Link to video about division using Dienes blocks and algebra tiles https://www.youtube.com/watch?v=LHrhFFEMEXo

What is the same and what is different about these questions?

Knowing to … I can know-that something is true as a fact. I can know-how to do something. I can know-why those techniques work or why those facts are true. But these “knowings” when taught are hard to move beyond book-knowledge, knowledge-about. The knowing- that can be memorised; the knowing-how can be routine, and the knowing-why a collection of “incantations” and learned phrases. What really matters in our increasingly problem- oriented culture is knowing-to use this or that technique, this or that way of thinking, this or that approach, in a given situation as and when it arises. When solving word problems it is often it is not the calculation that children can’t do – rather they are not sure what calculation they need to do. (From “Questions and Prompts for Mathematical Thinking”, Watson and Mason, ATM, 1998).

Solving Problems Tom has 20 sweets and shares some of them with his sister. He gives his sister 5. How many does he have? The strategy of highlighting key words to discern what needs to be done to solve a word problem can become mechanistic. Such a strategy based as it is on analysing language does not always offer insight into the structure of the problem.

Now ask participants to solve this problem and discuss how the representation of the bar(s) might provide greater opportunity for success

Laura had $240. She spent 5/8 of it. How much money did she have left? Overall percent correct: Singapore: 78%, United States: 25% Why were Singapore so successful? They used a particular representation which enabled pupils to access the structure of the mathematics This is taken from a research article, which can be accessed on the web http://math.coe.uga.edu/tme/issues/v14n1/v14n1.Beckmann.pdf Reference: Beckman, Sybilla. (2004). Solving algebra and other story problems with simple diagrams: A method demonstrated in grade 4–6 texts used in Singapore. The Mathematics Educator, 14(1), 42–46. Ask participants to solve the problem any way they wanted. You might wish to illustrate how it could be solved using the bar, or you might save this till later

Introducing the bar model The bar model is introduced within the context of part/whole relationships to solving problems involving the concepts of addition and subtraction. It exposes the relationships within the structure of the mathematics, which are used to find the unknown elements and thus supports the development of algebraic thinking. Relationships within the

Addition and Subtraction

Addition: Aggregation There are 3 footballs in the red basket and 2 footballs in the blue basket. How many footballs are there altogether? This illustrates part whole relationships Concrete to abstract In Singapore there would be a progression from concrete objects, in this case footballs, to pictures of objects to spaces on the bar(s) to represents objects Refer back to the structures of addition explored in residential 1 In this model aggregation can be seen in the two sets of balls coming together and being combined in the one strip There is also the potential to sow the seeds and link to proportional reason in seeing 3 balls as a proportion of the whole The importance of making connections in mathematics!

Addition: Augmentation Peter has 3 marbles. Harry gives Peter 1 more marble. How many marbles does Peter have now? Augmentation can also be seen in the representation From concrete to pictorial to abstract where the objects are represented by spaces Concrete Abstract

Subtraction: Comparison Peter has 5 pencils and 3 erasers. How many more pencils than erasers does he have?

Moving to the abstract Peter has 5 pencils and 3 erasers. How many more pencils than erasers does he have? Suggest that children are encouraged to visualise the objects in the spaces

Generalisation Generalising the structure of the mathematics to any context “The shift from model of to model for concurs with a shift in the students thinking, from thinking about the modelled context situation, to a focus on mathematical relations. This is a major landmark in mathematical development”. (Twomey and Fosnot citing Gravemeijer 2000).

The relationship between addition and subtraction a = b + c a = c + b a – b = c a – c = b Part / whole relationships

Problems to solve Tom has a bag of 64 marbles. His friend gives him 28 more. How many does he have now? Kelsey was running a 26 mile marathon. After 18 miles she felt very tired. How many more miles did she have to run? Carly bought an apple for 17p and a banana for 26p. How much has she spent? Ali had £10. He bought a DVD for £6.70 and a CD for £2.90. How much money did he have left? Ask participants to use the bar(s) to solve the problems, allow them to record in their own way and then compare how they drew the bars For problem 2 you might discuss whether 1 or 2 bars were drawn to illustrate the concept of difference

Developing a sense of scale

Take a strip and a paperclip Explain that the strip of card is the bar

Your strip represents 10p Show me 5p Show me 2p Show me 8p Show me 7p Note the following few slides are animated so that each question appears one at a time To represent cardinal numbers (the last number in the count represents the total size of the group) slide the paper clip from zero (the left hand end of the strip) to the middle, to represent a space representing 5p as a proportion of 10p Going from 2p to 8p some might turn their strip over, showing the relationship between 2 8 and 10

Your strip represents £1 Show me 50p Show me 20p Show me 80p Show me 70p Link to previous slide

Your strip represents 1 metre Show me 50cm Show me half a metre Show me 20cm Show me 80cm Show me 70cm Link to previous slide

Your strip represents £5 Show me £3 Show me £4 Show me £3.50 Show me £3.59 What would the half way mark represent? Link to previous slide

Multiplication, Ratio and Scaling

Multiplication Peter has 4 books Harry has five times as many books as Peter. How many books has Harry? How might you represent the problem? 4 Discuss the representation, including how it represents multiple addition and scaling for multiplication The squares on the second line might be pushed together to form one bar You might also discuss that multiplication always starts at zero (i.e. we don’t start with the original 4 and add 5 more to it, although altogether there are 6 times 4 since Peter has one times, and Harry has 5 times 4 4 4 4 4

Multiplication Problems Henry ate 10 meatballs at the Christmas party. Shane ate 3 times as many meatballs as Henry. How many meatballs did they eat altogether? Helen has 9 times as many football cards as Sam. Together they have 150 cards. How many more cards does Helen have than Sam? The sum of 2 numbers is 60. One number is 9 times as big as the other. What is the bigger number? The sum of 2 numbers is 64. One number is 7 times as big as the other. What is the smaller number? Participants draw bars/blocks and calculate the answers to the problems

Sam had 5 times as many marbles as Tom. If Sam gives 26 marbles to Tom, the two friends will have exactly the same amount. How many marbles do they have altogether? This is a complex problem, made easier with the bar. Judge whether or not you wish to use it at this stage

Tim and Sally share marbles in the ratio of 2:3 If Sally has 36 marbles, how many are there altogether?

Ratio Tim and Sally share marbles in the ratio of 2:3 Additional slides, explore ratio if time Tim and Sally share marbles in the ratio of 2:3 If Sally has 36 marbles, how many are there altogether?

A herbal skin remedy uses honey and yoghurt in the ratio 3 : 4. How much honey is needed to mix with 120 g of yoghurt? A health bar sells desserts with chopped apricot and yoghurt In the ratio 2 : 5.How much chopped apricot will be mixed with 150 g of yoghurt?

Key Stage 2 SATS

GCSE higher paper 2012! Ralph posts 40 letters, some of which are first class, and some are second. He posts four times as many second class letters as first. How many of each class of letter does he post?

Ratio, Multiplication or Scaling? The farmer has 24 animals There are three times as many sheep as cows. How many Sheep? How many cows? The farmer has 42 animals There are twice as many ducks as cows and three times as many sheep as cows. How many ducks

Farmer Brown has a third of the sheep that Farmer Giles has Farmer Brown has a third of the sheep that Farmer Giles has. After 12 of farmer Giles sheep escape into Farmer Browns field they have the same amount. How many sheep do they have in total?

Developing Fluency Routine: Another and Another Tom and Sam share stickers. Keep the ratio 2:3, but change the number of stickers, produce multiple examples: What numbers are easier? Can you do the same for the ratio 3:5?

Division and Fractions

Division Mr Smith had a piece of wood that measured 36 cm. He cut it into 6 equal pieces. How long was each piece?

Division 24 cakes were shared between 6 children, how many did each receive? Sam likes to read fantasy stories. His new book is 48 pages long. Sam wants to finish the book in 4 days. How many pages should Sam read each day in order to reach his goal?

Fractions Principle: Try to keep all sections the same size. A fraction can only be identified when the whole is divided into equal parts. Problems might involve reasoning about unknowns, from known information. Known Unknown The whole A proportion(part) A proportion (part) Another proportion (part)

Find a fraction of a number: The whole is known the part is unknown Find 2/5 of 30 This activity could be a routine to develop fluency. It provides opportunities to practice multiplication and division and connect these operations to fractions

Developing Fluency 24 Find… Find 2 3 of… Other Routines? 1 3 2 3 of 36 18 1 3 2 3 of 36 33 2 3 7 9 of 81 42 3 3 3 4 of 110 54 3 8 8 8 of 71 93 7 8 3 5 of 100

Developing Fluency Find 2 3 of: 18 33 42 54 93

Developing Fluency Find: 1 3 2 3 3 3 3 8 24 Using the squared paper, draw the strip you think you need to draw

Developing Fluency Consider how the drawing of the divided rectangle might support pupils in seeing the structure of the mathematics and knowing the operations to perform. 2 3 of 36 7 9 of 81 3 4 of 110 8 8 of 71 3 5 of 100

Part / whole and part / part relationships Sam has 24 cupcakes to sell. He sold half on Monday and half of the remainder on Tuesday. How many does he have left to sell? I cycled from home to Lea Valley Park, a distance of 24 miles. By 10am I had cycled half way and at 11am I had travelled two thirds of the remaining distance. How far had I left to travel?

Reflection on Problem solving Consider the need to: Recognise relationships Keep parts the same size See and model the structure Make connections Reason from what I know

Being aware of standard misconceptions (such as acceptance only of an addition strategy) is very valuable in that it makes it possible to offer valuable comments and to respond to classroom situations with stimulating questions. Misconceptions or, possibly more correctly, conceptions that may lead to errors, can often provide the opportunity for rich learning experiences. They provide the motivation for devising activities which might bring pupils up against surprises or situations that seem to contradict their intuitions. Such questions and activities (known as probes) can help in revealing incorrect strategies and can offer the opportunity for pupils to analyse their own senses of ratio.

Children's early number sense is usually focussed on addition and difference; only later are multiplication and division introduced. So addition becomes established as the primary arithmetic operation and multiplication as secondary. Now this may seem logical because, with whole numbers, multiplication is seen as synonymous with repeated addition, but should this really be so? Certainly, when considering multiplication of fractions, this comparison is not possible; for example 2 3 × 1 6 is not repeated addition. By developing multiplication solely as repeated addition there is therefore a danger of stunting pupils' awareness, and pupils' undeveloped sense of comparison by ratio may well be a result of this. For example, consider the following situation:

The essence of ratio is multiplicative and so it is necessary to evoke in pupils a shift towards multiplicative comparison. The last two examples demonstrate how pupils may be helped to make a shift towards such a strategy. However, to see the connection between 3 cm and 7 cm as ‘ 2 1 3 times as big' requires not only an acceptance of finding a multiplicative comparison but also a willingness to accept multiplying by a fraction. There are inevitably many, varied strategies that pupils employ, the exact type depending on the context and the numbers involved in the question. However, if pupils can be encouraged to reflect upon the range of strategies that can be employed to give correct answers, the likelihood of getting trapped in one type of strategy all the time is reduced.

It seems therefore as if there are at least three shifts in attention which pupils need to make in order to appreciate the idea of comparison by a ratio for all types of problems: the addition strategy must be seen to be appropriate in only a certain number of cases, where the numbers to be compared are such that one is a whole number multiple of the other the idea of comparison using multiplication (not just successive doubling) must be seen to be appropriate the idea of multiplying by a non-integer must be seen to be sometimes necessary.

Sometimes, however, it is the raw state of the numbers rather than what the numbers represent that signals one particular comparison as opposed to another, and this can lead to errors. For example, in the context 'John is 25, Sheila is 32',the relationship 'is 7 years older than’ is evident but 'John is 16, Sheila is 32’ may evoke the relationship 'is twice as old as’ not because the quantities represent ages but because they are 16 and 32. And the example '2 tea-bags will make tea for 3 people’ may evoke the relationship 'number of people is 1 more than the number of tea-bags’, because the raw state of the numbers suggests this relationship much more strongly than '1 1 2 cups per tea-bag'. In both these cases there is a need to see through the numbers to the context within which they are set, as it is in the context that the essence of ratio is to be found.

One of the challenges in teaching ratio is to produce activities which highlight oddities and apparent contradictions. It is very useful to develop the skill of using short, puzzling, impossible, paradoxical activities in order to help pupils see the conceptions they already have about the topic. Here are some questions you might ask. Draw a rectangle. Increase each side by adding on the same amount and draw the resulting shape. Keep increasing the size of the rectangle in this way and each time draw the result. What do you notice about the shape? Consider the two numbers 5 and 10. One is twice the other. Perform the same operation on both of them so that this 'twiceness' is conserved. I am three times as old as my son. What happens to this relation 'three times as old' as the years go by? When my son's age doubles, what happens to mine? Make a list of situations in which doubling one thing doubles another.

IMAGE OF RATIO When you perceive a ratio between two quantities, say 10 sweets and 15 sweets, do you have a picture of how they are related? What is your sense or image of ratio? Imagine a problem which involves finding the ratio associated with 10 and 15. How do you solve such a problem? How much do you use your image of ratio in your solution?

What would you say to children who gave these explanations? Tom got 10 out of 15 for his test and Anne got 15 out of 20 for hers. So there wasn’t anything to choose between them because they both got 5 wrong. I got a 10% reduction on the price of my coat and I made quite a saving. A 15% reduction would have given me another £5 off. The two fractions 2 3 and 3 4 are the same. If you increase something by 10% and then decrease the answer by 10% you must end up with the original number.

Favourite tables… Cutting a line… Imagine a line, divide it into two pieces by cutting git one third of the way along. Compare the two lines you now have. What is the ratio between them? Person A – choose your favourite times table Me – Ready, lets practise it then. 3 A: 15 Me 8 A 40 What is A doing? Now lets try the other way round. You start (encourage a number in the correct times table) and divide. What am I doing? Then with B. forwards then backwards then B back to me then to A and write down. Again to B.