Lesson 29 Straight line graphs

# Episode 1

Reasoning Resources: Large sheets of paper, Worksheet 1 on A2/A3 or OHT,
Worksheet 1 for pupils
This part is accessible to pupils at all levels of attainment. It highlights the experiential need to adapt the convention of the Cartesian axes. This is an interactive episode, with whole class discussions punctuated by brief periods where pupils consult each other.
Whole class work on word descriptions of positions
The story of having a scene to describe in natural language clarifies the rationale for the conventions of axes and graphs to guide pupils to construct the axes and the scale. Pupils’ recognition of the need for such a convention comes through the challenge to describe positions on a plane unambiguously. This is found to be easier by relating points to referents such as special vertical and horizontal lines (axes), by using some standard measures of distance from these special referent lines, and by specifying directions such as left-right and up-down.

The subsequent formalisation of these seemingly common sense ideas into symbolic form and with the plus and minus signs should then also be seen as good sense.
You can tell a story about a scout exploring uncharted territory who manage to find a foothold and look over the precipice and try to explain to somebody below what | see in the vast plain in front. How can you help me? Ask some pupils to volunteer to turn their backs to the board and to draw on a large sheet of paper, in landscape format, what the class and the teacher tell them.

Draw on the board five points, one at a time, labelled A-E, describing what is at that point, for example, a tree, a rock, etc. Each time you mark a point ask the class how to describe its position to the pupils drawing with their backs to the board. Without drawing any axes or lines, arrange the points visually so that they are of some recognisable approximate distances from each other (e.g. two handspans) and from the undrawn vertical and horizontal middle lines you are helping the class to construct as axes.

Get the class to suggest that drawing a middle vertical line would help. For the drawings the pupils could use a scale such as one finger-width on the page for a hand-width on the board.

Point out that they are using some standard units of measures and everything is related to the middle vertical line, so any other point can now be referred to it.

Some pupils should recognise the advantage of a horizontal axis without you having to say it. Accept any position for the horizontal line, especially if it is in terms of hand-widths (translated into finger-widths for the volunteer drawers). In this diagram the horizontal line has been drawn one hand-width above point C.

Let the class struggle to describe the position of E, valuing all contributions but prodding for them to be given in relation to the two middle lines. At this point invite the volunteers to show their drawings, and get the class to appreciate the power and limitations of the descriptions given.

Point out that they have ended up with the mathematical convention of axes. Explain that we normally write the distance across first, with ‘right’ being + and ‘left’ being -, and the distance up/down second with ‘up’ being + and ‘down’ being -.
Describing vertical and horizontal lines
In some classes an open question for the teacher is how to avoid confusion between the Cartesian system and the geographical system of NSEW in describing position. If this issue crops up it could be recognised and the comparison between the two alternative systems made. Give out Worksheet 1 and ask pupils to work in pairs on completing it. Show how any one value on the vertical axis belongs to the whole horizontal line at that height, even though we use the height to label it, and similarly for the horizontal line. Discuss the infinite extension and continuity. Introduce the idea that x is conventionally used as the symbol for distance across, but that any other letter could be used, and similarly with y.

Display the grid from Worksheet 1 and show how any horizontal or vertical line can be described as: a line so many units to the right of the y-axis, a line so many units above the x-axis, referring back to the introduction activity.

Discuss briefly how useful it is to name lines algebraically in this way, and how it would be different if the lines were slanted.