). The Dyscovery Programme |Eleanor Machin 38

Overview and RationaleRevisiting, and following on from, session 7, learners are introduced to the concept of division both asrepeated equal grouping and inverse/reversed multiplication.Learners will be able to discuss and identify mathematical vocabulary related to division.Learners will be guided to discover how to use their multiplication square to complete straight-forwarddivision sums, and will identify that multiplication and division sums can be placed into number triads inthe same way as addition and subtraction sums in partitioning.ResourcesResource Sheet 9aMultiplication SquareCurriculum LinksN1/E3.6PreliminaryGive each learner three dot enumeration sheets according to their performance in session 5. Allow thelearners 20 seconds to complete as many as possible and make a note of their results.Ask the learners:What strategies did you/might you have used?Have you improved since last time? The Dyscovery Programme |Eleanor Machin 39

Part 1Part 1 is adapted from the Let’s Think Through Maths activity ‘Sharing Presents’. Split learners into pairs orgroups of 3. Give out one copy of Resource Sheet 9a per group and stick one on the board.Introduce the story that some students have been working on an enterprise project selling birthday cardsand the picture shows the money they have taken. There were two students serving on the stall. Howmuch do they receive each? Ask learners to sketch/describe to each other/note down how they wouldwork this out from the picture. You may get a range of ideas, but place emphasis on ideas that incorporatethe notion of equal groups as opposed to ‘one-for-me-one-for-you’ style sharing (although this is a validtechnique for this level, it will not work as well when working with parts of numbers, and may encouragefixation on seeing numbers as groups of ‘ones’). Students should now note down their answer as a sum inmathematical notation, i.e. 12÷2=6.Now explain that two other students actually made the cards and they want an equal share as well. Howmany groups of how many £s now?What if another two students, who did all the marketing and arranging the hire of the stall, wanted anequal share? How many groups of how many £s?Learners should keep a note of each sum they create.Part 2Ask the learners to get out their work from Session 7, Parts 4 and 5. What are the similarities anddifferences? What can they see when they compare the sums they created from their activity today withthe sums they created from Resource Sheet 7d? Learners should work in pairs or small groups for just afew minutes to share ideas, before group ideas are shared with the class.Learners should recognise the relationship between division and multiplication, either by way of how thedigits appear within the sums, or how the groupings appear on the Resource Sheet. If necessary, learnerscould be prompted by reminding them of the number triad they used for addition and subtraction… (seeFig 2). The Dyscovery Programme |Eleanor Machin 40

Part 3Using the knowledge they’ve gained this session, challenge the learners to work in small groups to figureout how they can use their multiplication square to solve simple division sums. The Dyscovery Programme |Eleanor Machin 41

Overview and RationaleThis session revisits and consolidates learning on the number line. It also moves on from theDiene’s/Cuisenairre version of the number line, which incorporates only whole numbers, to the continuousnumber line, which involves starting to look at parts of numbers.Students should begin to develop a concept of fractions and possibly decimals in the context of simplemeasures.Resources30cm ruler, tape measure, thermometer, number line, measuring strips, resource sheet, Base 10 materials.Curriculum LinksN2/E3.1, E3.2 and L1.1PreliminaryPlay 5 minutes of array and numeral Snap, as in session 5. Ask the learners, did they perform better thanlast time? Have they used any techniques? Are they any more confident about responding quickly?Part 1Discussion – where do we find number lines? Discuss with peers, share ideas, create a list of items.Use number line created in previous session- Compare this with ruler and thermometer. What is the same?What is different? Some learners may note that the cm on the ruler are split into mm. You can take theopportunity here to ask, ‘at what point do we reach ‘full’ numbers?’, or ‘where is the half-way point?’ The Dyscovery Programme |Eleanor Machin 42

Part 2Part 2 is adapted from the Let’s Think through Maths activities ‘Using Intervals and ‘How Much Bigger?’The learners should have their own number line. Give out a 4cm strip and an 11 cm strip. Give the learnersa minute to measure these. You should find that the majority of learners get this right.Now give out a 3 ½ cm strip and a 3 ¼ cm length and ask them to measure these. This time, you may getsome different answers. How did the learners come to their decision? How do they know they areaccurate? Learners should be encouraged to use a ruler marked with mm markings to measure the strips.They should notice that the 3 ¼ cm strip still doesn’t meet a mark. What does this mean? Students shouldbe encouraged to discover that a number line is continuous; that is, it can be split into an infinite numberof parts.Now ask the learners to mark ‘halves’ on their number line; they should estimate with pencil first, thenmark on accurately in pen, using a ruler. Draw an enlarged sketch of the number line on the board.Pose the question: how many halves in 1? Most learners will get this right. Ask, so how many times biggeris one than a half? If necessary, draw two loops from 0 to ½, then ½ to 1 to demonstrate (see Fig 1).Now ask, ‘How many halves in two and a half?’ There may be some errors. Ensure any misunderstandingsare ironed out before progressing. Most mistakes will be a simple case of miscounting. Students shouldnow spend 5 minutes, in pairs, challenging each other to find out ‘how many times bigger is x than a half?’Ask the class to share their questions and answers. Can anyone notice a pattern? (i.e. it is alwaysdouble/two times, whole numbers result in even answers, numbers with fractions result in odd answersetc.) Why does this pattern exist?Go through the same process asking learners to halve their halves. What is this fraction called? How manyquarters in 1? How many quarters in two-and-three-quarters? Do we notice any patterns?Write down the fractions in numerical form: ½ and ¼ . What do the learners notice about this notation?What if we halved a quarter? What would this fraction look like in number form (1/8)? The Dyscovery Programme |Eleanor Machin 43

Part 3Look again at the ruler and thermometer. How many parts is a cm split into? What about the inch? Whatabout the Celsius/Fahrenheit on the thermometer? What can you notice about the different sizes of thelines used to split the units? Why are they different?Hand out one Resource Sheet per pair. Learners should label as many fraction parts as they can (See Fig 2).Fig 2 – Some fractions used on everyday number linesExtensionMore able students may be able to quickly grasp the concept of fractions. In this case, you could ask themto model various fractions using Base 10 materials and, if particularly able, convert their fractions todecimals, i.e. place 5 cubes onto a rod to depict ½ and identify that this translates to 0.5 (See Fig 4). The Dyscovery Programme |Eleanor Machin 44

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Resource Session Page(s)Advanced Organisers (One copy per learner) AllDominoes (Cut out and laminate) 1Dice (Copy to card) 1Playing cards (Cut out and laminate)Dot enumeration cards – pattern (Cut out and laminate one set per two learners) AllDot enumeration cards – random (Cut out and laminate one set per two learners) AllDot enumeration template (Laminate one copy per learner) 1Dot enumeration test sheets (One copy per learner) 2, 4Blank Numberline (One copy per learner) 2,10Cuisenairre Rods (Cut out and laminate one set per two learners) 2,3,4Numeral cards (Cut out and laminate one set per two learners) 3,4,5,6Function cards ((Cut out and laminate one set per two learners) 4Number Triad Templates (Laminate one per learner) 4Base 10 images (Cut out and laminate one set per 4 learners) AllPlace Value Mat: Base 10 materials (Enlarge to A3, one per 4 learners) 5,6Number Strips (Laminate and cut out 1 set per two learners) 5,6Place Value Mat: Number Strips (One copy per two learners) 5,6 The Dyscovery Programme |Eleanor Machin 46

Place Value Slider (Cut out, laminate and assemble, one per 2 learners) 5,6Grouping Sheets (Laminate, one copy of each per 4 learners) 7,9Multiplication Square (copy one per learner) 8,9Bridge Method example sheets (copy one per learner) 8Bridge Method Work Mat (Enlarge to A3, one copy per 2 learners) 8Measuring Strips (Cut out and laminate, one set per learner) 10Number Lines Worksheet 10The Dyscovery Programme |Eleanor Machin 47

•Numerals or words Number Patterns •Recognising small numbers•Symbols/representations •Creating bigger numbers•Collections of Objects •Representing numbers from smaller ones using dots What are •Numbers as patterns in Counting numbers? everyday life. Number PatternsFor each section, write or draw your own thoughts and ideas (use the back of this sheet if you need more space). What are numbers? Number Patterns Counting Number Patterns

• Numbers as sizes Making • Creating a• Numbers as numbers numberline. collections • Making numbers by • What can we use a adding smaller numberline for? Understanding numbers. Numbers The • Using 1s, 5s and 10s Numberline to make any numberFor each section, write or draw your own thoughts and ideas (use the back of this sheet if you need more Numbers as sizes or collections Making numbers What is a numberline for?space). The Dyscovery Programme |Eleanor Machin 49

•Splitting a large number into Addition using the •Recognising the relationship two smaller numbers number line between the numberline, dot•Identifying the answer to a patterns and rods. simple sum using dot patterns •Stepping forwards along the •Imagining dot patterns, rods or number line from any starting numberline whilst doing Adding two small point mental addition. numbers using dot •Using the number line to to answer a simple sum. Developing a concept patterns of additionFor each section, write or draw your own thoughts and ideas (use the back of this sheet if you need more space). Adding numbers with dot patterns Adding numbers with the numberline Visualising addition The Dyscovery Programme |Eleanor Machin 50