In this unit we use rulers and compasses to construct perpendicular lines and to bisect angles. These constructions are used to make a variety of polygons, nets and to illustrate loci.
 construct perpendicular bisectors of lines
 construct bisectors of angles
 use these skills to construct equilateral triangles and squares with a given side length, parallel lines, parallelograms and trapeziums, and regular polygons with a small number of sides
 use construction techniques, given defined parameters, to produce nets and to illustrate loci.
This unit gives students an opportunity to use mathematical skills to construct a variety of shapes. These skills will build upon geometrical knowledge of basic shapes, symmetry, similar triangles, angles, and polygons, as well as Pythagoras’ Theorem. In particular students should know how to find the interior angles of regular polygons for the last session of this unit.
Students should be able to use a protractor to check the angles in some of their constructions.
This work provides an opportunity for them to explore and to bring together a number of pieces of knowledge to solve problems. It should strengthen their understanding of basic 2dimensional shapes, nets and loci.
Having established a feeling for two dimensions the students will later go on to ideas in three dimensions that will lay the basis for 3dimensional vectors and later the important generalisation of vector spaces.
Protractor
Compasses
Copymaster: Illustrated constructions
Rulers
Session 1
In this session we explore rulers and compasses and confirm what they can do.
Teachers’ Notes
All of the constructions that are used in this session are to be found in copymaster called Illustrated constructions.
In this session we encourage students to experiment with their rulers and compasses (not compasses) to make up a variety of shapes. Constructions that groups might suggest include:
 triangle with given side lengths
 equilateral triangles
 regular hexagons of given side length
 irregular pentagons with given sides
 squares of given size (can’t be made accurately until they know how to construct a right angle).
Teaching sequence

Lead a discussion on constructions.
What shapes can you make if you just use a ruler? Can you make a list of these?
They should be able to see that only shapes involving straight lines and measured lengths can be made here. This means that they might be able to construct any kind of polygon.
What shapes can you make if you just use a compass? Can you make a list of these?
We can certainly make circles but we can’t specify their radii if we don’t have a ruler.
What shapes can you make if you use both a ruler and a compass? Can you make a list of these?
The two together should enable us to get a lot more shapes. 
Let the students work together in pairs to produce a list of shapes that they can make using rulers and compasses. They should be able to convince you that they can actually make these shapes. Try to keep them clear of the constructions that you are going to introduce in later sessions.

The groups should also make up problems for the other groups. Again, these problems should only require applications of the instruments.

Work with each group to keep them on track and help them where necessary. Don’t worry too much at this stage if the list is not totally exhaustive or accurate. The aim of this exercise is to get them thinking and to encourage their creativity.

Bring the class back together and let various groups pose their problems. Give different groups/students the opportunity to either say how to solve a particular problem or to say that the shape cannot be made.

Come to a conclusion about what can be accurately made using ruler and compasses in a simple way. This will include any triangle of given side lengths; regular hexagons with given side length; and polygons with given side lengths that are not necessarily regular.

Give the class the opportunity to construct two of these figures by giving them specific examples from the students’ problems of step 3.

Give a practical application of the skills covered in this session. This task could be: A tetrahedron is a platonic solid made from four equilateral triangles. Use a ruler and a compass to construct a net for a tetrahedron with 6 cm long edges.
Session 2
In this session we introduce the method of constructing a right angled triangle and use this to construct squares and right angles.
Teachers' Notes
All of the constructions that are used in this session are to be found in copymaster called Illustrated constructions.
Discuss making a right angle without a protractor.
Talk about how to develop this (guided discovery) so that it isn’t just the teacher telling them what to do. Try these steps:

Where can we start? What can we construct? Certainly we can construct triangles.

Do all triangles have a right angle? Not necessarily but they do have height. How can this help?

How can we construct their height?

It’s easier with two similar triangles. To get a perpendicular we need to have another copy of the first triangle underneath it.

So use rulers and compasses to construct a triangle wiht one horizontal side. Now draw the same triangle reflected on the horizontal side.

Join the top vertex of the first triangle to the bottom vertex of the bottom triangle. This line is the height of the two triangles and therefore it’s perpendicular to the base of both.

But how do we put the perpendicular where we want it? With the method we are using it can end up anywhere.
Demonstrate by using different triangles. 
How can we get control over this? How about we use an equilateral triangle?

Are equilateral triangles the only ones that will work for us? Actually any isosceles triangle will do.

So do we need to measure the sides of the triangle? It turns out that this perpendicular bisects the original line! And that puts it right where we want it.
Here are some quadrilaterals to construct:
Squares with side lengths 5 cm, 7 cm and 10 cm.
Rectangles with side lengths 3 cm and 5 cm; 4 cm and 6 cm ; and 5 cm and 8 cm.
Right angled triangles with side lengths 3 cm, 4 cm, and 5 cm; 6 cm, 8 cm, and 10 cm; and 5 cm, 12 cm, and 13 cm.
A right angled triangle can be made by using the right angle construction or by constructing a triangle with sides of length 3, 4, 5 or 5, 12, 13, etc.
Teaching sequence

In the first session we were able to make some shapes with rulers and compasses but we had trouble with squares because we were unable to make a right angle. So how can we make a right angle?

Have a class discussion that goes through the guided steps for making a perpendicular to a given line (see
Teachers' Notes steps 110). Go over this a couple of times so that they see the logic of it. Then go over the final construction again so that they can see how it works. 
Give the students ample chance to work on their own to make right angles. Insist that they do this starting with a line of a fixed length (chosen by the student) that should be at varying angles to the horizontal. (This will make them more flexible in their use of the construction.) They should do about three examples here. Ask them to measure the distance of the right angle from the end of their fixed lines. They should check their angles using a protractor.

Discuss with the whole class any problems that they had. Follow this by noting where the right angle is appearing. You might draw up a table that has two columns, one for the length of the original line and the other for the distance of the right angle from the end of that line. It should then be clear that the right angle is half way between the two ends of the line. The perpendicular bisects the line.

Now return to the question of making squares.
It’s easy to draw a line that is the first side of the square but how can we put the right angle exactly on the end of this line? (extend the line to twice its length) 
Then set the students to individually make squares with given side lengths. Insist that not all of the squares should have sides that are parallel to the edges of the paper they are using.

Check on their work. As students finish that task, challenge them to construct rectangles with given side lengths. They should do at least three of these.

Continue to check on their work. As students finish the rectangle task, challenge them to construct right angle triangles with given side lengths in two ways.

Give the students a loci problem that can be solved using the skills and knowledge covered in this session. An example of such a problem is: Construct the loci of the points that are 16mm from a rectangle of dimensions 45mm by 65 mm.
Session 3
Here we look at the problem of bisecting an angle and use this to construct angles of a given size.
Teachers’ Notes
All of the constructions that are used in this session are to be found in the copymaster called Illustrated constructions.
Encourage the class to think about how you might bisect an angle. Perhaps we could set up up similar triangles again. But how? (See diagram.)
We can construct these triangles by making equal arcs on the lines that make the angle with the point of the compasses on the angle itself; then makes arcs from these two points. They will meet at the other common vertex of the two similar triangles. The common side of the two similar triangles bisects the given angle.
To make a 30º angle, construct an equilateral triangle and bisect one of its angles. Now it’s easy to make 15º and 7.5º angles.
To make a 45º angle, construct a right angle and bisect it. Now it’s easy to make a 22.5º angle.
Is it possible to make angles that are not halves or halves and halves or … of 60º or 90º angles? How about 62.5º?
Note that the Greeks dearly wanted to trisect an angle with ruler and compasses. It turns out that this is impossible. Proving that is very difficult and requires quite a lot of maths.
Teaching sequence

Recall the construction of the last session.
What did it do? (produce a right angle, bisect a line)
Can we extend these ideas to other constructions? (bisect an angle)
How can we bisect any angle?
Discuss their suggestions. 
Go through the development of the construction as given in the
Teacher s’ Notes. 
Let students work individually to bisect a number of angles. Let them produce the angles using protractors and check their construction in the same way. Insist that the lines that make up the original angle are sometimes not both parallel to the edge of the paper. They should do at least three examples.

Check their work. As they finish their three free examples, challenge them to produce angles of 30º and 45º without using a protractor.

Continue to monitor their work. As they finish the last task ask them what other angles they can produce without using a protractor.

Have a class discussion on the last two challenges.
Are there any patterns?
Is it possible to trisect a line or an angle?
Discuss their ideas. 
Give students an application of all the skills and knowledge covered in these first three sessions. An example of such a problem is: An architect is designing a 5m by 8m rectangular room with a diagonal pipe, providing underfloor heating, running from two opposing corners. Two further pipes are to be added, each bisecting the angle made by the pipe and an 8 m wall.Construct a 1:100 scale diagram of the piping layout for this room.
Session 4
Now it is time to construct parallel lines and use them to make parallelograms and trapezia.
Teachers’ Notes
All of the constructions that are used in this session are to be found in the copymaster called Illustrated constructions.
Again encourage the students to think about how they might construct parallel lines. We show a method in the Construction section below.
But what if we wanted to make sure that the parallel line went through a given point not on the original line? How would that change things?
Now we have that construction we can tackle parallelograms and trapeziums.
Teaching sequence

Refresh the students memory about what has been done in the past three sessions. Get them to describe each of the constructions used so far and how they have been used to produce different figures.
What kinds of straightsided figures can’t we construct by the methods that we have used so far? (parallelograms, trapeziums)
What extra thing would be helpful to have in order to construct these figures? (parallel lines) 
Discuss what it would be good to be able to construct and how that construction could be used to make more figures.

Let the students go off in pairs to try to produce a pair of parallel lines using rulers and compasses. Tell them to record their method. Ask them to think about how they will check that the two lines they draw are parallel. If they can make parallel lines one way, ask them to see if they can find another method.

Have a reporting back session. Discuss the different ways that the students found to make parallel lines.
Which of these is the neatest?
Which is the most efficient?
Can any of these methods produce a line that is parallel to a given line and through a particular point? Why might this be important? 
Again send them off in pairs to construct any two parallelograms. Then restrict them to two parallelograms with given side lengths and a given angle between adjacent lines.

As soon as a pair has finished this task correctly and can explain what they have done, move them on to two arbitrary trapeziums and then two trapeziums with specific side lengths.

Give the students a loci problem that utilises these skills. An example is: A goat is tethered by a 3m chain which runs freely along a 12 m cable stretched between two waratahs. Construct a scale diagram of the area of the paddock in which the goat is able to roam.
Session 5
The students now apply what they have discovered to make as many different regular polygons as they can.
Teachers’ Notes
All of the constructions that are used in this session are to be found in the copymaster called Illustrated constructions.
In this session, students need to know the interior angle formula for regular polygons.
What regular polygons can you construct with rulers and compasses? Certainly you can get an equilateral triangle, a square, and a regular hexagon. But what others can be done? If you know how to divide a side in two you can get regular octagons and regular dodecahedrons. The general problem of what regular polygons can be made was solved by Gauss. It turns out that the answer is related to some interesting prime numbers.
Using the above techniques, students can now construct tessellations both regular and irregular. Perhaps this is something that they can do for homework.
Teaching sequence

Review what has been done so far with constructions. Get them to recall what a regular polygon is and how big the internal angles of a regular nsided polygon are in terms of n.
What angles can you construct? 
Discuss (180º – straight line; 90º – by perpendicular bisection; 60º – in an equilateral triangle; any angle that is half of any of these angles or half of half of any of these angles, etc.)
Which regular polygons can you construct? 
Discuss this. (So far they have constructed a regular polygon with three sides – equilateral triangle, with four sides – square, and with six sides – regular hexagon.)
How did you construct these polygons?
Discuss. 
Send them off in pairs to see if they can construct a regular octagon. Check that groups are on track. If they are taking a long time to produce a regular octagon, ask them what the interior angles of a regular octagon are. How might they construct such an angle?

Perhaps they should then construct another regular octagon of given side length, 5 cm say.

As students finish that task ask them to make a list of all of the regular polygons that can be constructed by ruler and compasses and all that can’t.

Discuss their results and Gauss’ role in the problem.

Introduce a practical application of the skills covered in this unit. An example is: Use construction techniques to produce a net for a dodecahedron that can just fit on an A4 sheet of paper. Or, for a large group activitiy, it might be a fun project to construct a regular polygon with about 100 sides on a large sheet of paper.