A. Types of lines
Materials: A string attached to two small spools
Presentation: The teacher, unseen, places one spool in each hand and closes his fists so that the spools cannot be seen. The string passes between the fingers of each hand. Placing the fists together, the children are invited to watch. The teacher unrolls the spools as the hands are separated.. this is a line, this is a line… As the line grows, the teacher changes its position continuously – horizontal, vertical, oblique – still identifying it only as a line.
Finally, with the string taut, this is a straight line; and with it drooping, this is a curved line. “A line” cannot exist by itself; it must be straight or curved or a combination of both. Any line that I make will have one of these qualities. The concept of a line goes infinitely in both directions.
Exercises: Classified nomenclature and commands
B. Parts of a straight line
…Two strings attached to spools
…Two scissors, a red felt pen
Presentation: Again the teacher prepares the spools in her hands and invites the child to watch as the line appears. What is it? A line…a straight line… a straight line…. Invite the child to find a point on this straight line and mark it in red. The child then takes the scissors to cut the line at the point… this is a straight line… (cut). Taking one spool and the point, the teacher extends the string… this is a ray… this is a ray. The teacher identifies the other ray in the same manner. These two rays are equal. a ray starts from a point and goes on to infinity. This red point is the origin. Therefore a point divides a straight line into two rays.
With the other spool the situation is repeated, with child identifying a straight line. This time the child is invited to make two red points on the line, and to cut the line at the two points simultaneously. Before the cut, the line is identified as a straight line. After the cut the teacher takes on piece at a time. This is a ray. This is another ray. this is a line segment. The two red points on this line segment are called endpoints, because we can’t tell which is the beginning and which is the end. Therefore two points divide a line into two rays and a line segment.
Exercises: classified nomenclature – after the child has put the cards with the corresponding labels, have him put them in order: origin, ray, endpoints, line segment.
C. Positions of a straight line
…A transparent pitcher and vase
…Water and red dye
…A level, two plumb lines, with a red line (cord)
…A red stick
Presentation: Dye the water in the pitcher red (because red is always used to highlight the subject of a presentation) and pour some into the vase, which is placed in the center of the table. Observe the surface of this water and describe it: it is still. This will be a point of reference.
Agitate the pitcher and place it next to the vase. Let’s wait without saying anything until the surface of this water becomes like the other. When it is exactly like the point of reference, the surface can be identified as horizontal. Place the red stick alongside the vase so that it aligns perfectly with the water. This stick represents a line which goes on in both directions. When it has the same position as the surface of the still water, it is a horizontal line.
Drop the stick in the water and wait until it is still (like the point of reference). This stick represents one of the lines that make up the surface of the water. This is a horizontal line. Remove the stick.
A straight line is horizontal when it follows the direction of still water (horizontal < horizon: Greek horizon < horas, boundary, limit; thus the horizon is the boundary of the visible earth in all directions, where it seems that the sky touches the water. Bring the children up the hill to see the horizon.
Hold the plumb line until it is still. This will be the point of reference now. Get another plumb line and wait without touching it, without a word, until it is exactly like the point of reference. Place the red stick along the red cord, so that it coincides just as the stick on the surface on the water. This straight line which goes on in both directions infinitely is vertical, because it follows the direction of the plumb line. This is a vertical line (vertical: Latin verticalis < vertex, whirlpool, vortex, crown of the head, summit, highest point, <vertere, to turn; therefore vertex can be applied to anything which turns like a whirlpool, or to the highest point, like the crown of the head or the summit). A vertical line is one which points to the vertex, that is, the topmost point in the sky over our heads (zenith). It passes through the center of the earth and on to the nadir (opposite of zenith). Take the plumb line and hold it still again. Let’s imagine that this is a straight line which goes on in both directions – up to the zenith and down through the center of the earth to the other side. Use the globe to show that a vertical line is relative to the position of the observer.
Place the points of reference for the two opposite elements in front on the child. What is the median? Hold the red stick horizontally. When a straight line follows the direction of the surface of still water, what is it? A horizontal line. Hold the stick vertically. When a straight line follows the direction of the plumb line what is it? A vertical line. Hold the stick obliquely. Is this straight line like the surface of water? The plumb line?
When a straight line is neither horizontal or vertical, it is oblique. Turn the stick 360o identifying its position as it turns – horizontal, oblique, oblique, oblique….vertical, oblique, oblique, oblique, oblique, horizontal, oblique …etc. (oblique: Latin obliquus, slanting, sloping, not straight, not right, devious) So what is straight, right, and normal? The horizontal and vertical line. The oblique line runs contrary to the true, contrary to vertical or horizontal.
Exercises: Classified nomenclature and commands. Demonstrate use of the level for determining lines in the environment.
D. Horizontal line – curved or straight?
…Frame of smallest circle inset
Presentation: Invite the child to identify their hometown on the globe. Place the inset frame on the globe so that the town coincides with the center of the circle. This curve (the rim of the frame) represents the horizon for everyone living here in and around the town. Because we are standing outside the earth we see the entire horizon as a circle, instead of as an arc-part of the circumference.
With the chalk mark a point on the floor. Draw a circle around this center point. Invite the child to stand at the center. What do you see? Without turning the child can only see an arc, a part of the circumference.
Reinforce the facts that in these demonstrations the child is much bigger than the circle on the globe or on the floor, when in reality it is the reverse. The child is a tiny, tiny point in relation to the earth which is huge. The curvature of that arc would be so slight that you would only be able to see a straight line.
With the knitting needle, hold it so that it forms an arc on a horizontal plane. Invite the child to identify what he sees as he lowers his body….. a curved line…. a curved line… a straight line. At eye level this curved line looks like a straight line.
E. Straight line in a horizontal plane
Materials: Box of sticks, supplies, board
Presentation: The teacher tacks a stick onto the board and identifies the board as a plane; a straight flat surface that continues in all directions infinitely. Imagine that this stick is a straight line that goes on infinitely in both directions.
Hold the plane vertically and ask the child to identify the position of the line as the plane is rotated: 1800…horizontal..oblique…oblique…vertical…oblique…etc.
Hold the plane obliquely. The plane in space could be in any position, but to facilitate your work, the plane will always be horizontal, like the surface of your work table.
Tack on two other sticks so that the three positions are represented: -, /, |. If these three lines were considered in space they would all be horizontal (hold the plane at eye level to show this). Let’s consider them on the plane surface. When a straight line follows the direction of the viewer’s body, it’s vertical. When a straight line doesn’t form a cross or follow the same direction of the viewer’s body, it’s oblique.
1. Leave only one stick on the board. As the plane is rotated (always horizontally) the child identifies the position of the line.
2. To understand that these positions are relative to the viewer, seat two children so that a right angle is formed between their bodies and the plane. As the plane is rotated, the children simultaneously identify the line as they view it.
3. The child draws the lines on the blackboard. The criteria hold true even if the plane is vertical (or oblique) like the blackboard.
F. Two straight lines lying in a plane – coplanar lines
1. Parallel lines
…Box of sticks, supplies, board
…Figures of children
First Presentation: Identify the board as a plane. Place a stick on the board and identify it as a line belonging to the plane. We’ve already explored everything we can about this straight line. Let’s see what happens when we add another straight line. Put another stick on the board so that it neither touches or crosses the first. We have two coplanar straight lines; they both belong to the same plane (coplanar: Latin con, together, planus, plane; thus lying in the same plane).
One stick is fixed to the board horizontally. Taking two small equal sticks, these are the key to the story. Place them perpendicularly along one side of the first stick. Move the second stick toward the first until it meets the guide sticks and fix it there. Remove the guide sticks, but leave them nearby to remind the child of their importance.
Place the two indifferent children on either sides of the lines so that they are walking in the same direction. The expressions on their faces show indifference. It is as though they don’t even know each other. Move the figures along to the end of the line and turn them over; make them walk back. They are like two people walking on opposite sides of the street. They don’t care to know each other. each one stays on his own sidewalk and they will never have the chance to meet.
We can extend these straight lines to infinity (add sticks of the same color, fixing them with the guide sticks until the lines go off the board in both directions) but these two lines will never meet. Substitute the red arrows for the two children. These two lines are parallel. They never meet no matter how far we follow them because they are always the same distance apart (parallel: Greek parallelos < para, beside, and allelon, of one another; thus one thing beside another)
Exercises: find parallel lines in the environment – door frames, fence rails, telephone wires, rows in the garden, railroad tracks.
Second Presentation (Parallel lines are parallel independently of their position): Invite the child to construct two parallel lines and then to identify their position: horizontal. Ask the child to construct two vertical parallel lines and then two oblique parallel lines using the same process as before. All are parallel regardless of their position.
Remove two pairs of parallel lines. Rotate the plane in its horizontal position as the child identifies the position…horizontal…oblique….vertical…etc.
Whenever we draw two parallel lines, the lines are also horizontal, vertical, or oblique.
Construct a series of parallel lines, using the same guide sticks or a pair of longer guide sticks. These are called “fascial lines” because this was the symbol of fascism, first used be Julius Caesar and later by Benito Mussolini.
G. Divergent and Convergent Lines
Materials: Box of sticks, supplies, board
Also 4 figures of children: 2 happy, two sad and 4 one-way red arrows
Presentation: The two parallel sticks may be left on the board for comparison. The teacher fixes one stick horizontally. Two small, but different guide sticks are used to position the second stick. The guide sticks are set aside.
Place the two unhappy children on the lines. These two children are very sad. They used to get along very well, but as they went along in life, the distance between them becomes greater and greater. (move the figures along the lines) That’s why they look so unhappy.
Replace the figures with one way arrows. These lines go only in one direction, the distance between the lines keeps increasing. Place extra sticks at the wide end, showing that the guide sticks would also need to increase in length. These are divergent lines.
(divergent < diverge: Latin di-apart, separatelym and vergo – to incline 0r – Latin divergare < devergere, de-opposite of con (together) and vergare, to direct oneself; thus to move away from each other). This term was coined in 1611 by Kepler to give the opposite of convergere which means to direct towards each other.
Exercise: Find divergent lines in the environment
Presentation: Position one stick horizontally on the board. As in the preceding presentation use two different guide sticks to position the second stick. Fix the second stick and set the guide sticks aside.
Place the two happy children at the wide end. As these two go along, they become closer and closer and happier and happier, knowing that in the end they will meet.
Replace the figures with one-way arrows. These lines go only in one direction – toward each other. These are convergent lines. (convergent: Latin con– together, and vergera – to incline) These lines come from two different points toward each other to one point.
Love stories in geometry, like those in real life can change. Place extra sticks at the narrow end to see how these line continue in their one direction. What happens? After the point of convergency, these two lines become divergent.
Exercise: Find convergent lines in the environment
H. Oblique and perpendicular lines
…Box of sticks, supplies, board
Presentation: Take two pairs of sticks with holes along the length and connect each pair with a brad at the center. Let’s see how two straight lines can meet. Two straight lines can meet this way X or (rotate the second pair from an overlapping position, through the position shown so that the child may see that they are equal and then on to a perpendicular position) two straight lines can meet this way + (Note: each pair started form a horizontal position).
Invite the child to measure the four angles of the first pair to see if they are right angles. None are right angles. In the second pair, all are right angles.
When two straight lines meet and do not form angles that are right angles, the tow straight lines are oblique to each other. Review the meaning of oblique (deviated, slanting, not right).
When two straight lines meet and form all right angles they are perpendicular to each other (perpendicular: Latin perpendicularis < perpendiculum, a plumb line < per, through and pendere, to hang). this perpendicular line hangs and goes through the other Note: the Old English word for plumb line is perpendicle.
Three period lesson with child constructing them.
1. Place a pair of overlapping sticks horizontally with the measuring angle positioned at the vertex. Ask the child to identify how the lines are in relation to one another as the top stick rotates … oblique, oblique… perpendicular, oblique …. as they overlap again – silence) …. oblique … etc.
2. the child is asked to take three pairs of sticks and unite them with brads in this way:
1st pair – both have hole along the length; united at the center
2nd pair – one has holes, the other is normal; united at the center of the one with holes
3rd pair – both have only end holes; united at one end.
The sticks are lain overlapping in horizontal positions. Using the measuring angle the child makes the first pair perpendicular and counts the right angles formed (4). The number is written on a piece of paper and is placed by the pair. The same procedure is followed for the second and third pairs. When two lines meet and are perpendicular to each other, they create four right angles, or two right angles or one right angle. Invite the child to try o arrange two perpendicular lines that create three right angles. It is not possible.
The first pair are two straight lines; the second are a line and a ray; the third are two rays.
3. With one pair of sticks with holes joined at the center and placed horizontally on the board, the child is asked to make the two line perpendicular, checking with the measuring angle. These lines are perpendicular. The teacher turns the whole thing 450 and measures the angles to check. How are these lines in relation to each other? Still perpendicular. Before the lines were horizontal and vertical, now both are in an oblique position. Do the same with the second and third pairs from the previous exercises. With the measuring angle, show that right angles are always formed, regardless of the position of the lines. Therefore all of these lines are still perpendicular to each other because the amplitude of the angle didn’t change.
I. Two straight lines crossed by a transversal
…Box of sticks, supplies
…Board covered with paper
Presentation: Place one, then another like stick on the board, having the child identify the number of straight lines on the plane. Then place a third stick (a different color with holes along the length) so that it crosses the other two. Now there are three straight lines on our plane; the third crosses the other two.
Remove the sticks. Place one horizontally and tack it down reminding the child that this straight line goes on in both directions to infinity. Into how many parts does it divide the plane? Indicate these two parts with a sweeping hand. Place the second stick on the plane so that it is not parallel. Even this straight line goes on to infinity. With a black crayon, draw lines to demonstrate this. Identify the three parts into which the plane has been divided. The part of the plane which is enclosed by the two straight lines is called the internal part which we can shade in red. Above and below the straight lines are the external parts of the plane because they are not enclosed by these two lines.
Place the third stick across the other two and tack it down where it intersects. This is a transversal (transversal < transverse: Latin trans, across, and versus, turned; thus lying crosswise). Two straight lines cut buy a transversal on a plane will determine a certain number of angles – how many? Using non-red or non-blue tacks, identify and count the angles. First conclusion: Two straight lines cut by a transversal will form eight angles.
Some of these angles are lying in the internal part of the plane, while others are lying in the external part. Remove the tacks. Identify and count the angles in the internal part, using red tacks (same color as the plane). These four angles are interior angles because they lie in the internal part of the plane. Do the same, identifying the exterior angles. The four angles are exterior angles because they lie in the external part of the plane. Second presentation: Two straight lines cut by a transversal form four interior and four exterior angles.
We need to divide these eight angles according to different criteria. Remove the red and blue tacks and identify two new groups using two other colors: four angles formed by one straight line and a transversal; and four angles formed by the other straight line and a transversal. All of the work that we’ll be doing involves pairing an angle from one group with an angle from another group. We won’t be working with two angles from the same group because that would mean only two straight lines were being considered, not three. Let’s examine these pairs.
Remove the tacks. Using two tacks of the same color, the teacher identifies two angles. These two angles are a pair of alternate angles. Recall the meaning of alternate. One is on one side; the other is on the the other side of the transversal. On what part of the plane are they? Internal, therefore they are also interior angles. We combine these two characteristics into one name: alternate interior angles. Invite the child to identify the other pair with two tacks of a different color. The child draws these and labels them.
Remove the tacks. The teacher identifies another pair of angles. These are a pair of angles that lie on the same side of the transversal. On what part of the plane do they lie? Internal, therefore they are also interior angles. We can call these interior angles that lie on the same side of the transversal. Invite the child to identify another pair with two tacks of a different color. The child draws the angles and labels them appropriately.
Remove the tacks. The teacher identifies another pair of angles. These are alternate angles because they lie on on one side one on the other side of the transversal. The child identifies in what part of the plane they lie – external – and their corresponding name – exterior. These are alternate exterior angles. Invite the child to look for another pair and identify them with two tacks of a different color. The child draws the situation and labels it accordingly.
Remove the tacks. The teacher identifies two angles. These are a pair of angles that lie on the same side of the transversal. The child identifies in which part of the plane they lie – external – and recalls their subsequent name – exterior. Therefore these angles can be called exterior angles that lie on the same side of the transversal. The child is invited to identify another pair using two tacks of a different color. The child copies this situation and labels it.
Remove the tacks. This time an exterior angle will be paired in a relationship with an interior angle. The child chooses an angle, identifying it with a tack. The other angle must be formed by the other straight line, as you remember, so that three lines will be involved. The teacher identifies the other angle of the pair. These are corresponding angles, because they follow a certain order. Both angles lie on the same side of the transversal, and each angle lies above its straight line. Invite the child to identify other pairs using different color tacks for each pair of angles. All eight angles are used. The child copies the situation and labels it accordingly. Note: These angles have only one quality, since the pair is divided among the two different parts of the plane.
Finish with classified nomenclature and commands. A command might ask the child to identify the other member of a given pair of angles.
J. Two parallel straight lines crossed by a transversal
…Box of sticks, supplies, board
…Twelve xeroxed forms showing above
…Paper, pencil, and ruler
Presentation: With the sheet of paper from the board and the sticks arranged as they were for the last series of presentations, nearby, construct two parallel (horizontal) lines crossed by a transversal. Recall construction of parallel lines from before using the guide sticks. Position the transversal and fix it with tacks.
Verify that the nomenclature used in the last situation is also applicable here: internal and external parts of the plane (red and blue shading is no longer necessary); interior and exterior angles, and so on, until the four pairs of corresponding angles have been identified. Since the same nomenclature is used, and the same angles exist, the old plane and its lines can be removed. What we must discover now is how the angles which constitute a pair are related.
Invite the child to choose a pair of alternate angles, identifying them with a pair of tacks which are the same color. The child names the pair he has chosen: either alternate interior or alternate exterior angles. The child then colors these angles on one of the forms. Note: This is only a sensorial demonstration of congruency.
Cut the form along the transversal. This will always be the first of the two cuts we must make. Choose one of the two resulting parts and identify the sides of the angle to isolated. Cut the form along the side of the angle. Superimpose this angle over the other angle, sliding it along one side until the vertices and other sides meet, showing that the two angles are congruent. Repeat the procedure with the other pair of alternate exterior angles, and the two pairs of alternate interior angles. Upon completion, it will be evident that all possibilities for alternate angles are exhausted, because the angles on the board will each have a tack.
Use a new form for each pair of angles considered. After each pair, make a conclusion about the congruency of that pair of angles. At the end we can make this generalization: There are four pairs of alternate angles. The angles of each pair are congruent to each other.
Invite the child to choose a pair of corresponding angles, identifying them with a pair of tacks. Remember that corresponding angles are not differentiated by the name interior and exterior, because one of the pair is lying in the internal part of the plane while the other is in the external part of the plane. The teacher controls the choice of angles; both angles lie on the same side of the transversal; both have the same position with respect to their line; i.e. they are both above their lines. The child transfers this situation to a form by shading in the angles.
As before, make the first cut along the transversal. Notice that this time both angles are still on the same part of the form. Make the second cut along the line that will divide the two angles, so that we will be able to superimpose them. Superimpose the angles as before to demonstrate congruency. We can conclude that: The first pair of corresponding angles are congruent.
Identify the second, third, and fourth pairs of corresponding angles. Follow the same procedure, using a new form each time, to demonstrate that the angles are congruent. Again, in the end the angles will all be identified with tacks. Generalization: There are four pairs of corresponding angles. The angles of each pair are congruent to each other.
This time, instead of demonstrating that a pair of angles are congruent, we must demonstrate that they are supplementary. This time we will again have two pairs of interior angles and two pairs of exterior angles. Invite the child to identify two angles that lie on the same side of the transversal with a pair of tacks. The child names the pair which he has chosen: interior or exterior angles; and transfers them to a form. As before cut along the transversal to find that both angles are still on the same piece of paper. We must separate the angles by the second cut. Place the two angles side by side so that their transversal sides are adjacent and their non-adjacent sides form a straight line; thus demonstrating that the two angles are supplementary. We can conclude that the first pair of (i.e.) interior angles that lie on the same side of the transversal are supplementary.
As before, repeat for the second, third, and fourth pair, making a conclusion with each pair. In the end, all of the angles have been identified with tacks. We can make this generalization: These are four pairs of angles which lie on the same side of the transversal. The angles of each pair are supplementary to each other.
Exercises: The teacher prepares forms with only the parallel lines. The child in his work completes the form by drawing the transversal either way. The direction of the transversal cannot be changed by changing the position of the form.
1. The child uses the forms to demonstrate that each pair of angles is congruent or supplementary. For each pair he writes the appropriate conclusion. At the end, he writes the generalization (as in the presentation).
2. The child works from command cards. For example: Demonstrate that the angles that constitute any pair of alternate angles are congruent. Note: In order to do this, the child must have realized that the two straight lines cut by the transversal must be parallel. He may choose interior or exterior angles.
Age: After 9 years
Direct Aim: Knowledge of the theorems involved regarding congruent angles and supplementary angles formed by two straight parallel lines and a transversal
Indirect Aim: Preparation for a more advanced study of parallelograms