Louisa Ingram is a primary mathematics adviser for HfL
To begin with, pupils need to become familiar with assigning a value to a rod and finding the fractional value of the other rods. A good starting point is to find the value of the white rod as this then allows you to find the value of all other rods. When the brown rod equals 1; the white rod is one eighth. Compared to dark green, the white rod’s value is one sixth. Against blue, it is one ninth and against orange one tenths etc. You can then start to apply this such as assigning the brown rod a value of 2. Through this you can also draw attention to fractions such as which rod is one half, one quarter, one third the length of etc.
Naturally, pupils will then start to notice that there may be rods which have fractional values larger than 1. For example, if the brown rod is 1 then the orange rod will be 1 and 2/8. This could then be used to introduce pupils to mixed numbers and improper fractions. This can be modelled by aligning 10 of the white pieces above the orange rod to represent 10/8.
Through visually overlaying the rods, you can start to draw attention to equivalent fractions. For example, ½ is equivalent to 2/4 as you can physically overlay the rods on top of each other. Using the same example of the brown rod having the assigned the value of 1, two of the red rods (quarters) can be placed over the purple rod (half) to show the equivalence between two eighths and one quarter. Similarly, two of the white rods (1/8) can be placed over the red rod (1/4) etc.
Adding & Subtracting Fractions
Conceptually, pupils need to understand that when they are adding or subtracting fractions the denominator does not change. They should be able to explain this in relation to the part-whole relationship. This can be modelled using the rods. For example, if the blue rod has the value of 1, then you can model the adding of thirds using the lime green rods.
When adding or subtracting two fractions with different denominators, the Cuisenaire rods can also help them see why we need to find a common denominator.
The process for modelling the multiplying of fractions and whole numbers is:
– Make the whole number a fraction by ‘putting it over’ 1
– Multiply the top numbers (the numerators)
– Multiply the bottom numbers (the denominators)
– Convert to a mixed number if needed
– Simplify the fraction if needed e.g. 2/5 x 3 = 2/5 x 3/1 = 6/5 = 1 and 1/5
Do pupils truly understand why this rule works?
As you can see here, Cuisenaire helps pupils to understand why the rules work. See if you can make the links from the instructions above to what is being represented by the Cuisenaire and the number line. Note the use of concrete, pictorial and abstract notation in the same piece of learning.
There is a significant amount of research that encourages the use of manipulatives, such as the Cuisenaire rods, to support pupils’ conceptual understanding. For example, Bruner’s work of the 1960s describes the stages of enactive, iconic and symbolic ways of working and suggests that the most effective way for pupils to develop a coding (number) system is to discover it and construct their knowledge by comparing it to other structures they have worked with. The key is that pupils should be immersed in the key manipulatives that a school decides to use and that the classroom ethos supports that discovery and exploration.