An idea for the classroom - making parallel rulers
Drawing instruments fascinate me. I like the cases in which they sit, and the beauty of the instruments, old or new. I use them in the classroom to motivate my students, who can experience some history of mathematics by exploring the development of the technology of the time. Making these instruments gives students a sense of achievement when they see how well they perform.
I start with an instrument that has connections with mathematics and navigation - the parallel ruler.
The photograph shows a large double parallel rule, a small parallel rule and one handmade from an A4 sheet of card, all placed on a sea chart showing a compass rose.
The most common parallel rulers look like two normal, usually unmarked, rulers joined with two bars of equal lengths. The smallest one I have is 6 inches long and 1 3/8 inches wide when closed. It is made of a black wood (ebony?) and joined by two brass bars. There are no markings, but there is a small brass peg in the middle of each rule, presumably to assist moving the rule. I think this one probably came from a case of several mathematical instruments. The edges of the rulers have parallel chamfers so that you can run a pen or pencil along an edge that either touches the paper or is above it.
By the year 1600, parallel rulers were common. The older bar types used to be made in ivory or ebony with brass, silver or electrum cross-links.
The double-barred parallel rule, illustrated below, was an improvement on the plain parallel, as the ruling edge moves a greater distance from the fixed rule, and also moves in a direct line. The difference between this and the plain parallel is the addition of an extra rule and pair of bars, which are joined at reverse inclination to the first pair. As this was more difficult to make it is seldom as true as the plain parallel.
In the classroom
I’ve used parallel rulers in a few ways in the classroom.
The first way was with a Year 10 class of middle-achievers when we were studying bearings.
I decided that rather than spend too much time drawing bearings from those given in a text book, I would set a lesson where the class were working in groups of 3 or 4 with real maps. I borrowed maps of the Solent area, the area around Southampton, as that area is known by my students, and asked them to plot a sailing course round the Isle of Wight.
I first showed them how to use the parallel ruler to get straight lines parallel to the given north line at the side of the map – so that they could measure the bearings on which they would have to sail to get round the Isle of Wight.
The work involved measuring angles and using a scale of 1:50 000. One student, a sea-cadet, said that it was the most relevant work he had ever done in mathematics.
Making a set of parallel rulers
Sir Isaac Newton just folded a piece of paper to make a straight edge: you can challenge your students to make a set of parallel ruler from card and split pins.
Steps for success!
- take an A4 piece of card and put it landscape format in front of you
- draw a line 5cm from the top edge of the card
- draw a line half way down this thin strip, perpendicular to the drawn line
- draw a line 8cm from the bottom edge of the card. Your card should look like this:
14. round off the ends of the short pieces if you wish.
- cut out the four pieces and fold them all in half to make thin rectangles
- on one side of each piece draw the long line of symmetry
- put the two larger pieces together with the folded edges at the top and bottom
- on one of the larger pieces, mark two points on the line of symmetry, 2cm and 17cm from the left hand edge
- on the smaller pieces mark points on the lines of symmetry 1.5cm from each end
- use a sharp point to make small holes on the marks you have made
- use the fasteners to join the small pieces to the large piece already marked and holed
- put a pencil point in the other holes on the short pieces, and locate where you need to make holes on the other large bit
- make the holes and use the fasteners to complete the instrument
I’ve also used parallel rulers with students in Years 7 and 8 – when we have studied parallel lines and locus.
There’s something about the movement that adds dynamism to the concept of angle. The students can see the corresponding angles remaining the same, and also the alternate angles.
With locus work you can consider the locus of a point on the hinge (what can you say about its distance from the pivot?), or on the moveable rule. The locus can be traced out using dynamic geometry software.