Angles

A.Types of Angles and the Parts of an Angle

Materials:
…Box of sticks, supplies, board
…Red felt pen (with a long thin head)
…The measuring angle, in its envelope (both sides are colored)

Presentation: The teacher takes two different sticks – a long one with holes along its length and a shorter one with just two end holes. Placing the longer one on top of the other at the center of the board the two sticks are fixed at the lower end with a red upholstery tack. At this point both are mobile, so place the shorter one in its vertical position and fix it there. Now only one moves.

Place the red pen in the last hole of the top stick and begin drawing a line in a clockwise direction… this is an angle, this is an angle…. (indicate the angle with one hand moving around between the sides of the angle) … (at 3600) this is a whole angle. It is called a whole angle because I went all the way around to the same point again.

Watch where I stop this time. Repeat the same procedure. This is an angle, this is an angle… (at 1800) this is a straight angle. The line it has made is a straight line.

Repeat the procedure as before, starting from zero. This is an angle, this is an angle … (at 900) this is a right angle. Take out the measuring angle and identify the red part which is the “measuring angle”. Note: this is the child’s first protractor). The measuring angle is a right angle. Demonstrate that the angle just constructed is truly a right angle by sliding the measuring angle along one stick until it meets the other line… a perfect match.

Leave the measuring angle there. Repeat the procedure as before, starting from zero. This is an angle, this is an angle … (before 900) this is an acute angle. It is less than the measuring angle.

Leave the measuring angle in its place. Repeat the procedure as before, starting from zero. This is an angle, this is an angle … (after 900) this is an obtuse angle. It is more than the measuring angle.

This red nail is the vertex of the angle. These two sticks are the sides. They are rays which continue to infinity in one direction. The inside part of the angle is the size of amplitude.

Using the measuring angle the child measures the size of each of the angles constructed. The acute angle is less than the measuring angle; the right angle equals the measuring angle; the obtuse angle is more; the straight angle is twice the measuring angle; the whole angle is four times the measuring angle.

To demonstrate the theorem that the size of an angle does not vary with the length of its sides, place the pen in the second hole and draw a right angle again. Measure it to see that it is the same.

Exercises:
1. classified nomenclature
2. Draw angles and cut them out. Classify them using the measuring angle.
3. Look for the various types of angles in the environment. The child will notice that most angles are right angles.

Age: Seven years

Aim: To give the first concept of angle, to explore the different types of angles, and to give nomenclature of the types and parts of an angle.



B. Study of angles

Materials: Box of sticks, measuring angle

Presentation: Take four sticks, of which two are alike. Separate the four into pairs having one each of the like sticks. Join the pairs with brads to form two angles in such a way that one of the like sticks is on top and one is on the bottom.

These are two different angles; they have nothing in common., Place the angles so that the like strips are superimposed. Now these two angles have one side in common. To emphasize that these two angles have three sides, disjoin the angles, remove one of the like sticks and rejoin the three with one brad. There are two angles and three sides; they have one side in common, and therefore a common vertex.

The two sides which are not in common are not opposite rays (opposite rays form a straight angle).

These angles are called adjacent (consecutive) angles (adjacent: Latin ad, to, near and jacere, to lie; thus to lie near each other) (consecutive: Latin consequa, one thing that comes after another). Put these angles aside, intact for later use.

Take four sticks, two of which are alike. As before, form two angles so that one like stick is on top; the other is on the bottom. These are two angles; there are two vertices and each angle has two sides. Superimpose the like sticks, then separate the sticks and remove one like stick, then rejoin the three. Now there are two angles, but they have one side in common and thus a common vertex.
The two sides which are not in common are opposite rays. These are adjacent angles. Put them aside for later use.

Take four different sticks and connect them to form two different angles. The teacher then places them, side by side but not overlapping. There are two angles, two vertices, and each angle has its own two sides. They have one special characteristic (place the measuring angle under the sticks to demonstrate), the sum of their angles is equal to the measuring angle. These two angles are complementary (complementary: Latin complementum, that which completes or fills up the other; thus they complete a right angle).

Take four different sticks and join them to form two different angles. Position the angles. there are two angles, each having their own vertex and two sides. However there is one special characteristic. Their sides form a straight angle. The sum of their angles is 1800. Place the measuring angle to show that the angle formed is the double of the measuring angle, and therefore it is a straight angle.these are supplementary angles (supplementary: Latin supplea, to fill up, more than). The etymology doesn’t help much. The angle formed is “more than” the measuring angle, in fact it is the double.

Take the sample of complementary angles made previously. The child identifies them and recalls their characteristics – sum is equal to 90o. Take also the adjacent (consecutive) angles. The child identifies them and recalls their principal characteristic – one side in common.

Take four sticks, two which are alike and join them to form two angles as before. Slide the two towards each other to make a 90 angle and verify this with the measuring angle. The child identifies these angles: only complementary. Disconnect them, remove one like stick and rejoin them. Position them again and verify with the measuring angle. The child identifies them again as having both characteristics: adjacent and complementary. These two angles are complementary adjacent angles.

Take four sticks, two of which are alike. On the board, the supplementary angles and adjacent (case 2) angle previously made, have been placed for reference. The child identifies them and recalls their principal characteristics.

As usual the four new sticks are joined to form two angles: one like stick on top; one like stick o the bottom. These two angles are only supplementary. Remove one like stick and rejoin the angles. These two angles are supplementary (because their sum is a straight angle) and adjacent (because they have one side in common) They are supplementary adjacent angles.

Finish with a three period lesson. The child works with classified nomenclature. They may draw, cut, paste and label the angles.



C. Vertical or Opposite Angles

Materials: Box of sticks, supplies

Presentation: The child brings four like sticks and constructs two angles connecting them with brads. These two angles have nothing in common. Let’s give them a common vertex. (simply superimpose them without disjoining and rejoining them). Now let’s move the sides of one angle. This side is the opposite ray to this side. In the same way indicate that the other two are opposite rays. Since this set-up isn’t too steady, we can reconstruct them using two sticks with holes along their lengths. Join them with a brad somewhere near the center and tack two of the ends down. (note: they are oblique lines – not perpendicular)

Identify the four angles that have been formed using 4 tacks of the same color from the box. This angle is on the opposite side of the vertex from this angle. These two angles have a common vertex. In the same way identify the other pair of angles as being opposite the vertex from one another. The characteristics of opposite angles have been identified: they have a common vertex and their sides are opposite rays.

Note: At three different age levels, there are three different demonstrations to show the equality of opposite angles:

1. Sensorial demonstration (7 1/2 years) On a piece of paper repeat the situation of the sticks, using these to trace two lines in black which go off the page. This line represents one stick; this line represents the other. Note the two angles indicated by red thumbtacks and color them with crayon on the paper. Do the same for the other two angles. Cut our plane (the paper) along one of the black lines, thus dividing the plane into two semi-planes. Take one of these and cut along the ray, dividing the plane into three pieces.

We must satisfy that this (loose) angle, which is colored red is equal to its opposite angle, shaded in the same color. Superimpose the loose angle at the edge, matching the sides. Slide the piece along that side until everything meets perfectly. In the same way demonstrate that the other pair is equal.

2. (approx. 8 1/2 years) Take the envelope entitled “vertical angles” from the box of supplies which contain four cards on which 1 – 4 are written. Remove the tacks and number the angles. We must show that angle 1 = angle 3, and angle 2 = angle 4. When writing we can use this notation ^1 to say “angle 1”. If we add ^1 and ^2, since they are adjacent and supplementary they will total 1800 (if the child has not learned how to use a protractor we say ^1 + ^2 = 2 measuring angles). Likewise, ^2 and ^3 form a straight angle. Placing a straight edge along the sticks isolates this characteristic, making it more visible to the child. Indicate the angles emphasizing the common angle – the common addend – angle 2. There for ^1 = ^3. Continue in the same way.

3. (approx. 11 years) given a diagram of the angles, we mark them with an arc to show they are equal; a double arc for the second pair. Simply state the textbook declaration: Angles 1 and 3 are both adjacent to angle 2. These are supplementary to the same angle, angle 2. therefore angle 1 and angle 3 are equal to each other. Finish the lesson with classified nomenclature and drawing of the angles.



D. Measurement of Angles

Introductory Presentation:
Note: Up until this point the child has only measured angles with the measuring angle, and before that with reference to the right angle (in the geometry cabinet drawer of triangles). Here we will introduce the concept of degrees.

When we measure things we use certain units of measurement. For water, we measure the length of an object in inches and feet. In other parts of the world they use liters, grams and meters. But when it comes to measuring angles there is a universal system that was invented a long time ago by Babylonian priests.

“The Story of the Star”
In ancient times there were some Babylonian priests who were very interested in astronomy and the calendar. These were priests who were accustomed to sleeping during the day and staying awake at night to look at the sky. They studied the paths of stars, and the constellations.

They discovered that in order to see a star in the exact same place as before, a certain number of days would pass. They counted and counted again, and by trial and error they arrived at the conclusion that 360 days would pass from the time a star was visible in a certain place until it returned to that same place again. They called this period of time a year.

They made a slight error in their calculations; they were off by 5 days, 5 hours and 49 minutes. But considering their instruments that they used for measuring, this wasn’t much of a mistake.

This time period of a year was too large to be practical; they needed a smaller space of time. They counted the dawns from the appearance of this star to its reappearance and divided the year into 360 days.

The priests thought that the paths followed by the stars made a circle. So they divided the circle into 360 parts and called each part a day. We call these parts degrees.

From that day onward no one has ever changed that measurement, even though we have found out with our more modern instruments that they had made a slight mistake.

From this path of the star in the sky, the Babylonian system of numeration was developed. Their system of numeration was based on 60. The priests discovered that the circle could be divided into six parts. A regular hexagon could be inscribed in this circle, then divided into six equilateral triangles( where two sides are formed by radii, and the third by a cor equal to the radius). Taking 1/6 of the path of the star (circumference), they obtained 60 which is the base of their system.

The Babylonians also gave us their symbol for days which is our symbol for degrees. Instead of writing the word degree(s) after a number we simply use this symbol 0. A little circle to remind us of the path of a star in the sky. Now instead of talking about angles in terms of their size: being wide, big or small, we will talk of the amplitude of an angle.



Materials
:
…Montessori protractor and other protractors
…Circle, square and triangle fraction insets
…Ruler and compass

1. Presentation of the Montessori protractor
This is the instrument used to measure angles. Its rim, like the path of a Babylonian star, is a circle. The Babylonians had determined that it took 360 days for the star to go around and come back to the same point; therefore we have divided this circle with little lines into 3600 (degrees).

With the child start at zero, where the star started, and count by ones up to 20, by 20’s around to 340 and by ones to 360. But 360 is not written there, because we have reached the same place from where we started- zero. There’s a line that runs from zero to the center of the circle. The center represents the point where the priests were standing in order to see this circle in the sky.



2.
 Use of the Protractor

Bring out the circle fraction insets of the thirds, ninths, sixths and halves. Taking 1/3, identify the angle to be measured: the only true angle on the piece. Recall the nomenclature and identify each part of the angle: angle, vertex, 2 sides.

Holding the knob of the piece, place the vertex on the red dot which is the center. Place one side down along the black line so that the side touches zero. Then place the inset piece flat, so that the side will touch one of the degrees. From zero count by 20’s around to the other side – 120. Therefore one third is 120 degrees. The child writes 1/3 = 1200.

Try with 1/9 placing the inset piece on the protractor as before – verte, side along the black line, surface 1/9 = 400 Go on to 1/6 : 1/6 = 600. This is the subdivision that established their system of numeration. The star had followed its path for 60 days of the 360 days which is the whole angle.

With 1/2, identify for the child the angle and the vertex, and the two sides of the angles. The vertex cannot be placed at the center ( in the priests room) as before. However the shape helps to line up the sides. We already know that this is a straight angle. 1/2 = 1800
Examine a second group of fraction. Bring out the fourths, eigths, fifths, and tenths. For each of these the second side meets a little line that is not numbered. Following the circumference of the circle from zero, count by 20’s then by 10’s, 5’s or 1’s to reach the second side.

The last group consists of the whole and the sevenths.

Remove the whole from the frame. Identify the angle, which is all of the interior, the vertex and the sides, which extend from the vertex in all directions. When this inset is placed into the frame, we cannot determine where the angle begins or end. The important thing to note is that everything is covered. Therefore the unit is 3600 – the whole angle. It follows the complete path of the star.
In measuring 1/7, we find that the side does not meet one exact mark. We can say that 1/7 is approximately 510.

Examine the triangle fraction insets. Begin with the whole triangle. Choose one angle, position its vertex on the center, one side on zero and read the measure on the other side. Continue for the other two angles. All angles are 600 therefore this equilateral triangle is equiangular as well.

The 1/2 piece has angles of 300, 600, 900. 1/3 has 1200, 300, 300. 1/4 has all 600 angles, just like the whole.

Examine the square fractions insets which are triangles formed by subdividing the square: 1/2, 1/4, 1/8, 1/16. All of the triangles have angles of 450, 450, 900. With 1/16 use a ruler to prolong the side,

Examine the rectangles and squares formed by subdividing the square: 1, 1/2, 1/4, 1/8, 1/16. All have all 900 angles. Again a ruler must be used to extend the sides of 1/8 and 1/16.

Ages: around 8 years

Aims: Measurement of angles
Indirect preparation for the sum of interior and exterior angles of polygons



Operations with Angles

a. Addition

Materials:
…Circle fraction insets
…Montessori protractors

Presentation: The teacher proposes an example: 1/2 + 1/4. The child isolates the two pieces, measures them one at a time and notes their measurements 1/2 = 1800, 1/4 = 900. We could restate the addition in terms of degrees: 1800 + 900. Place the 1/2 piece in the frame of the protractors and add the 1/4 piece. The result can be read on the protractor where the non-tangent side meets the frame. 1800 + 900 = 2700.



b. Subtraction

Materials:
…Circle fraction insets
…Montessori protractors

Presentation: The teacher proposes an example 1/3 – 1/6. The child isolates the two pieces, measures them one at a time and notes their measurements 1/3 = 1200, 1/6 = 600. Place the 1/3 piece in the protractor. From this we must take away 1/6.

1st method: Slide the 1/3 piece counter-clockwise the number of degrees corresponding to 1/6. Read the result where the second side of the angle meets the frame – 600

2nd method: Place the two pieces in the frame – minuend first. Slide the two pieces counter-clockwise until the non-tangent side of the second angle meets the measurement of the first angle – 1200. Take away 1/6. Read the result where the second side of the angle meets the frame – 600.



c. Multiplication

Materials:
…Circle fraction insets
…Montessori protractor

Presentation: The teacher proposes an example: 1/10 x 8. The child isolates the piece 1/10, measures it and notes its measurement. 360 x 8. Take the fraction 8 times placing them in the frame starting at zero and reading the result where the second side of the last fraction meets the frame.

Another example is 1/10 x 12 which is 360 x 12. Place all of the 10 pieces on; the angle is 3600. Remove them and place only two pieces on the protractor; the angle is 720. Add 3600 + 720 to obtain the results. In terms of fractions rather than angles, this result would be a mixed number.


d. Division – Bisecting an Angle

Materials:
…Full sheets of paper
…Compass, ruler

Exercise
1) An angle can be divided into two equal parts by folding the paper so that the two sides of the angle (paper) meet. Recall the nomenclature of an angle. This fold is called the bisector, it divides the angle into two equal parts.
2) Draw an angle. With a compass, place the point on the vertex and mark off two equidistant points on the sides. Open the compass wider and with the compass point on one side point, then the other, draw two arcs. A line drawn from the vertex to the intersection of these two arcs is the angle bisector. Therefore the two angles are equal.

Aim: Preparation for geometric constructions



e. Exercises in measurement and drawing of angles – other protractors

Materials: Various protractors – circular protractors (small enough to fit in Montessori protractor), those having clockwise numeration, counter-clockwise; also semi-circular protractors of various sizes

Presentation: Introduce the other protractors, comparing the first to the Montessori protractor by placing it inside the frame and aligning the degree marks. Show the others, reinforcing the theorem that the amplitude of the angle does not vary with the length of its sides.

Demonstrate the use of the protractor in measuring angles. Place the center hole over the vertex of the angle. Align one of the sides to zero, by sliding the compass around like a wheel. From zero, read the numeration progressively until the place where the second side corresponds to a degree mark. The child writes the angle measurement inside the angle, with degree symbol in red. the teacher can prepare many angles out of cardboard which the child traces and measures.

Demonstrate how to draw an angle. Make a point in red. this will be the vertex. From this point draw a ray; this will be one of the sides. Place the protractor so that the vertex corresponds to the center hole, and the side corresponds to zero. Make a mark at the number of degrees desired. Draw a line from the vertex through this mark to make the second side. Write its measure inside. The teacher can prepare command cards which tell the child to construct an angle of a stated number of degrees.

Age: About 8 years

Aim: Use of standard protractors, measuring and constructing angles, and operations with angles, including constructing a bisector



E. Convex and reflex angles

Materials: Box of sticks, supplies, board
Red pen (with a long narrow head)

Note: Up to this point an angle has been defined: “that part of a plane lying between two rays which have the same origin”, and five angles were identified: acute, right, obtuse, straight and whole.

Presentation: (By size) Invite the child to choose two sticks and fix them on the board as usual for making angles (the longer stick on top of the shorter and secured at one common end with the upholstery tack; the shorter is fixed at the other end as well). This pair of sticks should be toward the left side of the board, to leave room for the next pair.

Ask the child to make an acute angle or a right angle or an obtuse angle using the red pen. Identify the angle that is constructed. This angle is a convex angle. Color the angle. Because the child was given the choice of three angles, we can say that all three angles – acute, right, or obtuse are convex angles.

Choose two sticks identical to the first pair and fix them on the board. This time draw an angle which is greater than a straight angle, but less than a whole angle. This is a reflex angle. Color the angle with a different color (convex: Latin convexus, rounded or bent, curved, crooked) (reflex: Latin reflexus, bending back). Do a three-period lesson.

Presentation: (By sides): Begin from where the last lesson left off. Recall the nomenclature of the convex angle: vertex, side, side. ask the child to take two sticks just like the ones on the board and place them so that the sides of the angles are extended (prolonged). The existing sticks represent rays, therefore they extend the sides in the opposite direction. Place the loose sticks in position without fixing them. This green stick is a prolongation of this side of the angle, etc.., for the reflex angle as well.
Consider the first angle. Indicate the colored region. What type of angle is this? Convex. Do the prolongations of the sides fall inside this angle? No. We can conclude that the prolongations of the sides of a convex angle do not lie within the angle itself.
Consider the second angle with the same questions. Conclude that the reflex angle does contain the prolongation of the sides. Do a three-period lesson.

(Organization of the definitions) Combine the two viewpoints of size and sides. An angle whose amplitude is less than that of a straight angle is a convex angle. It does not contain the prolongation of its sides. An angle whose amplitude is less than a whole angle but greater than a straight angle is a reflex angle. It does contain the prolongation of its sides.


Classified nomenclature

Exercise: With only one pair of sticks fixed on a clean surface, slowly construct the spectrum of all of the angles while on child identifies them in the old way, the other in the new way:

Child One: “Acute, acute ….. right, obtuse …. straight-silence ..whole
Child Two: “Convex, convex …convex, convex .. silence,reflex ..silence

Note: With this exercise the child realizes that the straight angle and the whole angle do not fit the definition of convex or reflex angles.


F. Research in the environment

Here always the classification will depend on one’s point of view. which is more common in the environment; make a list
convex: the corner of a table, shelf or cupboard
reflex: a chair, sofa, a corner of the library or of the room
Try to form different angles using your body.

Age: After nine years

Direct Aim: Knowledge of convex and reflex angles

Indirect Aim: Preparation for convex and concave polygons




G.
 A new definition of an angle

Presentation: (A new definition of an angle) Invite child to draw an angle on a piece of paper, the sides reaching the edge, effectively dividing the paper in half. Review definition of a plane and on a slip of paper write, “An angle is”, and below it a label reading, “a part of a plane”. Planes go on for infinity but this plane is limited by what? The angle. And what constructs this angle? Two rays. Write on a new slip “limited by two rays” and place below previous slips. Can these two rays exist anywhere on the plane? No, they must share a common vertex. Write this on a final slip thus finishing the new definition: “An angle is a part of a plane, limited by two rays sharing a common point of origin”. Stack sheets of paper above and below to show the child that the angle exists in that plane only.

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