Teaching Guidance. For. Fractions, Decimals and Percentages

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1 Teaching Guidance For Fractions, Decimals and Percentages (Overcoming Barriers Moving Levels 1 to 2, 2 to 3, 3 to 4, 4 to 5)

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3 1 of 2 The National Strategies Primary Overcoming barriers in mathematics helping children move from level 1 to level 2 Can I use the position of both hands to tell the time to the quarterhour on a clock face? Teaching guidance Key vocabulary time, clock, watch, analogue, hour (h), minute (min), o clock, quarter to, quarter past, half past, hands, long (minute) hand, short (hour) hand Models and images, resources and equipment Use a demonstration clock with geared hands Analogue clock faces Ensure that children have opportunities to read the time on a variety of clock faces. Tell time ITP Use the ITP to set the clock to a specific time, or to leave the clock running in real time. When using the ITP ensure that children also have their own clock faces to manipulate CDO-EN Crown copyright 2009

4 2 of 2 The National Strategies Primary Overcoming barriers in mathematics helping children move from level 1 to level 2 Teaching tips When demonstrating times on an analogue clock, use a clock with geared hands. Children should also have small geared clocks for their own use. Simple clock faces without geared hands do not help children to understand how the long hand moves over an hour. Provide plenty of opportunities to tell the time during the routine of the day. Put children in charge of letting everyone know when it is lunch time, time to go to assembly, etc. Ensure that children apply their knowledge of half and quarter turns to reading the time to half and quarter hours. o clock quarter to quarter past half past Give opportunities to relate times on clock faces to recognisable events in the day. This would be particularly supportive of recently arrived EAL learners to help them gain familiarity with and understanding of the school routine and day. Activities might be based around matching significant events of the day to a basic timeline or asking questions and modelling responses so that children hear and get the opportunity to practise particular language structures, for example language structures used to describe a point in time ( What time does?, When does? ) and language structures to describe the length of time ( How long is? ). Emphasise the key physical features of the clock face, i.e. the significance of the 6, when past times become to times and the position of the hour hand as the change-over occurs. Make a large circle on the floor and place the numbers 1 to 12 around the edge to represent a clock face. Ask two children to be the hands of the clock and lie down on the clock face to show a given time. Take photographs for display. Produce sets of matching cards that enable children to match pictures of everyday events to clock faces showing the time, or to cards showing how the time is written. While asking questions about the position of the hands at various times, include those that cross the hour boundary, for example: It is a quarter to three now. School will finish in half an hour. What time will that be? As children become more confident with telling the time and positioning the hands, questions like these will begin to establish the concept of time intervals and lay the foundations of solving problems involving time CDO-EN Crown copyright 2009

5 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 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 DOC-EN-04 Crown copyright

6 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? DOC-EN-04 Crown copyright

7 The National Strategies Primary Overcoming barriers in mathematics helping children move from level 2 to level 3 Can I find a unit fraction of a shape, number or quantity by splitting it into the correct number of equal parts? Teaching guidance Key vocabulary fraction, part, equal parts, one whole, parts of a whole, number of parts, divide, one half, one third, one quarter, one fifth, one sixth, one tenth, unit fraction Models and images Link fractions of amounts and fractions of shapes ⅓ ⅓ ⅓ Create shapes divided in halves, quarters, thirds, etc. using card or paper plates. Share objects onto the appropriate shape to find fractions of an amount. 1 / 3 of 6 = 2 Ensure that children can also talk about and explain the inverse of this operation: 2 multiplied by 3 = 6 Counting stick / 4 of 20 Relate fraction notation to division 1 whole 5 1 whole DOC-EN-27 Crown copyright

8 The National Strategies Primary Overcoming barriers in mathematics helping children move from level 2 to level 3 Can I find a unit fraction of a shape, number or quantity by splitting it into the correct number of equal parts? Teaching guidance Teaching tips Check that children are able to read and write fractions. (See the section: Can I read and write fractions and explain their meaning?) Children are frequently able to read the fractions 1 / 2 and 1 / 4 because they have seen them regularly, but may struggle to read and write other fractions. Ensure that children understand that the bottom number of a fraction (the denominator) tells you how many pieces the whole is divided into. So 1 / 5 can be read as one whole divided into five equal parts. Explore what happens when these five equal parts are recombined to make the whole. Reinforce that when a shape (or number) is divided into fractions such as quarters, each piece must be the same size. This shape is divided into four pieces but the pieces are not quarters because they are not the same size. Give children practical experience of dividing shapes into fractions. For example: Give children a rectangle that is 10 cm long and access to a ruler. Explain that you want them to draw lines to divide the rectangle into fifths. Ask: How many fifths make one whole? How many pieces must we divide the rectangle into? Ask children to think about how to make sure that each piece is the same size. Use practical resources to link fractions of shapes and fractions of amounts. Make a set of card shapes/paper plates divided into halves, thirds, quarters, etc. To find, for example 1 / 3 of 12, ask children which shape shows the appropriate fraction (thirds). Take 12 counters/objects and ask a child to place these onto the shape so that there is the same number of counters/objects on each third. Ask: What is 1 / 3 of 12? Ask children to describe how they found 1 / 3 of 12. Ask: What number sentence could we write? Stress the link between fractions and division: to find 1 / 2 of an amount, divide it into two equal sets; to find 1 / 3 of an amount, divide it into three equal sets; etc. Give children opportunities to find fractions of shapes where the shape is divided into small pieces. These should not always be regular. For example, say: Shade 1 / 3 of this shape. How many squares is it made from? What is 1 / 3 of 15? How do you know? How many squares do you need to shade? Give children opportunities to find fractions of quantities, for example asking them to divide a metre stick into tenths. Ask: What is 1 / 10 of 1 metre? Ensure that children include an appropriate unit of measure in their answer DOC-EN-27 Crown copyright

9 The National Strategies Primary Overcoming barriers in mathematics helping children move from level 2 to level 3 Can I read and write fractions and explain their meaning? Teaching guidance Key vocabulary fraction, part, equal parts, one whole, parts of a whole, number of parts, one half, one third, one quarter, one fifth, one sixth, one tenth, two thirds, three quarters, unit fraction Models and images Use a range of representations ⅔ ⅔ ⅔ 0 1 Fractions ITP This interactive program allows children to create and compare fractions. Number lines Counting in, for example, tenths along a number line helps children to understand that tenths represent one whole divided into ten equal pieces DOC-EN-03 Crown copyright

10 The National Strategies Primary Overcoming barriers in mathematics helping children move from level 2 to level 3 Can I read and write fractions and explain their meaning? Teaching guidance Teaching tips Use practical activities to give children opportunities of making halves, quarters, etc. Make sure that children know how many of each fraction make one whole. Ask children to cut a piece of paper into quarters. Emphasise that quarters should be exactly the same size. Ask: How many quarters make one whole? Ask children to make a strip of paper 1 metre long. Use a metre stick to help divide the strip into ten equal pieces. Ask: What fraction is each piece? How many tenths make one whole? Ensure that children know that the bottom number (denominator) of a fraction tells you how many equal parts the whole is divided into, for example a denominator of 5 tells you that the pieces are fifths and so five pieces of that size would make one whole. Take equal strips of paper. Fold one into halves, one into quarters (half and half again) and one into eighths (half, half and half again). Label each half, quarter and eighth. Use this to discuss how many halves make one whole, how many quarters make one whole, etc. Ask: How can you tell how many of each fraction make one whole? (The denominator tells you.) Which is bigger, 1 / 4 or 1 / 8? Label the fractions on a fraction wall or use the Fractions ITP to create strips that are divided into halves, thirds, quarters, etc. Compare the size of unit fractions ask: Which is bigger, 1 / 2 or 1 / 3? Which do you think would be bigger, 1 / 50 or 1 / 100? How do you know? Teach children that the top number (numerator) of a fraction tells you how many pieces you have. In 3 / 5, the 3 tells you that you have three pieces and the 5 tells you that each piece is one fifth of a whole. One way to read 3 / 5 is 3 out of 5. Count in fractions, e.g. 1 / 10, 2 / 10, using number lines, fraction walls or fractions of shapes. This helps children to understand non-unit fractions. Some children are able to write fractions without being able to read them, so plan activities that involve children in reading fractions aloud, for example the following. Matching game: Children pick a picture card, say the fraction that it shows and then pick a fraction card to see whether it matches. Give pairs of children a set of 1 9 digit cards. Children take turns to pick two cards, then place the smaller digit over the larger to make a fraction, e.g. 7 / 9, and say the fraction made. Use a fraction wall to compare the fractions. Whoever has the larger fraction wins a point. Replace the cards, shuffle them and continue playing. Make sure that children experience a variety of representations for fractions, including objects, shapes and number lines/strips. Where possible, avoid saying a fraction on its own. Instead, try to say 3 / 5 of a metre, 3 / 5 of the shape, 3 / 5 of the line, etc. Ask questions such as: Which is smaller a quarter of an elephant or a quarter of a mouse? DOC-EN-03 Crown copyright

11 The National Strategies Primary Overcoming barriers in mathematics helping children move from level 2 to level 3 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 half-division? Teaching guidance Key vocabulary measure, estimate, length, distance, height, depth, weight, capacity, temperature, measuring scale, interval, division, unit, standard unit, metric unit, approximately, close, about the same as, ruler, tape measure, balance, scales, measuring cylinder/jug, thermometer, centimetre (cm), metre (m), millimetre (mm), kilogram (kg), gram (g), litre (l), millilitre (ml), degree Celsius ( C) Models and images Counting stick 0 1 kg g Use sticky notes to partly label the divisions. Discuss the missing numbers. Change the labels to represent 0 1 litre. Hold the counting stick vertically to represent a thermometer or a measuring jug. Measurement ITPs Measuring cylinder ITP Use measurement ITPs to provide examples of different types of scales and demonstrate how to read them. Settings can be changed to show various ranges and intervals. Measuring scales ITP Thermometer ITP DOC-EN-19 Crown copyright

12 The National Strategies Primary Overcoming barriers in mathematics helping children move from level 2 to level 3 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 half-division? Teaching guidance Teaching tips Plan practical activities that involve children using different types of measuring equipment. Assess whether children have the skills they need to read scales accurately. Ensure that children have regular experience of placing numbers and reading numbers on partly marked number lines. A key skill that children need in order to read scales accurately is being able to work out the value of each interval. In order to do this, children need to: identify two labelled divisions such as 0 g and 50 g; count how many intervals lie between these amounts (Note: Make sure that children understand that they must count the number of jumps between each labelled division rather than the number of marks); use division to work out the value of one interval; count along the scale to check that this interval is correct. Ensure that children know relationships between units of measurement. See the resource: Can I explain the relationship between km and m, m and cm, kg and g, l and ml? It is important to select equipment that is marked clearly enough for children to read. For instance, ordinary kitchen equipment may have scales that are very difficult for young children to make sense of, and rulers marked in millimetres can lead to confusion. Offer children as many experiences as possible to measure for a purpose, so it is important to be accurate, for example following a recipe, carrying out an experiment in science, making a scale model. Look for cross-curricular opportunities. Images of scales on ITPs are useful for supporting explanations and as a basis for discussion but are no substitute for practical experience. Measuring scales ITP DOC-EN-19 Crown copyright

13 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 3 / 4 into other equivalent fractions. Overcoming barriers in mathematics helping children move from level 3 to level CDO-EN Primary National Strategy Crown copyright 2007

14 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 3 / 4 of them to establish that 3 / 4 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. Overcoming barriers in mathematics helping children move from level 3 to level CDO-EN Primary National Strategy Crown copyright 2007

15 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. Overcoming barriers in mathematics helping children move from level 3 to level CDO-EN Primary National Strategy Crown copyright 2007

16 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 digits ITP Teaching tips Help children to generalise correctly so that they can cope with decimals. Multiplying by 10 gives an answer that is bigger than the original number and all the digits move one place to the left. Dividing by 10 gives an answer that is smaller than the original number and all the digits move one place to the right. Discuss why is not the same as 4.60 and 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 a multiplication and division by 10, 100 and 1000 should be a mental calculation. Overcoming barriers in mathematics helping children move from level 3 to level CDO-EN Primary National Strategy Crown copyright 2007

17 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 is two thirds of 12 counters? What is 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 / 6 Find 5 / 1 / 1 6 of / 6 What is 1 / 6 of 18? (3 6 = 18) 1 / 1 6 / 6 If one sixth is 3, what is five sixths? 1 / 6 Overcoming barriers in mathematics helping children move from level 3 to level CDO-EN Primary National Strategy Crown copyright 2007

18 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 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 1 / 5 4, 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 is two sixths of 30 cm? What is 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 CDO-EN Primary National Strategy Crown copyright 2007

19 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 a 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. Overcoming barriers in mathematics helping children move from level 3 to level CDO-EN Primary National Strategy Crown copyright 2007

20 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 CDO-EN Primary National Strategy Crown copyright 2007

21 Can I read, write, partition and order decimal numbers? For decimal numbers with up to two places, use a 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 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 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 CDO-EN Primary National Strategy Crown copyright 2007

22 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 = Moving digits ITP Use number lines or the Fractions ITP to reinforce the decimal equivalent of fractions such as 2 5 or Fractions ITP Fractions ITP Overcoming barriers in mathematics helping children move from level 3 to level CDO-EN Primary National Strategy Crown copyright 2007

23 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 (2 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 Overcoming barriers in mathematics helping children move from level 3 to level CDO-EN Primary National Strategy Crown copyright 2007

24 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 10, 100 and Moving digits ITP Overcoming barriers in mathematics helping children move from level 3 to level CDO-EN Primary National Strategy Crown copyright 2007

25 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 Explore the language of units, for example, roots from which centi and milli are derived and where else they are used (e.g. century, centurion). Provide regular practical opportunities for children to learn and understand the relationship between units. Overcoming barriers in mathematics helping children move from level 3 to level CDO-EN Primary National Strategy Crown copyright 2007

26 Can I calculate simple percentages of whole numbers or quantities? Teaching guidance Key vocabulary hundredths, percentage, equivalent, %, tenths 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 Overcoming barriers in mathematics helping children move from level 3 to level CDO-EN Primary National Strategy Crown copyright 2007

27 Can I calculate simple percentages of whole numbers or quantities? Teaching tips When finding a percentage of a quantity, remind children that: they might find it helpful to 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 buy in comparison with 248? % 62 25% % % % % % % Overcoming barriers in mathematics helping children move from level 3 to level CDO-EN Primary National Strategy Crown copyright 2007

28 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 gives an answer that is bigger than the original number and all the digits move one place to the left, while dividing by 10 gives an answer that is smaller than the original number 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). 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. Overcoming barriers in mathematics helping children move from level 3 to level CDO-EN Primary National Strategy Crown copyright 2007

29 Can I use my tables to work out multiplication and division facts with decimals? Ensure that children meet and can interpret multiplication and division calculations that are written in a variety of different ways, for example: 0.8 = = = 6 Reinforce the division facts corresponding to multiplication facts; for example: = = = = 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. Overcoming barriers in mathematics helping children move from level 3 to level CDO-EN Primary National Strategy Crown copyright 2007

30 1 of 3 The National Strategies Primary Overcoming barriers level 4-5 Can I solve simple problems involving ratio and proportion? Teaching guidance Key vocabulary problem, pattern, relationship, ratio, proportion, in every, for every, to every, fraction, equivalent, simplify Models, images and resources Ratio and proportion ITP Use this program to set the a ratio for yellow : pink liquid to provide a visual image for the relationship between the two quantities. The yellow and pink liquid can be combined in a single measuring cylinder to explore what proportion of the total mixture is each colour. Number lines and scales Children need to be able to work out the value of each interval on a number line using the proportion that it represents of a known amount. 120 B 200 Scale drawings, models and scaled maps The ratio of a length on a drawing, model or map to the equivalent length on the real item is given by the scale. Westbury Northcott Eastfield Scale 1 cm : 5 km EPD-EN-01 Crown copyright 2009

31 2 of 3 The National Strategies Primary Overcoming barriers level 4-5 Line graphs Graphs can be used to compare related measurements. Where the measurements are in a constant ratio, for example, in conversions between units of measurement or currencies, the graph formed will be a straight-line graph. Teaching tips Proportion Proportion describes the relationship between part of a quantity or measure and the whole. For example: o What proportion of the class is female? The class contains 25 children and ten out of the 25 are girls. Therefore 10/25, or 2/5, of the class are female. Make sure children appreciate that where a proportion can be described as, for instance 3 out of 4, this can be written as the fraction 3 / 4. Ensure that children meet proportion described in different ways: o Using everyday language: ten out of 25 children are girls; ten in 25 children are girls. o In simplified form: two out of every five children are girls; two in every five children are girls. o As a fraction: 2 / 5 of the class are female. o As a decimal: 0.4 of the class are female. o As a percentage: 40% of the class are female. Rehearse scaling proportions up and down. This technique can be used to solve problems. Provide visual images, for example: one in four tiles is black two in eight tiles is black three in 12 tiles are black Ratio Ratio can describe a part to part relationship. For example: The ratio of girls to boys in a class is two to every three (represented as 2:3). Ratio can also describe the relationship between two comparable quantities/measures: The ratio of a distance on a map to the distance on the ground is 1: Ensure that children understand and can use ratios described in different ways: o Using everyday language: there is one black tile to three white tiles; there is one black tile for every three white tiles. o Using a colon (use everyday language first, then the colon form): The ratio of black tiles to white tiles is one to every three. The ratio of black tiles to white tiles is 1:3. The ratio of white tiles to black tiles is 3: EPD-EN-01 Crown copyright 2009

32 3 of 3 The National Strategies Primary Overcoming barriers level 4-5 Ensure that children can use and describe ratios in their simplest form, for example 1:3 is the simplest form of the relationship 3:9. Rehearse scaling ratios up/down. This technique can be used to solve problems: o 5 miles is approximately equal to 8 km o 10 miles is approximately equal to 16 km o 15 miles is approximately equal to 24 km. Ratio and proportion Where ratio is describing part to part, this can be linked to the proportion of the whole, for example: The ratio of black to white tiles on a wall is 1:3. This means that for every one black tile there are three white tiles. Therefore there is one black tile in every four tiles and so ¼ of the tiles are black EPD-EN-01 Crown copyright 2009

33 1 of 2 The National Strategies Primary Overcoming barriers level 4-5 Can I explain and use ratio notation? Teaching guidance Key vocabulary ratio, for every, to every, equivalent, simplify, problem, pattern, relationship, scale up/down Models and images and resources Ratio and proportion ITP This program enables you to set the ratio for yellow:pink liquid that is poured into two measuring cylinders. This provides a visual image for the relationship between two quantities in this ratio. A scale factor can be used that scales up the amount of each colour poured in. Fraction ITP This program allows you to break a strip into different parts and to colour some of the parts yellow. The program can display the ratio of yellow to green parts. Area ITP or tiles or squared paper Rows of coloured tiles or squares can be displayed in a given ratio to create a sequence of equivalent ratios. 3 orange:2 pink 6 orange:4 pink 9 orange:6 pink... Number lines or counting stick A sequence of equivalent ratios is produced when a given ratio is scaled up by a factor of one, then two, then three... This can be represented on a number line or counting stick. Encourage children to describe the patterns within the sequence EPD-EN-01 Crown copyright 2009

34 2 of 2 The National Strategies Primary Overcoming barriers level 4-5 Teaching tips Ratio can describe a part to part relationship. For example: The ratio of girls to boys in a class is two to every three (represented as 2:3). Ratio can also describe the relationship between two comparable quantities/measures: The ratio of a distance on a map to the distance on the ground is 1: Provide visual images for ratios then ask children to describe the scenario using the language and notation of ratio, and vice versa: Each cone has two scoops of chocolate ice cream to every one scoop of strawberry. Ensure that children understand and can use ratios described in different ways: Using everyday language: there is one black tile to three white tiles; there is one black tile for every three white tiles. Using a colon (use everyday language first, then the colon form): The ratio of black tiles to white tiles is one to every three. The ratio of black tiles to white tiles is 1:3. The ratio of white tiles to black tiles is 3:1. Ensure that children can use and describe ratios in their simplest form, for example 1:3 is the simplest form of the relationship 3:9. Demonstrate how ratios can be simplified in a similar way to fractions: 12 yellow:8 red 3 yellow:2 red The following activity/activities available via the NRich site can be used to support use and application of mathematics associated with this Can I sequence. Pumpkin Pie Problem EPD-EN-01 Crown copyright 2009

35 1 of 3 The National Strategies Primary Overcoming barriers level 4-5 Can I work out the whole, having been given the fraction? Teaching guidance Key vocabulary fraction, numerator, denominator, unit fraction, whole, equivalent Models and images and resources Fractions of a whole set Use objects to model fractions of a whole set. Show two groups of cubes and explain that this shows 2 / 5 of the whole set. Encourage children to use and extend this model to explain how they can work out the number of cubes in the whole set. Counting stick or Counting stick with further options spreadsheet Attach cards onto a counting stick to create a sequence of fractions. Show a given fraction of the whole. Ask children what other fractions they can find using this piece of information. Alternatively, annotate the spreadsheet Counting stick with further options. Fraction number lines Ask children to use given information to add details to the number line. Masters of fraction number lines are provided in the Consolidation and practice section. Diagrams Model how to draw diagrams to show information given about the fraction of an amount and annotate these to find the whole. Encourage children to develop their own diagrams and explain their methods. 2 / 5 of a number is 20. What is the number? EPD-EN-01 Crown copyright 2009

36 2 of 3 The National Strategies Primary Overcoming barriers level 4-5 Pie chart spreadsheet Reveal the number of children represented by one segment. Discuss what fraction of the pie chart represents these children. Use this to work out the total number of children. Teaching tips Ensure that children are confident in reading and writing fractions, and in recognising fractions of amounts, shapes or objects. Clarify the role of the numerator and denominator of a fraction: in other words, the denominator tells you how many equal parts a whole is divided into; the numerator tells you how many of these equal parts are being considered. In mental and oral starters, count in fractional steps and place fractions on a number line. Use opportunities that arise to discuss equivalent forms of fractions: 0, 1 / 8, 2 / 8, 3 / 8, 4 / 8, 5 / 8, 6 / 8, 7 / 8, 8 / 8 or 0, 1 / 8, 1 / 4, 3 / 8, 1 / 2, 5 / 8, 3 / 4, 7 / 8, 1 Make sure that children have secure methods for finding fractions of amounts. For example, children should be able to explain how they would find 3 / 5 of 200. Encourage them to explain each step involved: 5 whole of 200 = / 5 of 200 = / 5 of 200 = Ensure that children can continue patterns in fractions of amounts: 1 / 5 of 200 = 40 2 / 5 of 200 = 80 3 / 5 of 200 = Initially, give children unit fractions of an amount and ask them to tell you the whole: 1 / 10 of a number is eight. What is the number? o Ensure that children can explain their reasoning. For example: 1 / 10 of a number is eight, so 2 / 10 of the number is 16, 3 / 10 of the number is 24, and so on; the whole number is 10 / 10, so it will be 10 8, that is, EPD-EN-01 Crown copyright 2009

37 3 of 3 The National Strategies Primary Overcoming barriers level 4-5 When they are given a non-unit fraction and asked to find the whole, encourage children to find the unit fraction first. For example: 3 / 5 of number = / 5 of number = whole number = 200 Encourage children to draw diagrams to show the fraction they are given and to annotate this to find the whole number. Use real-life contexts for problems, including those that involve approximation: o There are about people aged 16 to 65 in the UK. This is roughly 3 / 5 of the population. What is the approximate population of the UK? EPD-EN-01 Crown copyright 2009

38 1 of 2 The National Strategies Primary Overcoming barriers level 4-5 Can I solve multi-step problems involving percentages and/or fractions? Teaching guidance Key vocabulary per cent, hundredth, fraction, proper, numerator, denominator, equivalent, proportion Models and images and resources Fractions ITP This ITP can be used to compare fractions and find equivalent fractions. It can also show equivalence between fractions, decimals and percentages. Counting sticks and number lines Placing fractions, decimals and percentages on counting sticks and number lines helps children to understand that these are different ways of representing numbers. They provide a visual means of comparing and ordering fractions, decimals and percentages, and of identifying equivalents. Spider diagrams 10% = 3 50% = 15 25% = % = 6 100% = 30 1% = 30p 30% = 9 2% = 60p These can help children use known percentages or fractions of numbers to generate other percentages or fractions through scaling and combining EPD-EN-01 Crown copyright 2009

39 2 of 2 The National Strategies Primary Overcoming barriers level 4-5 Teaching tips When finding fractions of amounts, encourage children to find the unit fraction first and then to scale this up to find a non-unit fraction if required. For example, to find 3 / 5 of 40: 1 / 5 of 40 = / 5 of 40 = 24 Encourage children to draw diagrams to represent situations or problems involving fractions. Model how to do this, for example: 2 / 5 of a number is 20. What is the number? When finding percentages of amounts, encourage children to work out key percentages such as 50% and 10% to help them to find the required percentage. For example, to find 15% of 40: 10% of 40 = 4 halving gives 5% of 40 = 2 adding these gives 15% of 40 = 6 Model how to record the steps in a multi-step problem so that each stage is clear. Encourage children to develop confidence by writing down every calculation they do, even when they work them out mentally or on a calculator. For example: Charlie has saved 15 towards buying a computer game. This is 3 / 5 of the cost of the game. How much does the game cost? o We know that 3 / 5 of the cost = 15 o So 1 / 5 of the cost = 15 3 = 5 o If 1 / 5 of the cost is 5, then the whole cost = 5 5 = 25 o The game costs 25 Help children to be aware that there are two main types of problems involving fractions or percentages of amounts: (a)you are asked to find a given fraction or percentage of an amount. For example Ian scores 80% in a test. There were 40 questions. How many did he get right? Whole test = 100% = 40 questions 10% = 4 questions 80% = 32 questions (b) You are told an amount and asked to work out what fraction or percentage it is of another amount. For example, I score 30 out of 50 in a test. What percentage is this? Whole test = 50 questions = 100% 5 questions = 10% 30 questions = 60% Both types can be solved by writing what you know and then using proportional reasoning as shown above EPD-EN-01 Crown copyright 2009

40 1 of 2 The National Strategies Primary Overcoming barriers level 4-5 Can I extend my written methods for multiplying whole numbers to multiplying decimals by whole numbers? Teaching guidance Key vocabulary place value, digit, column, decimal point, tenth, hundredth, thousandth, partition, integer, method, strategy Models, images and resources Place-value cards Use place-value cards to model how to partition decimal numbers before multiplying each part and then recombining to get the final product. Grid method or Multiplication grid ITP Even where children are working with a compact written method for multiplication, it is sensible when first multiplying decimals to go back to an expanded form such as the grid method, to help them focus on the value of each digit in the calculation. Spider diagrams 42 6 = = = = 7 6 = = To be successful at multiplying decimal numbers using a written method, children need to be completely secure in using known multiplication facts to derive linked decimal facts. Spider diagrams provide a visual way of recording these facts EPD-EN-01 Crown copyright 2009

41 2 of 2 The National Strategies Primary Overcoming barriers level 4-5 Teaching tips Ensure that children know the value of each digit in decimal numbers and can state them as decimals and fractions, for example, the value of the six in 3.69 is 0.6 or 6/10. Help children to secure their understanding of place value in decimals. Use resources such as base ten apparatus or money to represent wholes, tenths and hundredths to provide them with a visual image. They need to know that ten thousandths = one hundredth, ten hundredths = one tenth and ten tenths = one whole. Teach them to use this understanding to appreciate that, for example, = 24 hundredths = two tenths + four hundredths, and so is written as Give children regular practice in using known multiplication facts to derive linked decimal facts. For example, children should be able to use 6 8 = 48 to derive: = = = = 0.48, and so on. Make sure that children can explain each step of their whole-number written method for multiplication before extending this into working with decimals. They should use the value of the digit they are working with as part of their explanation, for example, by saying two tens or 20 rather than just two. Compact methods for multiplication are efficient but often do not make the value of each digit explicit. When introducing multiplication of decimals, it is sensible to take children back to an expanded form such as the grid method where the value of each digit is clear, to ensure that children understand the process. Insist that children make an estimate for the answer to every written calculation before carrying it out. This can then be used to check that the answer they get is reasonable. Give children experience of calculations involving gaps. Ask children to work out the missing number or digit:? Build on children s understanding of written methods for whole numbers. Demonstrate multiplication of a decimal number alongside its whole number equivalent: For example: EPD-EN-01 Crown copyright 2009