Geometry

4.9 GEOMETRY

GEOMETRY LESSON IDEA 1

TITLE: DIFFERENT TYPES OF TRIANGLES
TOPIC: TRIANGLES
AIM:

1. That students will be able to recognise various types of triangles.
2. That students will be provided with concrete experience dealing with triangles.

RESOURCES:
The worksheet opposite with various types of triangles.

METHOD:
The students are introduced to a series of triangles on a worksheet, an example of which is presented opposite. The triangles comprise a mixture of isosceles, right angled, scalene and equilateral triangles. Their task is to determine the lengths of the sides, the magnitudes of the angles and consequently the type of each triangle presented. The results can be presented in tabular form, as shown. For ease of reference, angles and sides of each triangle may be labelled 1,2,3.

CLASSROOM MANAGEMENT IMPLICATIONS:
Students can work individually on the above exercise but should then be encouraged to share and discuss their results with their peers as the exercise goes on.

NOTE:

1. The worksheet opposite was constructed using a standard word processing package that incorporated a basic set of drawing tools.
2. Teachers familiar with dynamic geometry software will know that such packages can be used to prepare these worksheets also. An additional advantage is that they can also provide a teacher's master copy with a printout of the lengths of sides and angles in the given triangles.
3. Familiarity with isosceles (iso-skeles means "equal legs") and equilateral triangles can be further enhanced if students use cut-out triangles from the same sheet. The sides of each triangle can be compared by folding along a suitable axis.
4. The Irish phrases used to describe these different types of triangles can give an insight into their properties. Thus, for example, "triantán comhchosach" (equal legs) describes the isosceles triangle, while "triantán comhshleasach" (equal sides) describes the equilateral triangle.

GEOMETRY LESSON IDEA 2

TITLE: PROPERTIES OF TRIANGLES, PARALLEL LINES AND PARALLELOGRAMS
TOPIC: TRIANGLES, PARALLEL LINES AND PARALLELOGRAMS
AIM:

1. That students begin their study of geometry with a number of concrete experiences and spatial visualisations as geometry is the study of shape and space.
2. In particular that students will make a number of discoveries concerning triangles, parallel lines and parallelograms.

RESOURCES:
A set of triangular tiles constructed from cardboard

METHOD:

1. Mark off the corresponding angles in each triangle as shown.
2. The following is just a selection of some of the discoveries that students can be encouraged to make:
   - the equality of alternate, corresponding and vertically opposite angles
   - the external angle of a triangle is equal to the sum of the two interior opposite angles
   - opposite sides and opposite angles of a parallelogram are, respectively, equal in measure
   - a diagonal bisects the area of a parallelogram.

CLASSROOM MANAGEMENT IMPLICATIONS:
Students can work individually on the above exercise but should then be encouraged to share and discuss their results with their peers as the exercise goes on.

GEOMETRY LESSON IDEA 3

TITLE: BISECTING ANGLES AND CONSTRUCTING THE INCIRCLE OF A TRIANGLE USING A DYNAMIC GEOMETRY PACKAGE
TOPIC: GEOMETRY CONSTRUCTIONS
AIM:

1. That students will learn how to bisect an angle using dynamic geometry software.
2. That students will observe, by altering the size of the angle, how the angle bisector alters to divide the new angle into two equal parts.
3. That students will use the knowledge gained to bisect the three angles of a triangle and thus find the incentre.
4. That students will use the incentre to construct the incircle of the triangle and observe how the construction is preserved when a vertex of the triangle is dragged.

RESOURCES:
Computers with dynamic geometry software installed. Basic working knowledge of the package. (The method outlined opposite is based on Geometer's Sketchpad.)

METHOD:

1. On a new sketch with the Half-Line tool construct an angle, BAC.
2. Select each of the points B, A, and C making sure to select the common starting point second. With these points selected from the Construct Menu select Angle Bisector. Using the Display Menu show the bisector as a red broken line.
3. Click and drag on the point B and as the size of the angle changes notice how the angle bisector alters to divide the new angle into two equal parts. Construct a point D on the bisector. Record the measures of the angles indicated in the table. Click and drag on B a number of times to complete the table.
4. On a new sketch, with the line segment tool construct a triangle, ?ABC.
   Select each angle in turn, taking care to select the vertex point second in each case, and construct the bisector of each angle.
   Construct the point of intersection D of the bisectors by selecting any two of them and from the Construct Menu choose point of intersection.
   From the point D drop a perpendicular line to the line segment [AC]. Find the point E where thisperpendicular line intersects [AC].
   First select the point D, hold down shift and select the point E.
   From the Construct Menu choose Circle by centre and point.
   Display this circle by a thick green line.
   D is called the incentre and the circle is called the incircle.
   Click and drag on A, B and C in turn and notice how the incircle construction is preserved.
   What can you say about the incentre when the triangle is isosceles, equilateral ?

The skills acquired in the above lesson can be reinforced by asking students to carry out the following tasks.

1. On a new sketch,
   (i) Construct ?BAC measuring 120º.
   (ii) Construct the angle bisector.
   (iii) Construct a point D on the bisector.
   (iv) Measure ?BAD and ?CAD.
   (v) Check that ?BAC = ?BAD + ?CAD.
   Show all angle measurements on the screen.
   (vi) Save your work.
2. On a new sketch,
   (i) Construct a triangle.
   (ii) Bisect the angles.
   (iii) Construct the incentre.
   (iv) Construct the incircle.
   (v) Save your work.

CLASSROOM MANAGEMENT IMPLICATIONS:
Students might like to work in pairs on the initial skillbuilding exercise but they should then be encouraged to try out the exercises on their own and to share their mathematical experiences. There is ample opportunity here for enhancing their communication skills in mathematics.

GEOMETRY LESSON IDEA 4

TITLE: THE ANGLES IN A TRIANGLE SUM TO 180º - A DEMO!
TOPIC: THEOREMS - AN EXPERIMENTAL APPROACH
AIM: That students will demonstrate how the three angles in a triangle sum to 180º.

RESOURCES:
Paper or card, scissors, straight-edge.

METHOD:

1. Use a straight-edge to draw a triangle onto a sheet of paper or card (Fig. 1). (It is interesting to see how many of our students automatically draw an equilateral, isosceles or right-angled triangle - this gives a clue as to which kind and orientation of triangle they are most used to seeing.)
2. Cut it out carefully. Label the three corners A, B, and C (Fig. 2).
3. Now measure each angle with a protractor. The three angles may not add up to exactly 180º. Why is this? (Discussion on accuracy of drawing, accuracy of measuring, protractor is marked only in degrees, not minutes or seconds).
4. Now tear off each corner (tearing is better than cutting as it produces a jagged edge which makes the vertex easier to see) and position them with the vertices (pointed ends) together (Fig. 3).
5. They will form a straight angle measuring 180º (Fig. 4).
   Place a straight edge against it to see this. This will work no matter what size triangle you draw. What does this tell us about the measure of the three angles of a triangle?

CLASSROOM MANAGEMENT IMPLICATIONS:
Students can work individually on the above exercise but should then be encouraged to share and discuss their results with their peers as the exercise goes on. In addition, if time is an issue, the teacher can give a demonstration of the result on his/her own.

GEOMETRY LESSON IDEA 5

TITLE: AN EXTERIOR ANGLE OF A TRIANGLE EQUALS THE SUM OF THE TWO INTERIOR OPPOSITE ANGLES IN MEASURE
TOPIC: THEOREMS - AN EXPERIMENTAL APPROACH
AIM: That students will demonstrate how the exterior angle of a triangle equals the sum of the two interior opposite angles in measure.

RESOURCES:
Paper or card, scissors, straight-edge.

METHOD:

1. Use a straight-edge to draw a triangle onto a sheet of paper or card (Fig. 1).
2. Cut it out carefully. Label the three corners A, B, and C (Fig. 2).
3. Lay a straight-edge along the base of the triangle to produce the external angle D (Fig. 2).
4. Now tear off angles A and B (Fig. 3).
5. Position them with the vertices (pointed ends) together into angle D. They should fit exactly (Fig. 4).

CLASSROOM MANAGEMENT IMPLICATIONS:
Students can work individually on the above exercise but should then be encouraged to share and discuss their results with their peers as the exercise goes on. In addition, if time is an issue, the teacher can give a demonstration of the result on his/her own.

GEOMETRY LESSON IDEA 6

TITLE: USING A PHYSICAL MODEL TO SHOW THAT THE MEASURE OF THE ANGLE AT THE CENTRE OF THE CIRCLE IS TWICE THE MEASURE OF THE ANGLE AT THE CIRCUMFERENCE, STANDING ON THE SAME ARC
TOPIC: THEOREMS - AN EXPERIMENTAL APPROACH
AIM: That students will demonstrate, using concrete materials, how the measure of the angle at the centre of a circle is twice the measure of the angle at the circumference, standing on the same arc.

RESOURCES:
A wooden disc (hereinafter referred to as the bread board!) with small stud screws at the circumferenc at intervals of 10 or 20 degrees, rubber bands, scissors, coloured paper, protractor and set square. Many teachers of mathematics get their friendly materials technology colleagues to construct the wooden models.

METHOD:

1. The first idea that students need to experiment with is the concept of what an angle at the centre is. The teacher can use one elastic band to represent the diameter and this should remain for all steps below.
   Two additional bands are then used to make an angle at the centre as shown in Fig. 1 opposite.
   Students can construct various angles at the centre using the wooden model.
2. Next, the teacher can demonstrate what an angle at the circumference is as shown in Fig. 2. Again, various students can be selected to demonstrate different examples, helping to consolidate this idea of an angle at the circumference.
3. Now, the teacher can demonstrate the notion of both angles standing on the same arc ab as shown in Fig.3.
   Using the model, many different examples of this can be found by students. After each example, students should be strongly encouraged to turn the bread board around, showing that the diameter is not a fixed horizontal, or that the angle at the circumference is not fixed at the top, etc. This dynamic nature of the bread board is a huge advantage over the static model on the blackboard.
4. Two students can be asked to measure the angle at the centre with a protractor and then to cut this angle from coloured paper and place it at the centre angle as shown in Fig. 4. Pupils can then be asked then to explore how this angle at the centre compares with the angle at the circumference (both standing on the same arc ab).
   It is most useful, and makes quite an impact, if the angle at the centre is then taken, folded over, and superimposed on the angle at the circumference. Students can readily see in a concrete fashion that the latter angle is half the angle at the centre, thus laying a good foundation for the formal proof of this theorem.

CLASSROOM MANAGEMENT IMPLICATIONS:
If only one bread board is available the above steps can be effectively carried out in demonstration mode using different students for various steps. Alternatively, if several wooden models are available students can work in small groups of two or three but should then be encouraged to record, share and discuss their results with their peers as the exercise goes on. In addition, if time is an issue, the teacher can give a demonstration of the result on his/her own. Teachers who restrict this exercise to a demonstration often retain their cut-out coloured angles and keep them in a small paper pouch attached to the reverse side of the bread board for future use and indeed for revision purposes to consolidate the ideas learnt.

NOTE:
Students may use a similar experimental approach for the three associated deductions.

GEOMETRY LESSON IDEA 7

TITLE: THE THEOREM OF PYTHAGORAS - A DEMO!
TOPIC THEOREMS - AN EXPERIMENTAL APPROACH
AIM That students will demonstrate the Theorem of Pythagoras.

RESOURCES:
Paper or card, scissors, straight-edge, overhead projector.

METHOD:

1. Construct two squares of any size side by side (Fig. 1).
   Let the larger square have side length a, and the smaller square have side length b. The area of this figure is thus a2+b2.
2. Mark off a section of length b along the base of the larger square. Construct lines as shown, of length c.
   c is the length of the side opposite the right angle of a right angled triangle with other sides of length a and b (Fig. 2).
3. Cut out the two triangles thus constructed and label them X and Y. Rotate X anti-clockwise and rotate Y clockwise (Fig. 3).
4. Continue to rotate these pieces until a square with side c and area c2 is formed (Fig. 4).
5. The initial area of a2+b2 was preserved, so a2+b2=c2 where a, b, c are sides of a right-angled triangle, c being opposite the right angle

CLASSROOM MANAGEMENT IMPLICATIONS:
This is a complicated procedure and would probably be best done as a demonstration by the teacher using an overhead projector.

NOTE:
Once students have been led through one or two of these demonstrations in geometry lesson ideas 4-7, they can be given a result to demonstrate for homework. The resulting mound of cut-outs often proves interesting, particularly if students are given the opportunity to "explain" their demonstration to other members of their group or to the whole class. This gives students a chance to use geometrical language.

GEOMETRY LESSON IDEA 8

TITLE: DISCOVERING PYTHAGORAS
TOPIC: THEOREMS - AN EXPERIMENTAL APPROACH
AIM: To encourage students to try to "discover" what Pythagoras proved, by experimenting with squares of different areas.

RESOURCES:
Coloured cardboard sheets, graph paper, straight-edge and scissors.

METHOD:

1. Students (or the teacher) should construct a range of squares of side 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 units from different coloured cardboard sheets.
2. In groups (or individually, if resources permit) students have to try to make right-angled triangles from the given squares as shown in Figure 1.
3. It can be helpful to use a sheet of graph paper to confirm that the triangle is right-angled (see Figure 2).
4. Students can record their findings in a table.
5. Students should discover that squares of side 3, 4, 5 and 6, 8, 10 form right-angled triangles (the graph paper is a good help). Students then calculate the area of each of these squares and record their work as shown in figure 2.

NOTE:
Follow-up work could include a class discussion along the following lines:
Is there a pattern to the answers 9, 16, 25 and 36, 64, 100?
Students are encouraged to discover that 9+16 = 25 and 36+64 = 100.
Now add another square of side 13 to the squares already in use. Can students find one more right-angled triangle (5, 12, 13)? Does the pattern still hold?
Describe the pattern for the sort of squares that form a right-angled triangle.
Finally, this activity can be revised and assessed by the worksheet such as that used in geometry lesson idea 9.

CLASSROOM MANAGEMENT IMPLICATIONS:
Students can work individually (if resources permit) or in small groups.

GEOMETRY LESSON IDEA 9

TITLE: THEOREM OF PYTHAGORAS - RIGHT-ANGLED OR NOT?
TOPIC: THEOREMS - AN EXPERIMENTAL APPROACH
AIM: That students will apply the converse of the theorem of Pythagoras to check if triangles are right-angled or not. This lesson idea requires that lesson idea 8 has already been used in class.

RESOURCES:
The worksheet.

METHOD:
Recap on how the "3,4,5" triangle obeys the theorem of Pythagoras:
The area of the square on the side measuring 4 units was 4×4=4²=16 units squared.
The area of the square on the side measuring 3 units was 3×3=3²=9 units squared.
The area of the square on the side measuring 5 units was 5×5=5²=25 units squared.
Then it was noticed that the two smaller areas add up to the bigger area:
4² + 3² = 5²
16 + 9 = 25
By calculating the area of the squares on each side of the triangles in the worksheet find out which are right-angled and which are not. Here is an example.

EXAMPLE:
Area of square on side length 6 is 6×6=6²=36
Area of square on side length 8 is 8×8=8²=64
Area of square on side length 10 is 10×10=10²=100
Check:
Is 36+64=100? YES
So this triangle IS a right-angled triangle.

CLASSROOM MANAGEMENT IMPLICATIONS:
Students can work individually on the above exercise but should then be encouraged to share and discuss their results with their peers as the exercise goes on. The calculator can be used if the calculations become too cumbersome.

GEOMETRY LESSON IDEA 10

TITLE: PYTHAGOREAN TRIPLES
TITLE: THEOREMS - AN EXPERIMENTAL APPROACH
AIM: That students will revise and apply their knowledge of the theorem of Pythagoras.

RESOURCES:
The calculator can be used to accelerate the learning process.

METHOD:

1. From previous exercises and activities, Pythagorean triples like (3, 4, 5), (6, 8, 10) and (5, 12, 13) may well have been discovered by the students themselves. If so, they will remember them better.
2. Break the class into teams of "table quiz" type and see how many Pythagorean triples they come up with in a specified length of time, for example 7 minutes.

CLASSROOM MANAGEMENT IMPLICATIONS:
The class should be organised into small groups of three or four for the table quiz.

NOTE:
It is possible for teachers with a basic knowledge of a spreadsheet package to construct a spreadsheet file using elementary algebraic formulae, and so generate any desired number of Pythagorean triples. The columns in the spreadsheet - and the calculations to be performed on the chosen numbers x, y (with x > y) entered in the first two columns - are given by:

The final three columns yield Pythagorean triples.

GEOMETRY LESSON IDEA 11

TITLE: PYTHAGORAS AND WALLS!
TOPIC: THEOREMS - AN EXPERIMENTAL APPROACH
AIM: That students will verify the theorem of Pythagoras with the aid of walls in their classroom.

RESOURCES:
One or two lengths of timber, a measuring tape and calculator.

METHOD:

1. By varying where they position the length of timber, teams of students can take turns to measure the three sides of the triangle formed between the piece of timber and the two walls.
2. Using the calculator, the students can verify that the theorem of Pythagoras holds true.
3. This exercise can be related to workers who build a house and who want to check for right angles in the corners. Likewise the workers who do the internal plastering often have a large right-angled triangle with them to check that the walls form right angles at the corners.

CLASSROOM MANAGEMENT IMPLICATIONS:
Small teams of three students can be chosen for this activity, taking their turn at measuring and calculating. The activity can be repeated a number of times.

GEOMETRY LESSON IDEA 12

TITLE: SIMILARITY V. CONGRUENCE AND RATIO
TOPIC: GEOMETRY
AIM:

1. To give students an opportunity to discover the difference between similar and congruent triangles.
2. To give students an opportunity to calculate ratios and get a feel for the theorem which states that a line drawn parallel to one side of a triangle divides the other two sides in the same ratio.

RESOURCES:
Two A4 sheets of coloured card (different colours), compass, scissors, calculator, paper and pencil.

METHOD:

1. On one coloured A4 sheet students are asked to draw a scalene triangle, the larger the better. Next, they have to draw a line parallel to one side and cutting the other two (see diagram). There are many different ways of drawing parallel lines and much debate as to the best method ensues.
2. Now the students have to make an exact replica of the upper (smaller) triangle from the second coloured A4 sheet. Again, many methods might be used but the unsophisticated method of "pinning through" the vertices of the small triangle with the compass point onto the other sheet placed underneath works well. The new triangle is then cut out.
3. Placing the new triangle on top of the original one can show corresponding angles and emphasises "similarity" v. "congruence". Students should be allowed to discover, as they position the new triangle in each corner, that the "other side is always parallel". This configuration arises time after time in the mathematics programme and the students should be helped to recognise it instantly.
4. The ratio of the divided lines can now be investigated. The students need to learn to measure accurately in millimetres and to make sensible use of the calculator. They should be asked to investigate the link between the length of the corresponding sides of the small and large triangles.
   Within reasonable tolerances the ratios should be the same. Reasonable measurements, and sensible decimal places decided from the calculator result, will be the issues for debate in the final part of this exercise.
5. The new triangle can be repositioned and the process repeated. (It is perhaps easier for the students to see the equal ratios if the smaller value is divided into the larger.) As each student will have a differently sized scalene triangle the generalisability of this "rule" will be quite obvious, and the visual impact of the models will help the students to remember both the rule and the configuration.

NOTE:
From this work a number of mathematical terms can become part of the students' lexicon. These include congruence, similarity, scalene, vertex, ratio, proportion, corresponding, parallel, enlargement, reduction, translation, rounding up/down, accuracy, and estimation.

CLASSROOM MANAGEMENT IMPLICATIONS:
Students can work individually or in pairs when doing the investigative work.

GEOMETRY LESSON IDEA 13

TITLE: HIT OR MISS! [THIS ACTIVITY IS BASED ON THE CHILDREN'S GAME OF BATTLESHIPS, WHICH USES A COORDINATED GRID OF LETTERS AND NUMBERS.]
TOPIC: COORDINATE GEOMETRY
AIM: That students will learn an entertaining way to draw and read coordinate points.

RESOURCES:
Graph paper.

METHOD:

1. Students work in pairs for the game. Each student draws two grids as shown. One grid is to display the student's own battleships and the other to record guesses of the opponent's positions.
2. All students fill in their fleet of ships on their home grid. Each student has a fleet of 1 two-dot, 1 three-dot and 1 four-dot ships. Each ship can be placed on the grid in a horizontal or vertical position.
3. One student guesses a pair of coordinates and records them on the away grid and in the table of values. If this is a hit, the same student guesses again. When a hit is made, it is normal for the opponent to indicate if the battleship is a two-dot, three-dot or four-dot ship. If it is a miss, the opponent takes a turn.

Sample Game of Hit or Miss
I go first. I guess (2,2). It is a hit and I record this with a heavy dot (see table and away grid. I guess (2,1). It is a miss and I record this information with an x (see table and away grid).
My opponent guesses (0,-1) and I say that it is a hit and put a + onto my home grid. The next guess is (1,-1). Another hit. The third guess is (2,-1) - a miss which I record with an x on my home fleet grid.
I guess (1,2), which is a hit, and (0,2), another hit. These are recorded by two more heavy dots on the away grid. I have now sunk my opponent's 3-dot ship.
And so the game continues until one person has sunk all of the opponent's ships. The grids ensure that if a student is confusing (2,1) with (1,2), the opponent can check that the guesses correspond with correctly reported hits or misses once the game has finished.

CLASSROOM MANAGEMENT IMPLICATIONS:
Students play this game in pairs. If the grids are drawn in pen and the coordinates in pencil, the grids can be re-used.

NOTE:

1. If playing properly, students will check each other's knowledge of reading and plotting co-ordinate points.
2. The rules of the game can be changed to make the game move along more quickly or slowly as required.
3. It also works well with a positive number grid rather than the integer grid used above and this might be a more useful starting point with some students.

GEOMETRY LESSON IDEA 14

TITLE: CONGRUENT TRIANGLES
TOPIC: SYNTHETIC GEOMETRY
AIM: To introduce students to the idea of congruency with concrete materials. To give students practice at constructing triangles, given any of the following four sets of data: SSS, SAS, ASA, or RHS.

RESOURCES:
Light cardboard, scissors, geometry set and straight-edge.

METHOD:

1. The idea of congruent triangles can be conveyed to the students by getting them to construct triangles of light cardboard (such as that in cereal boxes), using given data, and cutting them out. All the triangles of the same measurements can be compared and seen to be identical.
2. For example, for homework, get each of the students to draw a triangle with side lengths 13cm, 12cm and 10cm (it needs to be reasonably big) on cardboard and bring it to class the next day. The teacher may have an accurate "template" against which to compare the triangles, and may discard inaccurate attempts. When all the accurate triangles are placed on top of each other, the students can see that it is impossible to draw a triangle with these measurements which is not identical to all the others. So SSS is seen to be enough to establish congruence.
3. The same can be repeated for three other triangles given SAS, ASA, and RHS.
4. The advantage of this method is that while students are improving their skills in constructing triangles, they are also becoming convinced of the fact that congruence in triangles can be established, given any of the four conditions.
5. A variation on the above exercise involves students being presented with a page of triangles on a worksheet and a cut-out triangle of light cardboard. The students are asked to compare the given cut-out triangle with those on the sheet and to test for congruency. Any computer drawing package can be used to prepare the cut-out triangle. The copy and paste commands can then be used to produce identical images which can be rotated into different positions. Finally, a number of non-congruent triangles can be added to the sheet before printing. This can be repeated a number of times using different triangles. A sample worksheet is presented opposite.

CLASSROOM MANAGEMENT IMPLICATIONS:
Students should be encouraged to work on their own with the worksheets and later to compare and discuss their results with their peers.

FURTHER TEACHING NOTES ON CONGRUENT TRIANGLES, EQUIANGULAR TRIANGLES, AREA AND PARALLELOGRAMS

NOTE 1:
Students may mix up the concepts of congruent triangles, triangles of equal area and equiangular triangles which they meet later. It should be emphasised that congruent triangles have equal areas but triangles of equal areas are not necessarily congruent.
"AAA" does not give congruence, obviously, as it is possible to draw two equiangular triangles of very different sizes (suggestion: give as an exercise to the students the task of drawing three different sized equiangular triangles).

Class Experiment:
Try to draw a triangle with sides of length 10cm, 3cm and 5cm. Why is this impossible?

NOTE 2:
The distance from a point to a line is the length of the perpendicular from that point onto the line.

NOTE 3:
Any of the three sides of a triangle may be called a BASE.
The perpendicular height of the triangle is the distance from the opposite vertex to a chosen base.
Similarly, any of the four sides of a parallelogram may be called a BASE, and the perpendicular height of the parallelogram is the distance to the chosen base from any point on the opposite side of the parallelogram.

NOTE 4:
Why is the area of a rectangle = Length x Breadth ?
Example: A rectangle measuring 3 units by 2 units.
By counting, this has 6 square units of the size shown on the left. Since multiplication is a quick way to add, we get 3 times 2 = 6 square units.
Why then do we not get the area of a parallelogram by multiplying length by breadth? For example, is the area of the parallelogram below equal to or less than the area of the rectangle which has the same side lengths? The answer will not be obvious to the student.
But if we continue to "squeeze" the parallelogram downwards, it begins to look like this:
We can see now that the area is definitely smaller and could eventually become close to zero.
The area of a parallelogram can be arrived at in the following steps:
(a) Start with the "fact" that the area of a rectangle = Length x Breadth.
(b) A triangle has half the area of a rectangle of the same height and base and therefore the area of a triangle = (1/2)Base x Height.
(c) Since a diagonal bisects the area of a parallelogram and divides it into two triangles of equal area, the area of a parallelogram = Base x Height.

NOTE 5:
The following properties of parallelograms may be interesting to investigate/prove:

• Opposite sides and angles are equal in measure
• Diagonals bisect each other (the proof is a nice exercise!)
• A diagonal bisects its area
• Area = Base x Height.
• Any pair of adjacent angles sum to 180° (easy to prove)
• The diagonals divide the parallelogram into four triangles of equal area (prove)
• Only in the case of a rhombus are the diagonals perpendicular (prove)
• Only in the case of a rhombus does a diagonal bisect the angle(s) through which it passes - this deceives a lot of students - (prove).