When letters are numbers
Mystery, magic and mumbo-jumbo
I have met people who really distrust letters as mathematical symbols - perhaps because they had a negative experience with algebra at school. Indeed, this seems to be one of the areas of maths that many adults are quite happy to confess to never quite having understood, and for which they could see little use in the grown up world! In this article we'll talk about what algebra actually is, why it is important, and how we can make really secure links from informal algebra at primary stages to the more formal algebra of the secondary curriculum.
What is algebra?
It's probably easier to describe algebraic thinking than algebra itself. Shelley Kreigler in her article 'Just what is algebraic thinking?' suggests three main ideas:
1. Algebra as abstract arithmetic
Algebra is sometimes referred to as generalized or abstract arithmetic. This means exploring numbers and how they combine together, and becoming familiar with ideas such as the inverse. This starts in primary school and when done well can lay a solid foundation for formal algebra. Teachers who help children to understand the underlying principles (for example moving from 'if I know 7 +3 = 10, what else do I know?' to 'If I know 359 + 763 = 1122, what else do I know?' give them networks of connections that they can draw upon when they begin the formal study of algebra 'If I know a + b = c, what else do I know?'.
So how do we begin introducing unknown numbers to be worked out and discovered? How do we begin teaching algebra?
Missing number questions are either based on known facts, or on known connections. Putting what needs to be found out anywhere except at the end of an equation often begins orally and is based on known facts.
'What do you need to add to seven to make ten?' 'What do you know that might help?'
'What do I need to take from six to leave four?' 'What do you know that might help?'
Gaps, boxes and shapes in NRICH
A Stage 1 problem, which comes in the form of a simple calculator game, is Secret Number. Yet another is Number Lines, which uses numbered strips rather than calculators. Both these are simply expressed algebraically in the form 'n + a = b' or 'n - a = b' and lend themselves to representing the unknown by a symbol or letter.
In Stage 2 there are several questions where shapes represent numbers. Super Shapes is a simple example. The first three questions could equally well be written
7+a+17 = 25
17+14+b = 21 and
14+2c+34 = 136.
Shape Times Shape is possibly more difficult because multiplication is usually harder than addition, less numerical information is given and all the examples are interrelated. Nevertheless this is a good example of where different symbols represent different numbers, but in each case the same symbol represents the same number.
All of these examples (and there are many more) help children to become used to representing an unknown by a symbol, whether that is a box, other shape, or letter. The next step is beginning to manipulate these shapes or symbols according to algebraic rules.
2. Algebra as a language
Algebra is the language of mathematics. Children will eventually need to be able to read, write, and manipulate both numbers and symbols in formulae, expressions, equations, and inequalities. Being fluent in the language of algebra means understanding its vocabulary (i.e. symbols and variables) and being able to use the correct grammar ie algebraic rules.
The meaning of the equals sign changes from being interpreted in the early years as "is" or "makes", or in the case of subtraction as "leaves", into an understanding of balance. When I did algebra at school (many years ago!) we were taught methods which took all unknowns, or all of a certain letter, to one side of the equation. The equals sign expressed a balance. Do the same thing to both sides and both sides will still balance. This idea is expressed neatly using a number balance, sometimes called a "Balance Bar" and sometimes an "Equalizer".
3. Algebra as a tool for mathematical modelling.
This means solving a real world problem using algebraic thinking, and later setting up equations or inequalities and solving them using algebraic rules. At early levels problem may be best solved using trial and improvement, but some children will already have more sophisticated (and potentially algebraic) ways of thinking about problems.
Modelling in NRICH
At Stage 1 I first came across Eggs in Baskets and Lots of Lollies. These are the kinds of problems which are best done practically with counters. This is also true of Heads and Feet which is simple using counters (heads) and sticks (legs). This is a version of a problem with very simple numbers which comes in many guises in many different places. A much more complicated variation, because the legs are not in pairs and less information is given, is Zios and Zepts. Some children may use symbols to represent different unknowns and some may even devise formal equations. Many others however will be using algebraic thinking even if they don?t record it using symbols.
There are plenty of Stage 2 problems that have unknowns that can be solved algebraically. Two examples are Cherry Buns and Buckets of Thinking. In the first it is the weight of an egg and in the second three different, but interrelated, amounts of water.
n is any number
There is another important way that letters are used in school mathematics. The letter is not an unknown to be discovered but a generalisation which covers many different examples. A few months ago I was working with a Year 5 and 6 class on a problem which called for a generalisation. It was an ideal time to use, in fact, it turned out it was to introduce, a letter to sum up what we were doing. Conventionally, I used "n". The class, including the teacher, looked bemused. Obviously, they had not met this before.
"n", I said, "stands for any number." I wrote 'any' on the board with a much enlarged 'n'. Everyone's faces cleared and the teacher smiled. I felt rather pleased. It seemed a happy, if somewhat erroneous, explanation of why I was using "n"!
There are some generalising problems at Stage 1. One example is Growing Garlic. At Stage 2 there are several. A simple one is Up and Down Staircases and slightly more complicated Break it Up! and even harder Sticky Triangles. All three of these would be suitable for older learners as an introduction or practice in generalising. Some of them lend themselves to working with symbols.
So what?
Secondary mathematics relies upon children having positive experiences with algebraic thinking in their earlier schooling. As primary practitioners, our role is to help children to realise that algebra is not just about putting letters in place of numbers, but about thinking in a different way.