Mathematics standards file Sachin – secure level...

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Mathematics standards file – Sachin

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Mathematics standards file Sachin – secure level 5Ma1 Using and applying mathematics

Ma2 Number

Ma3 Shape, space and measures

Ma4 Handling data




Ma1 Using and applying mathematics

Prime number investigation

Teacher’s notes

finds examples and counter-examples to justify answer

uses vocabulary accurately (prime number, sum of the digits) to explain and justify conclusion.

Next steps

investigate and explain other number patterns.




Primes and squares

Teacher’s notes

independently selects an appropriate method to record answers systematically

identifies complex patterns and relationships: for example, states that one of the squares has to be a multiple (or factor) of 4, and the other has to be an odd square




draws simple conclusions based on the investigation and gives an explanation of reasoning, supported by examples.

Next steps

justify methods (e.g. explain how systematic recording can be used to prove that all the possibilities have been found)

apply understanding of simple formulae and symbols to represent problems

make predictions and relate them to conclusions.




Mobile phone investigation




Teacher’s notes

discusses problem and possible approaches with a partner

identifies and obtains necessary information to solve the problem, recognising the information that is important

breaks a multi-step problem into simpler steps

checks as he works, reviewing methods with his partner and correcting any errors

chooses an appropriate, systematic way of recording findings and interprets results to draw simple conclusions.




Next steps

develop use of diagrams and graphs to communicate results and conclusions (e.g. create a line graph to show the set-up cost and subsequent monthly expenses for each network).

What the teacher knows about Sachin’s attainment in Ma1

Sachin independently interprets and solves problems in a range of mathematical contexts, applying existing knowledge and skills in a new context. He identifies and obtains the information required to solve a problem, deciding which information is important and which information is missing in order to solve the problem. He breaks down a problem or investigation into simpler steps when deciding how to approach a problem. Where appropriate he refers to previous work to support the development of problem-solving approaches. While working, he checks the reasonableness of his answers and reviews whether his approach is appropriate, making adaptations or corrections where necessary.

Sachin records his results and conclusions in a systematic way, using appropriate methods (e.g. a table, oral or written explanations). He gives detailed explanations to explain his solutions. He independently chooses precise and accurate mathematical vocabulary in his explanations and improves his explanations by redrafting them.

Sachin draws simple conclusions based on his mathematical investigations. He justifies his conclusions and supports his explanations with examples and counter-examples. He identifies complex sequences, patterns and relationships in his problem solving and investigations.

Summarising Sachin’s attainment at Ma1

Sachin’s attainment in Ma1 is best summarised as being securely at level 5. He independently solves mathematical investigations, recognising which information is important or which information is missing, and uses this to develop a line of enquiry. He breaks a problem down into simpler steps and confidently tries different approaches to reach a solution. Sachin draws conclusions based on his results and communicates his ideas using precise mathematical vocabulary. To progress further within this level, he needs to continue to use simple formulae and symbols to represent problems. He should represent his results in a wider variety of forms (e.g. graphs, diagrams). He also needs to make predictions at the start of his investigations and relate them to his conclusions. He is currently using and applying his mathematical skills from Ma2 and Ma3, but needs to develop this within Ma4, in particular when interpreting graphs and considering further questions to investigate.




Ma2 Number

Place value




Teacher’s notes

confidently multiplies and divides numbers by 10, 100 and 1000 to reach a target number

extends understanding by multiplying a number with four decimal places (0.4697) by 10 000 to reach the target number

understands the relationship between multiplication and division in the context of place value and multiplies or divides any number by powers of 10 (e.g. knows that 63 710 ÷ 1000 is the same as 63 710 ÷ 10 ÷ 10 ÷ 10).

Next steps

apply knowledge of place value to multiply multiples of 10, 100 and 1000 (e.g. 4.56 × 400 is the same as 4.56 × 100 × 2 × 2), reinforcing the use of factorisation to multiply and divide.




Multiplying decimals

Teacher’s notes

uses the grid method to multiply decimals by single-digit and two-digit numbers

uses his knowledge of table facts and place value to support calculations (e.g. question 7: calculates 6 × 0.4 = 2.4 and 6 × 0.05 = 0.3)

adds and subtracts numbers that do not have the same number of decimal places (e.g. question 11: 1900 + 760 + 38 + 5.7 + 1.14 = 2704.84).




Next steps

multiply decimals using a standard written method.

Exploring sequences

Teacher’s notes

calculates the number of dots required for some terms in the sequence, identifying larger numbers that would be in the sequence

finds the rules for the sequences, expressing them algebraically, and uses this to calculate the 100th term in the sequence

uses the correct notation, for example 6n + 2

expresses sequences in words and relates formulae to explanations.

Next steps

explore more complex sequences, with less scaffolding, and express these using algebraic notation.





Sachin confidently uses knowledge of place value to multiply and divide integers and decimals by 10, 100, 1000 and 10 000. He knows that multiplying by 1000 is the same as multiplying by 10 × 10 × 10. Sachin orders negative numbers in the context of temperature and money (e.g. bank balances). He is beginning to order them out of context. He explains what a prime number is and recognises whether a two-digit number is prime using knowledge of multiplication facts. He recognises when a three-digit or four-digit number is not a prime number using knowledge of divisibility rules (e.g. he knows that 1461 cannot be a prime number because it is divisible by 3) and using a calculator if necessary. He confidently rounds decimals to one, two and three decimal places.

Sachin reduces a fraction to its simplest form by cancelling common factors. He recognises common factors between different fractions. He orders fractions where the denominators are different by finding and converting them to their lowest common denominator. He recognises the equivalence of fractions such as and as well as the equivalence between fractions, decimals and percentages, for example = 0.6 = 60%. He knows that is represented on the calculator as 0.3333333 and understands that this can be recorded as 0.3 and called a recurring fraction. He confidently orders decimals with up to four decimal places.

Sachin uses his understanding of inverse operations to identify missing values in number sentences involving all four operations and to check results when solving problems. He understands how brackets can be used to affect the order of operations and applies this when solving calculations, but is not yet able to construct number sentences involving brackets given a word problem. When dividing, he expresses a remainder as a fraction; he uses a calculator to express a quotient as a decimal.

When working mentally, Sachin confidently uses all four operations with whole and decimal numbers. He estimates answers to questions by using appropriate approximations (e.g. to calculate 18 × 27 he approximates this to 20 × 25 to create an estimate). He mentally calculates decimal complements to 10 or 100, extending this to numbers with up to four decimal places, if he sees the written number (e.g. 63.0125 + __ = 100). He uses his knowledge of multiplication facts to multiply a two-digit number by a single-digit number. He applies his knowledge of tables and place value to multiply and divide decimals such as 0.05 × 8 and 24 ÷ 0.8. He mentally calculates fractions or percentages of a number or quantity (e.g. of 350 ml or 75% of £240).

Sachin solves multi-step problems involving all four operations in a variety of contexts, choosing to use a calculator or an efficient written, informal or mental method to perform calculations. He solves simple ratio and proportion problems and often uses diagrams or tables to support his calculations. He relates his understanding of proportion to fractions and percentages. Sachin understands simple expressions using symbols (e.g. 3n – 1) and evaluates these expressions by substituting numbers into them. With support, he constructs simple formulae, using two operations, and uses this to calculate the nth term in a sequence. He plots and interprets coordinates in all four quadrants.

Sachin uses written methods for all four operations with decimals to two places. He adds and subtracts numbers that do not have the same number of decimal places. He




multiplies decimal numbers by single-digit or two-digit numbers using the grid method. He uses partitioning or long multiplication to multiply a three-digit number by a two-digitnumber. When dividing, he employs the chunking method accurately, usually using the largest possible chunks. He uses the short division algorithm when dividing by a single digit.

Summarising Sachin’s attainment in Ma2

Sachin’s attainment in Ma2 is best described as working securely at level 5. He uses all four operations with decimals to two places, choosing appropriate non-calculator methods, including standard written algorithms. He applies his understanding of mental, written and calculator methods to solve problems. He reduces fractions to their simplest forms by cancelling common factors. To make further progress, he needs to apply his understanding of brackets to create number sentences from a two-step word problem. He should also solve more complex problems involving ratio using the correct notation.




Ma3 Shape, space and measures

Exploring shapes




Teacher’s notes

uses three triangles and a parallelogram to create new, different 2-D shapes

independently selects appropriate mathematical vocabulary to classify and describe the new 2-D shapes according to their properties, for example reflex angle, parallel, perpendicular, heptagon

uses correct notation to indicate parallel lines and sides of the same length

measures side lengths to calculate the perimeter of the irregular 2-D shapes.

Next steps

construct shapes with specific constraints: for example, draw a pentagon with three right angles, but no sides the same length; or draw a hexagon with three right angles and one line of symmetry

draw 2-D shapes of a given area on a square grid, for example a parallelogram with an area of 12 cm2.




Cube nets

Teacher’s notes

knows a variety of different nets will make a cube

visualises where arrows need to be drawn on the net to satisfy the constraint that the arrows must point to the right when the cube is constructed

checks solutions by constructing a net for each cube.




Next steps

explore and visualise patterns on nets of other 3-D shapes.

Doodle-stars

Teacher’s notes

approaches the investigation independently, selecting methods and equipment appropriate to the task

measures and draws angles to the nearest degree when neither edge is horizontal or vertical

draws simple conclusions based on results




checks results, considering whether or not they are reasonable

recognises that inaccuracies in measurements result in some totals appearing not to be 180.

Next steps

solve problems using angle properties of intersecting and parallel lines and explain these properties.

Who got the better deal?




Teacher’s notes

independently breaks into the problem, choosing appropriate calculations to compare like with like

converts between l and ml, and kg and g, to solve problems.

Next steps

begin to use the unitary method to compare prices.


Sachin knows that the sum of the angles in a triangle or on a straight line equals 180and uses this to calculate ‘missing angles’, including for right-angled triangles or isosceles triangles where only one angle is given. He uses knowledge that the sum of angles at a point equals 360 to solve problems. Using properties of shape, he explains his reasoning about special triangles and quadrilaterals (e.g. creates a tree diagram to sort quadrilaterals using properties such as pairs of parallel sides, lines of symmetry, right angle). He plots the missing coordinates of 2-D shapes when one or two coordinates are missing (e.g. when given the coordinates of three vertices of a parallelogram he plots the fourth vertex).

Sachin identifies line symmetries of common 2-D shapes and draws shapes with a fixed number of lines of symmetry (e.g. an octagon with four lines of symmetry). He recognises when a shape has been translated, reflected or rotated. He translates shapes along an oblique line and describes the translation in terms of movement along the x-axis and y-axis. He reflects shapes in two mirror lines where the shape is perpendicular or parallel to either mirror line. With the help of tracing paper, he rotates shapes clockwise or anticlockwise through 90, 180 and 270 when the centre of rotation is a vertex of the shape.

Sachin converts between metric units and knows the rough metric equivalents of imperial units still in common use. He applies this in problem-solving contexts. He reads and interprets scales, including values between labelled divisions, in a variety of contexts including mass, capacity and length. He measures and draws angles to the nearest degree, but when constructing 2-D shapes is not always accurate to 1 mm and 2. He finds the area and perimeter of compound shapes composed of rectangles given some of the edge lengths.

Summary of Sachin’s attainment in Ma3

Sachin’s attainment in Ma3 is best described as working securely within level 5. He uses appropriate mathematical language to describe the properties of 2-D shapes. He knows the angle sum of a triangle and the sum of angles of a point. He converts between different units of measure, including metric and imperial units, and applies this




in problem-solving contexts. To progress within the level, Sachin should become more accurate when constructing shapes and reflect shapes by measuring the distance of vertices from the mirror line.




Ma4 Handling data

Exploring forces




Teacher’s notes

understands a question can be addressed by collecting and analysing data

independently repeats measurements for an experiment, recognising that different outcomes may result from repeating an experiment




constructs an appropriate table to record results

chooses to find and use the modes (of the lengths of the elastic bands) to construct a line graph (in one case uses the median when the mode cannot be found)

creates a line graph, independently choosing a suitable scale, title and appropriate axis labels.

Next steps

draw conclusions from data collected and identify further questions to explore

recognise that where units are indicated on the axis labels they do not need to be included on the graph scale

understand which average is most appropriate (mean, median or mode) to understand a set of data.


Sachin understands that the outcome of an experiment may not be the same if repeated (e.g. rolling a dice, results of a spinner) and that the more an experiment is repeated, the more accurate the overall results. He decides which data he needs to investigate a question and recognises how to source this data effectively (e.g. measuring, observing and counting). He independently designs ways to collect and record information and recognises when to group data into equal intervals.

He understands and uses the terms continuous and discrete data. He compares discrete data by calculating the mode, median or mean. He compares two spinners, for example explaining which one is more likely to show an even number.

He interprets bar graphs using grouped data and interprets the scale on bar or line graphs where he has to read or estimate between labelled divisions. He draws conclusions from a wide range of graphs, charts and diagrams using ICT where appropriate, and is beginning to consider further questions to investigate. When analysing possible outcomes of a spinner, he explains whether it is equally likely or unlikely to obtain certain outcomes (e.g. with a regular seven-sided spinner labelled 1 to 7, he recognises there is a greater chance of obtaining an odd number). He is beginning to express probability as a fraction (e.g. there is chance of rolling a 5 on a dice).

Summary of Sachin’s attainment in Ma4

Sachin’s attainment in Ma4 is best described as the middle of level 5. He understands that different outcomes may result from repeating an experiment and why it is important to do this to increase the accuracy of the overall results in contexts such as science experiments. He uses the mean, mode and median of discrete data, but does not




always understand the most appropriate average to choose. He finds and justifies simple probabilities using the language of chance or likelihood. He creates and interprets bar and line graphs, choosing appropriate scales and labels. To make further progress in this level, he needs to understand and use the probability scale from 0 to 1, interpret pie charts and recognise when data could be represented in misleading ways (e.g. in bar graphs and pie charts).




Overall assessment summary for Sachin

Sachin’s teacher judges his attainment across mathematics as working securely at level 5. His particular strengths are Ma2 AF Fractions, decimals and percentages, Ma2 AF Mental methods and Ma3 AF Properties of shape.

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