Teaching Algebra KS2: A Guide For Primary School Teachers From Year 3 To Year 6
At Key Stage 2, pupils are introduced to algebra as a curriculum objective for the first time. Here is a great way to introduce algebra KS2 to your pupils by use of pictorial representation, particularly Cuisenaire rod models.
This being said, when you look through the objectives for this section there is a high chance that some objectives would have been touched on in previous years in KS2 maths; teaching the formula for finding the area of different shapes, for example.
This is no excuse to not teach these objectives in the context of it being algebra. It is, however, possible that pupils have been using algebraic thinking even before KS2.
- Where could pupils have encountered algebraic thinking before?
- Cuisenaire rods: Algebraic thinking in EYFS
- Algebra KS2: what Year 6 pupils should be taught
- Algebra lesson ideas for Year 6 pupils
- Theory behind teaching algebra KS2
- Algebra questions and word problems in Year 6
- Addition and subtraction: reasoning and problem solving in algebra KS2
Where could pupils have encountered algebraic thinking before?
Algebra can be interpreted as the generalisation of relationships between symbols. Whereas purely numerical equations show particular equivalences and transformations, by using letters to represent unknown values, algebraic equations seek to generalise.
Simple function machines may have been used in KS1 when looking at the four operations – this is where you put a number into the machine (the input), the machine provides instructions about what happens to that number, and delivers it as an output.
It is possible still that pupils’ experience of algebra goes back further than this, all the way back to the Early Years Foundation Stage (EYFS) in fact, when working with Cuisenaire rods (and if your reception teacher does not have a set, invest in them).
Before examining the algebra curriculum in Y6, it is worth examining the algebraic thinking that can happen in an Early Years classroom.
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Cuisenaire rods: Algebraic thinking in EYFS
Cuisenaire Rods are a set of 10 rods that are of different lengths and different colours as can be seen by the picture below (note that the actual physical rods do not have the letters printed on).
There is not enough time to explain the usefulness of these resources in the entirety of these blogs but one thing to note about their use is that they do not come predefined with a value.
While adults may look at the rods and quite quickly assign the values of w =1, r = 2 g = 3 etc, pupils in the Early Years will not. The unknown or unstated values given to different colour Cuisenaire rods can be thought of as children’s first exposure to algebraic generalisation.
They will, however, be able to put them into order. Ordinality comes before cardinality. It is from this that the teacher can leverage some algebraic thinking from the pupils across the whole year.
Take, for example, the following:
Children of reception age are quite happy in understanding that a green rod and a red rod are the equivalent length of a yellow rod or, to write it symbolically, y = g + r.
Or when they construct a set of trains like below:
Pupils can see the length of p and b is equal to the length of g, d and r and understand when this is presented as p + b = g + d + r.
Finally, with this example:
Using the rods as a manipulative, the pupils are happy that five red rods are equivalent to one orange rod. This can be written as r + r + r + r + r = O. Or, an even quicker way would be 5r = 0.
Here we are dealing exclusively in unknown values. As pupils progress through the year and they learn their numbers and become secure in them, these unknowns can begin to manifest themselves into cardinality.
y = g + r quickly becomes 5 = 3 + 2, p + b = g + d + r can become 4 + 7 = 3 + 6 + 2 and 5r can become 2 + 2 + 2 + 2 + 2 = 10 or even 5 x 2 = 10.
The fact that mathematical symbols are used in conjunction with letters that represent the rods, the value of which are unknown, does not provide such a cognitive challenge as it does when children are introduced to algebra in Year 6 when x is first introduced as an unknown variable.
It cannot be overstated that this type of thinking will NOT happen within a lesson or by the end of one sequence of lessons. Pupils in the Early Years Foundation Stage will need plenty of time to play, experiment and explore with the rods as well as developing their sense of cardinality in other ways before the level of understanding demonstrated above will be realised. Many may not reach this understanding until KS1 and that is okay.
If you believe your pupils have not been introduced to Cuisenaire Rods or that it may have been some time since they have used them, then beginning the algebra unit in this way would be a successful way to begin pupils’ thinking.
Algebra KS2: what Year 6 pupils should be taught
In the national curriculum for maths in England, for each area of maths outlined, there is both a statutory requirement and a non-statutory requirement. The statutory requirement is as follows:
- Use simple formulae
- Generate and describe linear number sequences
- Express missing number problems algebraically
- Find pairs of numbers that satisfy an equation with 2 unknowns
- Enumerate possibilities of combinations of 2 variables
The non-statutory notes and guidance suggests:
- Missing numbers, lengths, coordinates and angles
- Formulae in mathematics and science
- Equivalent expressions (for example, a + b = b + a)
- Generalisations of number patterns
- Number puzzles (for example, what 2 numbers can add up to)
Algebra lesson ideas for Year 6 pupils
A fun way to introduce algebra into Year 6 is to look at the etymology (history) of the word itself. ‘Algebra’ comes from the Arabic ‘al-jabr’ which roughly translates as ‘the reunion of broken parts.’ This is a nice visual representation to give pupils as it promotes the idea that algebra is like a puzzle to be solved by reuniting an unknown number to an equation.
The objective I will focus on here will be:
‘Find pairs of numbers that satisfy an equation with two unknowns.’
It is worth remembering that this objective from the National Curriculum is the end result. In order to reach this, it may be appropriate to break this down into smaller steps, and fulfil the other objectives first.
For this example, I will look at satisfying equations with one unknown. As this is Y6, this should be done using all four operations. Using a bar model or Cuisenaire Rods can be invaluable here.
I would first begin to get pupils to link the abstract to a concrete or pictorial representation, as demonstrated below.
If your pupils are familiar with bar models then this will look very familiar. It is important, however, to ensure that you approach these models from an algebraic perspective and ensure that your pupils are too.
Relating the abstract to the pictorial is key to this so that pupils can see 4x means 4 lots of the same unknown, for example.
Once pupils are comfortable, move on to models that have an unknown and a numerical value which equate to another numerical value.
Notice this time that all the bars are of the same colour and length, yet the numerical values are different. This is done purposefully so that pupils can see that we are merely representing the problem, not using them to actually perform the arithmetic.
When they have successfully matched the representations to the algebraic expressions, the pupils can then find the value of x.
Theory behind teaching algebra KS2
I am a great believer that the more experiences pupils have had with manipulatives and bar modelling, the easier they will find algebra. Where pupils’ misconceptions and difficulties arise is when they do not have these mental models to draw upon.
Coupled with the general fear the word strikes into young mathematicians who may have heard it spoken at home from older siblings with much disdain, algebra seems to be a topic within mathematics where pupils have developed negative preconceptions.
Our role, as educators and proponents of mathematics, is to make children realise they have been performing algebra, hopefully, since they were in reception, before they even began to play with number cardinality.
Algebra questions and word problems in Year 6
A typical example of a word problem that pupils may be expected to solve by themselves would look like this:
Jason is 9 years old
Jack is 12 years old
Gran is x + 12 years old.
The sum of their ages is 100.
Demonstrate this algebraically and find x.
The pupils are expected to use the information in the question to show the representation. This would look like this:
Pupils could then calculate the numerical values of the ages that are known – 9 + 12 + 12 = 33. Then subtract this from the total sum to get 67. Pupils now know that x = 67, they can then calculate that 67 + 12 = 79. Gran’s age is 79.
Addition and subtraction: reasoning and problem solving in algebra KS2
There is, of course, more to the learning of maths than just learning these objectives, and reasoning and problem solving should not just be limited to word problems. We want pupils to be able to conjecture, experiment (and dare I say have some fun) with algebraic thinking.
A question like this does just that:
Because the value of the triangle and rectangle are consistent throughout, the content is similar to the previous questions but in a slightly different context. With the sum of the equations all being within the tens or hundreds, it allows pupils to focus on the algebra and not get confused due to their lack of understanding of larger numbers.
The teacher could model the use of converting the shapes to bar models if they think pupils need some more scaffolding to access this task. For example:
The above can become the following bar model:
From here pupils can see that x =1 as 17 + 7 = 24 and 1 is needed to make 25.
The questions gradually get harder but are scaffolded in such a way that the subsequent questions only get slightly harder making this a very accessible problem for pupils to solve.
a) Red shape = 1
b) Red shape = 10
c) Red shape = 44
d) Red shape = 16
e) Red shape = 32.5
Looking for some more ideas of how to do this? You can find plenty of free resources and algebra KS2 worksheets on the Third Space Learning maths hub. For guidance on other KS2 subjects, check out the rest of the series:
- Teaching Decimals KS2
- Teaching Place Value KS2
- Teaching Fractions KS2
- Teaching Percentages KS2
- Teaching Statistics KS2
- Teaching Ratio and Proportion KS2
- Teaching Multiplication KS2
- Teaching Division KS2
- Teaching Addition and Subtraction KS2
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