Lesson 29 Straight line graphs

# Episode 3

Reasoning Resources: Worksheet 3, Graphical calculators (optional)
Whole class preparation
Pupils are asked to conceptualise the two parameters of a linear relation - slope and intercept — and then relate these to a representation of what can be imagined as a real problem. Give out Worksheet 3, and set out the story: A rocket moving along a straight path on a landscape is to be zapped by a defensive missile system starting from two different points.

Given the equation of the line that the rocket travels and the points where two missiles are launched, we need to find equations of lines that each missile can travel on to hit the rocket. Direct the class to choose a uniform range for the x and y variables, normally called ‘Window’ on calculators, so as to ease comparison and sharing. They could choose a range of -2 to 10 for both the x- and y-axes. Then they should find where to write the function y = 1 + 0.5x (or y=0.5x + 1) and where the other equations for other lines should be written. Explain the gradient of 0.5, if fractional values have not been covered earlier.
Pair work
Superficially the whole of the agenda appears to be quite elementary, but the way pupils are asked to address it is far from that. We are asking them to view both ‘graphs’ and ‘straight lines’ as hypothetical models which can be inspected from the outside and thought about. So the pupils are being asked to move from concrete generalisation to early, or even mature, formal reasoning in their modelling of reality. Direct the pupils to sketch lines that cross the line y = 1 + 0.5x in the interval x =6 to x = 8, starting from the intercepts given. They should then find the equations for these lines, by trial and improvement on the calculator or by calculation. The use of the graphing calculator Zoom function to read the coordinates of the intersection points and to improve the solutions should be encouraged, e.g. by posing narrower limits for the points of crossing, possibly leading to two missiles hitting the rocket at the same point.
Whole class sharing and discussion
In discussion elicit the thinking that different pupils have used, and their subsequent steps. The emphasis should be on descriptive accounts linking the visually available mathematical cues and checking for reasonableness and consistency. End with exploring the range of values for the multiplier of x that allow the paths to intersect in the interval specified, which excludes the end points, i.e. using inequalities.
Extensions
Three straight lines meeting in one point. Any two straight lines that are not parallel could be understood as simultaneous equations. For more able students this activity can be extended ina variety of directions, apart from further exploration of the field of linear graphs. One direction is the explicit linking of the intersections, arrived at through trial and improvement on the graphical calculator as above, with solving simultaneous equations on the page. Another is to explore simultaneous linear and quadratic functions. At all times focus on a small number of familiar mathematical forms and leave most of the time for thinking and discussion of the concepts involved in arriving
End of Lesson Reflection
Some pupils may recognise the value of looking at the same topic in mathematics from different points of view, or in different applications. Regardless of how far a class progresses in this activity, allow a few minutes at the end of the lesson for ideas on how different the experience of this lesson was compared with others dealing with algebra and graphs.

• What have you learnt about equations and graphs today?
• What helped your understanding move forward?
• How many points do you need to determine the line?
• What can you tell from the equation of a line?
• How can you find the equation of a line if I give you two points and you draw it?

An overview of the activity would cover how straight lines on the grid can be understood in relation to the axes, as linking points and having a pattern, and as equations that give a starting position on an axis and a value for the slope.