Study of Quadrilaterals
Materials: Box of sticks, supplies
Presentation: The teacher constructs the first figure using a yellow (20cm), a brown (12cm), a pink (8), and an orange (6 cm), uniting them so that the yellow and orange are not consecutive sides and are not parallel. This is a common quadrilateral (trapezium). the child identifies its principal characteristic; it has four sides (quadrilateral: Latin quadras < quator, side, and lateris, side).
Invite the child to build a figure exactly like this one, of sticks. Superimpose the second one on the first to show that they are equal. Move the two sides back and forth like the arms of a balance, stopping where the yellow and orange sticks are parallel. Use two small sticks as guides to check if they are parallel. This is a trapezoid. It is a quadrilateral, but it has something more. It is a quadrilateral which has at least one pair of parallel sides.
Note: It is important for the child to understand this concept; a trapezoid is a quadrilateral, but a quadrilateral is not necessarily a trapezoid.
Invite the child to choose two pairs of like sticks; join them to form two angles, then unite them to form a quadrilateral in such a way that two sticks of the same color do not touch. This is a common parallelogram. It has two pairs of parallel sides (indicate these pairs). The common parallelogram is a quadrilateral with two pairs of parallel sides. Is it also a trapezoid? Yes, but it has something more.
Invite the child to construct with the sticks the same figure as before – the parallelogram. Superimpose this figure on the other to show that they are equal. Stand the figure on end and using the measuring angle as a guide, straighten the sides until the side coincides with the measuring angle. If one angle of this quadrilateral is a right angle, then all of the others will also be right angles. This is a rectangle. It is a parallelogram with all right angles. The previous figure was just a common parallelogram, whereas this parallelogram is no longer common. It has all right angles.
Is the rectangle a parallelogram? Yes, it has two pairs of parallel sides. Is the rectangle a trapezoid? Yes, it has at least one pair of parallel sides. Is it a quadrilateral? Yes. Is the common parallelogram a rectangle? No.
Invite the child to unite four like sticks. This is a rhombus. It is a parallelogram with equal sides. Is the rhombus also a rectangle? No. Is it a parallelogram? Yes. Is it a common parallelogram? No. Is it a trapezoid? Yes. Is it a quadrilateral? Yes. Is the rectangle a rhombus? No.
Invite the child to reproduce the previous figure. Superimpose one on the other to show that they are equal. As before, stand the figure on end to straighten its sides, using the measuring angle as a gauge. This is a square. It is a parallelogram … (is that all I need to say? No) …. with equal sides … (is that enough?) …. with right angles. Repeat the definition. By saying that it has equal sides, the common parallelogram and rectangle are excluded. In order to exclude the possibility of being a rhombus, however, another quality must be added – right angles. Questions similar to those for the rhombus may be posed.
These are all the quadrilaterals of reality.
B. Quadrilaterals – Types of trapezoids
Materials: Box of sticks, supplies Geometry cabinet insets, reading labels -“square”, “rhombus”, “rectangle”, “trapezoid” Additional insets: common quadrilateral, common parallelogram, three trapezoids (all except the equilateral) Second level reading labels – “common quadrilateral”, “common parallelogram”, “isosceles trapezoid”, “scalene trapezoid”, “right-angled trapezoid”, “scalene trapezoid”, “right-angled trapezoid”, “obtuse-angled trapezoid”
Presentation: Isolate the trapezoid previously constructed with sticks. In order to extend our family of trapezoids, we must first know its parts. The two sides which are parallel are bases. Identify the other two as sides. Are the sides equal? No. This is a scalene trapezoid. Recall the meaning of scalene from the triangles.
Invite the child to take four sticks, two of which are equal. Unite them so that the two sticks are not touching each other. Arrange them so that the two bases are parallel. Identify the two bases and the two sides. Are the two sides equal? Yes. This is an isosceles trapezoid.
The teacher takes four sticks (yellow – 20cm, black – 10cm, orange – 6cm, brown 12cm) and unites them so that the yellow and black sticks form the parallel sides. Identify the bases and the sides. Use the measuring angle to show that a right angle is formed by the perpendicular side. This is a right-angled trapezoid.
The teacher takes four sticks (yellow – 20cm, green – 14cm, red – 8cm, brown 12cm) and unites them so that the yellow and green sticks form the parallel sides. Use the measuring angle to identify the two obtuse angles. They are opposite each other. This is an obtuse-angled trapezoid.
Note: It is not so much the number of obtuse angles, but the opposite position of the obtuse angles which is important. This is the only trapezoid which can be divided into two obtuse-angled triangles.
Bring out all of the quadrilateral formed with sticks (common quadrilateral, the trapezoids, common parallelogram, rectangle, rhombus, square) and organize them as the child recalls the name of each. Bring out the old labels and match them to the figures, This old label “trapezoid” will be of no use anymore, and discard it. Bring out the new labels and match them accordingly. Each member of the family of trapezoids has its own characteristic; therefore they each have a different name.
Match the insets of the geometry cabinet with the figures: square, rectangle, rhombus. Before the trapezoid was known only as a trapezoid. Identify its other characteristic: isosceles. Bring out the additional insets and match them as well.
C. Quadrilaterals – Classification according to set theory
Materials: All the quadrilateral insets (of previous presentation)
Board covered with paper
Blank labels, pen
Presentation: Like the portrait gallery to be found in the houses of noble families, the family of quadrilaterals has its own gallery, which has six portraits of the six members of the quadrilateral family. But instead of walking with our feet, we have strolled with our minds through this gallery of the quadrilateral. Now that we have examined these figures one by one, face by face, we must look at this family of quadrilaterals together, as a whole. We must look at the interrelationship of the members of this family. As we stroll along, we notice similar features in their faces. But in the family of quadrilaterals, we see increasing perfection as we go through the generations. This is like looking at the genealogical tree of the family. With the last descendant of this family the square, we have the perfect quadrilateral. It is the only quadrilateral which is a regular polygon. In this family of geometry we see increasing perfection in the last descendants. This is in contrast to some noble families where we see degeneration in the last descendants.
Place all of the insets on the board in a random group. Draw a circle around the figures. This is a gallery. All of these portraits belong to the family of quadrilaterals. Write a label “quadrilateral” and place it inside the circle. They are all quadrilaterals because they all have four sides.
Isolate the common quadrilateral and the label to one side of the circle, draw a circle around the remaining figures. These are all trapezoids. Place a label inside the circle. “trapezoid” All of these quadrilaterals have at least one pair of parallel sides.
Isolate the four trapezoids and the label to one side of the circle and draw a line around the remaining figures. These are all parallelograms. Place a label in the circle. All have two pairs of parallel sides.
Isolate the square and the rectangle and draw a circle around them. These are rectangles. Place a label in the circle. They are parallelograms with four right angles. Recall that a square was included in the drawer of rectangles.
Place the rhombus and draw a circle in such a way that the rhombus and the square are included in the circle. These are rhombi. Place the label on the rhombus side of the circle. They are parallelograms with four equal sides.
Place a label for the square in the intersection of these two sets. This means that the intersection of the set of rhombi and rectangles is the square.
It is interesting that at the end of our visit to this portrait gallery, we see one family member who is perfect and marvelous. His portrait is placed between the portraits of his father and his mother. As it sometimes happens in nature, this child has inherited the best qualities of both parents.
In the set of quadrilaterals, the child is the square. His father and mother are the rectangle and the rhombus. The square has the distinctive characteristics of the rhombus: equal sides. The square is the perfect quadrilateral; it is the only regular polygon among all of the quadrilaterals.
D. Parts of the Quadrilateral
Notes: The nomenclature of quadrilaterals, and other polygons with more than four sides, will present some difficulties, because, unlike the triangles, not all quadrilaterals have the same nomenclature. Also the stick figures must be used in order to identify the diagonal.
Materials: Quadrilaterals made previously with sticks
Insets of the quadrilaterals
Stand, small plumb line, perpendicular angle
Box of sticks, supplies
Presentation: (Square). Isolate the square made of sticks and invite the child to name it and recall its characteristics. Let’s examine the parts of this figure. the teacher points to each part as it is named: Surface …perimeter… Where there are more than one the teacher names and identifies each singularly, then gives the plural … side (sides), angle (angles), vertex, (vertices), base (bases).. to identify the base, stand the figure on its side. We can conclude that each side can serve as a base. Repeat with the inset.
Place the figure in the stand and drop the plumb line into the opening. Move the plumb line along until it coincides with one of the sides … “height” … Continue moving the plumb line along whispering … “height, height, height” ….. when the plumb line coincides with the other side, say aloud, “height”. Place the perpendicular angle beside the figure to determine that the height is perpendicular to the base. The principle heights are those two on the sides. all the segments in between are also heights but in reality they are all the same height; they are all equal. Repeat the experience with other bases.
Did you notice that I had to concentrate on keeping the square erect in this stand? If I let it slant, it is no longer a square. This figure needs a support. Line up the neutral sticks and find the one that corresponds to the sticks used for this square. Join it at two opposite vertices. This segment which connects two non-consecutive vertices is the diagonal (diagonal: Greek dia, through, and gonia, angle; thus the line that goes through the angle). We can say that for the quadrilaterals the diagonal is one of the most important elements.
Note: Dr. Montessori suggests that at this point we introduce the story of construction to show the importance of the diagonal. It keeps the roof from opening up.
Notice that the diagonal divides the square into two triangles. The triangle is the constructor of this figure. Place another stick along the other diagonal to show that the square is divided into four triangles. Again the triangle is the constructor.
(Rhombus). As before use the stick figure to identify … surface, perimeter, side (sides), angle (angles), vertex (vertices), base (bases) … In identifying the height, proceed as before this time staying silent until the plumb line coincides with the first top vertex … “height”. Continue, whispering “height, height, height”, until the plumb line coincides with the second bottom vertex. silence prevails until the plumb line coincides with the extreme top vertex … “height”. All of these heights are equal. The first principle one is internal. The second was external, and was perpendicular to the extension of the base. These are relative to this base.
Repeat the experience with the other bases and those of the inset. Lastly, identify the diagonals.
(Rectangle). Identify: surface, perimeter, side (sides), angle (angles), vertex (vertices), base (bases) … These four bases are equal in pairs. Identify the height as for the square. Notice that the heights are equal in pairs. If the base is long, the height is short, and if the base is short, the height is long. How many heights are there? Two. Identify the diagonals.
(Common Parallelogram). Identify the same parts as before. Notice that the bases are equal in pairs. Identify the heights as for the rhombus, and make the relative observations as for the rectangle. How many different heights are there? Two. Identify the diagonals.
(Trapezoid). Begin with the most general: the scalene trapezoid. Examine the parts as before: surface, perimeter, side (sides), angle (angles), vertex (vertices), diagonal (diagonals). Identify the two sides which can be bases. They are parallel. The other two can never be bases. Notice that in all of the parallelograms all of the sides served as bases. here, however, only two of the four sides can serve as bases.
Identify the longer (larger) base and the shorter (smaller) base. The other two sides which are not bases are called legs. (Recall the etymology and the story of the Greek table)
Since the number of bases has decreased in relation to the number of sides, the number of heights will also decreases. Identify the height as before, saying “height” when the plumb line coincides with the first top vertex, whispering “height, height” as it moves along, saying “height” as it coincides with the second top vertex and silence as it continues along. Repeat with the other base. How many heights are there? Only one, for they are all equal.
– the line segment which connects the midpoints of the two legs; and joining line of the midpoints of the parallel sides
– the line segment which connects the midpoints of the bases.
Identify the nomenclature of the isosceles trapezoid in the same way. Notice that the legs are equal. The joining line of the midpoints of the parallel sides corresponds to one of the heights.
Identify the nomenclature of the right-angled trapezoid. One of the legs is perpendicular, thus it corresponds to the height.
The nomenclature of the obtuse-angled trapezoid is the same as before. Since one height is external, identify the heights for the rhombus.
Note: For the following presentation, the common quadrilateral was enlarged so that the longest side was 20cm. #1 – plain, #2 – one diagonal drawn on each side, #3 – cut along one diagonal resulting in two triangles with the black line along one side each.
(Common Quadrilateral). Identify each part as before: surface, perimeter, side (sides), angle (angles), vertex (vertices), diagonal (diagonals) (use #2). seeing that nothing is new, the teacher asks after identifying each part – “Where’s the difficulty”?
Stand up the figure. Where is the base? Is it this? Try all four of them. We’ll realize which is the base when we find the height. Discover that each base has two different heights. It can’t be! The secret of this figure is that it has no base and therefore it has no height.
(Solving for the area). When we calculate the area we must multiply the base times the height. How can we calculate the area of this figure which has no base and no height? This is where the triangle becomes important. We must decompose the quadrilateral into triangles by tracing the diagonal (use #2, then #3, superimpose the triangles on the quadrilateral to show equivalence). With these triangles, we have no problem calculating the area because each triangle has as many heights as bases. Calculate the area of each and add.
We can conclude that the common quadrilateral has neither a base, nor a height, thus the diagonal has an important role. When we must calculate the area of such a figure, we must divide the figure along the diagonal.