Tag Archives: bar models

Four add three is equal to two add something

Could Reception and Year 1 children solve this problem?

4 + 3 = 2 + □

Of course they could.  Here’s how.  First children will need to work on their understanding of 7.  Using a manipulative for 1:1 correspondence such as multi-link cubes, we can show how the whole of 7 can be made up of two parts (in the first instance, 1 and 6):

R1

It is important to model the language that will help children think clearly when manipulating the cubes: ‘One add six is equal to seven.  The parts are one and six and the whole is seven.’  It is equally important to talk about the cubes saying the whole first: ‘ Seven is equal to one add six.’  This will help to prevent the misconception developing that the equals sign means ‘the answer is next’.  Then show them how to systematically make seven with other sized parts, talking about the parts and the whole in the same way:

R2

R3

Children should also use the cubes to write calculations.  A little modelling of turning the language of ‘Three add four is equal to seven’ into 3 + 4 = 7, followed by plenty of practice, will be exactly what is needed.

Lots of quality talking, as well as using pictorial representations, will develop children’s fluency with number facts.  Showing different representations, for example Numicon, could strengthen their conceptual understanding:

R4

Some children will grasp this idea quickly, and some will need more practice to internalise the number facts and recall them more fluently.  Those quick graspers can be challenged to think more deeply about the number facts that they are working with.  We can start by returning to the multi-link cubes and looking at two facts:

R5

Here, we can model the talk required to think more deeply: ‘Three add four is equal to five add two.’  Children could repeat that task with different facts to 7 before we show them how to write that as 3 + 4 = 5 + 2.  When children have practised this and can do it reliably with manipulatives, they could draw a bar model of what is happening:

R Bar

A further challenge is to present cubes where there is an unknown:

R6

We could model how to talk about this as: ‘One add six is equal to three add something.’  To model how to work out what ‘something’ is equal to, we simply fill the gap with cubes to make the second row equal to seven, then counting the cubes to figure out what ‘something’ is equal to.  When children have practised and are becoming more fluent, the cubes could be replaced with bars, at first presented in that way but moving on to children drawing it themselves:

R Bar 2

All the while, children could be shown how this looks written down: 1 + 6 = 3 = □.  When they have seen the abstract alongside the pictorial and the concrete, we can try starting with the abstract and asking children to represent the problem with cubes or by drawing bars.

The sequence described, over time, should be enough of a scaffold for the vast majority of children to end up being able to solve such problems and in doing so, develop a deep understanding of early number.

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Enumerating possibilities of combinations of two variables

With Year 6 children expected to work on the objective ‘enumerate possibilities of combinations of two variables’, we should be clear on the difference between the underlying concept and the algebraic representation of it.  

2g + w = 10

For questions such as this, children should first have a secure understanding of the part, part, whole model.  We can show that 2 lots of something add one lot of something else is equal to 10 by using a concrete manipulative such as Numicon.  First, children represent the whole, in this case 10. Then they can speculate on the two equal parts (2g), trying out g=1 before finding the Numicon piece that fills the gap and therefore is equal to w:

  
Having found one solution, they can continue to work systematically to find alternative solutions.  Trying g = 2 is logical:

  
Lining up solutions beneath the whole reinforces the idea that the expressions are equivalent.  Children can continue to work systematically:

  
This also provides a scaffold for questions of greater depth, such as ‘What is the greatest number that g can represent?  Explain…’  
Subtraction?  Not a problem, although in this case, children must know that for subtraction, you always do so from the whole.

10 = 3g – w

In this question, the whole is 3g and the parts are 10 and w:

  
What is not clear from this model is the trial and error that went into it.  Children may well try 3 ones and quickly realise that it is already less than 10, so subtracting from it will not give a valid solution.  There is lots of scope here for discussion about the smallest number that g could represent.  

The use of Numicon leads nicely into children representing problems as bar models.  Here are the two examples used so far:

   
    
 

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On teaching additive reasoning

The question was on the screen:

AR13

One year 6 child said: ‘The empty box is in the middle so you do the inverse.  You have to add the numbers together’.

This got me thinking about how children build on their early concepts of number to be able deal with problems like this, which I’ll call ‘empty box problems’.

AR1

The underlying pattern of additive reasoning is the relationships between the parts and the whole.   Getting children to think and talk about the whole and parts using concrete manipulatives early on should lay the foundations for them to internalise this underlying pattern.  Every time children think and talk about number bonds, they can be practising identifying the whole, breaking it into parts and then recombining to make the whole once more.

AR2

Alongside talking about the whole and parts, children should begin to generate worded statements whilst manipulating cubes or Numicon, for example.  At this point it is important to experiment with rearranging the words in the statement.  They should get to know that ‘four add two is equal to six’ and ‘six is equal to four add two’ are statements that are saying the same thing.  Some discussion around what is the same and what is different about these two statements would be worthwhile.

When children are then shown how this looks abstractly with numerals and the equals sign, this would hopefully go some way towards avoiding the misconception that the equals sign means that ‘the answer is next’.

AR4

In the examples used so far, the whole and each of the parts have been ‘known’.  Using the same manipulatives and language patterns, children can be introduced to unknowns.  It seems sensible to begin with giving children the parts and using the word ‘something’ to show that the whole is unknown, i.e., four add two is equal to something.  Some modelling alongside a clear explanation followed by plenty of practice should see children get used to the language patterns needed to think about the concept with clarity.  The next step is to show children the whole and one of the parts, using the word ‘something’ to replace the unknown part.  All of this talk and manipulation of objects is intended to support children to develop a concept of additive reasoning where they do not have the misconception that ‘inverse’ means ‘do the opposite’.

AR5

More sophisticated additive reasoning is the understanding of the inverse relationship between addition and subtraction.  Children need to fully understand that two or more parts can be equal to the whole.  From this, they need to internalise the underlying patterns: that Part + Part = Whole and that Whole – Part = Part.  From this, they should be able to work out the full range of calculations that represent one bar model.  Again, it is important to vary the placement of the = sign.

AR6

One more way to get children to think about the whole and the parts is to use bar models for calculation practice rather than simply writing a calculation for children to work out.  When done like this, children have to decide what calculation to do to work out the unknown.  Children often exhibit misconceptions such as ‘when you subtract, the biggest number goes first’.  These can be addressed using the underlying patterns; adding parts together makes the whole and, when you subtract, you always subtract from the whole.  When unknowns are introduced, they can be substituted into these basic patterns:

 

Part + Something = Whole           Part + □ = Whole              35 + □ = 72

Something + Part = Whole           □ + Part = Whole              □ + 35 = 72

Whole – Something = Part           Whole – □ = Part               72 – □ = 35

Something – Part = Part                □ – Part = Part                   □ – 35 = 37

AR11

Knowing these patterns will help children to able to analyse problem types in order to decide on the calculation needed.  An additive reasoning bar model with one unknown generates both an addition statement and a subtraction statement.  Showing children empty box problems pictorially, they can talk through the calculations that can be read from the bar model, using the word ‘something’ to represent the unknown.  The next step is to show children abstract empty box problems and get them to map it onto a blank bar model.  They should be drawing on their knowledge that the whole is equal to the sum of the parts and that when you subtract, you always start with the whole.  Eventually, the hope is that the language alone should suffice to work out how to solve empty box problems, with children no longer needing the bars.

Which brings us back to that year 6 child.  Of course, children will develop misconceptions as they make sense of what is shown and explained to them.  By expecting them to think and talk about additive reasoning in the ways described above, it should go some way to building sound conceptual understanding.

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Multiplying fractions and integers

This is an account of a particularly successful topic from Y5 on multiplying fractions and integers. It’s fast becoming one of my favourite concepts to teach.
I showed multiple representations of a calculation, showing how to make it with cubes:

  
In this example, it was important to make four lots of two thirds first before rearranging the coloured cubes: later when they draw it on a number line, they’ll need to be thinking of four jumps of two thirds. The language of number of parts, size of each part and whole was supported by the bar model and came in handy for the empty box problems later.
Children needed to make other examples of calculations, talking about it with a partner:

  
Most children were fine with this and ready to move on to pictorial representation on a number line. Some needed a bit more consolidation with how the cubes represent a calculation. This is of a set of scaffolded questions where children made the calculation with cubes, then drew it:

  
For those ready for a number line, I explained a worked example before modelling how to do one from scratch:

  
The explanation had to be clear on how to construct the number line accurately, which involved making decisions about how many squares to leave between each whole number (using the denominator of the ‘size of each part’), and how many squares each jump needed to be (using the numerator of the ‘size of each part’). The final decision to make was how many jumps to draw, using the number of equal parts. The whole, and the answer to this particular calculation, was where the last jump ended up in the number line.

When children practised doing this, they chose from using a scaffold of minimally different questions or tackling varied questions:
  
Some children needed a few lessons to keep practising this and each time we returned to it, I started by showing multiple representations of the same calculation:

   

 Some children grasped this concept quickly and needed to work on deeper tasks. I made some subtle changes to the language I used to describe what was happening (bottom right corner of the above image) and used the same bar model to introduce inverse operations with aim of learning to solve empty box multiplication of fractions:

  
Bar models helped to explain what was known and what was unknown, using the language of whole, number of parts and size of each part. I showed how it looked on a number line too:

  
Some managed to get on to dividing fractions and whole numbers, using bar models (which they were familiar with for integer multiplication and division) and number lines to show what was happening:

   
 All the while, more children began to move from basic multiplication of fractions to some deeper thinking about multiplicative reasoning and fractions.

A few weeks later, children worked out the area of rectangles involving fractions and had internalised much of the language to be able to represent questions like this:

  

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Throwing out that old RUCSAC

Experts, say Hattie and Yates, see and represent problems on a deeper level, whereas novices focus on superficial aspects. With this in mind, take a closer look at one of the most prevalent strategies for solving problems:

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In some variants of the acronym, the U even stands for ‘understand the problem’! If children practise solving problems in this way, they can only get better at analysing the superficial structures of the problems. Some of the advice leads children to develop near useless strategies when problems get trickier. At its worst, I’ve seen (and probably set up myself) lessons where children are told that they are doing subtraction word problems. Every problem has the same sort of language pattern and children could feasibly get by simply by picking the numbers out and subtracting one from the other.

Take the ‘underline’ and ‘choose which calculation’ advice. Underlining key words may well be useful by there is often ambiguity in the wording used. ‘How many more than X is Y?’ is different to ‘What is X more than Y?’ Although the wording is similar, they require different calculations.

If children are to understand the deeper structures, then they need to know the deeper structures. And of course they’ll need to practise analysing problems to classify those structures.

In the example of multiplicative reasoning we can start by suggesting that there’ll be three basic structures.

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By modelling the thinking behind this and relating the wording used in the problem to the bar models, children can be shown the three deep structures. Certainly, they’ll need to practise over time analysing problems like this to become skilled at it. In the first instance, a little guided practice is on order: sort these problems by underlying structure:

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Note that a deliberate difficulty built into this practice is that the problems are all similarly worded. There’s a decision to be made about the work that children will do to practise further. Do they solely work on sorting by deep structure or do they solve the problems too? In the example below, some children worked on just the first column. Novices have comparatively weaker short term memories than experts so may only be able to deal with the sorting. It makes some sense to provide a scaffold to help them remember important information: the deeper structure of the problem.

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Here is a similar approach when showing children the structures of problems involving ratio:

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Once children are aware of the possible structures, the teacher can show them how to represent the problem. Bar models are great here. Plus, worked or partially worked examples are powerful in showing children how to grapple with a problem:

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Children can then have a go on their own. Here are some example questions for them to sketch out with bars and solve:

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And here is an example of what children who already understand the basics would be up to – much trickier problems with more layers that might not entirely fit the basic structures described to most children:

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This work on deeper structures could start from an early age. Too often, problem solving like this is bolted on to work on calculation with the assumption that if children know which operation a word or phrase means, they can solve problems. This however, is only superficial analysis. Children need knowledge of the structures of problems, just like we teach them the structures of stories. Perhaps we should approach problem solving like this more along the lines of teaching reading comprehension…

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