How to teach the bar model method to ace arithmetic and word problems in KS1 & KS2 Maths
How to use bar modelling or the bar model pictorial method for word problems in KS1 and KS2 Maths as well as addition, subtraction, multiplication, division and national assessments Key Stage 2 SATs reasoning.
If you want pupils to do well in word problems, particularly at end of Key Stage tests KS1 SATs and KS2 SATs, you have to adopt the bar model. This post will show you how to do that with a whole school approach. We will look at the four operators and a progression of bar model representations that can be applied across school in KS1 and KS2. Then we will look at more complex examples that can support pupils in end of Key Stage 2 tests, including how to apply bar models to other concepts such as fractions and equations.
What are bar models in Maths?
Bar models are pictorial representations of problems or concepts that can be used for any of the operations: addition, subtraction, multiplication and division. In word problems, they hold the huge benefit of helping children decide which operations to use or visualise problems. Bar models will not, however, do the calculations for the pupil. The bar modelling approach to Maths is much used in Singapore and Asian Maths textbooks and lessons.
Addition word problems with bar models
Pupils in Reception and Year 1 will routinely come across calculations such as 4+3. Often, these calculations will be presented as word problems: Aliya has 4 oranges. Alfie has 3 oranges. How many oranges are there altogether? With addition, subtraction and multiplication, to help children fully understand later stages of bar modelling, it is crucial they begin with concrete representations. There are 2 models that can be used to represent addition:
Once they are used to the format and able to represent problems in this way themselves (assigning ‘labels’ verbally), the next stage is to replace the ‘real’ objects with objects that represent what is being discussed (in this case, we replace the ‘real’ oranges with button counters):
The next stage is to move away from the concrete to the pictorial. As with all the stages, when pupils are ready for the next stage is a judgement call that is best decided upon within your school. However, a general rule of thumb would be that towards the end of Year 1 or start of Year 2, pupils should be able to understand and represent simple addition (and subtraction) word problems pictorially and assign written labels.
The penultimate stage is to represent each object as part of a bar, in preparation for the final stage:
The final stage stops the 1:1 representation. Each quantity is represented approximately as a rectangular bar:
As mentioned before, it is a judgement call for your school to make, but if you want pupils to use the bar model to support them in end of Key Stage 1 tests, they are going to need to have had a fair amount of experience of this final stage.
How does subtraction work with bar models in a word problem?
The same concrete to pictorial stages can be applied to subtraction. However, whereas with addition it is really down to the pupil’s preference as to which of the 2 bar representations to use, with subtraction the teacher can nudge to pupils to one or other. The reason? One represents a ‘part-part-whole’ model, the other a ‘find the difference’ model. Each will be more suited to different word problems and different pupils. Let’s examine those at the final stage of bar modelling:
Austin has 18 lego bricks. He used 15 pieces to build a small car. How many pieces does he have left?
Calculation: 18 – 15 =
Find the difference
Austin has 18 lego bricks. Lionel has 3 lego bricks. How many more lego bricks does Austin have than Lionel?
Calculation: 18 – 3 =
Multiplication word problem with bar models
Bar models of multiplication start with the same ‘real’ and ‘representative counters’ stages as addition and subtraction. Then moves to its final stage, drawing rectangular bars to represent each group:
Each box contains 5 cookies. Lionel buys 4 boxes. How many cookies does Lionel have?
Using bar models and division together in a word problem
Due to the complexity of division, it is recommended to remain grouping and sharing until the final stage of bar modelling is understood. Then word problems such as the 2 below can be introduced:
Grace has 27 lollies. She wants to share them into 9 party bags for her friends. How many lollies will go into each party bag?
Grace has 27 lollies for her party friends. She wants each friend to have 3 lollies. How many friends can she invite to her party?
What are the next steps in bar modelling?
Now that we have established a structure across school that allows for children to use bar models for key stage 1 tests, giving them time to develop a deeper understanding of complex problems during key stage 2, we need to know how to teach pupils to use them. The key question at any stage, at any age is what do we know? By training pupils to ask this when presented with word problems themselves, they quickly become independent at drawing bar models.
For example, in the problem: Egg boxes can hold 6 eggs. We need to fill 7 boxes. How many eggs will we need?
We know that there will be 7 egg boxes, so we know we can draw 7 rectangular bars. We know that each box holds 6 eggs, so we can write ‘6 eggs’ or ‘6’ in each of those 7 rectangular bar. We know we need to find the amount of eggs we have altogether. We can see we will need to use repeated addition or multiplication to solve the problem.
Bar models in KS2 SATs
Let’s ramp up the difficulty a little. In the sample Key Stage 2 tests, pupils are asked: A bag of 5 lemons costs £1. A bag of 4 oranges costs £1.80. How much more does one orange cost than one lemon? Pupils could represent this problem in the below bar model, simply by asking and answering ‘what do we know?’
From here it should be straightforward for the pupils to ‘see’ or visualise their next step. Namely, dividing £1.80 by 4 and £1 by 5. Some pupils will not need the bar model to represent the next stage, but if they do, they would calculate and then allocate the cost onto the model:
Then those pupils that needed this stage, should be able to see that to answer the question, they need to calculate 45p – 20p. With the answer of 25p.
The importance of bar models in word problems with fractions
Here’s another example from the sample key stage 2 tests involving fractions. On Saturday Lara read two fifths of her book. On Sunday, she read the other 90 pages to finish the book. How many pages are there in Lara’s book? If we create our bar model for what we know:
Pupils will then see that they can divide 90 by 3:
As fractions are ‘equal parts’ – a concept they should be familiar with from key stage 1 – they know that the other 2 fifths (Saturday’s reading) will be 30 pages each:
Then they can calculate 30 x 5 = 150
Bar models to break down equations
There are lots of other areas bar models can assist pupil’s understanding such as ratio, percentages and equations. In this final example, we look at how an equation can be demystified:
2a + 7 = a + 11
Let’s draw what we know in a comparative model, as we know both sides of the equation will equal the same total:
The bars showing 7 and 11 could have been a lot smaller or larger as we don’t know their relative value to ‘a’ at this stage. However, it is crucial that the ‘a’ appearing first in both bars is understood to be equal (even if it is only approximately equal when drawn freehand in the bar). This allows the pupil to ‘see’ that to work out the second ‘a’ in the top bar, they can calculate 11-7.
So if that ‘a’ is 4, then both the other ‘a’s will also be 4. So each side of the equation will total 15. The below model shows all sections completed. This is not necessary for the pupils to do, the representation is merely useful until they can see the steps necessary to calculate whatever they are faced with:
Modelling bar models at primary school
Now that you have seen how bar models can help pupils solve questions in end of key stage tests and you have a structure that can be put in place across school to enable pupils to do so, what are you waiting for? Get a staff meeting booked, get your staff calculating difficult problems using bar models to support. They’ll be sold instantly. If you don’t believe me, use this problem to grab their attention:
Hussain wins first prize for his spectacular cake version of the Eiffel Tower. He generously gives three fifths of his winnings to his children and spends a third of what he had left. He has £80 left. How much money did he win?
With or without bar models? Which is easier? My guess is your staff will be hooked!