Algebraic expressions — the return! Guess the Misconception

This week on Guess the Misconception we tackle a tricky question that involves writing an algebraic expression.

What do you think the most popular wrong answer is, and why might a student select it?

View the question here: https://diagnosticquestions.com/Questions/Go#/23309

And here are the results:

View the data and insights here: https://diagnosticquestions.com/Data/Question/23309#/All///////

So, only 1 in 3 students gets the question correct, with answer A being the most popular, but answers C and D luring in alosot 405 of the students between them.

To get a better idea of exactly what is going wrong here we can turn our attention toward some actual student answers.

Answer A
Students going for answer A really struggled to get to grips with what the question was asking. One group of students seemed to be swept away by an irreistable urge to multiply 6 and w together:

I think this is the answer because w (pounds) is timsed with 6 to make 6w. Then you have to divide 6w by z (amount of rulers)
Because to find out how much z rulers cost you have to divide it by the amount of 6 rules with the cost of w pounds

Whilst another group of students seemed to identify this as being an inverse proportion question and tried to apply some half-remembered rule to the situation:

6 rulers = £w. This question is basically “work out z when w=6”. It’s inverse proportion as the equations are all fractions k = 6 k has to be on the top of the equation, so A is correct
For example, if w is 12, then it’ll make sense that it will be 6w/z. This question is inverse proportion.

Answer C
Here we have a different type of confusion. This time students seemed to have correctly identified the process needed to find the cost of z rulers, but have come unstuck with the algebraic interpretation of that, specifically when it comes to what multiplying by z looks like:

I think this because 6 rulers =w, so to find how much one Z ruler cost, you divide w by 6 to find the price for one ruler, then you times that by Z.
w divides by 6 shows you how much one ruler would cost so then times the answer by z and that tells you how much z amount of rulers costs

Answer D
More confusion here, this time with which letters should be placed together:

Because it says that 6 rulers cost w pounds so they are together and you have got to find how many z cost so, they will be on there own

And possibly my favourite explanation:

This is a dumb question and i am just guessing because there is no way to properly work out the answer. It is dumb

So, what are we to do?

Well, the students who started by thinking in terms of numbers definitely had the correct idea. As it stands, this is a very abstract question, but by letting w and z stand for numbers that are easier to work with, such as £30 and 6 rulers, then we can start to make better sense of it.

And students can be introduced to this technique slowly and systematically at the start. So, a sequence of questions like the following may help ease them into the algebra:

As ever, this would be followed up by questions like this appearing in daily low-stakes quizzes and homeworks.

Why not try the question out on your students, either in class or as part of a homework, and see how they get on? Talk about the correct answer, and also the wrong ones. Better still, you can ensure students receive a regular diet of quality questions like this — together with all the teacher insights you can ever want-by setting up our free schemes of work. We have free maths schemes from Year 1 to GCSE, with all the awarding bodies represented. Just click here to get started.

And if the intelligent sequencing of examples is of interest to you, then I discuss it further in Chapter 7 of my book How I wish I’d taught maths. The book also contains an entire chapter dedicated to the practicalities, benefits and considerations when using diagnostic questions in the classroom, is available to buy from Amazon and John Catt Educational Ltd.

Have a great week
Craig