By Anne Watson
Why am I talking about linear equations? Thinking about them takes me to several important hidden threads in school mathematics, wondering how learners might pull ideas together meaningfully. It is also helpful that there has been some research about ‘solving’ them.
What is linear anyway?
I started by searching textbooks and the internet for ‘linear equation’ and ‘linear function’.
An equation that makes a straight line when it is graphed. Often written in the form y = mx + b. | A linear function is a function whose graph is a straight line. |
Examples of linear equations: y = 3x – 6 y – 2 = 3(x + 1) y + 2x – 2 = 0 5x = 6 y/2 = 3 | For a fixed change in the independent variable, there is a corresponding fixed change in the dependent variable. |
A polynomial equation of the first degree, e.g. x + y = 7. | A polynomial function of degree at most one. |
Linear function definition: a mathematical function in which the variables appear only in the first degree, are multiplied by constants, and are combined only by addition and subtraction | |
A linear equation is obtained by equating to zero a linear polynomial over some field, from which the coefficients are taken (the symbols used for the variables are supposed to not denote any element of the field). The solutions of such an equation are the values that, when substituted for the unknowns, make the equality true. In the case of just one variable, there is exactly one solution. a 1 ≠ 0 ) {\displaystyle a_{1}\neq 0)} Often, the term ‘linear equation’ refers implicitly to this particular case, in which the variable is sensibly called the ‘unknown’. |
When learners search online, can they distinguish between definitions, descriptions, explanations, examples and properties? What do learners need to know in order to understand these?
In the results above there are possible confusions. Some talk about functions and some equations, e.g. (1) and (2). A useful distinction would be to use the word ‘equation’ when the equality allows you to find all the unknown values. But it is also useful to have the word ‘function’ in the vocabulary so that linear equations can be understood within the class of graphable functions. (5), (6) and (8) talk about polynomials but this is only useful if you have met other polynomials and can therefore see how ‘linear’ is different from ‘non-linear’. (1), (3) and (5) touch on representation, but while one of them offers y = mx + b (in the UK we usually use ‘c’ for the constant) another offers a range of possibilities. (7) is explicit about the operations that can be involved, while these are implicit in others. (4) is the only one that encourages a covariation approach, in which the straightness of the graphed line and change in variables are connected.
Of course it is possible to be simplistic and offer a description that is temporarily adequate. The textbooks I have looked at tend to see ‘solving linear equations’ as an extension of finding missing values in number sentences. It is worth looking at some of the research about this connection. A seminal paper by Eugenio Filloy and Teresa Rojano in 1989 took the view that there was a clear gap between finding missing numbers using number facts and solving equations using algebraic equivalence. They thought this gap occurred when the unknown appeared on both sides of the equation as this forced people to think about the equality of expressions, and/or resort to rules of manipulation. However, further research in 1994 by Nicolas Herscovics and Liora Linchevski showed that solving various forms of linear equation was often a dance between arithmetical knowledge and algebraic thinking. By algebraic thinking I mean that the focus is on the structure of the expressions involved, and the use of inverses to simplify them while maintaining the equality.
Missing number problems
Here are some examples of missing number problems, or linear equations to solve, that illustrate these issues.
4 + Δ = 9
When I gave this example to a passing ten-year-old the instant answer was ‘13’. I have not found research that recognises the pressure to ‘do something quickly’ that some children feel, although there is plenty of knowledge about not appreciating the meaning of the equals sign. In this case, the child understood equality and could demonstrate it with rods, but acting on ‘+ means add’ dominated all that knowledge.
Here are some examples which, I would argue, are easier to solve using arithmetic as a first resort.
15 = 37 – Δ. I solved this by asking ‘what number would give me 5 if I subtracted it from 7; what number would give me 1 when subtracted from 3?’
3Δ = 36 can be solved by using the three times table so long as you understand the symbol system.
4 + Δ – 2 + 5 = 11 + 3 – 5 can be simplified by calculating the right hand side to be 9 and seeing 4 + 5 as 9 on the left. How you would describe that move depends whether you think of order of operations as arithmetic or algebra, but ‘seeing’ 9 is about familiarity with number facts.
In all three cases you have to understand the use of symbols and what the equation is telling you.
11Δ + 14Δ = 175 depends on knowing how to ‘read’ the equation. The words ‘I think of a number …’ often help to direct the way an equation in one unknown can be understood.
Using strategies
It is useful to see what strategies the Year 8 students used in the examples from Herscovics’s and Linchevski’s research.
n + 15 = 4n
4n + 9 = 7n
5n + 12 = 3n + 24
In these three cases, all done correctly, with the unknown on both sides, the main method was substitution with adjustments and in many cases the first number tried was correct. It seems that if they understood the equality and the numbers ‘spoke’ to them, there was no need to manipulate even though they had been taught to use inverses.
In the following cases, which were all correct, the main method was to reason with inverses, but around 20% of students were successful using substitution and adjustment of likely values. The inverse reasoning, I would argue, works because they understand the linear structure. In UK this might mean they are thinking of function machines or ‘I think of a number …’.
13n + 196 = 391
16n – 215 = 265
420 = 13n + 147
6 + 9n =60
The following two cases were not fully successful, being correct for 95% and 86% of students respectively. In the first, inverses were used mostly; in the second it seemed that complements and number facts were used.
188 = 15n – 67
63 – 5n = 28
So it is not true that having the unknown on both sides makes the difference between using number knowledge and using algebraic manipulation. The position of the unknown and the familiarity of the numbers involved also make a difference. These competent learners seem to make choices about method that depend on them understanding equality and also having confidence with number relations.
This reminds me of a lesson I once watched on solving linear equations where the textbook had suggested using BODMAS by multiplying brackets out first before manipulating. The example was something like this:
6(x + 5) = 6(2x – 1)
The teacher was incandescent with fury about the textbook, and indeed the students sensibly decided to focus on dealing with the common multiple first by thinking about what the equation was telling them.
What is the equation telling you?
I am more and more convinced that linear equations should be meaningfully read before anyone dives in to doing things to them.
What sort of activities might help learners do this?
As always, I cull examples from practice where I have seen them used with understanding. One method would be to ask people to actually read the equation out loud with meaning, not merely voicing the symbols but explaining what they say, so ‘3x’ becomes something like ‘3 lots of x’. Another is to focus on equivalence such as complexifying equations by acting legally on both sides to find out what else we can know. So knowing x = 3 means that we also know: 2x = 6; x + 2 = 5; 2x = 3 + x; etc.
A graphical and a physical approach
Currently, I am convinced that a graphical approach shows the connection between linear functions, graphs, and the solution of linear equations by assigning particular values for y and understanding the answers obtained by reading from the graph. This paves the way for solving other kinds of equation and also for simultaneous equations.
‘I have a linear function in mind; when x = 2, y = 3. What else could I know?’ could be an exploration that combines linear graphs, equations and arithmetic progressions.
Using Cuisenaire rods: ‘If 4x + 2y = 3x + 5y, what else do we know?’ could be an exploration that uses the property that equal lengths of rods represent equal values, and maintaining equality is done by maintaining equal lengths. For this one, dark green and red do the work. The balance model works similarly, but diagrams in some resources fail to be true to the laws of physics.
As always, the choice of examples is crucial. If learners understand what an equation tells them, and then can solve it using number facts or substitution, there is no point in trying to use the more abstract rules of manipulation. If learners do not understand what an equation tells them, why are they being expected to ‘solve’ it?
Why do we expect learners to solve linear equations anyway? It seems to me that there are two main answers to this (in addition to ‘you need it for your exams’). One answer is to prepare the ground for understanding more complex functions, equalities and inequalities. Another is for applications in science, finance, or engineering where relationships might be actually or approximately linear. In both of these answers it is likely that information will be graphical and/or numerical or structural and that linearity might not be obvious until after some manipulation, approximation and simplification. These are all reasons to think about meaning before acting.
References:
Filloy, E., & Rojano, T. (1989). Solving equations: The transition from arithmetic to algebra. For the learning of mathematics, 9(2), 19-25.
Herscovics, N., & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra. Educational studies in mathematics, 27(1), 59-78.
Anne Watson has two maths degrees and a DPhil in Mathematics Education, and is a Fellow of the Institute for Mathematics and its Applications. Before this, she taught maths in challenging schools for thirteen years. She has published numerous books and articles for teachers, and has led seminars and run workshops on every continent.
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