Thanks to mathematics consultant, trainer and author, Caroline Clissold, for this article that looks at the importance of ensuring the basic concepts of fractions are understood from an early age to establish firm foundations for progress.
If you are interested in hearing more on this topic, please register for a free webinar on 14th June where Caroline will go into more detail and will be able to answer questions. Sign up for the free webinar.
In my previous blog post on preventing 'catch up' in Years 5 and 6 for fractions, I recommended that children should begin adding and subtracting fractions right from the beginning of their introduction to fractions in Year 1. Children should have been introduced to commutativity and inverse when learning about addition and subtraction of whole numbers. Ideally they would have been introduced to both concepts at the same time so they could explore the relationship between the two operations before focussing on one of the operations. Ideally they would check an addition using subtraction and check a subtraction using addition, therefore embedding their understanding of this relationship.
The rules of commutativity and inverse apply to adding and subtracting fractions, so they should be reinforced when learning about fractions. Using strips of paper or fraction cards similar to the ones below from the NCETM, most Year 1 children will be able to tell you that ½ + ½ =1 and therefore 1 – ½ = ½. They should be able to tell you that ½ + 2/4 = 1 and therefore 2/4 + 1/2 = 1, 1 – ½ = 2/4 and 1 – 2/4 = ½. This is how we can help children build connections, embed reasoning and deepen understanding.
Fractions in Singapore
In Singapore, children focus on the part whole model of fractions in Years 2 and 3 in order to give them a deep understanding of the relational side of fractions. They add, subtract and explore equivalences from Year 2. It is not until Year 4 that they find fractions of numbers and quantities, which is something that is expected for Year 2 over here. Equivalence is touched on when children need to recognise that 2/4 is equivalent to ½, but nothing more. If we focus on the part whole model of fractions in enough depth that children understand that:
the vinculum shows the whole is divided into parts
the denominator shows the number of parts there are
the numerator shows the number of parts we are considering;
then the children should understand that fifths are when there are five equal parts, sixths are when there are six equal parts.
Fractions in action
In a Year 2 lesson that I delivered, we explored half of lots of different types of objects. As we worked through the examples we used the bar model. The children drew bar models on the board explaining that the whole has been broken into two equal parts.
After we had explored halves, I wrote 1/3 on the board and asked someone to draw the bar model to represent this. They knew there would be three equal parts. We talked about these being thirds. We then repeated this for a variety of other fractions. If they understand this, they should be able to add and subtract fractions with the same denominator. Cubes are a good resource for this.
These six cubes represent the whole so that each part is one sixth. 2/6 are red and 4/6 are green. We can say that 4/6 + 2/6 = 1, 2/6 + 4/6 = 1, 1 – 2/6 = 4/6 and 1 – 4/6 = 2/6. I strongly believe that Year 2 children in the UK can understand this just as they do in Singapore! Of course they will need manipulatives and visual representations.
I have recently delivered a fractions lessons on adding and subtracting and finding equivalences in Year 2 using the fractions cards mentioned above. It was amazing what the children could do. We were looking at ways to make one whole. Pairs of children had a pack of cards and a dice. They had four green cards to represent four wholes. They threw the dice to select the fraction cards. If they threw 1 or 2 they picked a half card to put onto their whole. If they threw 3 or 4 they picked a quarter card and if they threw 5 or 6 they picked an eighth card. They used the cards to make one whole, drew a picture of what they had made and wrote an addition statement, for example, ½ + ¼ + 2/8 = 1. We also talked about the relationship between the different fractions. The children could see that half was half the size of one whole, one quarter was half of one half and one eighth was half of one quarter. I asked them what they thought half of one eighth would be. They saw the pattern that the denominators made and came up with one sixteenth. We talked about half of one sixteenth, they knew it would be one thirty second. We went on and talked about half of one thirty second. What was particularly impressive was that they knew that 1/64 is smaller than 1/8 of the same whole because 1/64 of one whole would be tiny because the whole would be divided into lots and lots of pieces. No misconception!
Somewhere along their journey with fractions, a few children had developed a misconception that the larger the denominator, the larger the fraction. It was lovely to see these Year 2 children engage with the activity and really reason about fractions. I observed a teacher using the cards in Year 1 to look at equivalence. It was equally successful. These children could confidently use the word equivalent, explain what it meant and explain, for example, why 2/8 are equivalent to 1/4, 4/8 are equivalent to 1/8 and 8/8 are equivalent to one whole. Sometimes we too readily put ceilings on children’s learning when sometimes it is just great to lift them in order to find out what they are capable of understanding.
So if this is possible for Year 2, surely we need to increase our expectations for Years 3 and 4. The national curriculum states that children spend two years adding and subtracting fractions with the same denominator. We should be asking them to add and subtract fractions with denominators that are multiples of each other, for example, halves, quarters and eighths and thirds, sixths and twelfths. It makes sense to do this within one in Year 3 and greater than one in Year 4. Year 3 need to develop an understanding of equivalence because we need to find equivalent fractions to add and subtract. In Year 4 they need to develop an understanding of improper fractions and mixed numbers because they are adding fractions that have sums greater than one whole. Again, to deepen their understanding they need representations. These fraction walls are helpful.
From these visuals, children in Year 3 can clearly see that, for example, 4/8 and 2/4 are equivalent to 1/2. This means that they can easily add ½ and 1/8 by converting ½ to 4/8 and adding the other 1/8. They can then begin to look for the generalisation for finding equivalent fractions by looking for a pattern: ½ = 2/4, ½ = 4/8. After exploring many examples they should be able to generalise that to find equivalent fractions they multiple the numerator and denominator by the same number.
In Year 5, the national curriculum requires them to add and subtract fractions with denominators that are multiples of each other. By this stage they should be adding more random fractions, such as 2/5 and 3/8 by finding common denominators. This shouldn’t be difficult if understanding has been built up from Year 1. Year 5 can then focus on the other areas of fractions such as percentages. By Year 6, they should have mastered addition and subtraction of any fractions and therefore can focus on exploring multiplication and division of fractions and developing an understanding of how these operations work without simply being told rules. They can also focus on other areas of fractions such as ratio. Ratio comes up for the first time in Year 6, yet children should be given opportunities to explore the beginnings of ratio from Year 1. But that is another subject for another blog post!
If you want to hear more about this topic, join our webinar on 14th June at 4:30. Sign up for free.
Looking for resources for Fractions? Take a look at Fluency with Fractions for Years 1 to 6.