Mathematics standards file Rebecca high level 5 Ma1 Using and applying mathematics Ma2 Number Ma3 Shape, space and measures Ma4 Handling data
Ma1 Using and applying mathematics Newspaper investigation Teacher s notes identifies and collects the relevant information to solve a problem breaks a multi-step problem into smaller steps independently decides how best to represent findings and conclusions using ICT constructs and interprets grouped data bar graphs interprets results to find conclusions and presents these using mathematical vocabulary.
Next steps recognise shortcomings of her data, for example sample size. What the teacher knows about Rebecca s attainment in Ma1 Rebecca makes decisions and choices to create her own tasks, answers questions and presents her findings. She identifies and obtains the relevant information required to solve multi-step problems and independently breaks tasks down into more manageable steps (e.g. Turning triangles in Ma3). She uses efficient methods, checks her own work for mistakes and approximates to see if answers are reasonable. Rebecca makes connections to previous work and suggests her own strategies to solve problems. She uses her mathematical skills when solving problems across the curriculum. Rebecca shows her understanding of situations by describing them using words, symbols and diagrams. She presents her work in a systematic way in order to show that she has found all possibilities (e.g. Scratch cards activity in Ma4). Rebecca decides how best to represent data and conclusions through using tables, graphs, diagrams or words. She is beginning to choose to use formulae and algebraic symbols to solve problems. Rebecca draws conclusions of her own and is beginning to make justifications using mathematical vocabulary. She recognises that using a systematic recording method informs her working and helps when trying to explain conclusions or when going back to check findings. Summarising Rebecca s attainment in Ma1 In Ma1 Rebecca is best described as working at the top end of level 5. She has started to move into level 6 in AF Problem solving. Rebecca identifies and obtains the relevant information in order to solve a mathematical problem. She explains her findings, is beginning to justify her reasoning and presents conclusions orally. To progress to level 6, she needs to solve more substantial tasks (including those involving the use of a calculator) and continue to develop her mathematical justifications. She should use precise mathematical language, particularly when writing, and use formulae and symbols to represent problems.
Ma2 Number Ordering fractions Teacher s notes orders fractions where the denominators are different by finding a common denominator recognises equivalence between fractions such as and since both are equal to. Next steps extend equivalence to more complex fractions where the denominators have a higher common multiple, for example order numbers such as,,, and.
Word problems
Teacher s notes uses appropriate operations, with decimals to two places, to solve multi-step problems in volume, fractions and percentages calculates fractional and percentage parts of quantities without a calculator (e.g. of 2 litres, 12% of 35) multiplies and divides decimal numbers by a single digit, 10 or 100 expresses remainders as fractions divides a three-digit number by a two-digit number using short division. Next steps use a calculator when appropriate and be consistent in checking answers present work in a logical and ordered way solve complex multi-step problems in a variety of contexts. Ratio problems Teacher s notes solves simple ratio problems calculates using ratios in appropriate situations divides quantities into given ratios. Next steps solve more complex ratio problems, using correct notation.
Algebra Teacher s notes understands the use of symbols to represent unknowns or variables substitutes numbers into equations to find solutions understands the use of brackets. Next steps solve more complex problems such as Find x when (5x 36) 8 3.
What the teacher knows about Rebecca s attainment in Ma2 Rebecca has a good understanding of place value and uses this to multiply and divide decimals by 10, 100 and 1000. She orders, adds and subtracts negative numbers in context and uses this to make generalisations about sequences. She rounds decimal numbers to one, two and three decimal places. She finds prime numbers beyond 100 and prime factors of numbers to 100, for example 70 = 2 5 7. She checks solutions by applying inverse operations or estimating using approximations. Rebecca reduces fractions to their simplest form by cancelling common factors. She compares fractions with different denominators and orders them by finding a common denominator. She calculates equivalence between fractions, decimals and percentages, with or without a calculator, for example = 0.375 = 37.5%. She divides a quantity into two or more parts in a given ratio. Rebecca checks her work using inverse operations. She knows the order of operations and uses brackets correctly and appropriately in solving problems. Rebecca expresses remainders as fractions or decimals, for example 845 25 = 33.8 or 33. Rebecca adds and subtracts numbers with one decimal place mentally. She knows number complements to 10, 100 and 1000, for example 546.7 + = 1000. Rebecca has a sound knowledge of tables facts and uses this to calculate the corresponding division facts. She multiplies and divides numbers such as 0.8 6 and 3600 60 using her knowledge of tables facts and place value. Rebecca calculates mentally fractions and percentages of numbers or quantities, such as of 75 or 60% of 30 kg. She estimates the square root of 85 as between 9 and 10 but closer to 9. She makes and justifies estimates and approximations of calculations. Rebecca solves proportion problems, sometimes using drawings to support her thinking. She calculates using ratios in different situations. She uses simple formulae in order to solve multi-step problems. She uses substitutions in algebraic expressions, including those that contain brackets. Rebecca uses coordinates in all four quadrants and identifies the missing coordinate when given three vertices of a parallelogram. She checks her answers by using inverse operations and using approximations to check whether an answer is reasonable. When asked to find the side length of a square with a given area, for example 8 cm 2 Rebecca uses a trial and improvement method to four decimal places. When exploring simple number sequences, she finds and describes the rule for the nth term. Rebecca uses all four operations confidently in her work with numbers up to three decimal places and chooses to use a calculator when appropriate. She adds and subtracts numbers that do not have the same number of decimal places and multiplies and divides decimal numbers by single digits. Rebecca multiplies and divides threedigit decimal numbers by two-digit numbers, using either a formal or an informal method. Rebecca calculates fractions and percentages of numbers or quantities both with and without a calculator, for example of 512 and 28% of 36.
Summarising Rebecca s attainment in Ma2 In Ma2 Rebecca is best described as working at the top end of level 5. She uses her understanding of place value to multiply and divide numbers by 10, 100 and 1000 and orders negative numbers in context. Rebecca uses efficient methods to calculate answers using the four operations. However, in AF Fractions and AF Mental methods she has already moved into level 6 (e.g. she adds and subtracts fractions by writing them with a common denominator; she understands and uses the equivalences between fractions, decimals and percentages and calculates using ratio in appropriate situations). She constructs, expresses in symbolic form and uses simple formulae involving one or two operations; but to progress into level 6 she needs more experience of algebra both in mathematics and in other subjects, including using spreadsheets to create formulae.
Ma3 Shape, space and measures Quadrilaterals Teacher s notes classifies special quadrilaterals, including trapezium and kite, using their properties explains that trapeziums (trapezia) don t always have a line of symmetry so will not always be located in the intersection. Next steps consider diagonal properties of quadrilaterals.
Turning triangles
Teacher s notes knows that the angle sum of a triangle equals 180º and around a point is 360º calculates missing angles in triangles, including isosceles triangles where only one angle is given reasons about special triangles and explains and justifies reasoning solves more complex problems involving finding angles in shapes composed of triangles. Next steps solve problems using angle properties of intersecting and parallel lines.
Coordinates and rotation Teacher s notes given the coordinates of three vertices of a parallelogram, finds the fourth plots and uses coordinates in all four quadrants rotates a shape through 90º when the centre of rotation is a vertex of the shape. Next steps rotate shapes where the centre of rotation is not a vertex of the shape.
Areas and perimeters Teacher s notes understands and uses the formula for the area of a rectangle finds the length of a rectangle given the area and width finds all the whole number possibilities (>1 cm) for the length and width of a rectangle, given the area calculates the length of a rectangle, given the perimeter and width. Next steps understand and use the formula for finding the area of a triangle.
What the teacher knows about Rebecca s attainment in Ma3 Rebecca applies reasoning (using her knowledge of the angle sum of triangles, the sum of angles on a straight line and the sum of angles at a point) to calculate missing angles in triangles, including isosceles and right-angled triangles, when only one angle is given and employs this knowledge in a range of multi-step problem-solving activities. She understands parallel and perpendicular in relation to edges of 2-D shapes and some 3-D shapes. Rebecca identifies missing coordinates, such as when given three of the four vertices of a parallelogram. Rebecca identifies line symmetry in 2-D shapes and recognises where patterns drawn on a 3-D shape will occur on its net. She translates shapes along an oblique line and reflects them in the x-axis and y-axis. She reflects shapes in two mirror lines and shapes not presented on grids by measuring distances to and from the mirror line. Rebecca rotates shapes clockwise or anticlockwise through 90º, 180º and 270º when the centre of rotation is a vertex of the shape. Rebecca measures and draws angles to the nearest degree and uses the associated language to describe them, for example obtuse, reflex. She constructs triangles given the length of two sides and the angle between them, accurate to the nearest mm and to 2º. She converts between metric units and uses this knowledge to solve multi-step problems. She finds the length of a rectangle when given the perimeter and width and finds the area and perimeter of compound shapes when given some of the lengths. Rebecca knows and uses the formula for calculating the area of rectangles and has developed this into finding the area of right-angled triangles by drawing the rectangle and halving the area. She calculates the volume of a cuboid given the lengths of its edges. Rebecca reads scales on a range of measuring instruments and interprets points between labelled divisions. She makes sensible estimates of a range of measures and relates these to her work. Summarising Rebecca s attainment in Ma3 Overall in Ma3 Rebecca is best described as working at the top end of level 5. She uses her knowledge of 2-D shape and angle properties to solve problems. She appreciates the formula for the area of a rectangle, using it to find the area of rightangled triangles. She reflects shapes not presented on grids by measuring distances to and from the mirror line and reflects shapes in two mirror lines where the shape is not perpendicular or parallel to either mirror line. To progress into level 6, she needs to extend her knowledge of angle properties to solve problems involving parallel lines and develop her repertoire of transformations by including enlargement.
Ma4 Handling data Conversion graph Teacher s notes creates a line graph where intermediate values have meaning chooses suitable scales and labels axes interprets scales on line graph to read or estimate between the labelled divisions answers questions such as If 4.20 was changed into Rupees how many would there be?
Pie charts Teacher s notes interprets pie charts to answer the question: Which team would you prefer to play for, Year 5 or Year 6? draws conclusions by comparing two pie charts where the sample sizes are different compares pie charts by describing events using fractions. Next steps collect data and construct own pie charts to represent them interpret more complex pie charts using a protractor.
Scratch cards Teacher s notes finds all of the combinations (with one minor error) and presents results systematically finds simple probabilities and presents them as fractions.
Next steps justify and explain findings in more detail, using mathematical vocabulary. What the teacher knows about Rebecca s attainment in Ma4 Rebecca understands that the results of an experiment may not be the same if it were repeated, for example rolling a dice 30 times. She understands that the more an experiment is repeated, the better the estimate of the probability. She asks questions and uses a range of methods to collect data, such as measuring, observing and counting. Rebecca collects data from primary and secondary sources and produces bar graphs with grouped class intervals. She constructs line graphs and bar charts, choosing suitable scales and labelling axes, using ICT where appropriate. She is beginning to create simple pie charts where divisions are already made on the circle: for example, when the circle is split into eight sections, Rebecca represents data for 24 people. She compares two probability spinners and explains which is more likely to show an odd number, identifying all of the outcomes, using diagrammatic or tabular forms to communicate her findings. When solving problems, Rebecca knows that the total probability of all the mutually exclusive outcomes of an experiment is 1. She constructs spinners that match particular criteria, such as for the question Given an eight-sided spinner with the numbers 1 4 where there is a 50 : 50 chance of spinning a 1, a chance of spinning a 2, impossible chance of spinning a 3, what is the probability of spinning a 4? She reads complex scales, such as 7:10 am on a scale that goes up in half hours and is marked every hour. Rebecca uses the mode, median, range and mean of sets of data in order to describe and compare them. Rebecca interprets bar graphs showing grouped data and draws conclusions when comparing two sets of data. Rebecca explains events using the correct language of chance or likelihood, for example certain, impossible, equal chance, likely, unlikely, fair, unfair. Summarising Rebecca s attainment in Ma4 In Ma4 Rebecca is best described as attaining at the top end of level 5. In AF Representing data she is working at level 6. She has a good understanding of the probability scale and predicts events using fractions and percentages. She compares two sets of data and draws conclusions from them. To move to the next level, she needs to construct and interpret more complex pie charts using a protractor. When constructing graphs, she should use the conventional zigzag when the axes do not start from zero. Rebecca also needs to improve her explanations and conclusions by using more mathematical vocabulary in her written work.
Overall assessment summary for Rebecca Rebecca s teacher describes her attainment across mathematics as the top end of level 5. She is beginning to work on and achieve aspects of level 6, particularly in Ma1 AF Problem solving, Ma2 AF Fractions, decimals and percentages, Ma2 AF Mental methods, Ma2 AF Solving numerical problems and Ma4 AF Processing and representing data. Assessment guidelines