Teaching Guidance For Counting and Partitioning Numbers (Overcoming Barriers Moving Levels 1 to 2, 2 to 3, 3 to 4, 4 to 5)
1 of 2 The National Strategies Primary Overcoming barriers in mathematics helping children move from level 1 to level 2 Can I tell someone how to order two-digit numbers? Teaching guidance Key vocabulary compare, order, size, bigger, smaller, biggest, smallest, before, after, between, count on/back, number line, position, digit, two-digit number, one-digit number, ones/units, tens Models and images, resources and equipment 100-square Help children understand the structure of a 100-square and see it as a rearranged number track. Use a range of different grids, including those that do not have all the numbers represented. Bead strings and number lines Help children use labelled number lines and bead strings to help them order numbers before progressing to partly labelled number lines. 00021-2009CDO-EN Crown copyright 2009
2 of 2 The National Strategies Primary Overcoming barriers in mathematics helping children move from level 1 to level 2 Teaching tips Provide plenty of opportunities for children to continue to gain confidence with the number system and the order of numbers by counting forwards and backwards to and from different numbers. Look out for children who may not know, for example, what comes after 29 or before 61, and for children who have to count from 1 to find the number before or after a given number as they are insecure when counting from other starting numbers. Check that children can read, write and partition the numbers you want them to order. Look out for children who may not, for example, distinguish 13 from 30 when spoken and distinguish between 17 and 71 when written. Help children gain confidence using the vocabulary of comparison, such as bigger, smaller, biggest, smallest and in between to help them describe how they are ordering a set of three or more numbers. Challenge children to provide instructions for a partner to order a set of two-digit numbers. Support EAL pupils with the time connectives they may need to use to sequence their explanation, such as first, next, after that and finally. Provide plenty of opportunities for children to order from largest to smallest. They are likely to be more confident ordering from smallest to largest. Initially, children may be more confident ordering sets of consecutive numbers. However, they also need to appreciate that sets made up of more widely dispersed numbers such as 73, 9, 38, 16 can also be ordered. Use a wide range of questioning that encourages ordering numbers, for example: o Give me a number between 15 and 21. Is it closer to 15 or 21? Show me why on a number line. o What number is halfway between 15 and 21? How did you work it out? o Tell me four numbers which are bigger than 50 but less than 100. When ordering numbers on a number line, help children appreciate that the spaces between the numbers on the number line are less important than the order of the numbers. 00021-2009CDO-EN Crown copyright 2009
1 of 3 The National Strategies Primary Overcoming barriers in mathematics helping children move from level 1 to level 2 Can I partition a two-digit number into tens and ones (units) and use this to create related addition and subtraction sentences? Teaching guidance Key vocabulary number, tens, ones/units, digit, value, partition, split, recombine, place value, two-digit number Models and images, resources and equipment Practical resources that partition into tens and ones Place value cards Have a large teacher set and smaller class sets for children to pick up and use. Use practical equipment alongside place value cards (e.g. straws bundled in tens and ones) to help children appreciate the size of numbers. Place value chart Point to a tens number and a units number and ask the children what two-digit number this makes. Explore what happens with the teens numbers. 00021-2009CDO-EN Crown copyright 2009
2 of 3 The National Strategies Primary Overcoming barriers in mathematics helping children move from level 1 to level 2 Teaching tips Use practical resources and equipment that you can partition into tens and ones alongside the use of place value cards to help children appreciate the relative size of each digit in a two-digit number. For example, you could use art straws bundled into tens and ones, rods of tens and ones, 10p and 1p coins or a 100-bead string to help children appreciate the difference in size of the 3 digits in a number such as 33. Use place value cards to give children practical experience of selecting, combining and partitioning two-digit numbers. Experiences of ICT images alone are not sufficient. o Make or buy sets that are different colours so that children sitting next to each other do not muddle their sets. o Have a large teacher set and smaller class sets for children to pick up and use. o Ask addition and subtraction questions such as: 26 subtract 6? (Put the 6 card behind your back.) 20 add 6? (Put the 6 card back.) 26 subtract 20? (Put the 20 card behind your back.) 6 add 20. (Put the 20 card back.) o Occasionally stop and show how you can represent these questions mathematically by writing the number sentences: 26-6 = 20 and 20 + 6 = 26 26-20 = 6 and 6 + 20 = 26 Model how place value questions can be recorded as number sentences. Help children to recognise that all of the following are examples of place value questions and that counting should not be needed to solve them. 14 = 10 + 4 14 - = 4 10 + 4 = 14 - = 10 10 + = 14-4 = 10 4 + = 14-10 = 4 + 10 = 14 Look out for children who: o record incorrectly numbers they hear, for example hear 31 but reverse the digits and record 13; o interpret the two digits in a two-digit number as separate single-digit numbers, for example associate 47 with 4 and 7 not with 40 and 7, as they do not have a secure understanding of place value; o associate combining with joining together rather than with addition, for example write 40 combined with 2 as 402 rather than 40 + 2 = 42; o recognise from saying a number how it can be partitioned, for example that 73 can be partitioned as 70 and 3, but are confused by the teens as how they are said does not help with partitioning. 00021-2009CDO-EN Crown copyright 2009
3 of 3 The National Strategies Primary Overcoming barriers in mathematics helping children move from level 1 to level 2 Help children to recognise when they can use knowledge of place value in calculation, for example: o to add 40 to 38; o to subtract 23 from a number they can count back 2 tens and then 3; o when calculating 14 + 8 the last jump can be seen as a known place value fact (recombining 20 and 2 to make 22). +6 +2 14 20 22 00021-2009CDO-EN Crown copyright 2009
1 of 2 The National Strategies Primary Overcoming barriers in mathematics helping children move from level 1 to level 2 Can I partition one- and two-digit numbers in different ways? Teaching guidance Key vocabulary number, tens, units, ones, digit, value, partition, split, recombine, place value, one-digit number, two-digit number Models and images, resources and equipment Use practical equipment to partition a one-digit number in different ways 2 + 5 = 7 1 + 6 = 7 Use practical equipment to partition tens and ones in different ways Partition tens and ones in different ways using bead strings 00021-2009CDO-EN Crown copyright 2009
2 of 2 The National Strategies Primary Overcoming barriers in mathematics helping children move from level 1 to level 2 Teaching tips Provide children with practical experience of partitioning a small number of objects into two groups, for example by: o creating all the possible dominoes with 7 spots; o putting out 8 biscuits onto 2 plates; o making two jumps to land on 9 on a number line. Model the different ways in which a partitioned number can be recorded, for example: 8 = 5 + 3 8 = 3 + 5 5 + 3 = 8 3 + 5 = 8 Use equipment that helps children to see that numbers can be partitioned in many different ways, for example: o Use a 100-bead string to partition number 46 into tens and ones in different ways. Model how these can be recorded as number sentences, for example: 46 = 40 + 6 46= 30 + 16 46 = 20 + 26 46 = 10 + 36; o Use bundles of 10 straws and single straws to help children see a number such as 35 as: IIIIIIIIII IIIIIIIIII IIIIIIIIII I I I I I 30 + 5 IIIIIIIIII IIIIIIIIII IIIIIIIIII I I I I I 20 +15 and so on. Explore with children how to use partitioning of a one-digit number and knowledge of number facts to 10 to add and subtract numbers that cross the tens boundary. Ask questions such as: o What is 37 + 8? What number facts might you use to help you work this out? How many do you need to add to 37 to get to the next multiple of 10? How might you partition 8 to help you? How could you show that on a bead string or number line? Understanding of partitioning numbers in different ways (e.g. partitioning 74 into 70 + 4 or 60 + 14) supports children s understanding of future calculation methods, for example decomposition and informal methods of division using partitioning. 00021-2009CDO-EN Crown copyright 2009
1 of 2 The National Strategies Primary Overcoming barriers in mathematics helping children move from level 1 to level 2 Can I round a two-digit number to the nearest ten? Teaching guidance Key vocabulary digit, ones/units, tens, one-digit number, two-digit number, round to the nearest ten, approximate, multiple of ten, before, after, between, explain Models and images, resources and equipment Bead strings Use bead strings to help children identify the multiples of ten that a given number lies between and decide which multiple the number is closer to. Number lines Position numbers on number lines to help children see the multiple of ten they are closest to. 0 10 20 30 40 50 60 70 80 90 rounded to the nearest ten 32 30 Revert back to numbered number lines if children find using a partly numbered line challenging. Vertical number lines Presenting a number line vertically can help some children with the language of rounding up and down. 00021-2009CDO-EN Crown copyright 2009
2 of 2 The National Strategies Primary Overcoming barriers in mathematics helping children move from level 1 to level 2 Teaching tips Use resources such as bead strings and number lines to help children: o identify the multiples of 10 that the number lies between; o decide which multiple the number is closer to. 100-squares are not the most appropriate resource to use when helping children round numbers to the nearest 10. Ask children to explain which digit they need to look at to decide whether to round a number up or down. If necessary, remind them that to round to the nearest 10, they should look at the units digit. If the units digit is below 5, round down. If it is 5 or above, round up. Make sure that children meet examples where numbers round to 0, for example ask them to round 2 to the nearest 10. Ask open questions such as: o A number rounded to the nearest 10 is 30. What could the number be? o I think of a number and round it to the nearest 10. The answer is 50. What could my number be? Discuss practical situations to help children understand that rounding is used when only approximate numbers or amounts are needed, for example a pot containing 27 pencils could be described as containing about 30 pencils. 00021-2009CDO-EN Crown copyright 2009
1 of 2 The National Strategies Primary Overcoming barriers in mathematics helping children move from level 1 to level 2 Can I show where a whole number is on a 0 to 100 number line? Teaching guidance Key vocabulary multiple of ten, two-digit number, compare, order, count on/back, number line, position, locate, near, between, halfway, more, less Models and images, resources and equipment Locate numbers on a 100-bead string. Bead string Use number lines with an appropriate degree of scaffolding The Ordering numbers ITP helps children make links between beaded number lines and partly numbered number lines. Counting stick Count forwards and backwards along the counting stick from 0 to 100. Place numbers in the correct places along the stick using sticky notes. 00021-2009CDO-EN Crown copyright 2009
2 of 2 The National Strategies Primary Overcoming barriers in mathematics helping children move from level 1 to level 2 Teaching tips Provide plenty of opportunities for children to continue to gain confidence with the number system and the order of numbers by counting forwards and backwards to and from different numbers. Look out for children who may not know, for example, what comes after 29 or before 61, and for children who have to count from 1 to find the number before or after a given number because they are insecure when counting from other starting numbers. Use bead strings to provide practical experience of locating numbers up to 100. Encourage children to locate numbers quickly, for example show them how to find 32 by finding 30 and then counting on 2. Take the children into the playground and use a 10m length of rope to represent a 0 100 number line. Ask children to position the multiples of ten on the number line. Invite a child to select a number between 0 and 100 and without letting anyone know the number, place themselves on the number line. Can the rest of the class work out their number? Physically make a number by jumping along an imaginary number line, using a big jump forward to represent a jump of ten and small hops forward to represent ones, for example model a number such as 22 by two big jumps forward and two small hops forward. Ask children to locate the number you have created on a 100-bead string or on a number line. Using a counting stick, count in tens forwards and back. Point to a division and ask the children to name the multiple of ten. Point to various places along the counting stick (e.g. 45, 99, 21) and ask the children to say what number you might be pointing at. 00021-2009CDO-EN Crown copyright 2009
The National Strategies Primary Overcoming barriers in mathematics helping children move from level 2 to level 3 Can I use and explain decimal notation for tenths and hundredths? Teaching guidance Key vocabulary place value, partition, digit, ones, units, tens, hundreds, one-digit number, two-digit number, three-digit number, tenths, hundredths, compare, order, read, write Models and images Number lines Position decimal numbers on number lines to get a sense of the size and order of decimals. 1.6 2.4 Decimal number line ITP Use the Decimal number line ITP to explore and expand the divisions between numbers to get a sense of what the decimal parts of the number represent (e.g. 3.6). Decimal place value cards and charts Use decimal place value cards to illustrate what the different digits represent in decimal numbers (e.g. the 3 in 2.38 represents 3 tenths and is written as 0.3). Ask children to suggest what number is formed on the place value chart by combining units and tenths. Alternatively, children can identify the parts to make a given number. 00099-2008DOC-EN-04 Crown copyright 2008 1
The National Strategies Primary Overcoming barriers in mathematics helping children move from level 2 to level 3 Can I use and explain decimal notation for tenths and hundredths? Teaching guidance Measuring stick or ruler Relate decimals to measuring length by using rulers, tape measures and measuring sticks with different scales to become familiar with decimal numbers written as units of length. Bead string If the bead string represents one whole then each set of ten coloured beads represent a tenth and each individual bead represents a hundredth. Teaching tips Make sure that children understand that the decimal point is used to separate whole amounts and parts of the whole. You might use place value cards and the Decimal number line ITP, or pounds and pence with money notation, to illustrate this point. Help children become aware of the relative size of decimal numbers by ordering a set of amounts of money or lengths. Link this to ordering numbers on a number line. Make sure that you include numbers to overcome misconceptions such as mistaking the length of the number with its size, for example thinking that 4.05 is larger than 4.5. Ensure that as children are introduced to decimal notation they hear and use the language of tenths and hundredths, i.e. they can read 3.6 as three units and six tenths and not just as three point six. Use a simple number line marked in divisions of 0.5 to familiarise children with counting forwards and backwards in steps of 0.5. Extend this to other number lines to develop counting in other step sizes (e.g. 0.2). Use money and length as practical examples of decimals to place decimal numbers in context and compare size of numbers. For example: Which is the larger amount, 0.75 or 90p? Which is longer, 3.06 m or 3.6 m? 00099-2008DOC-EN-04 Crown copyright 2008 2
The National Strategies Primary Overcoming barriers in mathematics helping children move from level 2 to level 3 Can I order two-digit and three-digit numbers and position these on a number line? Teaching guidance Key vocabulary compare, order, count on/back, difference, number line, interval, position, landmark, size, scale, two-digit, three-digit Models and images Counting stick Count forwards and backwards in regular steps. Place numbers in the correct places along the stick using sticky notes. Use a bead string to represent 0 100. Bead string Number lines Children need to use marked and partly marked number lines to order and position numbers. They should use number lines with different start points, end points and intervals. Scales The skills and understanding involved in positioning numbers and counting along number lines are linked to those needed to read scales on measuring equipment and to interpret scales on graphs and charts. 00099-2008DOC-EN-01 Crown copyright 2008 1
The National Strategies Primary Overcoming barriers in mathematics helping children move from level 2 to level 3 Can I order two-digit and three-digit numbers and position these on a number line? Teaching guidance Teaching tips Ensure that there is progression in children s experience of number lines. They need experience of marked then partly marked number lines. 0 100 Check that children have all of the necessary skills and understanding to order numbers on a number line. For example, ask children to: locate numbers in relation to landmark numbers (e.g. multiples of 10); read, write and partition numbers, explaining their place value. Once they have experience with partly marked number lines, demonstrate how to draw and order numbers on unmarked number lines. Initially start at 0, then show children that they can choose to draw only the relevant segment of the line. Provide experience of a range of number lines with different start numbers, end numbers and intervals. It is important that children understand how to use marked numbers to find the value of each interval. Children can then use these skills to read scales on measuring equipment and on graphs and charts. See the linked section of this resource: Can I explain what each division means on a numbered or partly numbered line and use this information to read a scale to the nearest division or halfdivision? 300 400 When children meet new types of numbers, such as negative numbers, fractions or decimals, placing numbers onto a number line will help them to incorporate these into their existing understanding of the number system. 00099-2008DOC-EN-01 Crown copyright 2008 2
The National Strategies Primary Overcoming barriers in mathematics helping children move from level 2 to level 3 Can I round whole numbers to 1000 to the nearest 10 or 100? Teaching guidance Key vocabulary explain, digit, ones, units, tens, hundreds, one-digit number, two-digit number, three-digit number, round, approximate, multiple Models and images Number lines Position numbers on number lines to clarify the multiple of 10 or 100 they are closest to. 0 100 200 300 400 500 600 700 800 900 1000 rounded to the nearest 100 329 300 Note: 100-squares are not appropriate images to support rounding. Round to the nearest 100 Position a three-digit number on a 0 1000 number line. Which multiple of 100 it is closest to? Round to the nearest 10 Expand your chosen interval and position the number between appropriate multiples of 10. Which multiple of 10 it is closest to? Decimal number line ITP Reading scales Children should use a range of instruments to measure objects to the nearest 100 units or to the nearest 10 units as appropriate. The Measuring scales and Measuring cylinder ITPs can be used to practise reading scales. 00099-2008DOC-EN-02 Crown copyright 2008 1
The National Strategies Primary Overcoming barriers in mathematics helping children move from level 2 to level 3 Can I round whole numbers to 1000 to the nearest 10 or 100? Teaching guidance Teaching tips To round a number to the nearest 10/100, encourage children to: identify the multiples of 10/100 that the number lies between; decide which multiple the number is closer to. If children struggle, ask them to locate the number on an appropriate number line. Ask children to explain which digit they need to look at to decide whether to round a number up or down. If necessary, remind them as follows. To round to the nearest 10, look at the units digit. If the units digit is below 5, round down. If it is 5 or above, round up. To round to the nearest 100, look at the tens digit. If the tens digit is below 5, round down. If it is 5 or above, round up. Ensure that children meet examples of rounding numbers that round to 0 or a multiple of 100 or 1000. Ask them to: round 25 to the nearest 100 (0); round 198 to the nearest 10 (200); round 965 to the nearest 100 (1000). Help children to realise that: rounding a number to the nearest 10 gives a multiple of 10 (and might also be a multiple of 100); when rounding a number to the nearest 100 the answer is a multiple of 100 (and might also be a multiple of 1000). Some children may not appreciate initially that multiples of 10 continue beyond 100: 100, 110, 120 200, 210, 220 300, etc. Ask questions such as these: A number rounded to the nearest 10 is 340. What could the number be? What is the smallest number that would round to 700 when rounded to the nearest 100? How many even numbers would round to 820 when rounded to the nearest 10? Discuss practical situations or word problems to help children understand that rounding is used when only approximate numbers or amounts are needed, as in the following example. A local newspaper reports that 800 people attended a concert. Did exactly 800 people attend? Demonstrate how to use rounding to find approximate answers to calculations and how to use these skills to check that answers to written calculations are sensible. For example: 38 + 51 is close to 40 + 50, so the answer will be approximately 90; 196 4 is close to 200 4, so the answer will be approximately 800. 00099-2008DOC-EN-02 Crown copyright 2008 2
The National Strategies Primary Overcoming barriers in mathematics helping children move from level 2 to level 3 Can I read, write and partition whole numbers to 1000? Teaching guidance Key vocabulary place value, partition, digit, units, ones, tens, hundreds, one-digit number, two-digit number, three-digit number, compare, order, read, write Models and images Place value cards These cards allow children to partition numbers into hundreds, tens and units or to combine parts to form a number. This can be particularly useful for numbers that contain a zero digit. Place value grid Children suggest what number is formed by combining multiples of 100, 10 and 1. Alternatively, children can identify the parts that make a given three-digit number. 100 200 300 400 500 600 700 800 900 10 20 30 40 50 60 70 80 90 1 2 3 4 5 6 7 8 9 Beadsticks ITP Use this interactive abacus to explore place value. Demonstrate repeatedly adding or subtracting 1, 10 or 100, so that children can observe which digit changes. Model the effect of counting over a boundary. Number lines Say a number out loud, e.g. seven hundred and six, and ask children to write the number then estimate its position on the number line and write it in. Use this to tackle mistakes in writing and reading numbers, for example if they write seven hundred and six as 7006. 0 100 200 300 400 500 600 700 800 900 1000 00099-2008DOC-EN-05 Crown copyright 2008 1
The National Strategies Primary Overcoming barriers in mathematics helping children move from level 2 to level 3 Can I read, write and partition whole numbers to 1000? Teaching guidance Teaching tips Ensure that children practise reading and writing numbers that contain a zero digit, such as 502, or a teen number, such as 718. Calculators can be a useful resource. Give pairs of children a calculator each and ask them to display a three-digit number. Children should pass their calculator to their partner, asking the partner to change a given digit to zero (e.g. change 378 to 308). Try the following if children make mistakes writing large numbers, for example writing 123 as 10023. Demonstrate with place value cards how the 20 and the 3 are placed on top of the 100 to form 123. Ensure that children handle place value cards themselves. Plan activities where children use place value cards to partition and combine numbers. Focus particularly on numbers that contain zero as a place holder. Ask: Which cards are needed to make 301? Why is no 10 card needed? Children should practise partitioning numbers in different ways (e.g. 327 = 300 + 20 + 7 = 300 + 10 + 17 = 200 + 120 + 7). This is key to understanding future calculation strategies. Children need to appreciate that the position of a digit tells you about its value. Demonstrate this using an overhead or interactive whiteboard calculator. For example, enter the number 372 and ask children the value of the 7. Subtract 70 to show that this removes the 7 from the number. If children do not appreciate, for example, that the 3 in the number 231 has greater value that the 3 in the number 473, model this using the dots on the Place value ITP or using ones, tens and hundreds bundles of craft straws. Use base 10 apparatus or money to demonstrate that ten ones make one ten and ten tens make one hundred. Use the Beadsticks ITP to show children which digit changes as you add or subtract ones, tens or hundreds. Focus particularly on what happens as you count over a boundary, e.g. 274, 284, 294, 304. Build in regular opportunities for children to position and order numbers using number lines. These skills underpin calculation methods such as finding the difference between two numbers by counting on from the smaller number to the larger. 00099-2008DOC-EN-05 Crown copyright 2008 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, count, sequence 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. 5 1 7 11 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, divide the difference of 12 by 4. 6 6 Overcoming barriers in mathematics helping children move from level 3 to level 4 00695-2007CDO-EN Primary National Strategy Crown copyright 2007
Can I count on from any given number in whole-number steps, extending beyond zero when counting backwards? 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 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 with positive numbers, and that sequences involve counting back as well as counting forward. Overcoming barriers in mathematics helping children move from level 3 to level 4 00695-2007CDO-EN Primary National Strategy Crown copyright 2007
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 Overcoming barriers in mathematics helping children move from level 3 to level 4 00695-2007CDO-EN Primary National Strategy Crown copyright 2007
Can I read, write, partition and order decimal numbers? For decimal numbers 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 that have different numbers of decimal places for ordering, 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 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. Overcoming barriers in mathematics helping children move from level 3 to level 4 00695-2007CDO-EN Primary National Strategy Crown copyright 2007
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 ITP Use a number line and/or a counting stick to identify and locate numbers. Where would you place 2? 4 0 4 0 7 4 Overcoming barriers in mathematics helping children move from level 3 to level 4 00695-2007CDO-EN Primary National Strategy Crown copyright 2007
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 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. Overcoming barriers in mathematics helping children move from level 3 to level 4 00695-2007CDO-EN Primary National Strategy Crown copyright 2007
The National Strategies Primary Overcoming barriers in mathematics helping children move from level 2 to level 3 Can I order two-digit and three-digit numbers and position these on a number line? Teaching guidance Key vocabulary compare, order, count on/back, difference, number line, interval, position, landmark, size, scale, two-digit, three-digit Models and images Counting stick Count forwards and backwards in regular steps. Place numbers in the correct places along the stick using sticky notes. Use a bead string to represent 0 100. Bead string Number lines Children need to use marked and partly marked number lines to order and position numbers. They should use number lines with different start points, end points and intervals. Scales The skills and understanding involved in positioning numbers and counting along number lines are linked to those needed to read scales on measuring equipment and to interpret scales on graphs and charts. 00099-2008DOC-EN-01 Crown copyright 2008 1
The National Strategies Primary Overcoming barriers in mathematics helping children move from level 2 to level 3 Can I order two-digit and three-digit numbers and position these on a number line? Teaching guidance Teaching tips Ensure that there is progression in children s experience of number lines. They need experience of marked then partly marked number lines. 0 100 Check that children have all of the necessary skills and understanding to order numbers on a number line. For example, ask children to: locate numbers in relation to landmark numbers (e.g. multiples of 10); read, write and partition numbers, explaining their place value. Once they have experience with partly marked number lines, demonstrate how to draw and order numbers on unmarked number lines. Initially start at 0, then show children that they can choose to draw only the relevant segment of the line. Provide experience of a range of number lines with different start numbers, end numbers and intervals. It is important that children understand how to use marked numbers to find the value of each interval. Children can then use these skills to read scales on measuring equipment and on graphs and charts. See the linked section of this resource: Can I explain what each division means on a numbered or partly numbered line and use this information to read a scale to the nearest division or halfdivision? 300 400 When children meet new types of numbers, such as negative numbers, fractions or decimals, placing numbers onto a number line will help them to incorporate these into their existing understanding of the number system. 00099-2008DOC-EN-01 Crown copyright 2008 2
The National Strategies Primary Overcoming barriers in mathematics helping children move from level 2 to level 3 Can I round whole numbers to 1000 to the nearest 10 or 100? Teaching guidance Key vocabulary explain, digit, ones, units, tens, hundreds, one-digit number, two-digit number, three-digit number, round, approximate, multiple Models and images Number lines Position numbers on number lines to clarify the multiple of 10 or 100 they are closest to. 0 100 200 300 400 500 600 700 800 900 1000 rounded to the nearest 100 329 300 Note: 100-squares are not appropriate images to support rounding. Round to the nearest 100 Position a three-digit number on a 0 1000 number line. Which multiple of 100 it is closest to? Round to the nearest 10 Expand your chosen interval and position the number between appropriate multiples of 10. Which multiple of 10 it is closest to? Decimal number line ITP Reading scales Children should use a range of instruments to measure objects to the nearest 100 units or to the nearest 10 units as appropriate. The Measuring scales and Measuring cylinder ITPs can be used to practise reading scales. 00099-2008DOC-EN-02 Crown copyright 2008 1
The National Strategies Primary Overcoming barriers in mathematics helping children move from level 2 to level 3 Can I round whole numbers to 1000 to the nearest 10 or 100? Teaching guidance Teaching tips To round a number to the nearest 10/100, encourage children to: identify the multiples of 10/100 that the number lies between; decide which multiple the number is closer to. If children struggle, ask them to locate the number on an appropriate number line. Ask children to explain which digit they need to look at to decide whether to round a number up or down. If necessary, remind them as follows. To round to the nearest 10, look at the units digit. If the units digit is below 5, round down. If it is 5 or above, round up. To round to the nearest 100, look at the tens digit. If the tens digit is below 5, round down. If it is 5 or above, round up. Ensure that children meet examples of rounding numbers that round to 0 or a multiple of 100 or 1000. Ask them to: round 25 to the nearest 100 (0); round 198 to the nearest 10 (200); round 965 to the nearest 100 (1000). Help children to realise that: rounding a number to the nearest 10 gives a multiple of 10 (and might also be a multiple of 100); when rounding a number to the nearest 100 the answer is a multiple of 100 (and might also be a multiple of 1000). Some children may not appreciate initially that multiples of 10 continue beyond 100: 100, 110, 120 200, 210, 220 300, etc. Ask questions such as these: A number rounded to the nearest 10 is 340. What could the number be? What is the smallest number that would round to 700 when rounded to the nearest 100? How many even numbers would round to 820 when rounded to the nearest 10? Discuss practical situations or word problems to help children understand that rounding is used when only approximate numbers or amounts are needed, as in the following example. A local newspaper reports that 800 people attended a concert. Did exactly 800 people attend? Demonstrate how to use rounding to find approximate answers to calculations and how to use these skills to check that answers to written calculations are sensible. For example: 38 + 51 is close to 40 + 50, so the answer will be approximately 90; 196 4 is close to 200 4, so the answer will be approximately 800. 00099-2008DOC-EN-02 Crown copyright 2008 2
The National Strategies Primary Overcoming barriers in mathematics helping children move from level 2 to level 3 Can I read, write and partition whole numbers to 1000? Teaching guidance Key vocabulary place value, partition, digit, units, ones, tens, hundreds, one-digit number, two-digit number, three-digit number, compare, order, read, write Models and images Place value cards These cards allow children to partition numbers into hundreds, tens and units or to combine parts to form a number. This can be particularly useful for numbers that contain a zero digit. Place value grid Children suggest what number is formed by combining multiples of 100, 10 and 1. Alternatively, children can identify the parts that make a given three-digit number. 100 200 300 400 500 600 700 800 900 10 20 30 40 50 60 70 80 90 1 2 3 4 5 6 7 8 9 Beadsticks ITP Use this interactive abacus to explore place value. Demonstrate repeatedly adding or subtracting 1, 10 or 100, so that children can observe which digit changes. Model the effect of counting over a boundary. Number lines Say a number out loud, e.g. seven hundred and six, and ask children to write the number then estimate its position on the number line and write it in. Use this to tackle mistakes in writing and reading numbers, for example if they write seven hundred and six as 7006. 0 100 200 300 400 500 600 700 800 900 1000 00099-2008DOC-EN-05 Crown copyright 2008 1
The National Strategies Primary Overcoming barriers in mathematics helping children move from level 2 to level 3 Can I read, write and partition whole numbers to 1000? Teaching guidance Teaching tips Ensure that children practise reading and writing numbers that contain a zero digit, such as 502, or a teen number, such as 718. Calculators can be a useful resource. Give pairs of children a calculator each and ask them to display a three-digit number. Children should pass their calculator to their partner, asking the partner to change a given digit to zero (e.g. change 378 to 308). Try the following if children make mistakes writing large numbers, for example writing 123 as 10023. Demonstrate with place value cards how the 20 and the 3 are placed on top of the 100 to form 123. Ensure that children handle place value cards themselves. Plan activities where children use place value cards to partition and combine numbers. Focus particularly on numbers that contain zero as a place holder. Ask: Which cards are needed to make 301? Why is no 10 card needed? Children should practise partitioning numbers in different ways (e.g. 327 = 300 + 20 + 7 = 300 + 10 + 17 = 200 + 120 + 7). This is key to understanding future calculation strategies. Children need to appreciate that the position of a digit tells you about its value. Demonstrate this using an overhead or interactive whiteboard calculator. For example, enter the number 372 and ask children the value of the 7. Subtract 70 to show that this removes the 7 from the number. If children do not appreciate, for example, that the 3 in the number 231 has greater value that the 3 in the number 473, model this using the dots on the Place value ITP or using ones, tens and hundreds bundles of craft straws. Use base 10 apparatus or money to demonstrate that ten ones make one ten and ten tens make one hundred. Use the Beadsticks ITP to show children which digit changes as you add or subtract ones, tens or hundreds. Focus particularly on what happens as you count over a boundary, e.g. 274, 284, 294, 304. Build in regular opportunities for children to position and order numbers using number lines. These skills underpin calculation methods such as finding the difference between two numbers by counting on from the smaller number to the larger. 00099-2008DOC-EN-05 Crown copyright 2008 2