Algebra now appears as a section of its own in the Year 6 programmes of study. In these programmes of study the children need to be taught to:
- use simple formulae
- generate and describe linear number sequences
- express missing number problems algebraically
- find pairs of numbers that satisfy an equation with two unknowns
- enumerate possibilities of combinations of two variables
The notes and guidance unpick these statements a little:
Pupils should be introduced to the use of symbols and letters to represent variables and unknowns in mathematical situations that they already understand, such as:
- 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 two numbers can add up to).
This new section makes it appear that this is the first time the concept of algebra is introduced in the primary curriculum. It’s not! It appears in various forms in all year groups.
In this article we will explore some of the concepts in the National Curriculum where early algebra appears.
From Year 1 primary school children should be developing a conceptual understanding and then mastering the concepts of inverse and commutativity. In later years they develop a conceptual understanding and mastery of the distributive and associative concepts. These aspects of mathematics are crucial if a child is to succeed in mathematics and take it further in their future education and potentially work.
The four operations of addition, subtraction, multiplication and division allow a child to combine numbers and perform calculations. Certain operations possess properties that enable them to manipulate the numbers in the problem, which comes is useful, especially when you get into higher mathematics like algebra. The important properties students need to know are the commutative property, the associative property, and the distributive property. Understanding what an inverse operation is also helpful. Therefore the children need to exposure to these in their early learning.
The commutative rule or law: 24 + 19 = 19 + 24, 8 x 9 = 9 x 8
The distributive rule or law: 67 × 8 = 60 × 8 + 7 × 8
The associative law (4 × 7) × 5 = 4 × (7 × 5)
Algebra often deals with unknowns so the children should be given plenty of opportunities to solve missing number statements which could make use of the properties mentioned above or balancing. It might be worth using the term algebra when tackling these types of problems. In KS2 why not use letters instead of shapes?
n + 23 = 75 – 48
One possible way would be to work out 75 – 48, which is 27. Then use the inverse operation of subtracting 23 from 27 to give 4.
Another possible way would be to balance the statement. Take 23 away from both sides and then complete the remaining subtraction:
n = 52 – 48 = 4
Algebra also entails exploring patterns which then lead to generalisations.
Here is a simple example, using the bar model
Leading to the generalisation:
This area of algebra begins in the EYFS when they make, for example, repeating patterns. In Year 1 there is a requirement in ‘number and place value’ that they should recognise patterns in the number system, for example odd and even numbers. Exploring these using Numicon or multilink made into Numicon shapes enables the children to see that adding two even or two odd numbers will result in an even number and one of each will result in an odd number.
In the notes and guidance section it suggests that children create repeating patterns with objects and shapes. This could also include patterns of numbers, for example, 1, 2, 3, 1, 2, 3 and then you could ask the children to predict what the 8th number would be, the 12th etc. You could begin a pattern, for example 1, 2 and ask the children to make up their own pattern from this starting point.
In ‘addition and subtraction’, Year 1 children need to solve missing number problems such as 7 = □ – 9. This again, could involve balancing using a set of balance scales. The children could put seven cubes on one side and nine on the other. The children then work out how many cubes need to be placed in the side with nine so that when the nine are taken out the scales balance.
Algebra includes sequences. In the ‘measurement’ section of the National Curriculum, the children are required to recognise and use language relating to dates, including days of the week, weeks, months and years – simple sequences that need learning.
In Year 2 under the section ‘number and place value’, the children need to compare and order numbers from 0 to 100 using <, > and =. An effective way to show the children the meaning of these symbols is this:
No crocodiles or tricks! This is the approach they use in higher performing jurisdictions, such as Shanghai.
In ‘addition and subtraction’, Year 2 children should be taught to recognise and use the inverse relationship between addition and subtraction and use this to check calculations and solve missing number problems. They also need to explore the property of commutativity. As mentioned at the beginning of this article, this is an important part of later algebra.
In ‘multiplication and division’ in Year 3 we are required to teach the children to solve problems, including missing number problems, involving multiplication and division, including positive integer scaling problems and correspondence problems in which n objects are connected to m objects.
The guidance suggests that children should solve simple problems in contexts, deciding which of the four operations to use and why. These include measuring and scaling contexts, (for example, four times as high, eight times as long etc.) and correspondence problems in which m objects are connected to n objects (for example, 3 hats and 4 coats, how many different outfits?; 12 sweets shared equally between 4 children; 4 cakes shared equally between 8 children).
‘Multiplication and division’ in Year 4 requires that the children are taught to solve problems involving multiplying and adding, including using the distributive law to multiply two digit numbers by one digit, integer scaling problems and harder correspondence problems such as n objects are connected to m objects.
The guidance suggests that children write statements about the equality of expressions (for example, use the distributive law 39 × 7 = 30 × 7 + 9 × 7 and associative law (2 × 3) × 4 = 2 × (3 × 4)). They combine their knowledge of number facts and rules of arithmetic to solve mental and written calculations for example, 2 x 6 x 5 = 10 x 6 = 60
In ‘measurement’ the children need to measure and calculate the perimeter of a rectilinear figure (including squares) in centimetres and metres. The notes and guidance suggest the children express the perimeter algebraically as 2(a + b) where a and b are the dimensions in the same unit.
In Year 5’s ‘multiplication and division’ notes and guidance, it suggests that the children consider the idea of distributivity being expressed as a(b + c) = ab + ac. The requirements require the children to use and explain the equals sign to indicate equivalence, including in missing number problems. These could include examples such as 33 = 5 x + 3.
In ‘measurement’ the children are required to measure and calculate the perimeter of composite rectilinear shapes in centimetres and metres and to also to calculate and compare the area of rectangles (including squares), and including using standard units, square centimetres (cm2) and square metres (m2) and estimate the area of irregular shapes.
By this year the children could be developing the formula for these.
As well as working on algebra as an area of mathematics in its own right, in the ‘measurement’ section for Year 6 children need to recognise when it is possible to use formulae for area and volume of shapes and also to calculate the area of parallelograms and triangles.
In the notes and guidance it suggests that they relate the area of rectangles to parallelograms and triangles, for example, by dissection, and calculate their areas, understanding and using the formulae (in words or symbols) to do this.
In the notes and guidance for ‘properties of shape’, it suggests that children describe the properties of shapes and explain how unknown angles and lengths can be derived from known measurements. These relationships might be expressed algebraically for example, d = 2 × r; a = 180 – (b + c).
In the notes and guidance for ‘position and direction’ it suggests that children draw and label rectangles (including squares), parallelograms and rhombuses, specified by coordinates in the four quadrants, predicting missing coordinates using the properties of shapes. These might be expressed algebraically for example, translating vertex (a, b) to (a – 2, b + 3); (a, b) and (a + d, b + d) being opposite vertices of a square of side d.
We have just touched on the possibilities for introducing algebra into all year groups in the primary school. If, as teachers, we explore these suggestions and other topics that we need to teach, we can enrich algebraic learning and help our children become confident in an area of mathematics that can often cause fear and confusion.
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