Twists and Turns with Straight Line Graphs

Feature Image

This blog post looks at straight line graphs for high school students.  In particular, it focuses on questions that ask students to “Write down the equation of the straight line that…” and then gives them some information to work with.  This often involves them finding the gradient and y-intercept and then writing an equation in the form y=mx+c.  But how in many different ways can the clues about the line appear?

For the majority of this post, I’m going to use examples where the equation of the line would be y=2x+3 and consider a selection of ways to present the information that leads to the equation.  This isn’t a sequence of questions that I would necessarily give to students, but it aims to highlight the variety of ways that the question can be posed.

Let’s first consider information that leads to the y-intercept.  In Examples 1-10, the information about the gradient is kept the same so that we can focus on the different ways of presenting details about the y-intercept.  How obvious or obscure can we make the y-intercept? And what different phrasings can we use?

01 Explicit y-intercepts

In Examples 1-4, the y-intercept is given fairly explicitly; students don’t need to do any calculations to work it out.  However, even with these fairly straight forward cases the wording can differ in many ways.

Close to the y-axis

In Examples 5-8, the wording is the same each time but the level of obscurity about the position of the y-intercept differs.  In each case, the y-intercept could be found by substituting the coordinate into the equation y=2x+C and then rearranging it to calculate the value of C.  However, this isn’t entirely necessary for Examples 5-7 because the coordinate given each time is very close to the y-axis.  Students may be able to work out the y-intercept for these mentally through mathematical reasoning with the gradient.

For instance, in Example 5 a student could think, “The coordinate (1,5) is one step to the right of the y-axis and the gradient is 2.  So to travel along this line to the y-axis, I would take two steps down and one step left.  That will be the coordinate (0,3).”  It’s not a very neat way of finding the y-intercept!  But it’s an approach that students could take if they haven’t yet learned the more formal method of substituting a coordinate into the partly formed equation.  It also exercises their understanding of the gradient too.

Further from the y-axis

Example 7 is a little trickier because they need to think about taking two steps to the right and so take four steps up.  The calculations become more arduous to perform mentally when the point is further away from the y-axis (such as in Example 8) but can still be done with a combination of multiplication and subtraction.  But still these calculations would be the same as the ones they’d do if they used the substitution method instead (see below).  Either way, they would start by multiplying 17 by 2 and then would subtract that answer from 37.  Nonetheless, it’s probably a good indicator at this point that a slicker written method (such as substitution) might be preferable, especially before getting to questions with fractions and negatives.

021 Working

A similar argument could be made for cases where this information is presented on a graph.  In Example 9, it’s fairly straight forward to visualise ‘stepping back’ to the y-axis from Point A.

03 step back to the y-intercept

However, this method would be tricky if the y-intercept wasn’t on the part of the graph that was visible.

04 Can't Step Back

Personally, I prefer to begin by giving students some questions where the point is near the y-axis so they can first think about how to find the y-intercept by taking steps towards it, both with and without diagrams.  Then when it comes to introducing the written substitution method, I might redo those same questions again so that the class can see how the two methods lead to the same answer and can hopefully reason why.

So now we’ve looked a handful of ways of changing the information about the y-intercept, let’s do the same for the gradient.  In Examples 11-18, the information about the y-intercept is kept constant while the gradient is presented in different ways.

05 Explicit Gradient

Example 11 is the same as Example 1 from earlier, but we’re starting with it again because it gives the gradient very explicitly.  Example 12 still doesn’t require any calculations but the information about the gradient is not quite as direct. This kind of description though can provide a starting point and then scope for thinking about how to calculate the gradient between points that are more than one unit apart in the x-direction.  For instance, in Examples 13 and 14 students need to start thinking about sharing the vertical distance out amongst the horizontal distance.  Like with the stepping method from earlier, it’s a fairly loose way of thinking about something that can later lead to a more reliable formula (y2-y1/x2-x1).

06 Difference in y shared between the difference in x

These descriptions also become tricky when negative and fractional gradients are involved.  It can be very easy for someone to mistake the gradient for each question below as being 2 if they simply skim read the information.  Nonetheless, such questions can get students to pause and think carefully about what the gradient tells them about a line before they start calculating with the formula.

061 Fractional and negative gradients

For the next set of examples, students are not given any information about the gradient but they are given a second coordinate so that they can work out the gradient for themselves.

07 y-intercept and a point

Just like with Example 5 earlier, the first two provide points that are close to the y-intercept so students might be able to find the gradient through mentally reasoning.  “To get from (0,3) to (1,5) I’ve taken one step to the right and 2 steps up.  So the gradient must be 2.”  Once again, this becomes trickier when the two points are further away from each other (such as with Examples 17 and 18) and require multiple calculations.  Students can also get into big problems with this method when negative and fractional gradients are included.  Personally, I like to begin with questions like Examples 15 and 16 to get students to reason their way to the gradient and then use harder questions like Examples 17 and 18 to lead towards the desire for a more reliable formula.

So far we’ve looked at ways to vary the information about both the y-intercept and the gradient.  When we consider all the different combinations of these variants (plus others), along with situations where the gradient and/or the y-intercept are fractional and/or negative… the number of possibilities seem endless!  Also, in all the previous examples at least one out of the two parts of the equation is given very explicitly.  Let’s now look at situations where students have to work out both parts for themselves.  How many different ways can we mix this up?

08 Graph given

In Example 19, students are not told either the y-intercept or the gradient but they are given the graph.  In this case, it’s fairly straight forward to obtain both pieces of information by visually inspecting the line.  This is particularly so because the gradient and intercept are both integers and the scales on both axis are 1 square for every 1.

However, this is trickier when the scales on the x- and y-axis differ from each other.  It’s all too easy for someone to mistake the gradient in Example 20 for being 1 if they are in the habit of just counting squares without considering the scale.

09 Graph with different scales

Examples 21-23 demand more abstracting thinking about the gradient and y-intercept than the previous two (unless students use the coordinates to draw the graph).  Even though all three examples are the same kind of question, the number of negatives used in each question differ.

10 Two points given

We can make this even more abstract by including coordinates that contain algebraic expressions.  Here students can calculate the gradient by either thinking about how the positions of the second and third coordinates relate to each other (so long as they pay extra attention to whether the gradient is positive or negative), or by substituting the expressions into the gradient formula.

11 Algebra Coordinates

We could even present the question in a more problematic way like Example 26.  Here, students can see the line but they can’t see the on scales either of the axis.  They are told the coordinates of two points but neither of them are on the line.  To make matters worse, the gradient between those two points is not even the same as the gradient of the line.  However, they can use the points to work out the scale and then use that to find some coordinates on the line.

12 Points off the line

Take Example 27, for instance.  A question like this can encourage students to think about what information is required to form an equation for a straight line and how to obtain it.  With this question, I tend to notice that students find it easy to spot that the y-intercept is 12.5.  But explaining how they know for certain that it is 12.5 and not 12.4 or 12.6 requires them to think more carefully about the coordinates on either side of the y-axis.

13 Y-intercept 12.5

This then gives an opportunity to explore what happens to the y-intercept when those points are moved from side to side.

14 Y-intercept 12.75

The aim of this post has been to highlight the sheer variety of ways that students can be asked to find the equation of a straight line.  These examples are far from an exhaustive list and probably only scratch the surface.  On reflection, I feel that I may have occasionally skimmed over this topic too quickly in the past and could have explored it in far more depth.  There seems to be endless scope for getting students to reason and problem-solve with information about straight lines, and this post hasn’t even touched on using parallel and perpendicular lines!

If you would like to use or share any of the images from this blog, feel free to use this Power Point to help you: Twists and Turns with Straight Line Graphs

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‘Making Connections’ Series