Carl s mathematics (lower end of level 3)

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1 Ma KEY STAGE 1 Mathematics exemplification SEPTEMBER 2006 This is one of a series of six leaflets intended to exemplify children s performance in mathematics at levels 2A and 3. They illustrate how judgements can be made by taking into account strengths and weaknesses in performance across a range of contexts and over a period of time, rather than focusing on a single piece of work. They also show how a child s test result can be taken into account when making a judgement. The examples show work typical of the child s performance across mathematics. They are followed by details of how the teacher uses the work to make a judgement about the child s level. Carl s mathematics (lower end of level 3) Ma1 Using and applying mathematics Towers builds towers of three cubes in combinations of red and yellow records clearly using diagrams uses a pattern of opposites to find other possibilities for three cubes without building them predicts that there will be more possibilities using four cubes uses a similar approach and perseveres to find 14 of the 16 possibilities for four cubes compare results with others and check for any further possibilities When working on problems and investigations, Carl suggests how to begin. He is organised in his approach, for example when finding the capacity of differently shaped containers. He is systematic in estimating how many small cups will fill each container, recording his estimate, measuring with the cup and recording the measurement. He also estimates that the small cup holds about 100 ml and then measures each container by pouring its contents into a litre jug and reading the scale. Carl is developing his reasoning. In group discussion, when given a simple general statement such as, When you multiply numbers, either way round the answer is the same, Carl gives an example to match it: 5 x 10 = 50 and 10 x 5 = 50 QCA/06/2871 Qualifications and Curriculum Authority

2 Ma2 Number x completes two-step calculations mentally doubles numbers to 20 Rounding to the nearest 10 identifies the multiple of ten before and after any twodigit number places numbers at an appropriate point on number line rounds to the nearest 10 round three-digit numbers to the nearest 100 round three-digit numbers to the nearest 10 What fraction is shaded? shades one half or one quarter of a diagram beginning to recognise 5 /10 as 1 /2 and 2 /8 as 1 /4 identifies the fraction that represents a position between 0 and 1 on a number line Three-digit addition understands place value in three-digit numbers partitions and recombines to add numbers that do not involve bridging develop the method to include examples that involve bridging tens Carl finds missing numbers and continues sequences that go up or down in equal steps of up to 10 or 11. He understands place value in three-digit numbers. For example, using single digits 8, 5 and 9, Carl generates all six possible three-digit numbers. He says which of them is the largest, smallest and closest to 600. He shades diagrams to show halves and quarters. He is beginning to use simple fractions that are several parts of a whole and to recognise that fractions such as 5 /10 and 2 /4 are equivalent to ½. He uses fractions to label one of several equally spaced positions between 0 and 1. Carl knows addition facts to 20 and uses them to derive totals of multiples of 10 and 100. He adds three-digit numbers by partitioning and recombining. He understands subtraction as taking away and as the inverse of addition. He is beginning to use an empty number line to subtract a two-digit from a two- or three-digit number and as an introduction to subtraction as finding a difference. He quickly recalls multiplication facts in the 2x and 10x tables QCA/06/2871 Qualifications and Curriculum Authority

3 and counts in steps of 3, 4 and 5 to solve multiplication and division problems. He understands division as sharing but also uses multiplication facts or counting multiples of the divisor to reach an answer. For example, to calculate 15 3 Carl counts in threes, 3, 6, 9, 12, 15 and having reached the fifth multiple of 3, gives the answer 5. Carl understands doubling and halving as x 2 and 2. He doubles singledigit numbers and multiples of 5 and 10 mentally. He halves numbers with whole number answers. He finds a quarter of a number by halving and halving again. Carl solves word problems requiring one operation and some two-step problems such as finding the total cost of several items and calculating change from 1. Ma2 Number: Processing, representing and interpreting data (level descriptions in attainment target Ma4 Handling data) Popular fruit constructs a block graph, using one block to represent two votes, independently and draws a key interprets graph confidently to answer questions such as, Which fruit is most / least popular? How many more people voted for than? How many children voted altogether? knows 2x multiplication facts refine his numbering of the vertical axis to help read the height of columns use scales where one interval represents 2 or 5 on a range of graphs In handling data, Carl uses a tally to record class votes. He interprets a given tally chart of goals scored by different countries and draws a block graph where one block represents two goals, including examples of an odd number of goals. He numbers the spaces on the vertical axis to help count blocks. He needs to refine this by numbering the lines on the vertical axis to help read the heights of columns. Carl uses a Venn diagram to sort using two criteria. QCA/06/2871 Qualifications and Curriculum Authority

4 Ma3 Shape, space and measures Capacity investigation estimates how many small cups of water are needed to fill a variety of containers measures containers using the small cup as a non-standard unit when filling a large container, uses his measurement at about half way, to make a closer estimate estimates the small cup holds 100 ml and talks about how many millilitres each container holds Carl knows the names and properties of 2-D and 3-D shapes such as such as square, triangle, rectangles, pentagon, hexagon, octagon, cube, cuboid and cylinder. He uses vocabulary such as corners, edges, faces, straight and curved to describe properties. He counts the number of edges and corners to identify irregular pentagons, hexagons and octagons. He recognises right angles in shapes and can identify vertical and horizontal lines of symmetry. Although he talks about the properties of 3-D shapes he can handle, Carl finds it difficult to recognise 3-D shapes from drawings. He knows left and right and turns through a quarter, half, three-quarter or whole turn, clockwise and anticlockwise. He is beginning to relate a quarter-turn to one right angle. He measures lines to the nearest half-centimetre. He knows there are 100 cm in 1 metre and is beginning to use a mix of metres and centimetres to measure distances between whole metres more accurately. He uses millilitres to measure capacity and grams to weigh objects. He reads scales with intervals of 50 units and with 100s labelled. Carl reads the time from an analogue clock at the hour, half-hour, quarter past and quarter to. He can work out what the time will be one hour or a half-hour before or after these times. He is beginning to use the 5-minute intervals on the clock to read other times and calculate time intervals within an hour. He knows units of time: 60 seconds in one minute; 60 minutes in an hour; 24 hours in a day; and 7 days in a week. QCA/06/2871 Qualifications and Curriculum Authority

5 Reaching an overall judgement for mathematics from ongoing work Carl s teacher judges that he is best described as just into number line to subtract two-digit numbers. He multiplies level 3 across attainment targets for mathematics. using repeated addition and knows table facts for 2 and 10. He needs to develop quick recall of 3, 4 and 5 In Ma1 Using and applying mathematics, Carl is beginning multiplication tables. He also needs more experience of to organise work and use pattern to generate and check interpreting remainders in division problems to become results. He finds ways of overcoming difficulties, for more secure at level 3. In Handling data Carl uses example by numbering all of the blocks on his graph when information in tables and lists. He draws block graphs and is the labelling of the vertical axis did not help him to record beginning to draw and interpret graphs and pictograms accurately. He understands a simple general statement and, where one block or symbol represents more than one item. in group discussion, finds an example to match it. In Ma3 Shape, space and measures, Carl sorts 2-D and 3-D In Ma2 Number, he understands place value in three-digit shapes using criteria such as the number of edges, corners numbers but does not yet use decimal notation for money. or faces. He recognises reflection symmetry and right He is beginning to understand fractions that are several angles in shapes. He uses standard units such as parts of a whole. He adds three-digit numbers. He is less centimetres, metres, grams, seconds, minutes and hours. secure with subtraction but is beginning to use an empty Taking the test into account Carl just achieved level 3 using the 2005 level 3 test in May. He achieved 10 marks, which was the threshold mark. He attempted all of the questions in the test and scored his marks in questions 1, 2, 3, 5, 6, 7, 8, 9, 12 and 21. His teacher feels other questions are within Carl s grasp with a little more experience. For example, Carl is just beginning to use decimal notation for money and using zero as a place holder in the pence to answer question 4 correctly. He solves similar division problems to question 15 but in the test did not round up after the division to find the number of 10-seater tables needed to seat 43 children. Carl has not been taught to use letters and numbers to identify squares on a grid. In question 13 he had to tick the square exactly halfway between A1 and G7. He ticked squares between A and G in the bottom row and squares in the lower half of the centre column. His teacher feels he will have no difficulty with the convention when he is taught it. Carl found questions with two steps, or where he had to keep two constraints in mind, difficult. This is not untypical of children at the lower end of level 3. For example, in question 10, he chose to calculate six lots of 24 hours, not taking account of the shop being open for only eight hours in each of the six days. In question 17 he identified the fourth number but did not first engage with Kiz puts them in order from smallest to largest. In his class work Carl orders similar sets of numbers. In question 11, Carl did not engage fully with the Carroll diagram which used: has brown eyes/does not have brown eyes; and is shorter than 120 cm/is 120 cm/is taller than 120 cm. Carl seems to have interpreted Jane s green eyes as not brown eyes but did not understand how to use the three rows in the diagram for height. In class work Carl is more experienced in using a Venn diagram to sort using two criteria: a diagram with four regions rather than six. Carl needs more experience of breaking into problems involving larger numbers and deciding on the appropriate operation and method of calculation to answer the kinds of questions that appeared later in the test. Reviewing Carl s ongoing class work, his progress with the mathematics he has been taught most recently as well as his test result, his teacher feels that Carl is best described as working at the lower end of level 3. QCA/06/2871 Qualifications and Curriculum Authority