# Level 3-4 1. Can I position positive and negative numbers on a

```Primary National Strategy
Level 3-4 1. Can I position positive and negative numbers on a
number line and find the difference between them?
Teaching guidance
Key vocabulary
positive, negative, above zero, below zero, degrees Celsius (&deg;C), minus,
difference, integer
Models and images
Use the Number line ITP to show how number lines extend beyond 0.
Encourage children to identify and discuss what the unmarked numbers are.
Number line IPT
Use a number line and/or a counting stick to identify and locate numbers.
Where would you place –2?
–4
0
–4
–7
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0
–4
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Level 3-4 1. Can I position positive and negative numbers on a
number line and find the difference between them?
Use a thermometer or the Thermometer ITP to look at negative numbers, and
find the difference between negative numbers, within a context.
Thermometer ITP
Teaching tips

Give children a range of opportunities to position numbers on number
lines, including practical washing lines and individual number lines.

Emphasise that counting continues beyond zero and use a number line to
demonstrate that –4 is less than –2.

Help children to use the benchmark numbers to determine the position of
other numbers on the number line.

Help children to make connections between using benchmark numbers on
a number line and reading scales.
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Level 3-4 2. Can I count on from any given number in whole
number steps, extending beyond zero when counting
backwards?
Teaching guidance
Key vocabulary
positive, negative, above/below zero, multiple
Models and images
Demonstrate how to find a missing term in a sequence by finding the step size.
Find the difference between the numbers on either side of a missing term. In the
example below, the difference between the two numbers is 8 and there are two
jumps from the last known number to the next known number in the sequence.
Therefore, we divide the difference of 8 by 2.
Demonstrate that the number of missing terms in a sequence can be different.
In the example below, the difference between the known numbers is 12 and
there are four jumps from the last known number to the next known number in
the sequence. Therefore, we divide the difference of 12 by 4.
Teaching tips
 Ensure children have frequent practice in counting in steps of any size,
including starting points that are not multiples of the step size. Use
resources to support counting, for example a counting stick or a projected
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Level 3-4 2. Can I count on from any given number in whole
number steps, extending beyond zero when counting
backwards?
calculator that has been set to count in given steps using the constant
function.
 Children need frequent opportunities to practise their counting skills.
Practising counting in different step sizes underpins children’s
understanding of place value and their skills in calculation.
 Ensure that counting in sequences sometimes starts with negative
numbers as well as starting with positive numbers, and that sequences
involve counting back as well as counting forward.
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Level 3-4 3. Can I read, write, partition and order decimal
numbers?
Teaching guidance
Key vocabulary
decimal, decimal fraction, decimal point, decimal place, tenth, hundredth,
thousandth, significant digit
Models and images
Use the Decimal number line ITP to zoom into a number line and position decimal
numbers.
Decimal number line ITP
Use place-value charts to help identify the value of each digit in a decimal number.
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Level 3-4 3. Can I read, write, partition and order decimal
numbers?
For decimals number with up to two places use a 10 &times; 10 grid so that each square
represents 0.01 and each row represents 0.1 Discuss the effect of repeatedly adding
the same decimal number, for example 0.01.
Teaching tips
 Build on understanding of decimals in the contexts of money and
measures when working with decimal numbers with up to two places.
However decimal place value should also be planned for and taught in its
own right and not just in those contexts.
 Use number lines to help children order decimals. Present children with
numbers for ordering that have different numbers of decimal places, to
tackle the common misconception that the more digits there are after the
decimal point, the bigger the number.
 Focus on the vocabulary of decimal fractions, and encourage children to
read decimal numbers using the language of tenths, hundredths and
thousandths so that, for example, they know the number comprising, two
tenths, five-hundredths and nine-thousandths is written as 0.259.
 Reinforce the equivalence between fractions and decimals. Fraction
notation gives you the language to help understand place value, for
example knowing 0.01 is equivalent to 1/ 100 helps you to read this decimal
number as one-hundredth and not just as zero point zero one.
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Level 3-4 4. Can I express tenths and hundredths as
percentages?
Teaching guidance
Key vocabulary
percentage, per cent, %, tenths, hundredths
Models and images
Represent a percentage using practical resources such as money (&pound;1, 10p
and 1p coins) or images, for example a 10 by 10 square grid or the Area ITP.
Area ITP used to represent 35 shaded squares out of 100 or 35%.
Teaching tips
 Use ICT resources, such as the Fractions ITP, to explore and instantly
record the link between fractions and percentages.
 Use money to show how 10p can be expressed as a percentage and
fraction of &pound;1. Give children the opportunity to use coins to convince
themselves that, for example, 10p is 1/ 10 or 10% of &pound;1 because they need
ten 10p coins to make &pound;1.
 Refer to percentage as being the number of parts per hundred. Reinforce
that 100% represents 100 per 100 or a whole.
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Level 3-4 5. Can I find simple equivalent fractions?
Teaching guidance
Key vocabulary
numerator, denominator, fraction, proper/improper fraction, equivalent,
reduced to, cancel
Models and images
Model how a fraction wall can be used to find equivalent fractions.
Fractions ITP
Demonstrate how a multiplication board can be used to scale up fractions.
Discuss with children what needs to happen to change &frac34; into other equivalent
fractions.
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Level 3-4 5. Can I find simple equivalent fractions?
Teaching tips
 Children need the opportunity to practise finding equivalent fractions by
scaling simple fractions up or down.
 Use paper-folding to help establish equivalence, for example fold a strip of
20 squares into quarters and colour &frac34; of them to establish that &frac34; is the
same as 15 out of 20 or 15/ 20 .
 Focus on recognising the patterns in sets of equivalent fractions and
making links between multiplication and division.
 Represent fractions on a number line. This can help show that the same
point on the number line can have more than one label, for example 1
could also be labelled as 2/ 2 , 3/ 3 , 4/ 4 , 5/ 5 , etc.
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Level 3-4 6. Can I relate simple fractions to their decimal
equivalents?
Teaching guidance
Key vocabulary
numerator, denominator, equivalent, proper fraction, decimal fraction, decimal
place, decimal point
Models and images
Use the Moving digits ITP to make links between fraction and decimal equivalents
of tenths, hundredths, and so on. For example, 3⁄ 10 = 0.3 and 13⁄ 100 = 0.13.
Moving digits ITP
Use number lines or the Fractions ITP to establish the decimal equivalent of
fractions such as 2⁄ 5 or 1⁄ 20.
Fractions ITP
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Level 3-4 6. Can I relate simple fractions to their decimal
equivalents?
Teaching tips
 Use a calculator and the language of fractions to find decimal and fraction
equivalents. For example, 2/ 5 is keyed into the calculator as 2 divided by 5
and shows a decimal equivalent of 0.4 (four tenths).
 Children can look for fraction equivalents by finding decimal equivalents.
Encourage them to make general statements about equivalent fractions.
 Use resources such as equivalent dominoes or washing lines with fraction
and decimal equivalent cards. Invite children to peg the cards on the line
and justify their choice of location.
 Present children with commonly confused fraction and decimal
equivalents, for example 0.4 and 1/ 4 . Ask them to use images or practical
resources to investigate whether these are actually equivalent; for
example they could use the fractions ITP.
Fractions ITP
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Level 3-4 7. Can I use my tables to multiply and divide?
Teaching guidance
Key vocabulary
multiply, multiplied by, multiple of, times, array
divide, divided by, divisible by
factor, product, inverse
Models and images
Use the Multiplication facts ITP to make the link between arrays and
multiplication facts.
Multiplication facts ITP
Teaching tips
.
 Plan regular activities for children to learn, rehearse and use multiplication
and division facts rather than simply test their recall.

Reinforce multiplication facts and the corresponding division facts, for
example, 8 &times; 7 = 56, 7 &times; 8 = 56, 56 &divide; 7 = 8, 56 &divide; 8 = 7. When solving a
missing number question, it is helpful to write down the other three number
sentences and then decide which one to use to find the missing number.
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Level 3-4 7. Can I use my tables to multiply and divide?

Children need to understand and use the language of multiples and
factors.

Help children develop strategies for quickly deriving multiplication facts.
For example, knowing 7 &times; 2 = 14 helps you to work out that 7 &times; 4 = 28 and
7 &times; 20 = 140.

Ensure that children meet calculations written in different ways:
 &times; 8 = 56
9 = 54 &divide; 
3&times;8=6&times;
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Level 3-4 8. Can I use my tables to work out multiplication and
division facts with decimals?
Teaching guidance
Key vocabulary,
times, multiply, multiplied by, product, multiple of, divide, divided by, divisible by,
quotient, factor, inverse
decimal, decimal point, tenths, hundredths, thousandths
Models and images
Use the Number dials ITP to explore multiples of decimal numbers.
Explore known multiplication facts, such as multiples of 6, before exploring
related facts such as multiples of 0.6.
Number dials ITP
What multiplication table does this diagram represent? How do you know? What are
the missing numbers? What division facts do you know by using this diagram?
Teaching tips

Ensure children can confidently multiply and divide by 10 and 100; and
that they understand that multiplying by 10 makes a number bigger and all
the digits move one place to the left, while dividing by 10 makes a number
smaller and all the digits move one place to the right. (See the teaching
guidance ‘Can I multiply and divide by 10 and 100 and 1000?’ in the
Calculating strand).
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Level 3-4 8. Can I use my tables to work out multiplication and
division facts with decimals?

multiplication facts; for example count on and back in steps of 3 before
relating this to counting on and back in steps of 0.3. Encourage children to
explain the relationship between the two sets of numbers.

Ensure that children meet and can interpret multiplication and division
calculations that are written in a variety of different ways, for example:
( &times; .8 = 5.6
9 = 5.4 &divide; (
0.3 &times; 8 = 6 &times; (

Reinforce the division facts corresponding to multiplication facts; for example,
8 &times; 0.7 = 5.6, 0.7 &times; 8 = 5.6, 5.6 &divide; 0.7 = 8, 5.6 &divide; 8 = 0.7. When solving a
missing number question, it is helpful to write down the other three number
sentences and then decide which one is most useful to use to help find the
missing number.

Model the use of jottings and encourage children to use jottings to help
keep track of the stages within a mental calculation.
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Level 3-4 9. Can I multiply and divide by 10 and 100 and 1000?
Teaching guidance
Key vocabulary
digit, decimal, multiply, times, divide, share, scale up, scale down, increase,
decrease, factor, how many 100s in …?, tens of thousands, thousands,
hundreds, tens, units, ones, tenths, hundredths, thousandths
Models and images
Show children how multiplying a number by 10 moves the digits one place to the
left, and multiplying by 100 moves the digits two places to the left.
Demonstrate the effect of dividing a number by 10. Show children how the digits
move one place to the right, and when dividing by 100 the digits move two places
to the right.
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Level 3-4 9. Can I multiply and divide by 10 and 100 and 1000?
Use a calculator or the Moving digits ITP to model how the digits move when
we multiply or divide by powers of 10.
Moving digit ITP
Teaching tips

Help children to generalise correctly so that they can cope with decimals.
Multiplying by 10 makes a number bigger and all the digits move one
place to the left. Dividing by 10 makes a number smaller and all the digits
move one place to the right.

Discuss why 4.6 &times; 10 is not the same as 4.60 and 40.3 &divide; 10 is not the
same as 4.3.

Explore with children the relationships between the operations and how to
simplify combinations of operations. For example, multiplying by 10 then
dividing by 100 is the same as dividing by 10. Help children to recognise
that dividing by 200 is the same as dividing by 10, dividing by 10 again
and then halving, by using a calculator to explore different examples.

Emphasise that multiplication and division by 10, 100 and 1000 should be
a mental calculation.
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Level 3-4 10. Can I add and subtract two numbers in my head
quickly?
Teaching guidance
Key vocabulary
subtract, subtraction, minus, take away, difference between
how many more/less than… ?, inverse
Models and images
Use a 100-square or the Area ITP to provide an image of complements to 100.
Model how as the number of squares in one region is increased, the number in
the other region decreases by the same amount.
.
Demonstrate the use of an empty number line as a support for children’s
mental calculations.
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Level 3-4 10. Can I add and subtract two numbers in my head
quickly?
Teaching tips

When finding pairs of two-digit numbers that total 100, ask children what
they notice about the corresponding digits of each pair (when the two-digit
numbers are not multiples of ten, the number of tens total nine and the
ones total ten).

Reinforce addition facts and the corresponding subtraction facts.
For example if you know 24 + 76 = 100, there are three other number
sentences you also know. When solving a missing number question, it is
helpful to write down the other three number sentences and then decide
which one to use to find the missing number.

Children need to see that they can use facts they know by heart in order to
solve new problems without reverting to counting, for example knowing
70 + 30 helps when calculating 270 + 30.

Help children understand that when working mentally there are times
when it is useful to jot down some notes because you can’t always do the
calculation solely in your head. Make sure this is modelled on a regular
basis.

After they have completed a practice mental test, give children a written
copy of the questions. Ask children, in pairs, to read the questions
together and sort them into those they can answer and those they can’t.
Collect the ones that most children have difficulty with, to discuss or
provide further reinforcement.
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Level 3-4 11. Can I use my tables to multiply and divide?
Teaching guidance
Key vocabulary
multiply, multiplied by, multiple of, times, array
divide, divided by, divisible by
factor, product, inverse
Models and images
Use the Multiplication facts ITP to make the link between arrays and
multiplication facts.
Multiplication facts ITP
Teaching tips
.
 Plan regular activities for children to learn, rehearse and use multiplication
and division facts rather than simply test their recall.

Reinforce multiplication facts and the corresponding division facts, for
example, 8 &times; 7 = 56, 7 &times; 8 = 56, 56 &divide; 7 = 8, 56 &divide; 8 = 7. When solving a
missing number question, it is helpful to write down the other three number
sentences and then decide which one to use to find the missing number.
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Level 3-4 11. Can I use my tables to multiply and divide?

Children need to understand and use the language of multiples and
factors.

Help children develop strategies for quickly deriving multiplication facts.
For example, knowing 7 &times; 2 = 14 helps you to work out that 7 &times; 4 = 28 and
7 &times; 20 = 140.

Ensure that children meet calculations written in different ways:
 &times; 8 = 56
9 = 54 &divide; 
3&times;8=6&times;
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Level 3-4 12. Can I use a written method to subtract?
Teaching guidance
Key vocabulary
subtract, find the difference, minus, take away, more than, less than
calculate, count on, count back
Models and images
Bead strings and number lines help provide a visual representation of the
strategy the children have used, for example:
1470 – 174 = 1296
Teaching tips
 The aim is that children use mental methods of calculation where
appropriate, but for calculations they cannot do in their heads they use an
efficient written method accurately and with confidence.
 To subtract successfully, children need to be able to:
– recall all addition and subtraction facts to 20;
– quickly derive complements to 10 and 100 and multiples of 10 and 100;
– subtract multiples of 10, such as 160 – 70 (using the related subtraction
fact 16 – 7, and their knowledge of place value);
– partition two-digit and three-digit numbers into multiples of one hundred,
ten and one in different ways (e.g. partition 74 into 70 + 4 or 60 + 14).
 It is important that children’s mental methods of calculation are practised
and secured alongside their learning and use of an efficient written method
of subtraction.
 Help children to take ownership of written methods, for example, by having
a ‘my method for subtraction’ card that travels between home and school.
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Level 3-4 13. Can I use a written method to multiply?
Teaching guidance
Key vocabulary
array, partition, multiply, multiplied by
calculate, calculation, strategy, method, equation
Models and images
Create arrays and then partition them to provide a visual representation of
the grid method of multiplication. The Multi array ITP can be used to help
model this.
Multi array ITP
Teaching tips
 Children should use mental methods of calculation where appropriate, but
for those calculations they cannot solve mentally they need to be able to
use an efficient written method, such as the grid method of multiplication,
accurately and with confidence.
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Level 3-4 13. Can I use a written method to multiply?
 To multiply successfully, children need to:
– know or quickly recall multiplication facts up to 10 &times; 10
– understand the effect of multiplying numbers by 10, 100 or 1000.
– multiply multiples of 10, for example 20 &times; 40
– approximate; for example recognise that 72 &times; 38 is approximately
70 &times; 40 = 2800 and use this information to check whether their answer
appears sensible.
 Give children opportunities to work with numbers for which they could use
mental or written calculation strategies, in order to build up their
confidence in the written method.
 Check which methods have been taught earlier on in the school, to ensure
children receive consistent modelling and demonstration.
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Level 3-4 14. Can I calculate a fraction of a number or quantity?
Teaching guidance
Key vocabulary
fraction, equal parts, numerator, denominator, divide, division, multiply,
multiplication
Models and images
Use models and images alongside oral work. For example, display 12 small
objects such as counters.
What is one-third of these 12 counters? What are two-thirds of 12 counters?
What are three-thirds of 12?
Arrange the counters in ways that help children to see the process and
gradually reduce the reference to the counters as the children become more
confident. Record the steps with the children and encourage them to
recognise the underlying counting in 4s.
1
/ 3 of 12 is 4
2
/ 3 of 12 is 8
3
/ 3 of 12 is 12
Link finding fractions of amounts to fractions of shapes, for example:
1
1
/6
/6
1
/6
1
/6
1
1
/6
Find 5/ 6 of 18.
What is 1/ 6 of 18? (6 &times; 3 = 18)
If one-sixth is 3, what is five-sixths?
/6
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Level 3-4 14. Can I calculate a fraction of a number or quantity?
Ensure children have experience of finding fractions of a range of wholes.
Give children ‘bags of images to represent the whole’, of which they can then
find fractions, such as 5/ 6 . This could include shapes, numbers, an amount of
money, a length of string, a set of counters, and so on.
240
&pound;1.80
Teaching tips
 Children need to be able to relate fractions to division, for example to
understand that finding one-tenth is equivalent to dividing by 10.
 Help children to understand that they are finding a fraction of a whole
amount by using practical equipment to explore a variety of different
wholes (see image box above).
 Use different models and images to help children understand that a
fraction such as 4/ 5 is 4 &times; 1/ 5 , and finding 1/ 5 is the first step to finding 4/ 5 .
This will help them begin to associate finding 4/ 5 with ‘divide the whole into
five equal parts and then group together four of these equal parts’.
 Introduce a scale or length to support the process, for example a length
representing 30 cm. Use this to ask questions such as: What is one-sixth
of 30 cm? What are two-sixths of 30 cm? What are three-sixths of 30 cm?
Identify the steps of 5 cm and the counting process to establish that 1/ 6 of
30 is 5; 2/ 6 of 30 is 10; 3/ 6 of 30 is 15; 4/ 6 of 30 is 20; 5/ 6 of 30 is 25; 6/ 6 of
30 is 30.It is important to establish that 6/ 6 represents the whole. When
they are confident they can extend the idea beyond the whole to 7/ 6 , and
so on.
 When finding a fraction of a quantity, children will also need to consider
whether a unit of measurement is required in the answer. Ensure that they
know the relationships between familiar units of measure.
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Level 3-4 15. Can I calculate a percentage of a number or
quantity?
Teaching guidance
Key vocabulary
tenths, hundredths, percentage, equivalent, %
Models and images
Demonstrate how finding 10% can often be a useful starting point when finding
other percentages. For example you can find 20% by doubling 10%, find 5%
by halving 10% or find 15%, by adding 10% and 5%.
The diagram helps model how 20% of 50 is 10.
Area ITP
ITP. This can help children realise that finding 50% is the same as halving, to
find 25% they are finding one quarter, etc.
Fractions ITP
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Level 3-4 15. Can I calculate a percentage of a number or
quantity?
Teaching tips

When finding a percentage of a quantity, remind children that:
– they might find it helpful to first find 1% or 10% first;
– they may need to decide whether their answer needs rounding up or
down;
– if the question is in the context of money and measures, they will need to
remember to include the relevant unit in their answer
 Help children make links by creating webs of percentages of numbers and
then comparing the different amounts. For example, ‘What would &pound;2.48 look
like in comparison with &pound;248?’
&pound;186
75%
&pound;62
25%
&pound;2.48
1%
&pound;248
100%
&pound;124
50%
&pound;74.40
30%
&pound;24.80
10%
&pound;49.60
20%
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Level 3-4 16. Can I interpret the numbers on a calculator
display?
Teaching guidance
Key vocabulary
calculator, display, key, operation key, enter, decimal point, digit
approximately, estimate, round up, round down
pound, pence, metre, centimetre, litre, millilitre, kilogram, gram
Models and images
12.0
12.5
Use a projected calculator to show how you solve a question using a calculator and
discuss the interpretation of the display.
Teaching tips

Teach children how to interpret the displayed numbers, particularly large
numbers as there no gaps to help them read these correctly.

Give children opportunities to discuss the numbers appearing on their
calculator screen. For example, ‘How many different contexts can you
provide for 5.6?’

Promote checking strategies consistently, so that they become ‘second
nature’ for the children. Strategies you might use include:
– appointing a ‘checker’ when children are working in groups.
– using a projected calculator to model checking strategies.
– asking questions such as; ‘The answer to 32 &times; 27 was given as 288.
How do you know this is incorrect? What would an approximate answer
be?’ or ‘14.7 &times; 2.3 = 338.1. What error do you think has been made?
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Level 3-4 16. Can I interpret the numbers on a calculator
display?
How do you know the answer is incorrect? What would an approximate

Further guidance on the use of calculators in the teaching and learning of
mathematics can be found in the Renewed Framework Guidance paper.
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Level 3-4 17. Can I use a calculator to solve problems with more
than
one step?
Teaching guidance
Key vocabulary
operation, multi-step, estimate, calculate, check, calculation, method, show your
method
Models and images
Use a projected calculator to show how you solve a question using a
calculator.
Model how you would ‘show your method.’
Teaching tips
 Encourage children to use key steps when solving problems, such as:
1. Read the problem carefully (twice or more).
understand the question.
4. Decide what information you need and the operations you will use.
5. Make an estimate of the answer.
6. Record your calculation(s) and solution.
7. Check your answer and make sure you have used the correct units for
measures or money.
reasonable by comparing it with your estimate.
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Level 3-4 17. Can I use a calculator to solve problems with more
than
one step?
 Children need to be taught to distinguish between important and
redundant information in word problems. They should be encouraged to
annotate questions to help them find the key words and numbers. Good
examples of annotation, estimation, calculating and checking should be
displayed as part of the classroom learning environment.
 A commonly occurring error is for children to misinterpret values that
represent money; for example they may give an answer of &pound;12.5 instead
of &pound;12.50 or interpret 12.5 on a calculator display as &pound;12 and 5p. Ask
children to match the correct number of notes and coins to the calculator
display.
 Teach children, when they use a calculator, to record their calculations
together with the answers they obtain, at each stage in a multi-step
calculation. Emphasise that recording the ‘method’ is about recording the
number sentences or the calculations involved. Encourage children to
check whether each answer makes sense as they work through a
problem.
 Further guidance on the use of calculators in the teaching and learning of
mathematics can be found in the Renewed Framework Guidance paper.
Overcoming barriers in mathematics – helping children move from level 3 to level 4
00695-2007CDO-EN
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Level 3-4 18. Can I estimate and measure angles less than 180˚?
Teaching guidance
Key vocabulary
straight line, angle, right angle, acute angle, obtuse angle, reflex angle
degree, whole turn, half turn, quarter turn
angle measurer, protractor
Models and images
Use practical equipment such as strips of paper
or a fan to make angles and model that angles
are a measure of turn.
Teaching tips

Ensure children understand that an angle is ‘a measure of turn’.

Provide opportunities for children to classify angles and use the correct
vocabulary.

Demonstrate the use of a protractor and ensure children have plenty of
practical experience of reading the scale correctly and accurately. Aim for
an accuracy of within one degree of the precise measurement.

Discuss common misconceptions and errors with children, for example:
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Level 3-4 18. Can I estimate and measure angles less than 180˚?

Encourage children to annotate angles when estimating their size. Model
how to estimate the size of an angle using ‘benchmarks’ such as 90&deg;,
180&deg;, and 45&deg;.

Make relevant cross-curricular links in subjects such as design and
technology, art and PE, to reinforce measuring and estimating angles.
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Level 3-4 19. Can I calculate angles on a straight line and in a
triangle?
Teaching guidance
Key vocabulary
straight line, angle, right angle, acute angle, obtuse angle, reflex angle
degree, whole turn, half turn, quarter turn
parallel, perpendicular
Models and images
Explore and establish with children that the angles of a triangle total 180&deg; by
tearing off the three corners of a triangle and arranging them together to make a
straight line. Use a range of different triangles to help show that this is always
the case.
Use the Fixing points ITP to create a triangle. Move one of the points to create
a different triangle (e.g. drag ‘b’ to a different position on the grid) to illustrate
that the angles still total 180&deg;.
Fixing points ITP
Teaching tips

Ensure that children understand that ‘calculate’ does not mean ‘measure’
(children often measure an associated image even if the instructions
clearly say that it is not drawn to scale).
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Level 3-4 19. Can I calculate angles on a straight line and in a
triangle?

Rehearse and practise associated mental skills, for example complements
to 180.

Model how to annotate diagrams and encourage children to record extra
information on their own work.
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Level 3-4 20. Can I use coordinates to draw, locate and complete
shapes?
Teaching guidance
Key vocabulary
coordinate, axis, axes, origin, x-axis, y-axis, horizontal, vertical, parallel,
perpendicular
(also shape vocabulary, for example vertex, vertices)
Models and images
Use the Coordinates ITP to create the start of a shape. Ask children to identify
the coordinates of the labelled points and then suggest the coordinates for the
missing vertex or vertices.
Coordinates ITP
Teaching tips

Explain that the point with coordinates (0, 0) is called the origin and
discuss the meaning of this word.

Ensure that children understand that coordinates are used to describe
position in relation to the origin.

Children need to appreciate that the order of the numbers is important.
The first number tells you how far to go across while the second number
tells you how far to go up or down.
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Level 3-4 20. Can I use coordinates to draw, locate and complete
shapes?

Give children experience of physically moving an image of an object from
the origin across and then up to a desired position, describing the journey
the image makes.

Model annotation and encourage children to add information to the grid.
Good examples of annotation should be displayed as part of the
classroom learning environment.

When drawing shapes make sure that the sides are not always parallel or
perpendicular to the axes. This will add extra challenge.
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Level 3-4 21. Can I estimate a quantity (mass, length or capacity)
and choose the most sensible unit of measure and equipment to
use?
Teaching guidance
Key vocabulary
estimate, measure, compare, convert, approximate, accurate, relationship,
scales, kilometre, metre, centimetre, millimetre, kilogram, gram, litre, millilitre,
metric, unit, length, distance, mass, weight, capacity
Models and images
Provide children with plenty of practical experience of estimating quantities
and then choosing and using suitable measuring equipment.
Teaching tips

Plan for children to experience use of a variety of measuring equipment,
including spring balances, calipers, different sized cooking spoons, and so
on, so that they know what is available and can make sensible choices.

Ensure that equipment is readily available for children to draw upon across
the curriculum.

Exemplify the purpose of estimation, in terms of developing a feel for the
different units of measure, by providing children with benchmarks to help
them to estimate, for example the height of a door is approximately
2 metres, a pencil is approximately 20 cm long, a bag of sugar usually
weight one kilogram.
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Level 3-4 21. Can I estimate a quantity (mass, length or capacity)
and choose the most sensible unit of measure and equipment to
use?

Help children to improve and refine their estimations as they gain
experience. For example, after ten tries, can they get within 50 g of the
real value when they estimate the mass of a handful of grapes.
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Level 3-4 22. Can I convert between units?
Teaching guidance
Key vocabulary
kilometre, metre, centimetre, millimetre, kilogram, gram, litre, millilitre
Models and images
Use the converting measures spreadsheet to practise and reinforce the
conversion between different measures.
Use the Moving digits ITP to model the effect of multiplying and dividing by
100 and 1000.
Moving digits ITP
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Level 3-4 22. Can I convert between units?
Teaching tips
 Link converting units of measures to estimating, in order to help children
spot where they have made mistakes, for example 1.5 m cannot equal
15 cm because I know 1.5 m is about my height but 15 cm is half the
length of a ruler.
 Build the rehearsal of converting units into oral and mental starter
activities, and as a practical context in lessons focused upon calculation
where children are multiplying and dividing by 10, 100 and 1000.
 Explore the language of units, for example, roots from which ‘centi’ and
‘milli’ are derived and when else they are used (e.g. century, centurion).
 Provide regular practical opportunities for children to learn and understand
the relationship between units.
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Level 3-4 23. Can I read a scale on a thermometer, protractor,
ruler, weighing scale and measuring cylinder?
Teaching guidance
Key vocabulary
measure, measurement, standard unit, scale, measuring scale, division,
Models and images
on different scales, for example
mark the arrow in the correct place
on the second scale.
The measuring cylinder, measuring scales and thermometer ITPs allow
you to adjust the intervals on the scale.
Measuring cylinder ITP
Measuring scale ITP
Thermometer ITP
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Level 3-4 23. Can I read a scale on a thermometer, protractor,
ruler, weighing scale and measuring cylinder?
Teaching tips

Provide practical opportunities for children to select and use appropriate
units of measure and to consider the degree of accuracy within a given
context. For example measuring glass for a window will require greater
accuracy than measuring material for curtains that can be adjusted.

Explore a variety of types of scales. For example make a scale using a
length of hose and present it vertically, horizontally or as a loop for

Take opportunities to practise reading scales accurately across the
curriculum, for example within science, geography and PE lessons.

Ask children to collect photographs of scales within their own homes to
display. Who has the greatest number of examples? Who can find the
most unusual example? Ask children to find out how their parents use
scales in their everyday lives.

Model annotation of images of scales, particularly to help children interpret
readings that lie in between numbered divisions.
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Level 3-4 24. Can I work out the area and perimeter of a
rectangle?
Teaching guidance
Key vocabulary
area, covers, surface, square centimetre (cm2), square metre (m2),
square millimetre (mm2)
edge, perimeter, metre, centimetre, millimetre
Models and images
Use the Area ITP to create a rectangle, and discuss its area and
perimeter. Gradually increase the size of the rectangle. How does the
area increase? How does the perimeter increase?
Area ITP
Ask children to measure the rectangles in the Ruler ITP and then use
this information to find the area and perimeter of the shapes.
Ruler ITP
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Level 3-4 24. Can I work out the area and perimeter of a
rectangle?
Teaching tips
 Develop children’s understanding of perimeter while they are working on
length, rather than only ever linked it with work on area. Provide practical
tasks such as using art straws to make rectangles and then laying them
end to end to find the perimeter.
 Use equipment to work towards learning the formula for calculating
perimeters of rectangles, for example use straws to make
a rectangle and then place the lengths together and the widths together.
 Model the annotation of rectangles to keep an ongoing record when
working out perimeters.
 Link work on finding area with arrays and developing images for
multiplication and division facts.
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Level 3-4 25. Can I sort and interpret data in Venn and Carroll
diagrams?
Teaching guidance
Key vocabulary
Venn diagram, Carroll diagram, region, intersection, describe, explain, categories
Models and images
Carroll diagram
Venn diagram
Teaching tips
 Ensure children have experience of creating their own categories
for sorting numbers, objects, ideas or shapes onto Venn and
Carroll diagrams.
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Level 3-4 25. Can I sort and interpret data in Venn and Carroll
diagrams?
 Present children with unusual situations, such as examples in which it is
impossible for there to be anything in the intersection of a Venn diagram
(e.g. numbers that are divisible by 11, numbers less than 10), and ask
them to explain why the intersection will be empty.
 When exploring Venn diagrams, discuss the meaning of the section within
the rectangle but outside the ellipses (or whatever shape is used for the
set of enclosures), and ask children to generate examples that would go
into this section.
 Take opportunities to discuss properties of different items that are
represented in Carroll diagrams across the curriculum. Information is often
presented in this way within geography, history and science.
 Ask children to deconstruct a Carroll diagram and reconstruct it as a Venn
diagram and vice versa.
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Level 3-4 26. Can I explain what information a graph or chart is
showing?
Teaching guidance
Key vocabulary
graph, chart, table, axis, axes, describe, explain, categories
Models and images
Introduce children to a wide and rich variety of tables, charts and graphs.
Teaching tips

Introduce children to a wide and rich variety of tables, charts and
graphs within all curricular areas, taking opportunities to ask questions
about the way information is organised and what different
presentations show.

The interpretation of charts and tables should be a focus in the oral
and mental starter and the plenary, and not just in the main part of the
lesson.

Provide children with different charts and graphs without titles, labels
on the axes, or units, and ask them to ‘tell the story of the graph’.
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Level 3-4 26. Can I explain what information a graph or chart is
showing?

Expect children to summarise what any graph or chart shows before

Give children opportunities to take information presented in one format
and re-present it in another format, for example deconstructing the barline chart below to create the frequency table.
Number of throws
Dice-rolling experiment
Score
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Level 3-4 27. Can I draw a conclusion from a graph?
Teaching guidance
Key vocabulary
data, information, survey, questionnaire, graph, chart, table, scale, interval,
division, horizontal axis, vertical axis, axes, label, title, pictogram, bar chart, barline chart, line graph, pie chart, interpret, describe, explain
Models and images
Ensure children experience a wide range of different graphs.
Average temperatures
35
30
25
20
Degrees Celsius
Majorca
Corfu
15
10
5
0
April
May
June
July
Month
August
September
October
Temperature throughout the day
Teaching tips
 Encourage children to pose their own questions from looking at graphs as
well as responding to specific questions.
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Level 3-4 27. Can I draw a conclusion from a graph?
 Ensure children experience a range of graphs, including those with
unlabelled divisions, line graphs where intermediate points do and do not
have meaning, and different presentations of the same data.
 Give children conclusions drawn from graphs that cannot be true, and ask
them to explain why; challenge them to spot conclusions that are invalid
(e.g. on the line graph shown in the Models and images section, ‘the
temperature at 5pm will be 11 &deg;C’) and explain why this is not a valid
conclusion.
 Model the process of annotating and interacting with graphs in order to
answer questions and draw conclusions; use ‘shared thinking’ and ‘shared
writing’ techniques to make explicit how you analyse and explain the
information presented by a graph.
 Give children questions that require them to compare information from two
graphs, different categories on the same graph or different sets of data
represented on the same graph, for example:
What was the most unsuccessful year for space missions?
How do you know?
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Level 3-4 28. Can I understand the meaning of points between
labelled divisions on graphs?
Teaching guidance
Key vocabulary
data, information, survey, questionnaire, graph, chart, table, scale, interval,
division, horizontal axis, vertical axis, axes, division, unit, label, title, pictogram,
bar chart, bar-line chart, line graph, pie chart, interpret, describe, explain
Models and images
Use the Line graph ITP to
create a line graph. Change
the scale on the y-axis and
discuss the effect this has
on the line graph.
Line graph ITP
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Dice-rolling experiment
Nnumber of throws
Ensure children have opportunities
to work with graphs and charts
where results do not always occur
at the labelled divisions.
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Level 3-4 28. Can I understand the meaning of points between
labelled divisions on graphs?
Teaching tips
 Help children to recognise the link between reading a scale on a bar chart
or line graph and finding a number on a partly numbered line.
 Ensure children have opportunities to work with graphs and charts where
results do not always occur at labelled divisions, and where axes are
labelled in different increments from multiples of 2, 5 or 10.
 Model strategies for annotating graphs accurately to work out points
between labelled divisions. Display examples of annotation as part of the
classroom learning environment.
 Encourage children to annotate the charts and to use tracing paper where
appropriate.
 Use ICT to experiment with the effect that changing the labelled divisions
on the axes has on graphs (e.g. use the Line graph ITP). Make
predictions, and then increase the number of divisions to
check accuracy.
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Level 3-4 29. Can I interpret the sectors in a pie chart?
Teaching guidance
Key vocabulary
data, information, survey, questionnaire, graph, chart, division, sector, label, title,
pie chart, interpret, describe, explain
Models and images
Compare the information shown within two related pie charts to emphasise
that each sector represents a proportion of the whole.
Compare the information shown within two related pie charts that are different
sizes.
Teaching tips
 Help children to understand that each sector on a pie chart represents a
proportion of a whole set, and that the total number of items represented
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Level 3-4 29. Can I interpret the sectors in a pie chart?
by the pie chart has to be known in order to calculate how many each
sector represents.
 Ensure children encounter a range of charts, including pairs of pie charts
that represent different total numbers (such as the TV programme
example above).
 Encourage children to annotate the charts and to use tracing paper where
appropriate to compare different sections.
 Model how to provide structured explanations for conclusions you are
drawing from pie charts.
 Use pie charts as contexts when finding fractions of numbers and
quantities. Ask, for example: ‘Roughly what fraction is the “lost” sector on
the netball chart? If the chart represents 30 games, how many were lost?’
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Level 3-4 30. Can I work out the mode and range of a set of data
and use this to answer questions?
Teaching guidance
Key vocabulary
frequency, mode, maximum/minimum value, range, average, statistics, data,
information, measure, survey, questionnaire, interval, explain, justify
Models and images
Use a number line to represent data given in a frequency table, to draw
attention to where and how often each value occurs, for example.
Daily
temperature
(&deg;C)
–2
–1
0
1
2
3
5
6
Frequency
2
2
2
2
1
1
3
1
&deg;C
–2 –1
0
1
2
3
–2 –1
–2 –1
0
0
1
1
2
3
4
5
6
5
5
5
6
7
The range is from –2 to 6 = 8 and the mode is 5.
Use practical methods to help children actually see the range. For example ask
groups of six children to take measurements of their hand spans for both
hands, and to record the lengths on paper strips of two colours (one colour for
the right hand, another for the left hand). Then ask them to order the 12 strips
of paper by size of the measurement, smallest to largest, and record the range
of hand spans for their group as follows:
The range of hand spans for our group was….
The range of left-hand spans for our group was….
The range of right-hand spans for our group was….
This group data could then be pooled to produce the three ranges for the
whole class.
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Level 3-4 30. Can I work out the mode and range of a set of data
and use this to answer questions?
Teaching tips

Introduce frequency and mode by giving children pieces of paper with the
results of a simple survey written on them, and asking them to produce a
human bar chart – the frequency for any number is the number of children
in the queue and the mode is the number represented by the longest
queue.

Create a frequency chart to give a visual image of the data (see above).
This gives an image of both mode and data range.

Tabulate the data, marking each value in the data list as it is recorded.

A number line is helpful to work out the range of a set of data.

Make links between handling data in mathematics and other areas of the
curriculum, for example finding out the range of children’s ages attending
school in Victorian times and the present day.

Children need to understand that the range and mode describe the data
set they are using or have generated. For example, give children two dice,
one with even numbers 2, 4, 6, 8, 10 and 12; the other with the odd
numbers 3, 5, 7, 9, 11 and 13. Ask the children to roll the two dice, find the
difference between the two numbers displayed and place a cube above
this number on a number line. After five rolls ask them to record:
The range of results was from ... to ...
The mode was ...
Repeat the process after 10, 15, 20 and 25 rolls; then ask children to look
at the pattern the in results to try to predict the range and mode after 40
rolls.
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Level 3-4 31. Can I use the vocabulary of probability to predict
outcomes and discuss and explain events?
Teaching guidance
Key vocabulary
fair, unfair, likely, unlikely, likelihood, equally likely, certain, uncertain, probable,
possible, impossible, chance, good chance, poor chance, no chance, equal
chance, even chance, fifty-fifty chance, risk, doubt, biased, random
Models and images
Represent probability on a scale, for example
Next week will be 8 days long
impossible
The day after Sunday will be Monday
evens
certain
Use a variety of spinners.
Number spinners ITP
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Level 3-4 31. Can I use the vocabulary of probability to predict
outcomes and discuss and explain events?
Teaching tips
 Use cross-curricular opportunities to discuss events that have a good
chance of happening and those that have a poor chance. Give children the
opportunity to use the language of probability.
 Model how to predict the probability of an outcome by talking through the
number of different possible outcomes and how likely each outcome is.
Include outcomes that are impossible.
 Ensure children have opportunities to predict and investigate probabilities
using a variety of resources that generate random outcomes.
 Encourage children to annotate the charts and spinners where
appropriate, to compare different sections.
 Help children recognize the difference between the theory of outcomes
and the actual experimental results. For example, do they actually get
each number once when throwing a six-sided dice six times?
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