Visual Line Equations

Visualising Equations of Lines'Visualise equations of lines. Big counters, big boards, quick changes'

In a quick glance around the classroom this activity will allow you to see why algebra can indeed describe lines: their slope (gradient) and their position (y-intercept). Everyone's lines should be at the same angle and cross the y-axis at the same point (though they may be a rotation or reflection of yours, depending on where they are in the class relative to you!). Is there anyone's whose gradient or y-intercept you are not sure about? Ask them to explain. Extensions: What equation would produce a steeper line than the current one? A line parallel to it? A line perpendicular (at 90° to) it??


  • Laminated A4 (enlarge to A3) grids. -10 to 10 on one side and degrees (and radians), for trig functions, on the other). This activity only uses the -10 to 10 grid, but if you're going to make a class laminated set it's a good idea to print double sided, with the trig scale on the other ready for work with sine an cosine graphs etc. You may like this activity using trig laminated graphs: Wire Transformations
  • Equations of Lines - double click on the embedded slides above for a copy.
  • Teacher Notes


This resource can be used as an introduction to equations of lines or as a practice exercise.

  • Print off the A4 grids in the "Resources" section above, enlarge to A3 and then laminate yourself a class set (one per two children).
  • The presentation embedded above offers an "introduction to equations of lines" set of equations and a "practice" ("visualise") set of equations.
  • The "introduction" set starts with "place your counters so that each ones y-coordinate is the same as its x-coordinate". (B) "so that its y-coordinate is always one more than its x-coordinate" etc.
  • When the word statements stop, get your students to "translate the maths" e.g. the slide is showing y=-2, students say (in unison), "I have to place all my counters so that their y-coordinates are all at -2" etc.
  • The "practice" set assumes students have looked at equations of lines. The teacher may like to start the lesson with a quick recap of the different methods (as in the video in the "Teacher notes" section): (1) Table of values approach: plotting just two points (x=0 and y=0 being the easiest to calculate) (2) Gradient & Y-intercept approach: y-intercept and use grad to plot the second point approach OR use the equation to work out one point then use the gradient to find the second point. It's always advisable for students to check their line by calculating/working out the position of a fourth point.
  • On the 10 by 10 grid side of the laminated A3 sheet students have space to make calculations, create a table of values etc. to help them with their working. Mini-whiteboards pens (erasable felt tips) and erasers (kitchen roll) for students will prove very helpful.
  • Pairs can join up to form teams. Pairs can line up their counters in a line. As pairs, individuals, or teams, students can then go round working out the equation of the all the lines formed by each pair within the class. One point for the correct gradient, one point for the correct y-intercept. Winner is the individual/pair or team with the most correct equations to describe all the lines made by the class.
  • There are a wide variety of ways this activity can be run and questions that the teacher may wish to set based on the level of the class, or the discussions that arise. The teacher can quickly write up on the board these additional, or alternative, equations of lines for students to reproduce as and when required/ discussion arises / misconceptions need addressing. It is a very rich, and flexible, activity base.

Advantages of the activity

  • The teacher can quickly see what students are doing thereby getting an insight into what they are thinking thanks to the size of the counters and the board (compared to pencil&paper), making student's work visible from almost anywhere in the class.
  • Students can get help from each other very quickly and can learn from each other. I generally ask students to stand on their desks and look around the classroom: "Whose counters are positioned the same as yours?", "Is there anybody's work you're not sure is the same as yours/correct and that you'd like to ask them to explain? ".

Want more resources on this topic? This Equations of Lines resource provides a good introduction to the "concepts" using student's prior knowledge. This Straight Line Graphs is a lot of fun and can be used as an introduction or for practice (computers required).

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