From Irregular to Regular Polygons
Materials: Box “regular and irregular polygons” which contains:
Six special measuring angles: 1080, 1200, 1280 (approximately), 1350, 1400, 1440
Reading labels “equilateral and equiangular polygon”, non- equilateral and non-equiangular polygons, equiangular but non- equilateral polygon, equilateral but non-equiangular polygon, regular polygon, irregular polygon, non- equilateral polygon, non-equiangular polygon, equilateral polygon, equiangular polygon
Cardboard sticks (40 cm long) with a hole at one end only
Box of sticks and supplies, measuring angle
Geometry cabinet drawer of “polygons
Cards for 5-10, which have the greek roots on the other side
Note: For the child to understand the characteristics which determine regularity or irregularity of a polygon, we must examine more than one family of polygons, since the triangle will not be sufficient.
Presentation: Invite the child to build an acute-angled scalene triangle or any other triangle he’d like except the equilateral triangle. Classify the triangle which was constructed by calling attention to whether its sides or its angles are equal: Are the sides equal? no, this is a non-equilateral triangle. (write the label in black: non-equilateral triangle) Are the angles equal? no, this is a non-equiangular triangle. (write another label)
Invite the child to construct with the sticks the triangle which he was forbidden to build before. Ask the same questions and write the two separate labels: equilateral triangle, equiangular triangle. Now each triangle has two labels: one which refers to the classification of its sides and one which refers to its angles. Read them. We want to make one label for each. Tear off “triangle” on one of them. Read what’s left. We need the word “and”. Write “and” in red and place it between the adjectives. Repeat the experience for the second. Now we can make new labels for each one (no color distinction for and anymore). Read the labels. “non-equilateral and non-equiangular triangle”, “equilateral and equiangular triangle”. All of the goodness is in one, while all the badness is in the other. Two negative qualities give an irregular triangle. Two positive qualities give a regular triangle. Set these figures aside.
Invite the child to construct a common quadrilateral, any trapezoid, or a common parallelogram. As before classify the figure by asking: Are the sides equal? Are the angles equal? After each answer, make the appropriate classification and write a label. Unite the two adjectives with and in red as before. Read the parts and rewrite a new label (all in one color): non-equilateral and non-equiangular quadrilateral.
Invite the child to build a rectangle. Use the measuring angle to verify that one angle is a right angle, therefore all are right angles. Classify the figure as before writing two labels. Notice that this is the first time a figure has one positive quality and one negative quality. Tear off the adjectives and eliminate one noun. Decide on which one shall come first. Read the two adjectives; we need the work but this time. Write but in red and place it between the adjectives. Read it and write new label: equiangular but non-equilateral quadrilateral.
Note: this use of conjunctions and, but will aid the development of set theory.
Invite the child to build a rhombus. Proceed as before. The label reads equilateral but non-equiangular quadrilateral.
Lastly the child constructs a square. Proceed as before. Equilateral and equiangular quadrilateral.
Align the four quadrilaterals and their labels in the order of which they were presented. Two negative qualities produce an irregular quadrilateral. One positive quality and one negative quality still produce an irregular quadrilateral. Two positive qualities produce a regular quadrilateral. Bring to the child’s attention the progression towards perfection.
Place the six figures just built – triangles and quadrilaterals on the table with labels. In both families we have the two extremes. Each family has two opposite figures: one has two negative qualities, the other has two positive qualities. But only the family of quadrilaterals has two intermediary figures which constitute the passage from imperfect to imperfect.
At this point we can begin to make some generalizations: a polygon is regular when it is equilateral and equiangular at the same time.
Game: Examine the quadrilaterals. Why is this common quadrilateral not regular? This rectangle seems to be regular for it has equal angles. (mention this positive quality first to show the move toward perfection) Have the child find the bad quality – the sides are not equal. The sides are equal in pairs. That’s not enough. repeat the procedure with the rhombus. Finally the last one is perfect; it is regular.
Ask the child to choose five sticks at random and join them. Identify the figure: 5 sides, pentagon. As before classify the figure, write the two separate labels. Unite the qualities with and in red. Rewrite the label non-equilateral and non-equiangular pentagon”. Invite the child to unite five equal sticks and arrange the figure so that it is not equilateral. Proceed as before. The last label will read equilateral but non-equiangular pentagon.
The third figure which must have the opposite qualities of the last, will be more difficult to build. Present the special measuring angles. Using the drawer of pentagons insets, we can find the figure to go with each measuring angle. Lay out the numeral cards 5-10 in a row. Take any measuring angle and place it on the back of an inset, matching the angles. The child copies this information into his notebook.
Remove all of the measuring angles except that which pertains to the pentagon. Ask the child to take these sticks: red(8), black(10), brown(12), pink(16), blue(18); and unite them :red-pink-brown-black-blue. Check their angles with the pentagon measuring angle. Classify this figure as before. equiangular but non-equilateral pentagon. Ask the child to unite five equal sticks. Control the angles with the appropriate measuring angle. Proceed as before. equilateral and equiangular pentagon.
Observe that in the family of pentagons, the same thing happened as with the family of quadrilaterals. There are two extremes and two mediators. Identify the irregular and regular pentagons. The two intermediary figures demonstrate the passage from imperfect to perfect.
Proceed as before. In the examination of polygons having more than four sides, we use one with an odd number of sides and one with an even number of sides.
Notes: When the child is working alone, he may not remember the colors of the sticks for the third figure, or their order. For this reason the cardboard sticks are provided.
Unite two sticks and use the measuring angle to form the desired angle. As each successive stick is added, check the angle formed with the measuring angle. For the last side, try the sticks from the box. If one cannot be found, measure off a cardboard stick. Cut it, punch a hole and attach it. Check the angles with the measuring angle.
Materials for Exercises:
additional inset figures
envelope containing cardboard figures
3 irregular pentagons, hexagons, octagons, nonagons,decagons: one for each of the irregular classifications
box “regular and irregular polygons”
several cords or circumferences (for making sets)First Exercise: Ask the child to read the four long labels; observe that “polygon” has ben substituted for the name of the family (triangle, quadrilateral, etc.)
Isolate two of these labels “equilateral and equiangular polygon”, non-equilateral and non-equiangular polygon”. Take all of the triangles (from cabinet and from the box of additional insets) and classify them, making two groups under these two headings. Since two equilateral triangles are identical, one can be removed. Also one of the acute-angled scalene triangles can be eliminated.
Bring out the two smaller reading labels – irregular polygons and regular polygons and place them accordingly above the headings. The child can copy this into his notebook, tracing the insets, or substituting the reading labels and making two lists.
Proceed with the family of quadrilaterals. This time all four long labels will be needed. Place them in order: negative, 2 mediators, positive. The child takes out all of the insets and classifies them. (eliminate one square) Place the two small labels above the headings. “Irregular polygons” includes the first three columns.
Proceed with the pentagons. The four labels are needed again. But there is only one inset. Classify it. Bring out the envelope of Convex Polygons, and invite the child to find more pentagons. Use the measuring angle to verify classification. Place the two small labels above the headings. Proceed with the other polygons.
Second Exercise: (Triangles) Place the two circumferences on the table side by side. Place the labels “equilateral polygon”, equiangular polygon” in one circle, and “non-equilateral polygon”, “non-equiangular” in the other. Invite the child to place all of the triangle insets in their respective places (again eliminating the duplicates). Only one is perfect. Superimpose the two circumferences to make an area of intersection. What triangle has the qualities of both sets? None. Why? Because there are no mediators in the triangle family. Place the labels “regular polygon”, “irregular polygon” appropriately.
(Quadrilaterals) With the circumferences side by side, place the labels “equilateral polygon”, equiangular polygon”, one in each circle. Classify all of the quadrilaterals (excluding the duplicate rectangles). Only a few of our quadrilaterals have these characteristics. One figure is found in both sets – the square. Superimpose the circles and place one square in the intersection; eliminate the other. Place the labels “regular polygon” – in the intersection; “irregular polygon” – so that it touches both sets, though separately.
(Pentagons, etc…) Proceed as for quadrilaterals with other polygons.
The child copies his work by tracing the insets or substituting the names of the figures.