By Shannen Doherty
Here, Shannen Doherty explains her teaching methods and thought processes as she teaches students the topic of Fractions in Maths.
I love nothing more than planning and teaching fractions, so I am going to be taking you through my thought process and all the internal debates I have when planning to teach the objective ‘add and subtract fractions with the same denominator’. On the surface this objective may seem quite simple, but there is an underlying depth that must be exposed for our pupils to have a real understanding of operating with fractions.
First and foremost, this must be part of a wider process in which you have already mapped out how your fractions unit will be sequenced. You need to know when you’re teaching it and why it slots into your curriculum at that moment. This isn’t a one lesson job. I’d plan a sequence of lessons where I make subtle tweaks as we go along to ensure the children have a rich experience and deep understanding.
Some of the internal debates I have when planning:
- Do I introduce the concept with a problem and pull out the maths or start with the maths and apply it to a problem?
- Do I teach the rule first or do I generate this with the children?
- Do I show the children the obvious misconception of adding the numerators and denominators or do I wait for this to happen and then draw their attention to it?
- Do I start with bar models or circles?
- Do I show two fractions being added on one bar/circle or two bars/circles?
- Do I use one colour throughout or one colour per fraction?
- Do I start with bars/circles pre-filled in or do I expect the children to make the representation and do the calculation?
- When do I add in a third addend?
- Do I teach adding and subtracting simultaneously or one after the other?
So where to start?
Where you stand on this may differ from concept to concept, but it’s crucial that you consider what you’re doing and why. When I teach adding and subtracting fractions with the same denominator, I prefer to kick off with a problem such as ‘Tommy eats two quarters of a chocolate bar and Seema eats one quarter. What fraction of the chocolate bar has been eaten?’ Starting with a problem here means that what was quite an abstract idea is now rooted in a relatable context.
I am in favour of getting the misconceptions out there early on. I will often say to my class ‘This is what some of you might want to do and why it isn’t right.’ and then show them the misconception in action, and we’ll pick apart the maths to reveal why it doesn’t work. In this case, I will show them something like 1/4 + 2/4 = 3/8. Then I will draw this out (and they’ll be very familiar with drawing out fractions from all the work leading up to this point) and the penny will drop when they see that 3/8 is smaller than the two fractions we were adding. Getting this out in the open first thing will reduce the number of children who make this mistake and secure their understanding further. Why wait for a mini plenary when you can nip it in the bud early on?
The easy thing to do next would be to tell the children ‘When we add fractions with the same denominator, the denominator remains the same.’ and give them a stream of calculations where they will perform that very thing. I have no issue with teaching rules, but they must be accompanied by representations that expose the deeper structures. Rather than teach this as a rule, I would generate this will the class and use it as a stem sentence. The children can repeat the sentence again and again so it becomes ingrained in their verbal explanations but doesn’t become a meaningless crutch.
I have the stem sentence displayed on my maths wall and slides as we go along and will gradually remove it. The stem sentence needs a visual representation to go with it, so I would use the same problem that we started with and show the children that when we have two quarters and we add another quarter, it becomes three quarters because the unit I am working with is a quarter.
There are so many representations to choose from, some better than others and some easier to use than others. Ultimately, you want to give your pupils the best journey through these models to develop their understanding as they go. Fractions are often represented with circles (usually cake and pizza!) but this isn’t a user-friendly image. You must consider what has come before, which is why sequencing and knowing a unit of work’s journey is vital. Children in my class will be used to bar models from earlier work so it makes sense to start with bars. Then you might want to move onto circles later so they’ve been exposed to it, but you can’t expect children in Year 3 and 4 to accurately split a circle into equal parts so you’ll want to provide the circle already split! Another useful model to use is a number line. Again, this is only helpful if the class have a good understanding of number lines. Showing how a number line is split into intervals of 1/5 and how you can add fifths on the number line would strengthen your pupils’ understanding of both fractions and number lines.
When showing fractions being added on bars, I use one bar for each whole and different colours. Colour can be an incredibly powerful tool in the classroom. Showing the children a bar with 2/6 in blue and 3/6 in green means you can ask the class ‘What fractions have been added together here?’ and they will be able to see it instantly. I think colouring or drawing onto the same bar makes sense so the children get used to working with the whole, and adding another whole bar if the sum is greater than one. This is where I would tend to show a third addend and highlight that the process is the same regardless of how many fractions you are adding.
You want to keep the process as simple as possible. When I design tasks and questions, I think about what I want the children to learn. To start off, I would give them a bar already coloured in and simply ask them to identify the calculation that has been done. Then, I might move onto the reverse by giving them the calculation and asking them to colour in the bar.
Something I picked up when working with a teacher from Shanghai was the idea of showing the addition of the numerators above the denominator to reiterate that it’s only the numerators being added. Then to push the children’s thinking on and remind them of prior work, I would use part-whole models and missing number questions, rather than lots of the same style of question.
And then once all of this has been done… we move onto subtracting like fractions! This should be simpler because you’ve already laid the groundwork with addition. Stick to the same types of representations but show the process of shading the minuend and crossing out the subtrahend.
The main message here is regardless of how many complex internal debates we have when planning, we must keep it simple for the children.