What mistakes do students make when representing a situation as an algebraic expression?

Essential Skills Quiz 7 has revealed some important and fascinating misconceptions held by both the Year 11s of Thornleigh Saleisan College, Bolton, and Year 11s cross the whole country. They are the kind of mistakes that make teachers want to tear their hair out, or reach for the bottle. But as our maths department discussed on Monday through our tears, it is better to know about these misconceptions now, understand them, and work together to resolve them to ensure that students are not making these mistakes in the exam.

Many schools, including our own, are also setting these weekly quizzes for younger year groups, and that is great to see. Whilst the GCSE Maths exam may be changing, these essential skills will remain the same. So, if you haven’t got invovled yet, now is the perfect time. Either pick up from these week, or go back to Week 1 start to see where all the fun began. It is completely free, and always will be.

Quiz 7 is below, followed by the dramatic announcement of this week’s Insight of the Week. Looking at the questions in the quiz, can you guess what the worst answered questions were?

Here’s how our students performed compared to the rest of the world:

 

As you can see, four questions in particular caught our students out. They were on tree diagrams, properties of quadrilaterals (this topic keeps causing problems!), mode from a list of data, and the following classic on representing a situation as an algebraic expression:

Only 36% of our students got this answer correct, with options B and C tempting over half of them:

 

Algebra will always rank highly on many students’ “topics in maths I hate” list, and when you think about it, it is not hard to see why. It is an abstract topic, and yet in exams it is routinely applied to so-called “real life” situations and concepts where it has no real place. Take the now infamous “Hannah’s Sweets” from June 2015’s Edexcel GCSE exam, for example. And the question above is typical of the kind of bizarre situation students would need to represent as an algebraic expression. As one of my Year 11s said: “if we have found all this out about Colm,  Anton and Gaz, surely we can just ask them how much money they have?”.

Fair point.

When you break the question down, there are two separate skills required for success here. Firstly, there is representing the problem algebraically – correctly deducing the relevant amount of money for each person. Secondly there is the need to simplify the expression, or “collect the like terms”. From the explanations students gave, it is definitely the former that is causing the most problems here.

Let’s take a look at how our lot got on:

 

Incorrect Answer B (28% of our students chose this)

Students choosing this option have successfully calculated the number of y’s, but have not got their head around the information regarding the £5. From their explanations, you can see that in many cases this was due to not interpreting the information in the question correctly, crucially not getting Gaz’s total correct:

“Anton has £5 more than Colm. Together they have four lots of y + 5”

“There are 4 y’s: Colm has one, Anton has one, Gaz has two. Then there is the extra fiver that Anton has.”

 

Incorrect Answer C (30% of our students chose this)

Here we have a different misconception. This time students have the figures correct – clearly explaining that they understand if Anton has £5, then Gaz must have £10, giving the £15. But now the number of y’s is incorrect:

“Anton has 5 more and gaz has twice as much which is £10 more which is 15 then Colms ‘y’ will be added also”

“Colm has y. Anton has 5. Gaz has 10. Together they have y + 15.”

 

Incorrect Answer D (16% of our students chose this)

Not many students went for this option, but those that did suggested an inability to separate numbers from letters:

“You have y + y + 5y + 2y + 10y = 19y”

Finally, one student going for this option asked the question on everyone’s mind:

“I don’t know, but who is Colm?”

He is my colleague, and yes his name is a bit weird.

 

The Correct Answer!

One of my motivations for developing Diagnostic Questions was so that students all around the world could learn from each other. When your students finish a quiz, please encourage them to review their answers, reading through other students’ explanations, until they find the magic one that makes sense to them. So, what are our Year 11 students’ favourite correct explanations to help them resolve their own misconceptions?

“because colm is y, anton is y+5 and gaz is 2(y+5) so if you expand the brackets and then add them together you get 4y+15=105”

“y + 1(y+5) + 2(y+5) expanded y+y+5+2y+10 simplified 4y+15”

 

Tackling the Misconception in Class

As ever, in our Departmental Meeting on Monday, we discussed how we would tackle this major misconception that our students appear to have. Here are a few suggestions:

• As an obvious first step, structure the answer by writing down how much Colm has, then Anton, and then Gaz, all in terms of y.
• This will then reveal whether the issue students have is in writing the algebraic expressions or simplifying algebraic expressions.
• Obviously go ahead and solve the resulting equation, substitute it back in the to the question, and check everything else works out.
• Using substitution another way. Choosing an initial amount for Colm to start with, and then building up how much Anton and Gaz have using numbers, and checking we get the same answer by substituting into each of the algebraic expressions. Then trying again with a different starting amount.
• As a challenge, can students write a related question which make each of the other 3 answers correct?

Please add any extra ideas in the Comments section below!

 

The series of GCSE Essential Skills Quizzes are available here and will always be completely free.

Quiz 8 is available here

I hope these quizzes will prove useful to help your students develop the essential skills necessary for success at GCSE, and aid your teachers gain a deeper understanding of how your students learn.