Polygons – Triangles
A. Triangles – Classification
Materials: Box of sticks, supplies, board
Measuring angle (additional materials listed later)
Presentation: The teacher chooses three sticks randomly – a, b, c. The second group of sticks containing a, a, b. A third group of sticks are all equal – a, a, a. Invite the child to unite the sticks of each group to form a polygon. Notice the color of the sticks. These triangles are all related. Identify each triangle. This is a scalene triangle, because all the sides are different. This is an isosceles triangle, because two sides are the same. This is an equilateral triangle, because all three sides are the same.
Note: The child already knows the names and the etymology. The new experience here is the construction of the triangles.
The isosceles triangle has two equal sides. The equilateral triangle has two plus one… three equal sides. This means that the equilateral triangle is also isosceles, but an isosceles triangle is not equilateral. An equilateral triangle has two equal sides like the isosceles triangle, but it has something more – the third side is also equal, thus it is equilateral. Set the three triangles aside.
The teacher takes two sticks: orange (6cm) and red (8cm). These are united and placed on the board to form a right angle. Use the measuring angle to verify this, and leave the measuring angle in position. Let’s find the stick that will join these two sticks, without altering the position of the first two sticks, without altering the angle. The stick that fits is the black (10cm). Join the three sticks to form a triangle and place it on top of the measuring angle.
Note: This 6, 8, 10 triangle is one of the Pythagorean triples – a series of three numbers which satisfies the Pythagorean theorem: a2 + b2 = c2. The Egyptians discovered the first triple: 3, 4, 5. With the box of sticks only 6, 8, 10 and 12, 16, 20 are possible (others: 5, 12, 13 and 8, 15, 17).
The teacher takes two different sticks and unites them to form an angle greater than the measuring angle. The third stick is found to unite them. After checking the angle to be sure that it is greater than the measuring angle, remove the measuring angle.
The teacher takes three different sticks and unites them. With the measuring angle and acute angle is formed, and the third side is added and united. This time we must check the other two angles to be sure that they are also acute angles. Measuring just one angle is not enough. Since all three of the angles are smaller than the measuring angle, that is, acute angles, this is an acute-angled triangle.
Identify the others as well, stating the number of characteristic angles. Note that the right-angled triangle and the obtuse angled triangle each had two acute angles. An acute-angled triangle however must have three acute angles.
Geometry cabinet drawer of triangles, reading labels
Paper for making labels
Second level reading labels:
right-angled scalene triangle
obtuse-angled scalene triangle
acute-angled scalene triangle
right-angled isosceles triangle
obtuse-angled isosceles triangle
acute-angled isosceles triangle
The triangles just constructed are lain out in two rows as they are in the drawer of triangles: the top row is classified according to sides, and the second row according to angles.
Every triangle by its nature has two qualities. One quality refers to its sides and the other refers to its angles. If I ask you to draw a right-angled triangle; the triangle you construct will also be isosceles or scalene. Let’s label these triangles as we have always known them (using the geometry cabinet labels).
For each of these triangles we must add another quality. The first triangles has been classified according to its sides (copy the name onto a paper label). Let’s classify it according to its angles. Use the measuring angle to determine the classification. Write a new label. Place the two labels on top of the figure. Do the same for the other two figures. Use a protractor to measure the angles to determine that all of them are equal. Therefore it is also equiangular. Continue with the triangles that were classified according to their angles. Classify them according to their sides determining this by the sticks which are different colors; thus the sides are different lengths. Write two labels for each, as before. Now we no longer have some triangles classified according to sides and others according to angles. All have been classified by both characteristics.
Look over the triangles to see if there are any duplicates. remove one acute-angled scalene triangle and its labels. There are no other duplicates.
There is something missing from our series. The teacher constructs an obtuse-angled isosceles triangle. The child is invited to classify it according to its sides (write a label) and then by its sides (write another label). Be sure it is not a duplicate. There is only one other triangle missing. Invite the child to unite two like sticks and to form a right angle with the measuring angle. Leave the measuring angle in its position as you try to find a stick which will unite them. Allow the child to try a second pair. It is impossible.
Introduce the neutral sticks. Invite the child to lay them in a stair. This stick (indicating the first) will be used to close a right angle formed by a pair of sticks of the first series. Place a colored stick next to it. Go on up to ten.
Invite the child to try the neutral stick that corresponds to his pair. Unite the sticks and classify the triangle. Be sure it is not a duplicate.
There are no other triangles in reality. Arrange the triangles in three columns – scalene, isosceles, and equilateral; then in three rows – right-angled, obtuse-angled, and acute-angled.
Isolate the first triangle and place its labels below it as you read them. Ask the child to identify the functions of each word; there are two different adjectives and two like nouns. Since we only have one triangle we can eliminate one of these nouns. Tear off the word “triangle” from “right-angled triangle”. Place both adjectives in front of the noun and read the whole thing. Copy this onto one new long label. Remove the old labels. Continue in the same way for the others. The adjective describing the sides is always closest to the noun because it is the most important.
With the equilateral triangle, isolate the three adjectives and discard two of the nouns. Begin with “acute” and see if that quality would precisely indicate this triangle. No, there are other acute-angled triangles. With equiangular and equilateral, each one by itself is sufficient to identify this triangle. We’ll use the more common one. We can remember that equiangular refers to the same triangle.
Take the insets from the drawer of triangles and match them one by one to the figures made with sticks. Identify each inset as it was previously known and give it its second quality. By passing the inset through its frame backwards, we can prove that it is isosceles.
Two triangles of the geometry cabinet correspond to the same triangle, so we can eliminate one inset and its frame. However, there are two triangles which have no inset to match. Bring out the additional insets. Invite the child to identify each using the measuring angle. Use the surface cards as you would use the frame to classify the sides.
These are all of the triangles of the 6 – 9 classroom. We also have labels for the 6 – 9. Invite the child to match new labels as he puts the triangle insets away. “Equilateral” is an old label, because the name didn’t change.
Exercise: Trace each inset, copy its name and write the reason for its name.
B. Triangles – Parts of the triangle
Materials: Seven triangles constructed previously Drawer of triangles, now including two additional inset All other triangles in the environment Box of sticks, supplies, including perpendicular angle Triangle stand
Presentation: The teacher takes the equilateral triangle as a first example. Touching the surface, the teacher says – surface. The layer paint gives the concept of the surface.
Using the corresponding triangle of sticks, the teacher runs her finger around the perimeter. The sticks are the image of the perimeter (perimeter: Greek peri, about, and metron, measure; thus the distance around). the teacher indicates each side, naming each “side”. At the end of the plural form is given “sides”. The angles and vertices are identified in the same way. The triangle inset is standing on its side, perpendicular to the table as the teacher identifies “base:; the triangle is turned to identify each new base. The plural form is given “bases”.
Note: This is to prepare for the conclusion – that any side can function as a base.
Place the triangle inset upright in the stand with reverse side closest to the groove and facing the children. Hang the plumb line in the groove to that only the cord is visible. Move the plumb line along until the cord meets the vertex opposite the base. Holding the line at the vertex so that a line segment is formed, the teacher says “height”. This is the height of the triangle in relation to this base. Indicate the base in the groove.
Take out the perpendicular angle and identify it. Place it in the groove at the center of the triangle so that one side will coincide with the plumb line. Repeat the identification of the height. since the height coincides, we can say that the height is perpendicular to the base.
How many bases are there? Three. How many heights? Three, same as the bases. How many sides? Three. We can conclude that the number of heights is equal to the number of bases which are the same as the sides. The common characteristics of the height is that each is perpendicular to its relative base. Continue identifying the parts: perpendicular bisector, median (medians), angle bisector (angle bisectors).
Explore the nomenclature of other triangles. For two of the three bases of the obtuse-angled triangle, the height is external and is perpendicular to the extension of the base. For two of the three bases of the right-angled triangle, the height coincides with a side.
Note: This is why the initial presentation should deal with an acute-angled triangle.
C. Special nomenclature of the right-angled triangle
Presentation: Take the two right-angled triangles and identify their parts as before. In identifying its sides, we can give these sides particular names.
Right-angled scalene triangle – The side which is opposite the right-angle is called the hypotenuse. The other two bear the Greek name cathetus. One is longer (major cathetus); the other is the shorter (minor cathetus). All of the other nomenclature is the same.
Right-angled isosceles triangle – The longest side which is opposite the right angle is called the hypotenuse. the other two sides bear the name cathetus. since they are equal, there is no distinction between them.
Note: The presentation which should follow is “Points of Concurrency” which would be a study of the meeting point of the the three heights, that of the three medians, and the orthocenters.