Equations of Lines

'How do you define a line? What's the most effective definition? Why?' 

How do you define a line? It's important to differentiate the flight path of one plane, boat, missile etc. from another, if you don't, they may collide. Sketch 10 straight lines quickly on a piece of paper. How would you describe the picture you've produced to a friend on the phone as accurately as possible so that they reproduced your picture perfectly? That's what this activity is all about - see the video below:


Worksheet A has a picture that students, in pairs, will have to describe to their partners. Worksheet B has a space in which to draw the described picture. It may help to ask students to line up their chairs back-to-back for the first part of this activity. Any student caught looking at their partners work disqualifies both students.

Thereafter, students will need to use the paper worksheets to calculate gradients and y-intercepts. The geogebra applets below will allow students to check if their calculations are correct. If student's calculations do not match the lines on their worksheet, geogebra offers a visual aid that should help students refine/find the error in their calculations.

Equations of Lines A (with picture)

Equations of Lines B (without picture)

Equations of Lines TN

Find the gradient of each the segments (bonus question: what's the difference between a line and a segment?) A to I below (these segments are the same as on your worksheet). Once you've entered in the gradient you need to move the "parallel" slider to check if you're line's gradient (slope) can overlap that of the line.

You can ZOOM in and out of the applet if you need to adjust its width and/or height to your devices screen. Click on the 'reset' icon  (top right corner of the applet below) if you need to return to the original screensize or undo your zoom settings.

You can "drag&drop" the "Gradient" and "Parallel slider" to re-position them if required. 

Now you should know how, in the international mathematics community, the gradient (slope) and the y-intercept (intercept) of a line are used to define it's equation.

Work out the gradient and y-intercept for the line below.

When you have finished, click on the "reset" icon    (see top right of the below geogebra applet) for a new question: When you get the gradient or y-intercept right a message will appear to confirm a correct solution. Answer at least five different questions before moving on to the next activity.

Practice Zone

Think you undersand the most efficient way to define a line? Try and find the gradient and y-intercepts of these three lines (they are the same as on your worksheet). A message will appear when you arrive at the correct gradient and/or y-intercept. Your "test line" is the red dotted line in each of the three applets below.

Find the equation of the lines d, e and f below. This time you have to enter the entire equation each time into the box "line equation" e.g. type "-2x+1" in the box if you think this is the equation of one of the lines. A message will appear if you are correct.

Find the equation of the lines g, h and i below. Enter the entire equation each time into the box "line equation". A message will appear if you are correct.


This description of the activity mainly targets teachers. However, it is carefully worded so as not to give anything key away that may diminsh the interest or excitement of the activity for the students.

  • "How do you define a line?". Using worksheets A and B from the Resources section above, students will reflect on the key properties that separate one line from another and derive their own ideas for how best to describe these differences (see "Teacher Notes").
  • The "rise/run" or "staircase" and "y-intercept" conventions for defining lines are introduced and compared with the methods created by students (see "Teacher Notes"). The teacher should aim to draw out the pros and cons of each and establish arguments for which method is most effective and why.
  • Students then practice working out gradients of lines and using the applet on computers or on the interactive whiteboard, to check if they are correct.
  • The "Equation of Lines" applet above allows students to randomly generate lines for which they have to find the gradient and y-intercept. Feedback is given within the applet if students get either the gradient or the y-intercept (or both) correct.
  • Students then have three sets of three lines on their worksheets for which they are to find the correct function. The emphasis is on students working this out on paper and using the applet, "Practice Zone: Equations of Lines 1-3" to check visually how close they are and get feedback if they are correct. The equations get more demanding as students move from set 1 to set 3.
  • For the applet "Equations of Lines 1" students enter in the gradient and y-intercept. For "Equations of Lines 2" and "Equations of Lines 3" students have to enter in the entire function e.g. 3x - 2 etc.
  • The focus of this activity is for students to be able to find the gradients and y-intercepts of lines on paper. The Geogebra applets are there to help offer feedback and experimentation so students can check and make progress on their own whilst waiting for the teacher to be available (if required).

Further straight line graph resources

Looking for a game / investigative approach to finding equations of lines? Or you simply want more practice/extension work on equations of lines? Try Straight Line Graphs

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