# Gradient of Straight Line Graphs: Maths Diagnostic Question of the Week 16

What are the major misconceptions students have when it comes to working out the gradient of a straight line? Using real life data and explanations from students all around the world from my Diagnostic Questions website, we can find out! :-)

Have a look at the question below.

What percentage of students do you think get it correct?
What is the most popular incorrect answer?
What explanations do students give for the incorrect answers?

Give the question a go yourself, and try to come up with an explanation of the correct answer that would make sense to students:

At the time of writing, this question has been answered 179 times, and has been answered correctly 68% of the time. The two most popular incorrect answers are C and D.

Now, looking at the alternative answers, we can probably hazard a guess as to why students have gone wrong. But why do that when we can read actual student responses? :-)

Visiting the Data Question page (which is also accessible to all who have answered the question by clicking on the small graph icon in the bottom-right corner of the question window) is very revealing.

Correct Explanations for A

Students have lovely ways of explaining things. Here are some beautiful answers given to this question:

Because to work out the gradient you do the change in y divide by the change in x and so that is 3 divide 1.5 which is 2 but the graph has a negative correlation so the answer is -2

I counted how many squares it was from (2,-1) it was 4 then I counted how many squares across it was and it was two, then I did 4 divided by 2 which equaled -2 because the line slopes down so it’s a negative number.

if you left along the x axis by one from the line and then go up the Y axis by two then you will hit the line. One times by two equals two. The line is going backwards along the X axis so the gradient is negative. The gradient is therefore -2.

All it takes is for one on these explanations to make sense to your students at their gradient problems could be solved!

But, what can we learn from the explanations to the incorrect answers?

Incorrect Explanations for B

B is an interesting one. I included it to try to identify any students who simply assumed that lines sloping downwards had a gradient of -1. Here are some of their explanations:

It’s like y=-x

It moves downward.

It was a guess I can’t remember how to work it out

Incorrect Explanations for C

I thought C might catch a few students out, as the line crosses the x-axis at 1.5, and also there is the fact that 1.5 is 3 (the y-intercept) divided by 2. Here is what they made of it:

It goes through the x axis at point 1.5 and as it has a negative correlation the answer must be a – so the answer is -1.5

As it crosses the x axis at 1.5

I don’t really know how to find this but I saw that the line passed through 1.5 so I put that

Interestingly, the concept of negative correlation with regard to straight line graphs keeps cropping up in student explanations – even in the correct ones!

Incorrect Explanations for D

I hoped this would identify students who are dividing the change in x by the change in y, instead of the other way around. Here are some of their explanations:

the line is going down so its negative. it is 1/2 because for every one it goes down it only goes half across

Every 1/2 it goes up 1

if you draw a line down one number on the y axis, and across to the line it is half but the line has a negative correlation making it a negative

So, there you go. A minefield of misconceptions are doing the rounds in students’ heads when it comes to the gradient of straight line graphs. But forewarned is certainly forearmed!

If you want to assess your students’ understanding of straight line graphs even further, then why not try assigning the following quiz to them (full instructions how to do this can be found here):

Here is a video taking you through how you can access the pages discussed in this blog post on gradient of straight line graphs:

# Question of the Week 7: Straight Line Graphs

The following question is taken from my website Diagnostic Questions. Here you will find 1000s of high quality maths multiple choice diagnostic / hinge questions, ideal for assessment for learning, which have been created and shared by maths teachers all over the world.

This question was created by one of the delegates at the 2014 TSM Maths Conference where I ran a series of workshops on the effective use of Autographin the maths classroom.

I set my group of delegates a "homework" activity to create 3 diagnostic questions each using Autograph. I thought this would be a good way to test out their new found skills, and also getting them thinking about other ways to use the software.

The advantage of designing a diagnostic question on a dynamic graphing package (the same is true of Desmosor GeoGebra) is that once the question has been asked, answered and discussed, there is opportunity for further discussion and investigation by firing up the original graph file itself.

With the question above, once students have established how to work out the equation of the perpendicular line, there is perfect opportunity to ask questions such as:

If I moved Point A one square up, what would the equation of the green line be?

Where could I move Points A and B so that the equation of the green line was y = -2x + 1?

Can you give me the equation of two perpendicular lines that cross the x-axis at (-4, 0)?

The possibilities are endless, and it reinforces my view that it is far better to plan questions than tasks. Here you have a really great question for assessing students' understanding of gradients and the equation of perpendicular lines, and then a fantastic opportunity to extend their understanding further.