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Tracking children’s learning through addition and subtraction Reception key objective Begin to relate addition to combining two groups of objects, and subtraction to ‘taking away’ (NNS Framework for teaching mathematics, Supplement of Examples, Section 4, pages 14, 15, 16) Associated knowledge and Errors and misconceptions Questions to identify errors Notes skills and misconceptions Count along and back on a Can only begin counting at one; Ask the child to choose a starting number track to and from a given inaccurately counts objects when number. position. rearranged; has no consistent What are the next two numbers? recognition of small numbers of What number comes after the third Count objects set out in different objects; lacks systematic number? arrangements; begin to recognise approaches. What comes before your starting small numbers without counting 1 YR +/- number? and that the number of objects is not affected by their position. The contents panel says there are twelve. Let’s check. Tip them out Count objects that are out of and put them back. How many reach. are there now? 1 YR Can you tell me how many sweets there are here without counting them? How many spots are in this picture? Throw a small number of objects onto the table. Can you count them without touching them? How did you do it? How do you know you’re correct? Find one more and one less than a Misunderstands meaning of ‘one Here are four counters. How given number. more’ and ‘one less; does not many will you have I give you one 2 YR consistently identify the number more? before or after a given number. 2 YR +/- There are six spots showing on my dice. Imagine there is one less spot. How many spots would there be? What is one more than seven? … one less than seven? Say how many there are Does not relate the combining of How many spots are there on this altogether by counting all the groups of objects to addition blue card? objects when combining groups for and/or does not interpret the How many spots are there on this addition. counting of all of the objects as an red card? answer to the question ‘How many How many spots are there Separate a given number of are there altogether?’ altogether? objects into two or more groups 3 YR +/- and say how many there are in There are three spoons in this cup each group. and two spoons in this cup. How 3 YR many spoons altogether? How do you know? Listen to these claps. How many were there? Now tell me how many extra claps I make. How many claps is that altogether? There are seven grapes in my lunch box. How many red grapes and green grapes could there be? Use these cubes to show how you did it. Say how many are left when some Is not confident about when to There are six in this box. I take objects are taken away, by stop counting when taking away away two and put them here. How counting how many objects are (subtracting) in answer to the many are left? left. question ‘How many are left?’ 4 YR 4 YR +/- Here are eight cups. Three cups are full and the rest are empty. How many are empty? We ate four of the six cakes we made. How many are left? How did you work this out? D:\Docstoc\Working\pdf\2567ce12-08b6-4763-a729-09fa7dbeb8a4.doc 1 Tracking children’s learning through addition and subtraction Year 2 key objective Understand that subtraction is the inverse of addition; state the subtraction corresponding to a given addition and vice versa (NNS Framework for teaching mathematics, Supplement of Examples, Section 5, pages 25, 29, 35) Associated knowledge and Errors and misconceptions Questions to identify errors Notes skills and misconceptions Count on and back in ones and Makes mistakes when counting Choose a teen number and count tens. using teen numbers and/or backwards from your number in 1 Y2 crossing boundaries. ones. What is the next number, 1 Y2 +/- the number before? Count forwards in tens from 7. What number is ten more than 27? … ten less than 97? Identify pairs of numbers that add Has difficulty in remembering What is 3 + 5? to twenty and use known number number pairs totalling between ten What is 13 + 5? facts to add mentally. and twenty, resulting in calculation How did you work that out? 2 Y2 errors. 2 Y2 +/- Which pair of numbers adds to 18? Are there any other pairs? Find a difference by counting up Counts up unreliably; still counting How many do I add on to get from from the smaller to the larger the smaller number to get one too three to eight? number. many in the answer. 3 Y2 3 Y2 +/- I’ve got three sherbet dips and I want eight. How many do I need to buy? Recognise subtraction as taking Does not relate finding a What is the difference between 21 away, finding the difference and difference and complementary and 18? complementary addition. addition to the operation of How did you work it out? 4 Y2 subtraction. What operation did you use? 4 Y2 +/- What other ways can you think of for subtracting 18 from 21? Recognise, for example, that Is insecure in making links What is the answer to 30 add 20? subtracting 13 ‘undoes’ adding 13 between addition and subtraction If 30 add 20 is 50, what is 50 and vice versa, and this means and/or recognising inverses. subtract 20? that since 4 + 13 = 17,we can 5 Y2 +/- state the inverse that 17 – 13 = 4. What is 17 subtract 8? Write a 5 Y2 number sentence for this calculation. Use the three numbers to write an addition fact. Develop and recognise patterns to Does not readily use number What is 14 + 5? … 14 + 15? … 14 help deduce other addition and patterns to support calculating, for + 25? subtraction facts. example: Using this information, tell me 6 Y2 46 – 5 = 41, so what 24 + 25 is. 46 – 15 = 31, 46 – 25 = 21, etc. What is 6 – 4? … 16 – 4? … 26 – 6 Y2 +/- 4? Now what do you think is the answer to 56 – 4? How did you work that out? D:\Docstoc\Working\pdf\2567ce12-08b6-4763-a729-09fa7dbeb8a4.doc 2 Tracking children’s learning through addition and subtraction Year 4 key objective Carry out column addition and subtraction of two integers less than 1000 and column addition of more than two such integers (NNS Framework for teaching mathematics, Supplement of Examples, Section 6, pages 48, 50) Associated knowledge and Errors and misconceptions Questions to identify errors Notes skills and misconceptions Know the value of each digit in a Has insecure understanding of the Is 43 smaller or larger than 34? three-digit number, for example structure of the number system, What about 343 and 334? that the 3 in 437 has a value of 30. resulting in addition and How did you decide? subtraction errors and difficulty Recognise the significance of each with estimating. Which is the largest/smallest digit in numbers up to 1000 and 1 Y4 +/- number: 216, 612 or 162? find an estimate of the addition or Ho do you know? subtraction calculation. 1 Y4 What number would you add to 437 to make 477? How did you decide? Partition two- and three-digit Has difficulty in partitioning, for What number does 20 + 10 + 1 numbers in a number of ways. example, 208 into 190 and 18 and represent? 2 Y4 31 into 20 and 11. Is it the same as 20 + 11? 2 Y4 +/- What should I add to 190 to make 208? What other ways can you partition 208? Know when it is most appropriate Does not make sensible decisions What is 700 – 1? … 700 – 10? … to use column addition or about when to use calculations 700 – 9? subtraction rather than carrying laid out in columns. What about 30 + 20? … 30 + 21? out the calculation by another 3 Y4 +/- ... 30 + 27? … 296 – 57?... 328 + method. 187? 3 Y4 How did you do these? Recognise that the operation of Has difficulty with adding three Referring to a written addition column addition can apply to more numbers in a column, except by column calculation involving three than two numbers. adding the first two and then the numbers, ask the child to talk 4 Y4 last one. through their steps in calculation. 4 Y4 +/- D:\Docstoc\Working\pdf\2567ce12-08b6-4763-a729-09fa7dbeb8a4.doc 3 Identifying children’s misconceptions in Year 6 (addition and subtraction) Year 6 key objective Carry out column addition and subtraction of numbers involving decimals (NNS Framework for teaching mathematics, Supplement of Examples, Section 6, pages 49, 51) Associated knowledge and Errors and Questions to identify errors Notes skills misconceptions and misconceptions Apply knowledge of the number Has inefficient counting Imagine you have a money box system to enable efficient strategies and/or containing 2p and 1p coins. What do counting of a large number of insecure understanding you think would be a good way to objects. of the number system. count these quickly to find out how 1 Y6 +/– much money there is? Add and subtract multiples of ten, a What is 60 + 20? … 60 + 30? … 60 + hundred and a 40? thousand. What changed when you found 60 + 1 Y6 40? What is 40 + 40? … 400 + 400? Which answer is the larger? How is the calculation 40 + 400 + 4000 different from the others? What is 60 – 20? … 600 – 200? … 6000 – 2000? Explain how you worked these out. What is 6000 – 200? … 6000 – 20? Give an estimate by rounding, to Rounding inaccurately, Is 26 nearer to 20 or 30? determine whether particularly when the answer to a decimals are involved, Is 271 nearer 270 or 280? calculation is and having little sense of the size of sensible. the numbers involved. Is 1.8 nearer to 1 or 2? 2 Y6 2 Y6 +/– Draw a sketch to illustrate your answer and explain how you know. What is the value of the digit 1 in Partition whole Has difficulty in 3010? numbers and partitioning numbers with zero place … 201? … 6.1? decimal numbers, holders What do you notice about the value of and add and and/or numbers less than one, for the digit 2 in 4.2? … 0.2? … 0.25? subtract the example Write the following statements: constituent parts partitioning 0.45 as 0.4 507 is equal to 50 + 7 and recombine to and 0.05. 7403 is equal to 7000 + 40 + 3 complete the 3 Y6 +/– 0.75 is equal to 0.7 + 0.05 calculations. Which are true? How do you know? 3 Y6 Add and subtract a Has difficulty in What would you add to 37 to make pair of numbers choosing suitable the nearest multiple of 10? that involves methods for crossing calculations that cross What would you add to/subtract from boundaries, boundaries: addition 240 to make the nearest multiple of recognising when 4a Y6 +/– 100? to adjust, to Has difficulty in compensate and to carry choosing suitable Explain to me, using an empty numbers methods for number line, how you find 300 – 237. across the calculations that cross boundaries. boundaries: subtraction Using an empty number line, explain 4a Y6 4b Y6 +/– how you would work out 5016 + 3700. 4b Y6 What is 217 – 6? … 217 – 7? What happens when you subtract 8 from 217? Explain how you subtract 18 from 217. D:\Docstoc\Working\pdf\2567ce12-08b6-4763-a729-09fa7dbeb8a4.doc 4 Tracking children’s learning through multiplication and division Reception key objective Use developing mathematical ideas and methods to solve practical problems. (NNS Framework for teaching mathematics, Supplement of Examples, Section 4, page 20) Associated knowledge and Errors and misconceptions Questions to identify errors Notes skills and misconceptions Count in twos and talk about how Confuses numbers when counting Can you match this sock/glove to many pairs. in twos; has difficulty make a pair? 1 YR understanding a pair consists of Can you find some more pairs? two objects. How many pairs have you made? 1 YR x/÷ Say how many altogether in a Has difficulty with identifying Within role play, engage with child double; recognise that finding a doubles and adding a small and ask questions, for example: double means forming a pair or number to itself, for example 2 + 2, If there are two wheels on this adding a number to itself. to make twice as many. scooter, how many wheels on two 2 YR 2 YR x/÷ scooters? How many gloves make a pair? Solve simple problems (where Makes unequal groups and is Can you put the same number of there are no remainders) that unable to compare the groups. spots on each wing of the involve counting the numbers of 3 YR x/÷ ladybird? objects by organising them into Can you share out these biscuits groups of equal size, including fairly so there are the same more than two groups; talk about number of biscuits on each plate? the number of groups and objects, and record their answers in their own way. 3 YR Share objects out fairly into two or When sharing, can sometimes Can you share these sweets more groups, talk about whether make equal groups, but has no equally between you and a friend? there will be any left over. strategies to deal with any left Will there be any left over? 4 YR over. 4 YR x/÷ Can you share these marker pens between the group? Will there be any left over? What will you do with any left over ones? Count in tens forwards and Has difficulty with counting reliably Can you say all the numbers backwards from a tens number in tens from a multiple of ten. marked on this number line? For (i.e. multiple of ten), and identify 5 YR x/÷ example, ten, twenty… the tens number before and after a Can you put the cards in the given tens number. correct order so we can count in 5 YR tens to one hundred? Solve problems that involve finding When halving, makes two unequal How many sheep in the field? halves. groups or splits a single object Can you put half of them in the 6 YR unequally. pen? How many are in the pen? 6 YR x/÷ How many are in the field? D:\Docstoc\Working\pdf\2567ce12-08b6-4763-a729-09fa7dbeb8a4.doc 5 Tracking children’s learning through multiplication and division Year 2 key objective Understand the operation of multiplication as repeated addition or as describing an array, and begin to understand division as grouping or sharing; know by heart facts for the 2 and 10 multiplication tables; and know and use halving as the inverse of doubling. (NNS Framework for teaching mathematics, Supplement of Examples, Section 5, pages 47, 49, 53, 57) Associated knowledge and Errors and misconceptions Questions to identify errors Notes skills and misconceptions Carry out repeated addition, Still counts in ones to find how Show an array of six rows of two. recognise the relationship between many there are in a collection of multiplication and repeated equal groups; does not understand How many sets of two can you addition, and use the associated vocabulary, for example, ‘groups see? vocabulary of multiplication. of’, ‘multiplied by’. How many are there altogether? 1 Y2 1 Y2 x/÷ Repeat with six rows of ten. Multiply two or ten by a single-digit Does not link counting up in equal How can you quickly work out two number, by counting up in twos steps to the operation of multiplied by seven? and tens from zero. multiplication; does not use the 2 Y2 vocabulary associated with What steps could you count up in multiplication. to help you? 2 Y2 x/÷ How many steps do we need? Interpret an array as a repeated Does not focus on ‘rows of’ or Display six rows of two. How addition and as a multiplication, ’columns of’, but only sees an many rows are there? How many and recognise how the array can array as a collection of ones. dots are there in each? How can be described as, for example, 3 Y2 x/÷ you describe the array? 3 + 3 + 3 + 3, 3 x 4, or as 4 + 4 + 4, 4 x 3. Turn the array through 90°. How 3 Y2 has the array changed? Can you describe the array to me now? What’s the same and what’s different? Recognise that doubling and Has difficulty relating multiplying What is six multiplied by two? multiplying by two are the same, by two to known facts about How did you work it out? What is and use known multiplication facts doubles; records double four as 4 double six? What did you notice? and partitioning to double numbers + 4. to fifteen. 4 a Y2 x/÷ What is the answer to double ten? 4a Y2 …double four? 4b Y2 Does not use partitioning to find How can we use these answers to double twelve or double thirty-five. find double fourteen? 4b Y2 x/÷ How can we work out double thirty five? Recognises that when finding the Does not use knowledge of What do you think half of ten is? double of a number, half the doubles to find half of a number; How do you know? answer is the original number. for example, continues to find half Uses the inverse of double to find by sharing, using a ‘one for you’ Can you think of a number that’s halves of small even numbers to approach and cannot apply easy to halve? Why do you think ten, using known facts. knowledge of doubles. it’s easy to halve your number? 5 Y2 5 Y2 x/÷ Share a given number of objects Is not systematic when sharing Here are twelve counters, share out equally, recognise the into equal groups, using a ‘one for them out equally into these three relationship between sharing you’ approach; does not use the boxes. How many counters are equally and division and use the language of division to describe there in each box? Here are six vocabulary of division to describe the process. coins, can you divide them the process, for example ‘divide 6 Y2 x/÷ between these purses? How by’, ‘share equally’. many will be in each purse? 6 Y2 Altogether we had eighteen counters and there are six in each box, can you describe what we have done? Begin to understand division as Does not understand that ‘sets of’ There are fourteen children and repeated subtraction or grouping. or ‘groups of’ need to be the children are asked to work in 7 Y2 subtracted to solve the problem. pairs (twos). How many pairs are 7 Y2 x/÷ there? D:\Docstoc\Working\pdf\2567ce12-08b6-4763-a729-09fa7dbeb8a4.doc 6 Tracking children’s learning through multiplication and division Year 4 key objective Use informal pencil and paper methods to support, record or explain multiplications and divisions, develop and refine written methods for TU x U, TU ÷ U, and find remainders after division (NNS Framework for teaching mathematics, Supplement of Examples, Section 6, pages 56, 66, 68) Associated knowledge Errors and misconceptions Questions to identify errors and Notes and skills misconceptions Know multiplication facts for 2, Is not confident in recalling What is three multiplied by four? Draw 3, 4, 5 and 10 times tables. multiplication facts. me a diagram to show what this 1 Y4 1 Y4 x/÷ means. How do you know the answer to calculations such as three multiplied by zero and ten multiplied by zero? What is four multiplied by one? How do you know? Know the division facts which Is muddled about the Tell me the multiplication facts this four correspond with multiplication correspondence between by five array shows (support this facts listed above. multiplication and division facts, question with a diagram of the array). recording, for example: And the division facts? Understand the inverse 3 x 5 = 15 relationship connecting so 5 ÷ 15 = 3 If you know three multiplied by five multiplication and division. 2 Y4 x/÷ equals fifteen, what number sentences 2 Y4 using division do you also know? Draw a picture for three multiplied by six. What other multiplication and division facts can you find from the picture? Understand the effect of Describes the operation of What is the answer to forty-six multiplying whole numbers less multiplying by ten as ‘adding a multiplied by ten? … three hundred than a thousand by ten. nought’. and fifty-one multiplied by ten? 3 Y4 3 Y4 x/÷ How did you know? Can apply the distributive law Does not apply partitioning and What number would you partition to (but not by name) to multiplying, recombining when multiplying, for work out twenty-seven multiplied by using partitioning and example: three? recombining, for example: 14 x 3 is calculated as (10 x 3) + How would you recombine your 14 x 3 = (10 x 3) 4 = 34, calculations? + (4 x 3) = 30 + 12 = 42, or Can you spot the mistake in this so X 1 4___ calculation: X 10 4__ 3 3 12 15 x 7 3 30 12 = (10 x 7) + 5 14 x 3 = 312, = 75? Know how to record TU x U confusing the value of two-digit multiplication calculated by a numbers. partitioning method in a grid 4 Y4 x/÷ format. 4 Y4 Recognise that the commutative Assumes that the commutative How do you work out your answer to law holds for multiplication but law holds for division also, for three divided by fifteen? Take some not for division. example, assuming that cubes, or use a diagram to explain 5 Y4 15 ÷ 3 = 5, so 3 ÷ 15 = 5. what this calculation means. 5 Y4 x/÷ Understand the idea of Writes a remainder that is larger With 29p to spend, you want to buy as remainder, and when to round than the divisor, for example, 36 ÷ many sweets at 3p each as you can up or down after division. 7 = 4 remainder 8. afford. How many sweets can you 6a Y4 6a Y4 x/÷ buy? Show your working out on a 6b Y4 number line. 6c Y4 Discards the remainder; does not understand its significance. Which of these calculations is correct? 6b Y4 x/÷ 36 ÷ 7 = 4 remainder 8 Does not recognise when a 36 ÷ 7 = 3 remainder 15 remainder is significant in the 36 ÷ 7 = 5 remainder 1 decision about whether to round 36 ÷ 7 = 5. up or down. Which do you think is the best one? 6c Y4 x/÷ Why? There are thirty-two children and each tent takes three children. What is the least number of tents they will need? Know how to record division as Continues to subtract twos when Ask the child to look back at a division repeated subtraction, with calculating twenty divided by two calculation they have just completed. appropriate use of chunking. without using knowledge that two Can you think of a quicker way to find 7 Y4 multiplied by five equals ten. out an answer for this question? 7 Y4 x/÷ D:\Docstoc\Working\pdf\2567ce12-08b6-4763-a729-09fa7dbeb8a4.doc 7 Tracking children’s learning through multiplication and division Year 6 key objective Carry out long multiplication of a three-digit by a two-digit integer and short multiplication and division of whole numbers (NNS Framework for teaching mathematics, Supplement of Examples, Section 6, pages 67 and 69) Associated knowledge and Errors and misconceptions Questions to identify errors Notes skills and misconceptions Know by heart all multiplication Refer to Year 4 chart (1 Y4 and 2 Y4) where this is described in facts up to 10 x 10. connection with a limited range of multiplication and division facts. At Year 6 level, the number range will extend to include multiplication facts Derive division facts up to 10 x 10 and related division facts. corresponding to multiplication facts up to 10 x 10. Multiply and divide one-, two- and Misuses half-understood rules Can you tell me a quick way of three-digit numbers by ten and about multiplying and dividing by multiplying a number by one one hundred. powers of ten and the associative thousand? law, for example: Apply the associative law (but not 145 x 30 = 145 000 I have thirty-seven on my by name) to multiply up to three- 1 Y6 x/÷ calculator. What single digit numbers by multiples of ten multiplication should I key in to and hundred, for example: change it to three thousand seven 147 x 20 hundred? = 147 x 2 x10 1 Y6 What is 3 ÷ 1? … 30 ÷ 10? … 300 ÷ 100? Tell me about this pattern. If I had four thousand eight hundred on my calculator, what single division could I key in to change the display to forty-eight? 47 x 10 = 470 What do you think 47 x 20 equals? Present the remainder in a Has difficulty, when appropriate, There are twenty-six apples for quotient as a whole number or interpreting a remainder as a four horses, so they can have six fraction. fraction, for example: and a half each. 2 Y6 16 ‚ 3 = 5⅓ How do I know this? 2 Y6 x/÷ Dividing by numbers smaller than Interprets division as sharing but How many half tomatoes can you one, for example: not as grouping (repeated get from three whole tomatoes? 12 ÷ ½ = 24 subtraction) so is unable to 6 ‚ ⅓ = 18. interpret a calculation such as 12 ÷ How many quarters of pizza can 3 Y6 ½. you get from four pizzas? 3 Y6 x/÷ Explain how to work out 5 x ½, 5 ÷ ½. Judge whether the answer to a Is not confident in making 600 ÷ 30 = 2 multiplication or division reasonable estimates for Can this calculation be correct? calculation is reasonable. multiplication or division How do you know? 4 Y6 calculations. 4 Y6 x/÷ Try some more, such as: 540 ÷ 20 = 52 24 x 20 = 4800 15 ÷ ½ = 30 24 ÷ ¼ = 22. D:\Docstoc\Working\pdf\2567ce12-08b6-4763-a729-09fa7dbeb8a4.doc 8

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