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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
                                                                            How many spots are in this

                                                                            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

                                                                            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?

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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
                                                                             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?

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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
                                                                             What other ways can you partition
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 +/-

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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
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

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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?

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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
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
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?

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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
                                                                          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/÷
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/÷

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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 ÷
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.

C:\Docstoc\Working\pdf\785373da-405a-4e8a-a81f-6338b341da59.doc                                                                                     8

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