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):

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:

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:

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:

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:

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

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:

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.

This is almost exactly what my Y1 class will be doing next week. I want to introduce the = symbol as a representation of ‘is the same as’ or ‘is equal to’ early on so they can interpret empty box problems later on. Was going to use balance scales to reinforce the idea of equality too.

The Numicon scales are good. I’d be interested to hear if you’re doing anything different…

I’ve not really bought into numicon although I see the value in how you’ve used them. I’m involved in the Singapore textbooks project and am combining that approach with other good ideas I’ve gleaned from local training and MaST training. Trying to bring it all together to get it right for my class of Y1s!

Yes, I agree, Helen – I think balances are great for getting to what the equals sign is really about. I use them with my Y4 class for a similar reason:

http://year4atist.blogspot.fr/2014/05/number-balances.html

I’ve just ordered a number balance. Used the nrich one before but the real thing will be good. I have used cuisenaire but need to get them out more!

It’s a great sequence of experiences, Nick, and you’ve described it really clearly. I use Cuisenaire rods a lot and would add them into the same kind of sequence.

This is a really helpful blog post. I am finding it quite tricky to introduce bar model problems into my lessons. Thanks for this.

Do you find numicon better than the cubes? I’m just wondering.

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