Water everywhere

Discuss Dad’s plan.

• Would it work? (Yes.)
• What is wrong with it? (There are unnecessary sections of pipe, which will waste money and installation time.)

The students can then set about deciding how they will redesign the irrigation system, using as few pipes as possible. The most likely problem-solving strategy will be trial and error. The students may find it helpful to make a list of their various attempts, adding the lengths so that comparisons can be made. If they want to explore whether the best solution is to lay pipes in a completely different configuration (for example, a single trunk line with feeder pipes going off at right angles), they will need to construct a scale diagram. This will require careful use of a compass to draw the triangles accurately and would be a valuable exercise in its own right. (In reality, a trunk line does not minimise pipe length.)

Question 2 is a very open question. It may help if the students use square grid paper or if you give them a scale plan of the garden of a private home (real or imagined) and ask them to see how they can irrigate it using the shortest length of pipe. Gardening stores have brochures with quite detailed information on home irrigation systems. These will tell the students what coverage to expect from different fittings.

For able students, this activity leads naturally into investigations of networks and the discovery of Euler’s Law. Mathematical Investigations, A Series of Situational Lessons, Book Two, by Souviney et al. (California: Dale Seymour Publications, 1992) has three investigations looking at networks. Connected Networks is an investigation involving irrigation networks.

Alternatively, you could give the students various patterns and ask if they can draw the pattern without lifting their pencil off the paper and/or going over the same line twice.

• Can they work out why some patterns can be traced in this manner but others cannot? (The clue is in the nature of the “nodes” – the points where lines meet or cross.)

This could lead into an investigation of the famous Königsberg Bridge problem.

See also page 18 of At Camp, Figure It Out, Level 3.

1.

a. One possible answer, which uses 170 m of piping, is:

b. Answers will vary. A useful strategy is to take out the longest pipe you can without missing a crop. Repeat if possible.

c. Practical activity

2.

Practical activity