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Bar Model - Multiplication, Division, Fractions, and Ratio


Created on 24 April 2014 by ncetm_administrator
Updated on 14 May 2014 by ncetm_administrator

Multiplication, Division, Fractions, and Ratio

All of these concepts involve proportional and multiplicative relationships and the bar model is particularly valuable for representing these types of problems and for making the connections between these concepts visible and accessible.

Multiplication

Notice how each section of the bars in the problem below has a value of 4 and not 1. This many-to-one correspondence, or unitising is important and occurs early, for example in the context of money, where one coin has a value of 2p for example. It is also a useful principle in the modelling of ratio problems.

Peter has 4 books
Harry has five times as many books as Peter.
How many books has Harry?

five times four

4 × 5 = 20
Harry has 20 books

Division

When using the bar model for division it is the image of sharing rather than grouping which is highlighted in this representation.

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

36 divided by 6

36 ÷ 6 = 6
Each piece is 6 cm

Problems Involving Proportion

When modelling problems involving proportion it is useful to divide the bar into equal parts so that the proportional relationship and multiplicative structure are exposed.

Once the value of one part is labelled, the other parts can be identified as they are the same, for example in this KS2 SATS Question.

24

In a class, 18 of the children are girls.

A quarter of the children in the class are boys.

Altogether, how many children are there in the class?

show your working answer box

 

The Class

The bar represents the whole class

Boys

 

 

 

Folding the bar into quarters allows us to represent the boys as a proportion of the whole class

Boys

Girls

Girls

Girls

The rest of the class must be girls

Boys

Girls

Girls

Girls

There are 18 girls so each of the three girl sections must equal 6 and so the boy section must also be 6. 6 × 4 = 24, there are 24 children in the class.

Fractions

The bar model is valuable for all sorts of problems involving fractions. An initial step would be for pupils to appreciate the bar as a whole divided into equal pieces. The number of equal pieces that the bar is divided into is defined by the denominator. To represent thirds, I divide the bar into three equal pieces, to represent fifths I divide the bar into five equal pieces. A regular routine where pupils are required to find a fraction of a number by drawing and dividing a bar, using squared paper would be a valuable activity to embed both the procedure and the concept and develop fluency.

Find 15 of 30

bar model showing a fifth of 30

The same image can be used to find 25 or 35 of 30 etc.

Finding the original cost of an item that has been reduced in a sale is one that pupils find particularly tricky. The ease at which such problems can be solved is demonstrated below:

A computer game is £24 in the sale. This is one quarter off its original price. How much did it cost before the sale?

bar model showing £24 with a quarter added

The bar represents the original cost. It is divided into quarters to show the reduced cost of £24.

£24 ÷ 3 = £8, giving the value of three sections of the bar. The final section of the bar must also be £8, since it represents the same proportion as each of the other sections.

£8 × 4 = £32

The original cost of the computer game is £32

 

Percentages

Problems involving percentages are solved in a similar way to those involving fractions. The key is to divide the whole into equal parts.

A computer game is reduced in a sale by 30%. Its reduced price is £77. How much was the original price?

bar model showing £77 with 30% added

Dividing the bar into ten equal pieces allows us to represent 30% and keep the other pieces the same size.

£77 ÷ 7 = £11

The original cost (the whole bar) is £11 × 10 = £110

 

Problems involving Ratio

The ratio problem outlined below is represented with double sided counters. These can be used as an alternative representation to drawing a bar. The advantage of this is that they can be moved around to represent a change in relationship.

Sam and Tom have football stickers in the ratio of 2 to 3. Altogether they have 25 stickers. If Sam gives half of his stickers to Tom, how many will Tom have?

Sam 2 red counters

Tom3 yellow counters

25

25 counters

Step 1

Represent the ratio

Step 2

Recognise that if together the counters have a value of 25 then one has a value of 5

Step 3

Give half of Sam's stickers to Tom

 

Sam 1 red counter

Tom3 yellow counters and 1 red counter

25

Step 4

Multiply 5 by 4 to give the total value of Tom's stickers

Answer:

5 × 4 = 20

25 counters

Notice how the many-to-one correspondence, as discussed above, allows this problem to be modelled efficiently. Children may at first use a one-to-one correspondence (i.e. each counter having a value of 1) and represent the problem using 25 counters, setting them out in a two to three ratio as illustrated below.

25 counters set out 2 to 3 ratio in an array

Children can then be supported in transferring their understanding to many-to-one correspondence as illustrated above (i.e. that every 5 counters in each column can be replaced with one counter worth 5). The flexibility of appreciating that the value of one counter can change depending on the context and the total quantity helps to develop the pupils’ algebraic reasoning.

Notice how the strategy explored above supports this KS2 SATS question

tulip bulb illustrationA gardener plants tulip bulbs in a flower bed.

She plants 3 red bulbs for every 4 white bulbs.

She plants 60 red bulbs.

How many white bulbs does she plant?

3 red counters, each worth 20
60
4 white counters, each worth 20
Answer =
80 white bulbs

 


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Comments

 


03 December 2016 11:33
very helpful indeed, planning lesson on word problems using bar method
By LizCairo
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13 April 2016 12:26
NCETM stands for Nice, Caring, Educational, Terrific and Mega! Well done guys. You've nailed it!
By bradbeer
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24 January 2015 15:05
Works well with cuisenaire when demonstrating addition and subtraction. My Y6 clas are finding fractions of quantities, ratio and percentage change much easier now thanks to the bar.
By DGittins
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13 December 2014 11:16
This bar method is accessible to all children; simple yet effective.
By id85
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05 August 2014 11:46
Whoops! Did I forget to mention that the two-number-line tool is also handy for division (with due respect and refereence to the bar method described above): if one wants to see what 24 divided into three is, one can slide 3 along until it matches 24 and then look to see what the unit must be worth. It's all about zero and the single unit! :-) But +where+ is that tool? and can someone +please+ make sure it becomes available as a free android app so pupils can download it onto their phones/tablets/etc (as well as us being able to use it on whiteboards)!
By BW_2012
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05 August 2014 11:40
This is jolly lovely; but I'd like to see two number lines: one above the other; one flexible; one not. Why? So (using tablet-like pinch/expand controls on my interactive whiteboard) I can define what I want a unit to be. If I'm multiplying by 29, I want my unit to be 29 and hence to "squish" the flexible numberline accordingly so I can read it off against the the top number line when I scroll along both. If I'm multiplying by 1.357, I want my unit to be 1.357 so I can do pretty much the same thing. Overall, that approach exphasises the commonality of zero, the importance of the unit +and+ is recogniseable from primary school upward. One might be able to request algebraic terms on the flexible number line (e.g. -2a, -1.5a, -a, -0.5a, 0, 0.5a, etc) or fractions into improper/mixed fractions, or percentages, or...
By BW_2012
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25 June 2014 21:37
The website Singapore Teacher has many excellent examples of solving problems using bar modelling. I love this method for ratio quesitons.
By nickybrady
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13 June 2014 12:59
simple for children to visualise and very helpful, never thought of using to show commutative nature of additiona and subtraction1
By tridler
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