The Vegetable Plot

Programme Outline

The Guard, late for his duty shift, is presented with a puzzle by Lisa. At the Wizard's Vegetable Plot (where the onions and peppers are laid out on Cartesian principles), they look for a prize 'at the end of the line.' By following clues hidden in the flower-bed, they eventually track it down, exploring co-ordinate pairs and simple graph functions along the way.

Start to 01:10 - Setting the scene The Guard has slept in, but not to worry, Lisa is standing in for him. She administers the day's puzzle, which is set at the Vegetable Plot. The Guard is required to find a prize at the end of a line, though it is not made clear which line. Our heroes set off towards the gardens.
01:10 to 03:55 - The Vegetable Plot The Wizard's vegetable plot is, of course, a vegetable garden laid out on graph axes. Conveniently, all the vegetables are labelled with their co-ordinates. Lisa and the Guard investigate, and the Guard works out how the labels relate to position. They do not, however, work out along which line they must walk to find the Guard's prize.
03:55 to 04:27 - The Mathematical Encyclopaedia The Encyclopaedia tells us the story of René Descartes, and his invention of the co-ordinate pair. For Descartes' system to make sense, we must agree a point from which to start counting - the origin (0,0).
04:27 to 08:00 - Straight lines The Wizard arrives, looking for ingredients for his dinner. On his menu tonight: 'x = 3', which consists of all the vegetables he finds along the line x = 3. From this example, the Guard works out that 'x = 3' is a shorthand for all the points on the graph whose x co-ordinate equals 3: that is, a vertical line passing through the x-axis at (3,0).
08:00 to 10:30 - Gradients Buried in the flower-bed on the line x = 3 is a clue. Spurred on by this, Lisa and the Guard try the line y = x, which they conclude must pass through the points (1,1), (2,2), (0,0), (-1,-1), and so on. On this line, again buried in the flower-bed, is a further clue. They are urged to plot the line y = 2x - 1. The Wizard arrives to collect some onions, and he offers a vague clue: y = 2x - 1 is, he says, steeper than y = x.
10:30 to 11:20 - Functions Since one doesn't usually plot graphs with peppers and onions, the Wizard shows us the more normal method: he takes a function, makes a table of results, uses the table to plot points, and then joins the points up.
11:20 to end - The prize Lisa and the Guard have started using a rope, rather than the peppers, to plot lines. But they have plotted y = x, which isn't right. Having adjusted the rope, the Guard checks to make sure that it passes through the points he would expect. He heads off along the line to claim his prize, which he eventually finds after a false start. Then he returns to the bridge to do his guarding duty, which proves to be more complicated than he had expected.