Algebra is about putting real life problems into equations and then solving them. Although formal algebra does not really start until Key Stage 3, primary schools can lay the lay the foundations for algebra in Key Stages 1 and 2 by providing early activities based on algebra from which later work in algebra can develop.
These activities include:
When you are questioning your class you might at times ask them to give more than single word or single number answers. For example, you might sometimes expect the response to short questions such as: ‘What is 16 add 8?’ to be expressed as a complete statement: ‘sixteen add eight equals twenty-four’, which children can repeat in chorus. You might also invite a child to the board to write the same equation in symbolic form: 16 + 8 = 24.
By asking questions such as: ‘Complete 3 + ? = 10’ youcan introduce children to the idea that a symbol can stand for an unknown number. You can also ask questions in the form: ‘I double a number, then add 1, and the result is 11. What is the number?’ By considering equations with two unknowns, such as ? + ? = 17, or inequalities like 1 < ? < 6, you can lead children towards the idea that the unknown is not necessarily one fixed number but may also be a variable.
Another important idea in both number and algebra is the use of an inverse to ‘reverse’ the effect of an operation. For example, the inverse of doubling is halving, of adding 7 is subtracting 7, and of multiplying by 6 is dividing by 6. Once they have grasped this idea, pupils can use their knowledge of an addition fact such as 4 + 7 = 11 to state a corresponding subtraction fact: 11 – 7 = 4. Similarly, pupils should be able to use their knowledge of a multiplication fact such as 9 × 6 = 54 to derive quickly a corresponding division fact: 54 ÷ 6 = 9.
Identifying number patterns
Encourage children to look for and describe number patterns as accurately as they can in words and, in simple cases, to consider why the pattern happens. For example, they could explore the patterns made by multiples of 4 or 5 in a 10 by 10 tables square, or extend and describe simple number sequences such as 2, 7, 12, 17… and, where appropriate, describe and discuss how they would set about finding, say, the 20th term.
When discussing graphs drawn, say, in science, ask children to describe in their own words the relationships revealed: for instance, ‘every time we added another 20 grams the length of the elastic increased by 6 centimetres’. They can also be asked to use and make their own simple word equations to express relationships such as: cost = number × price
By Year 6, pupils should be ready to express relationships symbolically: for example, if cakes cost 25p each then c = 25 × n, where c pence is the total cost and n is the number of cakes.
As well as drawing graphs which display factual information, teach older pupils to draw and use graphs which show mathematical relationships, such as those of the multiplication tables, or conversions from pounds to foreign currency. Games like Battleships can be used to introduce the idea of co-ordinates to identify spaces and, later, single points. It is then possible to record graphically, for example, pairs of numbers that add up to 10.
Developing ideas of continuity
Another foundation stone for algebra is laid in Years 5 and 6 when you help children to appreciate that between any two decimal numbers there is always another, and that the number line is continuous. They also need to understand that quantities like heights and weights are never exact. In growing from 150 cm to 151 cm, say, every possible value in that interval has been attained because measures too are continuous.
Finding equivalent forms
You should emphasise from the very beginning the different ways of recording what is effectively the same thing. For example:
- 24 = 20 + 4 = 30 – 6;
- 30 = 6 × 5 = 3 × 2 × 5;
- 15 + 4 = 19 implies that 15 = 19 – 4, and 3 × 4 = 12 implies that 12 ÷ 3 = 4;
- 1⁄2 = 2⁄4 = 3⁄6 … and each of these is equivalent to 0.5 or 50%.
Factorising 30 as 2 × 3 × 5 is a precursor of the idea of factorising in algebra. It is also a useful strategy for multiplication and division.
For example, since 12 = 6 × 2, the product 15 × 12 can be calculated in two steps, first 15 × 6 = 90, then 90 × 2 = 180. Similarly, 273 ÷ 21 can be worked outby using the factors of 21, first 273 ÷ 3 = 91, then 91 ÷ 7 = 13. Encourage pupils to factorise numbers as far as is possible. To factorise 24 as 6 × 4 is not as complete as 2 × 2 × 2 × 3.
Understanding the commutative, associative and distributive laws
Pupils do not need to know the names of these laws but you need to discuss the ideas thoroughly since they underpin strategies for calculation and, later on, algebraic ideas.
Children use the commutative law when they change the order of numbers to be added or multiplied because they recognise from practical experience that, say: 4 + 8 = 8 + 4 and 2 × 7 = 7 × 2
The associative law is used when numbers to be added or multiplied are regrouped without changing their order: for example, (4 + 3) + 7 = 4 + (3 + 7) and (9 × 5) × 2 = 9 × (5 × 2)
An example of the distributive law would be a strategy for calculating 99 × 8: 99 × 8 = (100 – 1) × 8 = 100 × 8 – 8
Pupils who have a secure understanding of all these important ideas by the age of 11 will be in a sound position to start work on more formal algebra in Key Stage 3.