Bar Models A Problem Solving Tool Thursday 30th March 2017 Bar Models A Problem Solving Tool

Objectives To develop an understanding of bar modelling as an approach to problem solving. To develop an understanding of how bar models can be used to develop conceptual understanding. Raise awareness of resources to support ‘bar-modelling’

Making Connections Haylock and Cockburn (2008) Symbols Language Concrete Experiences Pictures Language 3 + 2 = 5 plus sum total Bar modelling supports the Connective model When introducing the bar model to children connections need to be made between the different representations of mathematics: symbols, imagery, concrete experiences and contexts, language. Teachers often go through these stages when teaching calculation. When we move onto word problems the same approach needs to be applied

3 1.Counting objects 2.Lining up objects in rows 3.Drawing bar around pictures 3 4.Recording the number of bars Children start by counting familiar things using objects... 5. Drawing a bar to represent an amount & labelling with a number

Step-by-Step Approach – moving from concrete to picture Sam bakes 20 cookies. If he gives away 8 cookies, how many would he have left?

Using Cuisenaire rods to support bar modelling https://www.youtube.com/watch?v=sbsPK3LXVLc

A little bit of history.. Cuisenaire rods were very popular in the latter half of the last century but seem to have fallen from favour with the increased use of ICT-based resources. Once you start to use them with the children you will quickly realise what an invaluable resource they are. The rods are named after their inventor, Georges Cuisenaire (1891-1976), a Belgian primary school teacher, who published a book on their use in 1952 called Les nombres en couleurs. The use of rods for both mathematics and language teaching was developed and popularised by Caleb Gattegno in many countries around the world. Use of colour rods is also advocated by Professor Mahesh Sharma and Ronit Bird for supporting children with dyslexia and dyscalculia.

Supporting early number work Early counting will start with developing an understanding of the relationship between the numbers and the colours. Your turn! 1.Make a staircase with the rods.

Your turn! 2.Find as many ways to make the same value as the orange rod. Calculation Use this technique to support making number bonds to 10. 8 + 2 6 + 4

Subtraction/Addition stories How many ways are there to make ten (orange) using the red and white rods? Subtraction/Addition stories Take an orange and purple rod and create a subtraction/addition story.

Finding the difference Colour rods are a particularly useful visual model to use to support the ‘difference’ conceptual structure of subtraction. Multiplication Add rods to make a rectangles e.g. 3 + 3 + 3 + 3 + 3 = 5 lots of 3 Multiplication calculations can be formed. 3 x 5 or 5 lots of 3 are equal to 5 x 3 or 3 lots of 5. The length of the rod is the constant.

Moving from rods to bar models A Bar Model is… not a tool for performing a calculation not a quick fix solution to problem solving most effective when used progressively over time effective as part of ‘concrete, pictorial, abstract’ Final reminder to staff

Progression to bar modelling Represent problem with concrete objects Line up objects in a row Draw a picture of objects with a box around each picture Draw a box to represent an amount

Arithmetical Proficiency Children who do not understand what they are doing often make errors when choosing and carrying out procedures, cannot reconstruct steps, cannot self-monitor their work, and cannot check for reasonableness. The ability to solve problems requires: Fluency and flexibility in applying the mathematics The ability to reason mathematically Conceptual understanding which allows the learner to see and express the structure of the mathematics Making connections between relationships Advisory Committee on Mathematics Education – reporting on arithmetical knowledge, skills and capabilities required by end of primary school 5

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 pupils in Singapore so successful? They used a particular representation which enabled pupils to access the structure of the mathematics

Bar Models Since the early 1980s, a distinguishing characteristic of the maths taught in Singapore is the use of “model drawing”. Students are taught to use rectangular “bars” to represent the relationship between known and unknown numerical quantities and to solve problems related to these quantities. The bar model is used in Singapore and other countries, such as Japan and the USA, to support children in problem solving. It is not a method for solving problems, but a way of revealing the mathematical structure within a problem and gaining insight and clarity as to how to solve it. It supports the transformation of real life problems into a mathematical form and can bridge the gap between concrete mathematical experiences and abstract representations. It should be preceded by and used in conjunction with a variety of representations, both concrete and pictorial, all of which contribute to children’s developing number sense. It can be used to represent problems involving the four operations, ratio and proportion. It is also useful for representing unknowns in a problem and as such can be a pre-cursor to more symbolic algebra .

Why rectangles? Rectangles are easy to draw divide represent larger numbers display proportional relationships Draw out from staff: Rectangles are easy to draw, divide, represent larger numbers, and display proportional relationships The rectangular bars are drawn roughly proportionally to one another, however there should not be an over-emphasis on drawing the bars to the exact proportion but estimate to the best of the learners’ ability.

A Bar Model Helps the child to: visualise the problem involved identify the unknown quantity and know what operation to perform see relationships/structures between and among variables in the problem view all problems from an algebraic perspective Bar modeling is a systematic method of representing word problems and number relationships. It needs to be explicitly taught beginning in KS1 as seen in the steps above. It can be used into secondary and above to represent the relationship between known and unknown numerical quantities and to solve problems related to these quantities.

Addition and Subtraction

Model 1

Model 2

Addition There are 3 footballs in the red basket and 2 footballs in the blue basket. How many footballs are there altogether? Once pupils can generalise the image, the footballs are replaced by the bars. Give staff one to do: There are 334 children at Springfield School and 75 at Holy Trinity Nursery. How many children are there altogether? 3 2 ?

Subtraction - reduction I had 10 pencils and I gave 6 away, how many do I have now? X X X X X X 10 To fully support pupils’ understanding of subtraction problems consider different models for reduction and difference until the outcome of the two models can be generalised (i.e. pupils realise the that the result of subtracting a smaller number from a larger one is the same as finding a difference) 6 ?

Subtraction: Comparison Model Peter has 5 pencils and 3 erasers. How many more pencils than erasers does he have? ? 5 3 Give staff the following to try: Gemma collected 293 badges but she gave 45 of them to her friend, Rebecca. How many badges did she have left? Gemma had 293 badges. Rebecca had 45. How many fewer did Rebecca have? Gemma had a collection of USA and UK badges. She had 293 altogether. 45 were USA. How many were UK? Have they used the same bar model for each? Ask them to discuss their bar models. Remind staff subtraction problems come in many forms.

Your turn! On paper, use bar models to solve the following: Gemma collected 293 badges but she gave 45 of them to her friend, Rebecca. How many badges did she have left? Gemma had 293 badges. Rebecca had 45. How many fewer did Rebecca have?

Bar model for empty number box addition and subtraction 15 + = 24 - 17 = 20 24 ? 17 20 ? 15

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

Multiplication and Division

Multiplication – repeated addition Mrs Morton has five 5p coins in her purse. How much money does she have? ? 5 ? This multiplication bar model is an extension of the addition bar model – the relationship between the two operations is clear. Cuisinere rods with 1cm number tracks/ multilink with 2cm number track are the concrete representation and can be used to support the actual calculation

Multiplication - scaling Peter has 4 books Harry has five times as many books as Peter. How many books has Harry? 4 P H 4 Discuss how the representation represents scaling for multiplication The squares on the second line might be pushed together to form one bar The model gives a clear picture of Harry’s collection of books being five times bigger/ Peter’s collection a fifth of the size and the fact that when their collections are compared there are 6 parts altogether ? P H 4 ? or…

Problems to solve A bookcase in the library has 5 shelves with 46 books on each shelf. How many books are there in the bookcase altogether? Amy has 20p in her purse. Her friend Lisa has 8 times as much money. How much do they have altogether? 8 children each download 59 songs to play on their iPod. How many songs do they have altogether? Emma buys seven markers for 30p each. How much change does she get from £3.00? Helen has 9 times as many football cards as Sam. Together they have 150 cards. How many more cards does Helen have than Sam? Give staff time to draw and discuss the bar models

Division - grouping How many 5p stickers can Alex buy with his 55p pocket money? Discuss with staff how the grouping structure of subtraction is recorded and generalised into the bar model – the unknown part to be calculated is the number of groups that can be made 55p 5p ?

Division - sharing Farmer Smith had a bag of 18 pony carrots. He shared them equally between 6 ponies. How many did each pony get to eat? 18 ? Building on equal sharing into two groups (half), four groups (quarters), three groups (thirds) children begin to find other ways of splitting groups. Folding paper strips with the images/ representations drawn on supports the idea of equal sharing and the size of each share.

Problems Mr Smith had a piece of wood that measured 36 cm. He cut it into 6 equal pieces. How long was each piece? Aiden has seven marbles and Harvey has fifteen. They decide to share them equally between them. How many do they get each? At the dressmakers, Debbie buys buttons weighing 3 grams each. If she has 81 grams of buttons, how many buttons does she buy? 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?

Ratio Problems Share £48 in the ratio 3 : 5 £48 ÷ 8 = £6 £6 x 3 = £18 £6 x 5 = £30 The ratio of boys to girls in Y5 is 3 : 2 There are twelve girls. How many children in this class? ? 12 ÷ 2 = 6 boys girls 12

A Bar Model is… not a tool for performing a calculation not a quick fix solution to problem solving most effective when used progressively over time Final reminder to staff

Your turn!

Answers

Resources for home

Resources for home For the fridge at home!

Progression Y1-6

Resources to support Thinking Blocks http://www.mathplayground.com/thinkingblocks.html Cuisenaire environment https://nrich.maths.org/4348 Nrich activities http://nrich.maths.org/thismonth/all/2005/10/?choice=3&showfulltier=yes NCETM https://www.ncetm.org.uk/resources/44565 Cuisenaire Rods the Way of Zen (video) https://www.youtube.com/watch?v=sbsPK3LXVLc

Resources to support Virtual Manipulatives Library http://nlvm.usu.edu/en/nav/vlibrary.html Math Learning Center APPs http://catalog.mathlearningcenter.org/apps Braining Camp APPs for i-pad https://www.brainingcamp.com/product/mobile.html The School Run http://www.theschoolrun.com/what-is-the-bar-model-method

‘Thinking Blocks’ app/website (only available on iPad/computer)

MyMaths & Bar modelling