Quadrilaterals

Purpose

In this unit we conduct a couple of investigations looking at the relationship between the angle between two diagonals of a quadrilateral, the sides of the quadrilateral, and the type of quadrilateral. The main emphasis is on rectangles.

Achievement Objectives
GM4-5: Identify classes of two- and three-dimensional shapes by their geometric properties.
Specific Learning Outcomes
  • investigate the relationship between the diagonals and lengths of a rectangle
  • investigate the relationship between the angle of the diagonal and length of rectangles sides
  • use rulers, compasses and protractors accurately
Description of Mathematics

In this unit the students will (i) see how to draw the circle that goes through all of the corners of a rectangle; (ii) explore the angles made by the diagonals of rectangles and see how this is related to the ratio of the lengths of the sides of the rectangle; and (iii) see what quadrilaterals arise when the diagonals meet at 90º and the sides have certain specific properties.

We give a quick outline of the basis for this unit. For (i) it is simply a matter of noting that diagonals of a rectangle bisect each other. Hence the point of intersection of the diagonals is the centre of the circle (the circumcircle, though we don’t use that word here). To draw the required circle simply put the point of the compasses on this point and open them up until the pencil meets a corner. The circle can then be drawn.

For the second point we simply want students to see that if the sides of two rectangles are in the same ratio, then the angle formed by the diagonals is the same. This works in reverse.

The material of the final session is best tackled by first drawing two lines to represent the diagonals and then trying to construct the quadrilaterals with the various conditions from there.

There are really three main reasons for doing this unit. These are (i) to explore quadrilaterals and have a better feel for them; (ii) to use mathematical instruments (ruler, compasses, protractor); and (iii) to be involved in an investigation.

Required Resource Materials

Rulers

Protractors

Compasses

Activity

 

Sessions 1 and 2
In these sessions the students review the names of basic geometric objects and learn to draw circles around rectangles.
  1. Show students in a whole class setting, a variety of polygonal shapes for them to identify.
  2. Assess their knowledge of basic concepts related to polygons such as square, rectangle, quadrilateral (any four sided polygon), kite, rhombus, vertices, edges, interior angles, diagonals.
    What is this? (Show a rectangle.)
    What is this part of a rectangle called (Point to vertices = corners, edges = sides.)
    Where would I draw in a diagonal?
    Describe a rectangle carefully.
    Describe a square carefully.
    What do these shapes have in common?
    Is a square a rectangle? (Yes.)
    What is a kite (rhombus)? What are its special features?
    What shapes have more than one name? (A square is a rectangle, kite, rhombus; a kite is a rhombus)
  3. Give the students time to make a poster with all of the different quadrilateral shapes on them.
  4. Now concentrate on rectangles. Ask:
    How would you draw any old rectangle?
    How would you draw a rectangle with given side lengths?
  5. Probably they would draw one side (with a given length); construct a right angled corner using a protractor; then measure the ‘vertical’ side and draw it in; repeat at the other side; join the two vertical edges to complete the rectangle. However, if they have used compasses they may be able to do this by constructing the right angles needed.
  6. Get one or two students to demonstrate on the whiteboard.
  7. Then let the class go away in pairs to draw three different rectangles. They need to record the lengths of these sides on their rectangles.
  8. While they are in their pairs, challenge them to find a way to draw a circle through all four vertices of their rectangles. (Quicker students should try to find a reason why their method works. They can also try to find a way of getting the diameter of their circles in terms of the sides of the rectangle.)
  9. Discuss the tasks that have been given to the quicker students. The class should come up with a good reason for what they say. (If they can’t settle this at this point allow them time later to come back to it. If they are still having trouble, tell them the answer and leave it for a while.)
  10. Now challenge them with the following questions:
    Can you draw a circle through the vertices of every kite?(No.)
    Can you draw a circle through the vertices of some kites?(Yes.)
    Can you draw a circle through the vertices of every rhombus?(No.)
    Can you draw a circle through the vertices of some rhombuses?(Yes but only if the rhombus is a square!)
  11. Give the students time to tackle these problems in pairs.
  12. Have a reporting back time. (The best way to do the second and fourth questions is to first draw a circle and then draw the special shape inside the circle.)
  13. Allow the students to produce posters on these results or put the results with justification into their maths book.
Session 3
In this session the students investigate the angle made between two diagonals of a rectangle.
  1. Remind the students of the work that they did yesterday.
  2. Then ask them:
    How would you draw a diagonal in your rectangle?
    How would you find the angle between the two diagonals of a rectangle?
  3. Note that there are two possible angles that could be measured. Make sure that you are all talking about the same one. (It doesn’t matter which but it might be consistent to always measure the smaller one.)
  4. Follow this up by:
    Do the diagonals of all rectangles have the same angle between them?
    If they do, why is it so? If they don’t what rectangles do have the same angles?
  5. Discuss this and write up on the board the general feeling(s) of the class.
    How would we test this?
  6. Discuss ways this might be done. Get them to see that it would help to be systematic. It might be useful to do a series of drawings with one side of fixed length. For instance, fix one side at 4 cm and draw rectangles where the other is 4 cm, 6 cm, 8 cm, 10 cm and 12 cm. Get them to see that they will need a table to write down their answers. A possible table is given below.

 

Side 1
Side 2
Angle between diagonals
 
 
 
 
 
 
 
 
 
 
 
 
  1. When you think that they have all got the idea, send them off in pairs to construct rectangles with a fixed side and variable other sides. It would probably be good to get some groups to draw the rectangles with sides listed above; to let other groups fix one side at 6 cm and draw rectangles where the other is 4 cm, 6 cm, 8 cm, 10 cm and 12 cm; and yet other groups to fix one side at 8 cm and draw rectangles where the other is 4 cm, 6 cm, 8 cm, 10 cm and 12 cm.
  2. When they have all completed their task bring the whole class together and combine all of their results on a class table. (Students who finish quickly might be asked to draw more rectangles to help a slower group; to guess the pattern; to see if they can find/guess the angle at the diagonals from the sides of the rectangle – even for some angles/sides; or to say if it is possible to draw a circle through the corners (vertices) of all of their rectangles.)
  3. Discuss their results. They should see that if the two sides of one rectangle are in the same ratio as the two sides of another, the angle between the diagonals is the same.
Session 4
In this session we tackle the reverse problem to Session 3 – given the angle between two diagonals, what are the lengths of the sides.
  1. Remind the class of what has happened in the last session.
  2. Let them investigate the problem: given the angle between two diagonals, what are the lengths of the sides of the rectangle?
  3. From session 3 they should realise that, at best, they will only be to find the ratio between the two side lengths. They should also tackle the problem by taking specific angles and determining the ratio by measurement. The best that they will be able to do will be to find approximate ratios for each angle (say from 10º to 90º in tens). The actual result is that tan θ/2 = a/b, where a and b are the lengths of the sides with a < b, but this will be a little beyond this level.
  4. They might also like to find out which angles come from rectangles where the sides have a ratio of 1, 2 and 3.
  5. Let the class agree on the various ratios and angles and make posters to illustrate what they have done. You might want to talk about the tan of an angle as an introduction to the work of the next level.
Session 5
Here we fix the angle between the diagonals of a quadrilateral and see what properties of sides give what quadrilaterals when their diagonals intersect at 90º.
  1. Recall the problems of the previous sessions and the methods used to solve them.
  2. Now look at quadrilaterals more generally. Ask and discuss each of the following in turn. Allow different students the chance to show (i) their answers, and (ii) their methods of construction, on the board to help the discussion:
    Is it possible to find a quadrilateral all of whose sides are different and whose diagonals intersect at right angles?
    Is it possible to find a quadrilateral all of whose sides are different and whose diagonals intersect at 60º?
  3. Send them away in their pairs to discuss the following questions. Tell them that in each case if their answer is ‘yes’ they will need to be able to construct one of the quadrilaterals. If the answer is ‘no’ they will need to be able to explain why. (However, all of these can be constructed. Some can be constructed in more than one way.)

    A. Is it possible to find a quadrilateral all of whose sides are different and whose diagonals intersect at 45º?
    B. Is it possible to find a quadrilateral with two adjacent equal sides and two adjacent sides different, whose diagonals intersect at 50º?
    C. Is it possible to find a quadrilateral with two opposite sides equal and two opposite sides different, whose diagonals intersect at 60º?
    D. Is it possible to find a quadrilateral with precisely three equal sides and whose diagonals intersect at 30º?
    E. Is it possible for you to make up your own question like this?  
     
  4. Call the class together for a reporting session. You might discuss what happens to the answers if you change the angle in the questions to any size you like.
  5. In the next set of questions the students should concentrate on quadrilaterals with diagonals that INTERSECT AT 90º. Send them away to consider these questions in pairs. Again a justification is needed. Measuring a lot of examples would be okay at this Level though you can say that a proof can be found.

    F. What can you say about quadrilaterals that have two adjacent sides equal? (They are kites.)
    G. What can you say about quadrilaterals that have opposite sides equal but not parallel?(There seems to be no pattern at all here.)
    H. What can you say about quadrilaterals that have opposite sides equal and parallel? (They are rhombuses.)  
     
  6. Discuss the results of the students' investigation. Give them a chance to write up their work.

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