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Ma2 Number

Equivalent fractions

Pupil's handwritten work to show working out for equivalent fractions.
The page features two sets of working out. The first set of working out is titled L1: understanding equivalent fractions is written at the top of the page. Below this 'eg' is written next to two pieces of working out. The first working out is a line of fractions separated by commas. These are 1 over 2, 2 over 4, 3 over 6 and 4 over 8. Above this are three curved lines connecting the first numerator (1) with the other three numerators; each marked with a multiplier. 1 is connected to 2 with the multiplier x2. It is also connected to the 3 with the multiplier x3, and connected to the 4 with the multiplier x4. The denominators are similarly connected. 2 is connected to 4 with the multiplier x2. It is also connected to 6 with the multiplier x3, and to 8 with the multiplier x4. The second working out is also a line of fractions, separated by commas. These are 5 over 8, 10 over 16 and 25 over 40. Curved lines with multipliers connect the first numerator (5) to the other numerators. 5 is connected to 10 with the multiplier x2, and connected to 25 with the multiplier x5. The denominators are similarly connected. 8 is connected to 16 with the the multiplier x2, and to 40 with the multiplier x5. The second set of working out is titled Li: work out equavalent fractions. Below this is �x3 x7 x9 x5� and four pieces of numbered working out. Working-out 1 A line of fractions, separated by commas. These are 3 over 4, 9 over 12, 21 over 28, 27 over 36, 15 over 20. Above this are four curved lines connecting the first numerator (3) with the other four numerators; each marked with a multiplier. 3 is connected to 9 with the multiplier x3. It is connected to 21 with the multiplier x7, to 27 with the multiplier x9, and to 15 with the multiplier x5. The denominators are similarly connected. 4 is connected to 12 with the multiplier x3, to 28 with the multiplier x7, to 36 with the multiplier x9 and to 20 with the multiplier x5. This is marked with a tick. Working-out 2 A line of fractions, separated by commas. These are 4 over 7, 12 over 21, 28 over 49, 36 over 63, 20 over 35. Above this are four curved lines connecting the first numerator (4) with the other four numerators; each marked with a multiplier. 4 is connected to 12 with the multiplier x3. It is also connected to 28 with the multiplier x7, to 36 with the multiplier x9, and to 20 with the multiplier x5. The denominators are similarly connected. 7 is connected to 21 with the multiplier x3, to 49 with the multiplier x7, to 63 with the multiplier x9 and to 35 with the multiplier x5. This is marked with a tick. Working-out 3 A line of fractions, separated by commas. These are 2 over 3, 6 over 9, 14 over 21, 18 over 27, 10 over 15. Above this are four curved lines connecting the first numerator (2) with the other four numerators. 2 is connected to 6. It is also connected to 14, 18, and 10. The denominators are similarly connected, and marked with multipliers. 3 is connected to 9 with the multiplier x3, to 21 with the multiplier x7, to 27 with the multiplier x9 and to 15 with the multiplier x5. This is marked with a tick. Working-out 4 A line of fractions, separated by commas. These are 4 over 5, 12 over 15, 28 over 35, 36 over 45, 20 over 25. Above this are four curved lines connecting the first numerator (4) with the other four numerators; each marked with a multiplier. 4 is connected to 12 with the multiplier x3. It is also connected to 28 with the multiplier x7, to 36 with the multiplier x9, and to 20 with the multiplier x5. The denominators are similarly connected. 5 is connected to 15 with the multiplier x3, to 35 with the multiplier x7, to 45 with the multiplier x9 and to 25 with the multiplier x5. This is marked with a tick.

Teacher’s notes

  • Understands numerator and denominator.
  • Understands that both parts of the fraction have to be multiplied/divided by the same number.
  • Is aware that there can be more than one equivalent fraction.

Next steps

  • Use knowledge of equivalence to order fractions with different denominators.
  • Develop understanding of equivalence between fractions, decimals and percentages.

Carroll diagrams

Pupil's completed worksheet on Carroll diagrams.
The worksheet is titled Carroll Diagrams. Below this are 6 incomplete Carroll diagrams that the pupils has completed, with a question for each. Diagram 1 The text above this reads 'Put a number into each cell' The diagram is a table of two columns and two rows (each with a title), to create four empty cells. The column titles are 'Multiples of 4' and 'Not multiples of 4'. The row titles are 'Multiples of 3' and 'Not multiples of 3'. The pupil has filled in each cell as follows. Cell 1: handwritten 12 ('Multiples of 3'/'Multiples of 4') Cell 2: handwritten 3 ('Multiples of 3'/'Not multiples of 4') Cell 3: handwritten 4 ('Not multiples of 3'/'Multiples of 4') Cell 4: handwritten 2 ('Not multiples of 3'/'Not multiples of 4') Diagram 2 The text above this reads 'Put two numbers into each cell' The diagram is a table of two columns and two rows (each with a title), to create four empty cells. The column titles are 'Multiples of 2' and 'Not multiples of 2'. The row titles are 'Multiples of 3' and 'Not multiples of 3'. The pupil has filled in each cell as follows. Cell 1: handwritten 6, 12 ('Multiples of 3'/'Multiples of 2') Cell 2: handwritten 3, 9 ('Multiples of 3'/'Not multiples of 2') Cell 3: handwritten 8, 16 ('Not multiples of 3'/'Multiples of 2') Cell 4: handwritten 5, 11 ('Not multiples of 3'/'Not multiples of 2') Diagram 3 The text above this reads 'Put a number into each cell. Try to use numbers that others wouldn't use' The diagram is a table of two columns and two rows (each with a title), to create four empty cells. The column titles are 'Multiples of 5' and 'Not multiples of 4'. The row titles are 'Multiples of 3' and 'Not multiples of 3'. The pupil has filled in each cell as follows. Cell 1: handwritten 1500 ('Multiples of 3'/'Multiples of 5') Cell 2: handwritten 420003 ('Multiples of 3'/'Not multiples of ') Cell 3: handwritten 25 ('Not multiples of 3'/'Multiples of 5') Cell 4: handwritten 0.61 ('Not multiples of 3'/'Not multiples of 5') Diagram 4 The text above this reads 'Put a label into each empty space to complete the sorting diagram'. The text below it reads 'The labels are: multiples of 5, multiples of 3, not multiples of 5, not multiples of 3'. The diagram is a table of two columns and two rows (each with a title), to create four empty cells. The column titles are handwritten, using the supplied labels. The pupil has filled the column title cells with 'multiple of 3' and 'not a multiple of 3' and the row title cells with 'multiple of 5' and 'not a multiple of 5' Cell 1: 30 (handwritten 'multiple of 5'/'multiple of 3') Cell 2: 20 (handwritten 'multiple of 5'/'not a multiple of 3') Cell 3: 9 (handwritten 'not a multiple of 5'/'multiple of 3') Cell 4: 7 (handwritten 'not multiples of 5'/'not multiples of 3') Diagram 5 The text above diagram five reads 'What do you think the missing labels are?' The diagram is a table of two columns and two rows (each with a title), to create four empty cells. The row titles are 'Greater than 20' and 'Not greater than 20'. The pupil has written in 'smaller than 70' and 'greater than 5' as the column cell titles. The cell contain the following. Cell 1: 36, 27, 60 ('Greater than 20'/handwritten 'smaller than 70') Cell 2: 31, 37, 40 ('Greater than 20'/handwritten 'greater than 5') Cell 3: 12, 18, 9 ('Not greater than 20'/handwritten 'smaller than 70') Cell 4: 14, 7, 11 ('Not greater than 20'/handwritten 'greater than 5') Diagram 6 The text above diagram six reads 'Put a number into each cell. Are there more posiibilties?' The diagram is a table of two columns and two rows (each with a title), to create four empty cells. The column titles are 'Only 4 factors' and 'More than 4 factors'. The row titles are 'Less than 20' and 'Greater than 20'. The pupil has filled in each cell as follows. Cell 1: handwritten 6, 8, 10, 14, 15 ('Less than 20'/'Only 4 factors') Cell 2: handwritten 12, 16, 18 ('Less than 20'/'More than 4 factors') Cell 3: handwritten 21, 22 ('Greater than 20'/'Only 4 factors') Cell 4: handwritten 24, 30 ('Greater than 20'/'More than 4 factors')

Teacher’s notes

  • Understands the vocabulary multiple, factor and square.
  • Sorts numbers by comparing them and recognising properties.
  • Calculates mentally to generate examples.

Next steps

  • Use sorting conventions for Carroll diagrams more consistently, for example use ‘smaller than 70’/‘not smaller than 70’.
  • Search for different relationships and write more challenging labels, for example ‘multiples of 3’, ‘square numbers’, rather than ‘smaller than 70’.

Squares and circles sequences

Pupil's completed worksheet on sequences.
At the top of the page, it reads 'Here is a sequence of patterns made from squares and circles.' Below this is a table that corresponds to a three sequences of patterns. The table has two columns, one titled 'number of squares', one titled 'number of circles'. The first pattern is a square with a circle to its left, to its right, and below it. This is recorded in the table as 1 square and 3 circles. +2 is handwritten across the two columns. The first pattern is a square with a circle to its left, and to its right, and below it, followed by a square with a circle to its right and a circle below it. This is recorded in the table as 2 squares and 5 circles. +3 is handwritten across the two columns. +2 is written at the point between the first and second row of the table. The third pattern is a square with a circle to its left, to its right, and below it; followed by a square with a circle to its right and a circle below it; followed by another square with a circle to its right and a circle below it. This is recorded in the table as 3 sqaures and 7 circles. +2 is handwritten across the two columns. +2 is written at the point between the second and third row of the table. Text reads 'The sequence continues in the same way', then 'Calculate how many squares there will be in the pattern which has 25 circles'. The 25 is written in bold text. There is a framed space for working out, with the text 'Show your method'. In this, is a hand-drawn sequence of square and circle patterns, using 12 squares. The pupil has written 12. Below the space for working out, the pupil has written '1x2=2+1+3', '3-1=2 divided by 2=1', '2x2=4+1=5', and '25-1=24 divided by 2=12'.

Teacher’s notes

  • Continues the sequence using the pattern given.
  • Identifies differences between numbers by looking for patterns.
  • When prompted, identifies a rule for the sequence using multiplication and addition.
  • Uses the inverse operations to work backwards through a sequence, when prompted.

Next steps

  • Use the = sign correctly, in working.
  • Use simple algebra to express patterns and rules.
  • Evaluate expressions by substituting numbers.
  • Find the rule for the nth term.

What the teacher knows about Polly’s attainment in Ma2

Polly recognises and describes number patterns and sequences. She has a good understanding of multiples, squares and factors and recognises some prime numbers under 100. She continues sequences forwards and backwards and writes simple rules or general statements in words to explain them. Polly clearly understands the effect of multiplying and dividing whole numbers by 10, 100 and 1000.

Polly recognises proportions of a whole and uses simple fractions and percentages to describe them. She knows the place value of the digits within a decimal and how to order decimals with up to three decimal places, for example 0.706, 0.76, 0.67, 0.607. She recognises equivalences between commonly used fractions, decimals and percentages, for example 2 5 = 0.4 = 40 % ; and calculates simple fractions and percentages of given amounts, for example 3 8 of 48 or 60% of £40. Polly is beginning to understand and use ratio although she does not yet use ratio notation.

Polly understands the relationships between the four operations with whole numbers. She uses inverse operations to find missing numbers. She is less confident, however, when working to two decimal places. Polly understands the use of brackets to indicate the order of operations, and understands the role of = in showing that expressions on each side of the symbol have the same value, for example 36 × 4 = 150 - .

Polly calculates mentally using all four operations. She recalls multiplication facts up to 10 × 10 quickly and derives corresponding division facts. She uses her knowledge of complements to 100 and place value to calculate complements to 1000, for example 360 + = 1000 . She uses her knowledge of multiplication facts and place value to calculate with larger numbers, for example 80 × 7. She understands that halving then halving again is the same as finding 14 or dividing by 4.

When solving problems Polly deals with two constraints: for example, she finds a number that is a multiple of two and not a multiple of three. She works through both steps of a two-step problem, for example using multiplication and subtraction to find the change from £10 when buying three comics at £1.35 each. She reads temperatures and understands negative numbers in this context. When using a calculator, Polly appreciates the value of the digits in the display. She interprets the calculator display when solving money problems and knows to record 4.6 as £4.60. Polly checks the reasonableness of her answers and explains why she believes an answer to be correct. She is beginning to use simple formulae expressed in words, although she does not yet use letters and symbols. She plots and interprets coordinates in all four quadrants accurately.

Polly uses efficient methods to add and subtract four-digit numbers and decimals with up to two decimal places. Having previously used the grid method, Polly now multiplies a three-digit by a two-digit number using a standard efficient method. She multiplies decimal numbers by a single digit: for example, to calculate 7.5 × 8, Polly multiplies 75 by 8 and uses her understanding of place value to position the decimal point. She uses short division to divide whole numbers by a single digit.