Introduction to algebra

Algebraic thinking - part 2

Problem: The number of students in a school is s and the number of teachers is t. There are 20 times as many students as teachers. Write down an equation using s and t.

The temptation here is to write down the relationship between s and t incorrectly as 20s = t. When we do this we think that what we are saying is, ’20 students for 1 teacher’. Here we are thinking that the s stands for ‘a student’ and the t stands for ‘a teacher’. Again, we see the same misunderstanding. The symbols s and t are not abbreviations for a student and a teacher. They stand for ‘the number of students’ and ‘the number of teachers’. They are variables. The value of t can be any number and whatever number is chosen, the value of s is 20 times this. So the relationship is s = 20t. This means ‘the number of students is 20 times the number of teachers’ or, referring to the tabulation of values in Figure 14, ‘the number in column s is 20 times whatever number is in column t’.

Remember to emphasize the idea that a letter in algebra stands for ‘whatever number is chosen’, that is, a variable.

It is understandable that so many people get the algebraic statements in these problems the wrong way round. First, the choice of s and t as letters to represent the variables in the problem is actually unhelpful (deliberately, I have to admit). Using the first letters of the words, ‘student’ and ‘teacher’ does rather suggest that they are abbreviations for these things. (Fewer people get these relationships the wrong way round if other letters are used for the variables, such as n and m.)