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Primary National Strategy Level 3-4 1. Can I position positive and negative numbers on a number line and find the difference between them? Teaching guidance Key vocabulary positive, negative, above zero, below zero, degrees Celsius (°C), minus, difference, integer Models and images Use the Number line ITP to show how number lines extend beyond 0. Encourage children to identify and discuss what the unmarked numbers are. Number line IPT Use a number line and/or a counting stick to identify and locate numbers. Where would you place –2? –4 0 –4 –7 Level 3 – Level 4 Intervention Primary National Strategy 0 –4 DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 1. Can I position positive and negative numbers on a number line and find the difference between them? Use a thermometer or the Thermometer ITP to look at negative numbers, and find the difference between negative numbers, within a context. Thermometer ITP Teaching tips Give children a range of opportunities to position numbers on number lines, including practical washing lines and individual number lines. Emphasise that counting continues beyond zero and use a number line to demonstrate that –4 is less than –2. Help children to use the benchmark numbers to determine the position of other numbers on the number line. Help children to make connections between using benchmark numbers on a number line and reading scales. Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 2. Can I count on from any given number in whole number steps, extending beyond zero when counting backwards? Teaching guidance Key vocabulary positive, negative, above/below zero, multiple Models and images Demonstrate how to find a missing term in a sequence by finding the step size. Find the difference between the numbers on either side of a missing term. In the example below, the difference between the two numbers is 8 and there are two jumps from the last known number to the next known number in the sequence. Therefore, we divide the difference of 8 by 2. Demonstrate that the number of missing terms in a sequence can be different. In the example below, the difference between the known numbers is 12 and there are four jumps from the last known number to the next known number in the sequence. Therefore, we divide the difference of 12 by 4. Teaching tips Ensure children have frequent practice in counting in steps of any size, including starting points that are not multiples of the step size. Use resources to support counting, for example a counting stick or a projected Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 2. Can I count on from any given number in whole number steps, extending beyond zero when counting backwards? calculator that has been set to count in given steps using the constant function. Children need frequent opportunities to practise their counting skills. Practising counting in different step sizes underpins children’s understanding of place value and their skills in calculation. Ensure that counting in sequences sometimes starts with negative numbers as well as starting with positive numbers, and that sequences involve counting back as well as counting forward. Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 3. Can I read, write, partition and order decimal numbers? Teaching guidance Key vocabulary decimal, decimal fraction, decimal point, decimal place, tenth, hundredth, thousandth, significant digit Models and images Use the Decimal number line ITP to zoom into a number line and position decimal numbers. Decimal number line ITP Use place-value charts to help identify the value of each digit in a decimal number. Place value chart spreadsheet Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 3. Can I read, write, partition and order decimal numbers? For decimals number with up to two places use a 10 × 10 grid so that each square represents 0.01 and each row represents 0.1 Discuss the effect of repeatedly adding the same decimal number, for example 0.01. Increasing number grid generator spreadsheet Teaching tips Build on understanding of decimals in the contexts of money and measures when working with decimal numbers with up to two places. However decimal place value should also be planned for and taught in its own right and not just in those contexts. Use number lines to help children order decimals. Present children with numbers for ordering that have different numbers of decimal places, to tackle the common misconception that the more digits there are after the decimal point, the bigger the number. Focus on the vocabulary of decimal fractions, and encourage children to read decimal numbers using the language of tenths, hundredths and thousandths so that, for example, they know the number comprising, two tenths, five-hundredths and nine-thousandths is written as 0.259. Reinforce the equivalence between fractions and decimals. Fraction notation gives you the language to help understand place value, for example knowing 0.01 is equivalent to 1/ 100 helps you to read this decimal number as one-hundredth and not just as zero point zero one. Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 4. Can I express tenths and hundredths as percentages? Teaching guidance Key vocabulary percentage, per cent, %, tenths, hundredths Models and images Represent a percentage using practical resources such as money (£1, 10p and 1p coins) or images, for example a 10 by 10 square grid or the Area ITP. Area ITP used to represent 35 shaded squares out of 100 or 35%. Teaching tips Use ICT resources, such as the Fractions ITP, to explore and instantly record the link between fractions and percentages. Use money to show how 10p can be expressed as a percentage and fraction of £1. Give children the opportunity to use coins to convince themselves that, for example, 10p is 1/ 10 or 10% of £1 because they need ten 10p coins to make £1. Refer to percentage as being the number of parts per hundred. Reinforce that 100% represents 100 per 100 or a whole. Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 5. Can I find simple equivalent fractions? Teaching guidance Key vocabulary numerator, denominator, fraction, proper/improper fraction, equivalent, reduced to, cancel Models and images Model how a fraction wall can be used to find equivalent fractions. Fractions ITP Demonstrate how a multiplication board can be used to scale up fractions. Discuss with children what needs to happen to change ¾ into other equivalent fractions. Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 5. Can I find simple equivalent fractions? Teaching tips Children need the opportunity to practise finding equivalent fractions by scaling simple fractions up or down. Use paper-folding to help establish equivalence, for example fold a strip of 20 squares into quarters and colour ¾ of them to establish that ¾ is the same as 15 out of 20 or 15/ 20 . Focus on recognising the patterns in sets of equivalent fractions and making links between multiplication and division. Represent fractions on a number line. This can help show that the same point on the number line can have more than one label, for example 1 could also be labelled as 2/ 2 , 3/ 3 , 4/ 4 , 5/ 5 , etc. Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 6. Can I relate simple fractions to their decimal equivalents? Teaching guidance Key vocabulary numerator, denominator, equivalent, proper fraction, decimal fraction, decimal place, decimal point Models and images Use the Moving digits ITP to make links between fraction and decimal equivalents of tenths, hundredths, and so on. For example, 3⁄ 10 = 0.3 and 13⁄ 100 = 0.13. Moving digits ITP Use number lines or the Fractions ITP to establish the decimal equivalent of fractions such as 2⁄ 5 or 1⁄ 20. Fractions ITP Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 6. Can I relate simple fractions to their decimal equivalents? Teaching tips Use a calculator and the language of fractions to find decimal and fraction equivalents. For example, 2/ 5 is keyed into the calculator as 2 divided by 5 and shows a decimal equivalent of 0.4 (four tenths). Children can look for fraction equivalents by finding decimal equivalents. Encourage them to make general statements about equivalent fractions. Use resources such as equivalent dominoes or washing lines with fraction and decimal equivalent cards. Invite children to peg the cards on the line and justify their choice of location. Present children with commonly confused fraction and decimal equivalents, for example 0.4 and 1/ 4 . Ask them to use images or practical resources to investigate whether these are actually equivalent; for example they could use the fractions ITP. Fractions ITP Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 7. Can I use my tables to multiply and divide? Teaching guidance Key vocabulary multiply, multiplied by, multiple of, times, array divide, divided by, divisible by factor, product, inverse Models and images Use the Multiplication facts ITP to make the link between arrays and multiplication facts. Multiplication facts ITP Teaching tips . Plan regular activities for children to learn, rehearse and use multiplication and division facts rather than simply test their recall. Reinforce multiplication facts and the corresponding division facts, for example, 8 × 7 = 56, 7 × 8 = 56, 56 ÷ 7 = 8, 56 ÷ 8 = 7. When solving a missing number question, it is helpful to write down the other three number sentences and then decide which one to use to find the missing number. Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 7. Can I use my tables to multiply and divide? Children need to understand and use the language of multiples and factors. Help children develop strategies for quickly deriving multiplication facts. For example, knowing 7 × 2 = 14 helps you to work out that 7 × 4 = 28 and 7 × 20 = 140. Ensure that children meet calculations written in different ways: × 8 = 56 9 = 54 ÷ 3×8=6× Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 8. Can I use my tables to work out multiplication and division facts with decimals? Teaching guidance Key vocabulary, times, multiply, multiplied by, product, multiple of, divide, divided by, divisible by, quotient, factor, inverse decimal, decimal point, tenths, hundredths, thousandths Models and images Use the Number dials ITP to explore multiples of decimal numbers. Explore known multiplication facts, such as multiples of 6, before exploring related facts such as multiples of 0.6. Number dials ITP What multiplication table does this diagram represent? How do you know? What are the missing numbers? What division facts do you know by using this diagram? Teaching tips Ensure children can confidently multiply and divide by 10 and 100; and that they understand that multiplying by 10 makes a number bigger and all the digits move one place to the left, while dividing by 10 makes a number smaller and all the digits move one place to the right. (See the teaching guidance ‘Can I multiply and divide by 10 and 100 and 1000?’ in the Calculating strand). Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 8. Can I use my tables to work out multiplication and division facts with decimals? Start with known multiplication facts before relating these to decimal multiplication facts; for example count on and back in steps of 3 before relating this to counting on and back in steps of 0.3. Encourage children to explain the relationship between the two sets of numbers. Ensure that children meet and can interpret multiplication and division calculations that are written in a variety of different ways, for example: ( × .8 = 5.6 9 = 5.4 ÷ ( 0.3 × 8 = 6 × ( Reinforce the division facts corresponding to multiplication facts; for example, 8 × 0.7 = 5.6, 0.7 × 8 = 5.6, 5.6 ÷ 0.7 = 8, 5.6 ÷ 8 = 0.7. When solving a missing number question, it is helpful to write down the other three number sentences and then decide which one is most useful to use to help find the missing number. Model the use of jottings and encourage children to use jottings to help keep track of the stages within a mental calculation. Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 9. Can I multiply and divide by 10 and 100 and 1000? Teaching guidance Key vocabulary digit, decimal, multiply, times, divide, share, scale up, scale down, increase, decrease, factor, how many 100s in …?, tens of thousands, thousands, hundreds, tens, units, ones, tenths, hundredths, thousandths Models and images Show children how multiplying a number by 10 moves the digits one place to the left, and multiplying by 100 moves the digits two places to the left. Demonstrate the effect of dividing a number by 10. Show children how the digits move one place to the right, and when dividing by 100 the digits move two places to the right. Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 9. Can I multiply and divide by 10 and 100 and 1000? Use a calculator or the Moving digits ITP to model how the digits move when we multiply or divide by powers of 10. Moving digit ITP Teaching tips Help children to generalise correctly so that they can cope with decimals. Multiplying by 10 makes a number bigger and all the digits move one place to the left. Dividing by 10 makes a number smaller and all the digits move one place to the right. Discuss why 4.6 × 10 is not the same as 4.60 and 40.3 ÷ 10 is not the same as 4.3. Explore with children the relationships between the operations and how to simplify combinations of operations. For example, multiplying by 10 then dividing by 100 is the same as dividing by 10. Help children to recognise that dividing by 200 is the same as dividing by 10, dividing by 10 again and then halving, by using a calculator to explore different examples. Emphasise that multiplication and division by 10, 100 and 1000 should be a mental calculation. Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 10. Can I add and subtract two numbers in my head quickly? Teaching guidance Key vocabulary add, addition, plus, sum, altogether, how many more to make…? subtract, subtraction, minus, take away, difference between how many more/less than… ?, inverse Models and images Use a 100-square or the Area ITP to provide an image of complements to 100. Model how as the number of squares in one region is increased, the number in the other region decreases by the same amount. . Demonstrate the use of an empty number line as a support for children’s mental calculations. Overcoming barriers in mathematics – helping children move from level 3 to level 4 00695-2007CDO-EN Primary National Strategy © Crown copyright 2007 Primary National Strategy Level 3-4 10. Can I add and subtract two numbers in my head quickly? Teaching tips When finding pairs of two-digit numbers that total 100, ask children what they notice about the corresponding digits of each pair (when the two-digit numbers are not multiples of ten, the number of tens total nine and the ones total ten). Reinforce addition facts and the corresponding subtraction facts. For example if you know 24 + 76 = 100, there are three other number sentences you also know. When solving a missing number question, it is helpful to write down the other three number sentences and then decide which one to use to find the missing number. Children need to see that they can use facts they know by heart in order to solve new problems without reverting to counting, for example knowing 70 + 30 helps when calculating 270 + 30. Help children understand that when working mentally there are times when it is useful to jot down some notes because you can’t always do the calculation solely in your head. Make sure this is modelled on a regular basis. After they have completed a practice mental test, give children a written copy of the questions. Ask children, in pairs, to read the questions together and sort them into those they can answer and those they can’t. Collect the ones that most children have difficulty with, to discuss or provide further reinforcement. Overcoming barriers in mathematics – helping children move from level 3 to level 4 00695-2007CDO-EN Primary National Strategy © Crown copyright 2007 Primary National Strategy Level 3-4 11. Can I use my tables to multiply and divide? Teaching guidance Key vocabulary multiply, multiplied by, multiple of, times, array divide, divided by, divisible by factor, product, inverse Models and images Use the Multiplication facts ITP to make the link between arrays and multiplication facts. Multiplication facts ITP Teaching tips . Plan regular activities for children to learn, rehearse and use multiplication and division facts rather than simply test their recall. Reinforce multiplication facts and the corresponding division facts, for example, 8 × 7 = 56, 7 × 8 = 56, 56 ÷ 7 = 8, 56 ÷ 8 = 7. When solving a missing number question, it is helpful to write down the other three number sentences and then decide which one to use to find the missing number. Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 11. Can I use my tables to multiply and divide? Children need to understand and use the language of multiples and factors. Help children develop strategies for quickly deriving multiplication facts. For example, knowing 7 × 2 = 14 helps you to work out that 7 × 4 = 28 and 7 × 20 = 140. Ensure that children meet calculations written in different ways: × 8 = 56 9 = 54 ÷ 3×8=6× Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 12. Can I use a written method to subtract? Teaching guidance Key vocabulary subtract, find the difference, minus, take away, more than, less than calculate, count on, count back Models and images Bead strings and number lines help provide a visual representation of the strategy the children have used, for example: 1470 – 174 = 1296 Teaching tips The aim is that children use mental methods of calculation where appropriate, but for calculations they cannot do in their heads they use an efficient written method accurately and with confidence. To subtract successfully, children need to be able to: – recall all addition and subtraction facts to 20; – quickly derive complements to 10 and 100 and multiples of 10 and 100; – subtract multiples of 10, such as 160 – 70 (using the related subtraction fact 16 – 7, and their knowledge of place value); – partition two-digit and three-digit numbers into multiples of one hundred, ten and one in different ways (e.g. partition 74 into 70 + 4 or 60 + 14). It is important that children’s mental methods of calculation are practised and secured alongside their learning and use of an efficient written method of subtraction. Help children to take ownership of written methods, for example, by having a ‘my method for subtraction’ card that travels between home and school. Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 13. Can I use a written method to multiply? Teaching guidance Key vocabulary array, partition, multiply, multiplied by calculate, calculation, strategy, method, equation Models and images Create arrays and then partition them to provide a visual representation of the grid method of multiplication. The Multi array ITP can be used to help model this. Multi array ITP Teaching tips Children should use mental methods of calculation where appropriate, but for those calculations they cannot solve mentally they need to be able to use an efficient written method, such as the grid method of multiplication, accurately and with confidence. Overcoming barriers in mathematics – helping children move from level 3 to level 4 00695-2007CDO-EN Primary National Strategy © Crown copyright 2007 Primary National Strategy Level 3-4 13. Can I use a written method to multiply? To multiply successfully, children need to: – know or quickly recall multiplication facts up to 10 × 10 – understand the effect of multiplying numbers by 10, 100 or 1000. – multiply multiples of 10, for example 20 × 40 – approximate; for example recognise that 72 × 38 is approximately 70 × 40 = 2800 and use this information to check whether their answer appears sensible. Give children opportunities to work with numbers for which they could use mental or written calculation strategies, in order to build up their confidence in the written method. Check which methods have been taught earlier on in the school, to ensure children receive consistent modelling and demonstration. Overcoming barriers in mathematics – helping children move from level 3 to level 4 00695-2007CDO-EN Primary National Strategy © Crown copyright 2007 Primary National Strategy Level 3-4 14. Can I calculate a fraction of a number or quantity? Teaching guidance Key vocabulary fraction, equal parts, numerator, denominator, divide, division, multiply, multiplication Models and images Use models and images alongside oral work. For example, display 12 small objects such as counters. Ask questions such as: What is one-third of these 12 counters? What are two-thirds of 12 counters? What are three-thirds of 12? Arrange the counters in ways that help children to see the process and gradually reduce the reference to the counters as the children become more confident. Record the steps with the children and encourage them to recognise the underlying counting in 4s. 1 / 3 of 12 is 4 2 / 3 of 12 is 8 3 / 3 of 12 is 12 Link finding fractions of amounts to fractions of shapes, for example: 1 1 /6 /6 1 /6 1 /6 1 1 /6 Find 5/ 6 of 18. What is 1/ 6 of 18? (6 × 3 = 18) If one-sixth is 3, what is five-sixths? /6 Overcoming barriers in mathematics – helping children move from level 3 to level 4 00695-2007CDO-EN Primary National Strategy © Crown copyright 2007 Primary National Strategy Level 3-4 14. Can I calculate a fraction of a number or quantity? Ensure children have experience of finding fractions of a range of wholes. Give children ‘bags of images to represent the whole’, of which they can then find fractions, such as 5/ 6 . This could include shapes, numbers, an amount of money, a length of string, a set of counters, and so on. 240 £1.80 Teaching tips Children need to be able to relate fractions to division, for example to understand that finding one-tenth is equivalent to dividing by 10. Help children to understand that they are finding a fraction of a whole amount by using practical equipment to explore a variety of different wholes (see image box above). Use different models and images to help children understand that a fraction such as 4/ 5 is 4 × 1/ 5 , and finding 1/ 5 is the first step to finding 4/ 5 . This will help them begin to associate finding 4/ 5 with ‘divide the whole into five equal parts and then group together four of these equal parts’. Introduce a scale or length to support the process, for example a length representing 30 cm. Use this to ask questions such as: What is one-sixth of 30 cm? What are two-sixths of 30 cm? What are three-sixths of 30 cm? Identify the steps of 5 cm and the counting process to establish that 1/ 6 of 30 is 5; 2/ 6 of 30 is 10; 3/ 6 of 30 is 15; 4/ 6 of 30 is 20; 5/ 6 of 30 is 25; 6/ 6 of 30 is 30.It is important to establish that 6/ 6 represents the whole. When they are confident they can extend the idea beyond the whole to 7/ 6 , and so on. When finding a fraction of a quantity, children will also need to consider whether a unit of measurement is required in the answer. Ensure that they know the relationships between familiar units of measure. Overcoming barriers in mathematics – helping children move from level 3 to level 4 00695-2007CDO-EN Primary National Strategy © Crown copyright 2007 Primary National Strategy Level 3-4 15. Can I calculate a percentage of a number or quantity? Teaching guidance Key vocabulary tenths, hundredths, percentage, equivalent, % Models and images Demonstrate how finding 10% can often be a useful starting point when finding other percentages. For example you can find 20% by doubling 10%, find 5% by halving 10% or find 15%, by adding 10% and 5%. The diagram helps model how 20% of 50 is 10. Area ITP Links can be made between fractions and percentages using the fractions ITP. This can help children realise that finding 50% is the same as halving, to find 25% they are finding one quarter, etc. Fractions ITP Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 15. Can I calculate a percentage of a number or quantity? Teaching tips When finding a percentage of a quantity, remind children that: – they might find it helpful to first find 1% or 10% first; – they may need to decide whether their answer needs rounding up or down; – if the question is in the context of money and measures, they will need to remember to include the relevant unit in their answer Help children make links by creating webs of percentages of numbers and then comparing the different amounts. For example, ‘What would £2.48 look like in comparison with £248?’ £186 75% £62 25% £2.48 1% £248 100% £124 50% £74.40 30% £24.80 10% £49.60 20% Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 16. Can I interpret the numbers on a calculator display? Teaching guidance Key vocabulary calculator, display, key, operation key, enter, decimal point, digit approximately, estimate, round up, round down pound, pence, metre, centimetre, litre, millilitre, kilogram, gram Models and images 12.0 12.5 Use a projected calculator to show how you solve a question using a calculator and discuss the interpretation of the display. Teaching tips Teach children how to interpret the displayed numbers, particularly large numbers as there no gaps to help them read these correctly. Give children opportunities to discuss the numbers appearing on their calculator screen. For example, ‘How many different contexts can you provide for 5.6?’ Promote checking strategies consistently, so that they become ‘second nature’ for the children. Strategies you might use include: – appointing a ‘checker’ when children are working in groups. – using a projected calculator to model checking strategies. – asking questions such as; ‘The answer to 32 × 27 was given as 288. How do you know this is incorrect? What would an approximate answer be?’ or ‘14.7 × 2.3 = 338.1. What error do you think has been made? Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 16. Can I interpret the numbers on a calculator display? How do you know the answer is incorrect? What would an approximate answer be?’ Further guidance on the use of calculators in the teaching and learning of mathematics can be found in the Renewed Framework Guidance paper. Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 17. Can I use a calculator to solve problems with more than one step? Teaching guidance Key vocabulary operation, multi-step, estimate, calculate, check, calculation, method, show your method Models and images Use a projected calculator to show how you solve a question using a calculator. Model how you would ‘show your method.’ Teaching tips Encourage children to use key steps when solving problems, such as: 1. Read the problem carefully (twice or more). 2. Identify key words to help you think about what the problem is asking. 3. Put the problem into your own words or use pictures to help you understand the question. 4. Decide what information you need and the operations you will use. 5. Make an estimate of the answer. 6. Record your calculation(s) and solution. 7. Check your answer and make sure you have used the correct units for measures or money. 8. Read the question again and check whether your answer is reasonable by comparing it with your estimate. Overcoming barriers in mathematics – helping children move from level 3 to level 4 00695-2007CDO-EN Primary National Strategy © Crown copyright 2007 Primary National Strategy Level 3-4 17. Can I use a calculator to solve problems with more than one step? Children need to be taught to distinguish between important and redundant information in word problems. They should be encouraged to annotate questions to help them find the key words and numbers. Good examples of annotation, estimation, calculating and checking should be displayed as part of the classroom learning environment. A commonly occurring error is for children to misinterpret values that represent money; for example they may give an answer of £12.5 instead of £12.50 or interpret 12.5 on a calculator display as £12 and 5p. Ask children to match the correct number of notes and coins to the calculator display. Teach children, when they use a calculator, to record their calculations together with the answers they obtain, at each stage in a multi-step calculation. Emphasise that recording the ‘method’ is about recording the number sentences or the calculations involved. Encourage children to check whether each answer makes sense as they work through a problem. Further guidance on the use of calculators in the teaching and learning of mathematics can be found in the Renewed Framework Guidance paper. Overcoming barriers in mathematics – helping children move from level 3 to level 4 00695-2007CDO-EN Primary National Strategy © Crown copyright 2007 Primary National Strategy Level 3-4 18. Can I estimate and measure angles less than 180˚? Teaching guidance Key vocabulary straight line, angle, right angle, acute angle, obtuse angle, reflex angle degree, whole turn, half turn, quarter turn angle measurer, protractor Models and images Use practical equipment such as strips of paper or a fan to make angles and model that angles are a measure of turn. Teaching tips Ensure children understand that an angle is ‘a measure of turn’. Provide opportunities for children to classify angles and use the correct vocabulary. Demonstrate the use of a protractor and ensure children have plenty of practical experience of reading the scale correctly and accurately. Aim for an accuracy of within one degree of the precise measurement. Discuss common misconceptions and errors with children, for example: Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 18. Can I estimate and measure angles less than 180˚? Encourage children to annotate angles when estimating their size. Model how to estimate the size of an angle using ‘benchmarks’ such as 90°, 180°, and 45°. Make relevant cross-curricular links in subjects such as design and technology, art and PE, to reinforce measuring and estimating angles. Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 19. Can I calculate angles on a straight line and in a triangle? Teaching guidance Key vocabulary straight line, angle, right angle, acute angle, obtuse angle, reflex angle degree, whole turn, half turn, quarter turn parallel, perpendicular Models and images Explore and establish with children that the angles of a triangle total 180° by tearing off the three corners of a triangle and arranging them together to make a straight line. Use a range of different triangles to help show that this is always the case. Use the Fixing points ITP to create a triangle. Move one of the points to create a different triangle (e.g. drag ‘b’ to a different position on the grid) to illustrate that the angles still total 180°. Fixing points ITP Teaching tips Ensure that children understand that ‘calculate’ does not mean ‘measure’ (children often measure an associated image even if the instructions clearly say that it is not drawn to scale). Level 3 – Level 4 Intervention Primary National Strategy DSCF reference to go here © Crown copyright 2007 Primary National Strategy Level 3-4 19. Can I calculate angles on a straight line and in a triangle? Rehearse and practise associated mental skills, for example complements to 180. Model how to annotate diagrams and encourage children to record extra information on their own work. Level 3 – Level 4 Intervention Primary National Strategy DSCF reference to go here © Crown copyright 2007 Primary National Strategy Level 3-4 20. Can I use coordinates to draw, locate and complete shapes? Teaching guidance Key vocabulary coordinate, axis, axes, origin, x-axis, y-axis, horizontal, vertical, parallel, perpendicular (also shape vocabulary, for example vertex, vertices) Models and images Use the Coordinates ITP to create the start of a shape. Ask children to identify the coordinates of the labelled points and then suggest the coordinates for the missing vertex or vertices. Coordinates ITP Teaching tips Explain that the point with coordinates (0, 0) is called the origin and discuss the meaning of this word. Ensure that children understand that coordinates are used to describe position in relation to the origin. Children need to appreciate that the order of the numbers is important. The first number tells you how far to go across while the second number tells you how far to go up or down. Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 20. Can I use coordinates to draw, locate and complete shapes? Give children experience of physically moving an image of an object from the origin across and then up to a desired position, describing the journey the image makes. Model annotation and encourage children to add information to the grid. Good examples of annotation should be displayed as part of the classroom learning environment. When drawing shapes make sure that the sides are not always parallel or perpendicular to the axes. This will add extra challenge. Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 21. Can I estimate a quantity (mass, length or capacity) and choose the most sensible unit of measure and equipment to use? Teaching guidance Key vocabulary estimate, measure, compare, convert, approximate, accurate, relationship, scales, kilometre, metre, centimetre, millimetre, kilogram, gram, litre, millilitre, metric, unit, length, distance, mass, weight, capacity Models and images Provide children with plenty of practical experience of estimating quantities and then choosing and using suitable measuring equipment. Teaching tips Plan for children to experience use of a variety of measuring equipment, including spring balances, calipers, different sized cooking spoons, and so on, so that they know what is available and can make sensible choices. Ensure that equipment is readily available for children to draw upon across the curriculum. Exemplify the purpose of estimation, in terms of developing a feel for the different units of measure, by providing children with benchmarks to help them to estimate, for example the height of a door is approximately 2 metres, a pencil is approximately 20 cm long, a bag of sugar usually weight one kilogram. Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 21. Can I estimate a quantity (mass, length or capacity) and choose the most sensible unit of measure and equipment to use? Help children to improve and refine their estimations as they gain experience. For example, after ten tries, can they get within 50 g of the real value when they estimate the mass of a handful of grapes. Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 22. Can I convert between units? Teaching guidance Key vocabulary kilometre, metre, centimetre, millimetre, kilogram, gram, litre, millilitre Models and images Use the converting measures spreadsheet to practise and reinforce the conversion between different measures. Converting measures spreadsheet Use the Moving digits ITP to model the effect of multiplying and dividing by 100 and 1000. Moving digits ITP Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 22. Can I convert between units? Teaching tips Link converting units of measures to estimating, in order to help children spot where they have made mistakes, for example 1.5 m cannot equal 15 cm because I know 1.5 m is about my height but 15 cm is half the length of a ruler. Build the rehearsal of converting units into oral and mental starter activities, and as a practical context in lessons focused upon calculation where children are multiplying and dividing by 10, 100 and 1000. Explore the language of units, for example, roots from which ‘centi’ and ‘milli’ are derived and when else they are used (e.g. century, centurion). Provide regular practical opportunities for children to learn and understand the relationship between units. Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 23. Can I read a scale on a thermometer, protractor, ruler, weighing scale and measuring cylinder? Teaching guidance Key vocabulary measure, measurement, standard unit, scale, measuring scale, division, Models and images Read the same measurement on different scales, for example mark the arrow in the correct place on the second scale. The measuring cylinder, measuring scales and thermometer ITPs allow you to adjust the intervals on the scale. Measuring cylinder ITP Measuring scale ITP Thermometer ITP Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 23. Can I read a scale on a thermometer, protractor, ruler, weighing scale and measuring cylinder? Teaching tips Provide practical opportunities for children to select and use appropriate units of measure and to consider the degree of accuracy within a given context. For example measuring glass for a window will require greater accuracy than measuring material for curtains that can be adjusted. Explore a variety of types of scales. For example make a scale using a length of hose and present it vertically, horizontally or as a loop for children to read. Take opportunities to practise reading scales accurately across the curriculum, for example within science, geography and PE lessons. Ask children to collect photographs of scales within their own homes to display. Who has the greatest number of examples? Who can find the most unusual example? Ask children to find out how their parents use scales in their everyday lives. Model annotation of images of scales, particularly to help children interpret readings that lie in between numbered divisions. Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 24. Can I work out the area and perimeter of a rectangle? Teaching guidance Key vocabulary area, covers, surface, square centimetre (cm2), square metre (m2), square millimetre (mm2) edge, perimeter, metre, centimetre, millimetre Models and images Use the Area ITP to create a rectangle, and discuss its area and perimeter. Gradually increase the size of the rectangle. How does the area increase? How does the perimeter increase? Area ITP Ask children to measure the rectangles in the Ruler ITP and then use this information to find the area and perimeter of the shapes. Ruler ITP Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 24. Can I work out the area and perimeter of a rectangle? Teaching tips Develop children’s understanding of perimeter while they are working on length, rather than only ever linked it with work on area. Provide practical tasks such as using art straws to make rectangles and then laying them end to end to find the perimeter. Use equipment to work towards learning the formula for calculating perimeters of rectangles, for example use straws to make a rectangle and then place the lengths together and the widths together. Model the annotation of rectangles to keep an ongoing record when working out perimeters. Link work on finding area with arrays and developing images for multiplication and division facts. Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 25. Can I sort and interpret data in Venn and Carroll diagrams? Teaching guidance Key vocabulary Venn diagram, Carroll diagram, region, intersection, describe, explain, categories Models and images Carroll diagram Venn diagram Teaching tips Ensure children have experience of creating their own categories for sorting numbers, objects, ideas or shapes onto Venn and Carroll diagrams. Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 25. Can I sort and interpret data in Venn and Carroll diagrams? Present children with unusual situations, such as examples in which it is impossible for there to be anything in the intersection of a Venn diagram (e.g. numbers that are divisible by 11, numbers less than 10), and ask them to explain why the intersection will be empty. When exploring Venn diagrams, discuss the meaning of the section within the rectangle but outside the ellipses (or whatever shape is used for the set of enclosures), and ask children to generate examples that would go into this section. Take opportunities to discuss properties of different items that are represented in Carroll diagrams across the curriculum. Information is often presented in this way within geography, history and science. Ask children to deconstruct a Carroll diagram and reconstruct it as a Venn diagram and vice versa. Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 26. Can I explain what information a graph or chart is showing? Teaching guidance Key vocabulary graph, chart, table, axis, axes, describe, explain, categories Models and images Introduce children to a wide and rich variety of tables, charts and graphs. Teaching tips Introduce children to a wide and rich variety of tables, charts and graphs within all curricular areas, taking opportunities to ask questions about the way information is organised and what different presentations show. The interpretation of charts and tables should be a focus in the oral and mental starter and the plenary, and not just in the main part of the lesson. Provide children with different charts and graphs without titles, labels on the axes, or units, and ask them to ‘tell the story of the graph’. Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 26. Can I explain what information a graph or chart is showing? Expect children to summarise what any graph or chart shows before answering questions about detail. Give children opportunities to take information presented in one format and re-present it in another format, for example deconstructing the barline chart below to create the frequency table. Number of throws Dice-rolling experiment Score Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 27. Can I draw a conclusion from a graph? Teaching guidance Key vocabulary data, information, survey, questionnaire, graph, chart, table, scale, interval, division, horizontal axis, vertical axis, axes, label, title, pictogram, bar chart, barline chart, line graph, pie chart, interpret, describe, explain Models and images Ensure children experience a wide range of different graphs. Average temperatures 35 30 25 20 Degrees Celsius Majorca Corfu 15 10 5 0 April May June July Month August September October Temperature throughout the day Teaching tips Encourage children to pose their own questions from looking at graphs as well as responding to specific questions. Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 27. Can I draw a conclusion from a graph? Ensure children experience a range of graphs, including those with unlabelled divisions, line graphs where intermediate points do and do not have meaning, and different presentations of the same data. Give children conclusions drawn from graphs that cannot be true, and ask them to explain why; challenge them to spot conclusions that are invalid (e.g. on the line graph shown in the Models and images section, ‘the temperature at 5pm will be 11 °C’) and explain why this is not a valid conclusion. Model the process of annotating and interacting with graphs in order to answer questions and draw conclusions; use ‘shared thinking’ and ‘shared writing’ techniques to make explicit how you analyse and explain the information presented by a graph. Give children questions that require them to compare information from two graphs, different categories on the same graph or different sets of data represented on the same graph, for example: What was the most unsuccessful year for space missions? How do you know? Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 28. Can I understand the meaning of points between labelled divisions on graphs? Teaching guidance Key vocabulary data, information, survey, questionnaire, graph, chart, table, scale, interval, division, horizontal axis, vertical axis, axes, division, unit, label, title, pictogram, bar chart, bar-line chart, line graph, pie chart, interpret, describe, explain Models and images Use the Line graph ITP to create a line graph. Change the scale on the y-axis and discuss the effect this has on the line graph. Line graph ITP Level 3 – Level 4 Intervention Primary National Strategy Dice-rolling experiment Nnumber of throws Ensure children have opportunities to work with graphs and charts where results do not always occur at the labelled divisions. DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 28. Can I understand the meaning of points between labelled divisions on graphs? Teaching tips Help children to recognise the link between reading a scale on a bar chart or line graph and finding a number on a partly numbered line. Ensure children have opportunities to work with graphs and charts where results do not always occur at labelled divisions, and where axes are labelled in different increments from multiples of 2, 5 or 10. Model strategies for annotating graphs accurately to work out points between labelled divisions. Display examples of annotation as part of the classroom learning environment. Encourage children to annotate the charts and to use tracing paper where appropriate. Use ICT to experiment with the effect that changing the labelled divisions on the axes has on graphs (e.g. use the Line graph ITP). Make predictions, and then increase the number of divisions to check accuracy. Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 29. Can I interpret the sectors in a pie chart? Teaching guidance Key vocabulary data, information, survey, questionnaire, graph, chart, division, sector, label, title, pie chart, interpret, describe, explain Models and images Compare the information shown within two related pie charts to emphasise that each sector represents a proportion of the whole. Compare the information shown within two related pie charts that are different sizes. Teaching tips Help children to understand that each sector on a pie chart represents a proportion of a whole set, and that the total number of items represented Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 29. Can I interpret the sectors in a pie chart? by the pie chart has to be known in order to calculate how many each sector represents. Ensure children encounter a range of charts, including pairs of pie charts that represent different total numbers (such as the TV programme example above). Encourage children to annotate the charts and to use tracing paper where appropriate to compare different sections. Model how to provide structured explanations for conclusions you are drawing from pie charts. Use pie charts as contexts when finding fractions of numbers and quantities. Ask, for example: ‘Roughly what fraction is the “lost” sector on the netball chart? If the chart represents 30 games, how many were lost?’ Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 30. Can I work out the mode and range of a set of data and use this to answer questions? Teaching guidance Key vocabulary frequency, mode, maximum/minimum value, range, average, statistics, data, information, measure, survey, questionnaire, interval, explain, justify Models and images Use a number line to represent data given in a frequency table, to draw attention to where and how often each value occurs, for example. Daily temperature (°C) –2 –1 0 1 2 3 5 6 Frequency 2 2 2 2 1 1 3 1 °C –2 –1 0 1 2 3 –2 –1 –2 –1 0 0 1 1 2 3 4 5 6 5 5 5 6 7 The range is from –2 to 6 = 8 and the mode is 5. Use practical methods to help children actually see the range. For example ask groups of six children to take measurements of their hand spans for both hands, and to record the lengths on paper strips of two colours (one colour for the right hand, another for the left hand). Then ask them to order the 12 strips of paper by size of the measurement, smallest to largest, and record the range of hand spans for their group as follows: The range of hand spans for our group was…. The range of left-hand spans for our group was…. The range of right-hand spans for our group was…. This group data could then be pooled to produce the three ranges for the whole class. Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 30. Can I work out the mode and range of a set of data and use this to answer questions? Teaching tips Introduce frequency and mode by giving children pieces of paper with the results of a simple survey written on them, and asking them to produce a human bar chart – the frequency for any number is the number of children in the queue and the mode is the number represented by the longest queue. Create a frequency chart to give a visual image of the data (see above). This gives an image of both mode and data range. Tabulate the data, marking each value in the data list as it is recorded. A number line is helpful to work out the range of a set of data. Make links between handling data in mathematics and other areas of the curriculum, for example finding out the range of children’s ages attending school in Victorian times and the present day. Children need to understand that the range and mode describe the data set they are using or have generated. For example, give children two dice, one with even numbers 2, 4, 6, 8, 10 and 12; the other with the odd numbers 3, 5, 7, 9, 11 and 13. Ask the children to roll the two dice, find the difference between the two numbers displayed and place a cube above this number on a number line. After five rolls ask them to record: The range of results was from ... to ... The mode was ... Repeat the process after 10, 15, 20 and 25 rolls; then ask children to look at the pattern the in results to try to predict the range and mode after 40 rolls. Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 31. Can I use the vocabulary of probability to predict outcomes and discuss and explain events? Teaching guidance Key vocabulary fair, unfair, likely, unlikely, likelihood, equally likely, certain, uncertain, probable, possible, impossible, chance, good chance, poor chance, no chance, equal chance, even chance, fifty-fifty chance, risk, doubt, biased, random Models and images Represent probability on a scale, for example Next week will be 8 days long impossible Throwing heads on a coin The day after Sunday will be Monday evens certain Use a variety of spinners. Number spinners ITP Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007 Primary National Strategy Level 3-4 31. Can I use the vocabulary of probability to predict outcomes and discuss and explain events? Teaching tips Use cross-curricular opportunities to discuss events that have a good chance of happening and those that have a poor chance. Give children the opportunity to use the language of probability. Model how to predict the probability of an outcome by talking through the number of different possible outcomes and how likely each outcome is. Include outcomes that are impossible. Ensure children have opportunities to predict and investigate probabilities using a variety of resources that generate random outcomes. Encourage children to annotate the charts and spinners where appropriate, to compare different sections. Help children recognize the difference between the theory of outcomes and the actual experimental results. For example, do they actually get each number once when throwing a six-sided dice six times? Level 3 – Level 4 Intervention Primary National Strategy DSCF ref •••••• to go here © Crown copyright 2007