mathematics key learning indicators of performance

Mathematics

Key Learning Indicators of Performance: Exemplification for Year 5 (Secure)

© Lancashire County Council (2016)

Number – number and place value Number – addition and subtraction Number – multiplication and division

Count forwards or backwards in steps of powers of 10 for any

given number up to 1 000 000.

Child E can continue the counting sequence when counting in

steps of power of 10 from any number. He can confidently

identify which digits change in the count and why this is. He is

able to cross boundaries, for example when counting in steps

of 10 and starting at 49 661, he identified the continuation of

the sequence as 49 671, 49 681, 49 691, 49 701, 49 711 etc.

Count forwards and backwards in decimal steps.

Child E can count forward and backward in decimal steps

where the step size is either 0.1 or 0.01. As with whole

numbers, he is confident when crossing the boundaries. He

continued a count from 8.9 in steps of 0.1 beyond 10.

Read, write, order and compare numbers to at least 1 000 000

and determine the value of each digit.

Throughout his work in mathematics, Child E demonstrates his

ability to both read and write numbers with up to six digits

accurately.

See Sample A

Read, write, order and compare numbers with up to 3 decimal

places.

See Sample B

See Sample C

Identify the value of each digit to three decimal places.

Child E can confidently identify the value of a digit in a number,

either whole or decimal. He can identify the value of decimal

numbers as tenths, hundredths or thousandths. He can identify

19.46 as having 4 tenths and six hundredths but also knows

that the 0.46 can be identified as 46 hundredths.

See Sample D

See Sample E

Identify represent and estimate numbers using the number line.

See Sample C

See Sample F

Find 0.01, 0.1, 1, 10, 100, 100 and other powers of 10 more or

less than a given number.

Child E can find these numbers and can also explain which digit

changes in each and why. He knows it is not always one digit

and the language he used in the sample was clarified with him.

See Sample G

Choose an appropriate strategy to solve a calculation based

upon the numbers involved (recall a known fact, calculate

mentally, use a jotting, written method).

Given a calculation, Child E can identify whether he needs to

use a mental or written strategy. He prefers column methods

for addition, but can partition to add. When asked how he

would solve the calculation 3002 - 1995, he explained that he

would count on because it would be 5 to 2000, then 1002 to

3002 so the answer would be 1007.

See Sample M

Select a mental strategy appropriate for the numbers involved in

the calculation.

Recall and use addition and subtraction facts for 1 and 10 (with

decimal numbers to one decimal place).

Child E confidently recalls facts for both 1 and 10, linking them

to the pairs of numbers that make 10 and 100. He uses them in

calculating and solving simple word problems. When asked the

question ‘John had a piece of string 10cm long. He cut off a

piece 7.3cm long. How much was left?’ he answered 2.7cm

because 2 + 7 makes 9 and the 0.7 and 0.3 make 1 and that

makes 10 altogether.

See Sample N

Derive and use addition and subtraction facts for 1 (with decimal

numbers to two decimal places).

Child E can use the knowledge from the previous objective and

apply to these numbers.

Add and subtract numbers mentally with increasingly large

numbers and decimals to two decimal places.

Child E can add and subtract numbers mentally where it is an

appropriate strategy. When asked to work out 875 641 - 875

619, he was able to correctly identify that the answer was 22,

however, when faced with numbers of this magnitude, he often

defaults to written methods although if encouraged he will use

other strategies.

See Sample O

See Sample P

Choose an appropriate strategy to solve a calculation based

upon the numbers involved (recall a known fact, calculate

mentally, use a jotting, written method).

Given a calculation, Child E could say whether they would use a

mental or written strategy. When asked how he would

calculate 199 x 3, he explained that he would first calculate 200

x 3 and then take away 3 to give 597. He confidently uses

written methods where required.

Identify multiples and factors, including finding all factor pairs

of a number, and common factors of two numbers.

See Sample W

See Sample X

Know and use the vocabulary of prime numbers, prime factors

and composite (non-prime) numbers.

In conversation, Child E was able to define prime numbers as

‘numbers that have just two factors’. He explained that 9 is not

prime because it has more than two factors: 1, 3 and 9.

Establish whether a number up to 100 is prime and recall prime

numbers up to 19.

Child E can recite the list of prime numbers to 20. Initially he

created arrays using counters to establish whether or not a

number was prime. He can use his knowledge of rules of

divisibility for this. He was able to identify 83 as prime by

discounting immediately 2, 5, 4, 6, 8, 10 and 12 due to the units

digit and then used rules of divisibility to check 3, 7 and 9.

Recognise and use square (2) and cube (3) numbers, and

notation.

Child E used counters to create and investigate arrays and

square patterns. He identified that square numbers are a

number multiplied by itself ‘because it is length x width for the

array and they are the same so it must be a square’. He also

used the base 10 to identify cube numbers by creating cubes.

Use partitioning to double or halve any number, including

decimals to two decimal places.

During starter sessions, Child E uses partitioning strategies to

support his doubling and halving, for example, he was able to

halve 7.6 by partitioning it into 6 and 1.6 and halving each.

When asked why he chose this and not 7 and 0.6, he replied

‘because halving 6 is easier than halving 7.’

Mathematics



Number – number and place value (continued) Number – addition and subtraction (continued) Number – multiplication and division (continued)

Round any number up to 1 000 000 to the nearest 10, 100,

1000, 10 000 and 100 000.

See Sample H

Round decimals with two decimal places to the nearest whole

number and to one decimal place.

See Sample I

Multiply/divide whole numbers and decimals by 10, 100 and

1000.

Child E does this confidently and can explain the effect on the

digits. He can also answer questions requiring knowledge of

the inverse of this. When given the problem, ‘I’m thinking of a

number. I divide it by 100 and the answer is 3.46. What was my

number?’ He gave the answer 346 and explained that to find

what you started with you would have to multiply instead of

divide.

See Sample J

See Sample K

Interpret negative numbers in context, count on and back with

positive and negative whole numbers, including through zero.

Child E can continue counts that pass either forwards or

backwards through 0.

See Sample L

Describe and extend number sequences including those with

multiplication/division steps and where the step size is a

decimal.

Read Roman numerals to 1000 (M); recognise years written as

such.

Child E knows each of the symbols for Roman numerals. He

identified Roman numeral representations when playing a

bingo game with a group. He knows that MM represents 2000

and that 2016 is represented as MMXVI. When asked when the

next year would be that did not have an I in its representation,

he answered 2020 because that will be MMXX.

Solve number and practical problems that involve all of the

above.

Child E uses his place value knowledge to reason and solve

problems.

See Sample A

See Sample G

See Sample I

Add and subtract whole numbers with more than 4 digits and

decimals with two decimal places, including using formal

written methods (columnar addition and subtraction).

Child E confidently uses column methods for both addition and

subtraction. When adding, he can securely add more than two

numbers.

See Sample Q

See Sample R

Use rounding to check answers to calculations and determine,

in the context of a problem, levels of accuracy.

Child E can use rounding to make estimates about calculations

and will check his own and his partner’s work when working in

a pair. When estimating the answer to 9.4 + 3.89, he said 13

because he had rounded each number to the nearest whole (9

+ 4). When asked why he didn’t round to the nearest tenth he

stated that whole numbers were much quicker for him to

estimate.

Solve addition and subtraction multi-step problems in

contexts, deciding which operations and methods to use and

why.

See Sample O

See Sample P

See Sample S

See Sample T

See Sample U

Solve addition and subtraction problems involving missing

numbers.

Child E can use inverse to solve missing number problems with

addition and subtraction. He is also developing his knowledge

of identifying missing numbers in column methods.

See Sample P

See Sample V

Multiply and divide numbers mentally drawing upon known

facts.

Child E has a sound knowledge of times tables to 12 x 12. He is

also able to use these to help him with mental multiplication

and division alongside partitioning. He calculated 32 x 6

mentally and when asked how, stated ‘I know 3 x 6 is 18, so 30

x 6 is 180. 2 x 6 is 12 so I added them together and got 192.'

Solve problems involving multiplication and division including

using their knowledge of factors and multiples, squares and

cubes.

See Sample Y

Multiply numbers up to 4 digits by a one- or two-digit number

using a formal written method, including long multiplication

for two-digit numbers.

Child E can use the grid method to multiply. He is developing

his use of column method of multiplication.

See Sample Z

Divide numbers up to 4 digits by a one-digit number using the

formal written method of short division and interpret

remainders appropriately for the context.

Child E confidently uses the chunking method to carry out

division calculations. He is confident with subtracting larger

groups to make the calculation more efficient.

See Sample AA

Use estimation/inverse to check answers to calculations;

determine, in the context of a problem, an appropriate degree of

accuracy.

When asked to estimate the answer to 689 ÷ 4, Child E

suggested that ‘the answer would be near 170 because 68 ÷ 4

is 17 (I halved it and halved it again) and 689 ÷ 4 is ten times

bigger.’ When calculating, he uses inverse to check answers.

Solve problems involving addition, subtraction, multiplication

and division and a combination of these, including

understanding the meaning of the equals sign.

See Sample O See Sample P

See Sample S See Sample T

See Sample U See Sample Y

Solve problems involving multiplication and division, including

scaling by simple fractions and problems involving simple

rates.

Mathematics



Number – fractions Geometry – properties of shapes Measurement

Recognise mixed numbers and improper fractions and convert

from one form to the other.

See Sample AB

See Sample AC

Read and write decimal numbers as fractions (e.g. 0.71 = 71

100).

See Sample E

Count on and back in mixed number steps such as 11

2.

Child E joins in class counts in fraction and mixed number

steps. He can identify the sequence and continue it. When

asked to carry on the sequence 21

5, he said he splits the number

and counts on the whole number and then the fraction.

Compare and order fractions whose denominators are all

multiples of the same number (including on a number line).

Identify, name and write equivalent fractions of a given

fraction, represented visually, including tenths and hundredths.

See Sample AD

Recognise and use thousandths and relate them to tenths,

hundredths and decimal equivalents.

Through his work on place value and with measures, Child E

shows a good understanding of thousandths. He can identify

which digit is the thousandth digit in the number. He can write

the fraction and decimal equivalent between, for example,

0.326 and 326

1000. He knows that

300

1000 is equivalent to

3

10 and 0.3.

Add and subtract fractions with denominators that are the

same and that are multiples of the same number (using

diagrams).

Child E uses his knowledge of factors and multiples to help him

identify equivalent fractions to support his addition and

subtraction of fractions.

See Sample AE

Write statements > 1 as a mixed number (e.g. 2

5 +

4

5 =

6

5 = 1

1

5).

See Sample AB

See Sample AC

Multiply proper fractions and mixed numbers by whole

numbers, supported by materials and diagrams.

Distinguish between regular and irregular polygons based

on reasoning about equal sides and angles.

When given a set of assorted shapes, Child E was able to

sort them into two sets - regular and irregular. When

asked how he knew which shapes should go into the

regular set, he replied ‘On regular shapes, the sides are all

the same length and the angles are all the same too.’

Use the properties of rectangles to deduce related facts

and find missing lengths and angles.

Identify 3-D shapes from 2-D representations.

See Sample AH

Know angles are measured in degrees: estimate and

compare acute, obtuse and reflex angles.

See Sample AI

Draw given angles, and measure them in degrees (°).

See Sample AI

Identify:

- angles at a point and one whole turn (total 360°).

- angles at a point on a straight line and half a turn

(total 180°).

- other multiples of 90°.

See Sample AJ

Use, read and write standard units of length and mass.

Child E understands what the abbreviations mm, cm, m, km, g and

kg stand for and uses them in his work.

Estimate (and calculate) volume ((e.g., using 1 cm3 blocks to build

cuboids (including cubes)) and capacity (e.g. using water).

See Sample AK

Understand the difference between liquid volume and solid volume.

Child E knows that liquid volume is measured in ml and solid

volume is measured in cm3.

Continue to order temperatures including those below 0°C.

Convert between different units of metric measure.

Child E knows the relationships between the different units of metric

measure. He can very quickly mentally convert between mm and

cm, and uses his knowledge of multiplying and dividing by 1000

and his knowledge of decimals to mentally convert between litres

and millilitres; grams and kilograms; metres and kilometres.

Understand and use approximate equivalences between metric

units and common imperial units such as inches, pounds and

pints.

When given the conversions (i.e. 1 inch is approximately equal to

2.5cm), Child E can convert between measures, sometimes using

diagrams to support his work. He worked with his group to

convert a recipe from metric to imperial units.

Measure/calculate the perimeter of composite rectilinear shapes.

See Sample AL

Calculate and compare the area of rectangle, use standard

units square centimetres (cm2) and square metres (m2) and

estimate the area of irregular shapes.

See Sample AM

Continue to read, write and convert time between analogue and

digital 12 and 24-hour clocks.

Child E can tell analogue and digital time. When asked how he

converts between analogue and digital times he said ‘If it’s after

midday, you add 12 to the hour and there always has to be two

digits before the colon, so sometimes you have to put a 0 in, like

in 08:30 which is half past eight in the morning.’

Solve problems involving converting between units of time.

Use all four operations to solve problems involving measure using

decimal notation, including scaling.

See Sample AN

Mathematics



Number – fractions (continued) Geometry – position and direction Statistics

Recognise the per cent symbol (%) and understand that per

cent relates to ‘number of parts per hundred’, and write

percentages as a fraction with denominator 100, and as a

decimal.

Child E can identify percentages and understands they relate to

fractions with a denominator of 100. He can match percentage

cards to an appropriately shaded 100 square.

See Sample AF

Solve problems involving fractions and decimals to three places.

Solve problems which require knowing percentage and

decimal equivalents of 1

2, 1

4, 1

5, 2

5, 4

5 and fractions with a

denominator of a multiple of 10 or 25.

See Sample AG

Describe positions on the first quadrant of a coordinate

grid.

Using the strategy of tracing his fingers down from

the marked point to the x axis and across from the

marked point to the y axis, Child E can identify the

coordinate positions of marked points.

Plot specified points and complete shapes.

See Sample AO

Identify, describe and represent the position of a

shape following a reflection or translation, using the

appropriate language, and know that the shape has

not changed.

Complete and interpret information in a variety of sorting

diagrams (including those used to sort properties of numbers and

shapes).

Child E can use both Carroll and Venn diagrams. He can sort to

given criteria and his own. When he is familiar with the content,

he can also identify how information has been sorted.

See Sample X

Complete, read and interpret information in tables and

timetables.

See Sample AP

Solve comparison, sum and difference problems using

information presented in all types of graph including a line

graph.

Calculate and interpret the mode, median and range.

See Sample AQ

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A B

C

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D E

F

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G H

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I J

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K L

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M N

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O P

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Q R

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S T

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U V

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W X

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Y Z

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AA AB

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AC AD

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AE AF

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AG AH

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AI AJ

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AK AL

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AM AN

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AO AP

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AQ

mathematics key learning indicators of performance

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