Construction of Polygons
Note: This activity pertains to all polygons.
Materials: Different colored drinking straws, scissors, yarn, Upholsterer’s needle
Exercise: Following command cards, the child constructs the figures, using diagonals as needed for stability. For sides of equal length, the child should use straws of the same color.
Circle – Level One
Materials: Box of sticks, supplies, board, red pen
Fraction insets of the circle; whole, half, one other
Inset of the triangle inscribed in a circle
Two wooden circumferences (painted embroidery hoops of two different sizes, such that the sticks may serve as radii – large red hoop 20cm in diameter; small blue hoop 12cm in diameter)
A. Circle and its parts
Presentation: The teacher takes any stick and fixes one end to the board. remember that when we constructed an angle, two sticks were needed. Here, using one stick we’ll construct something different. Place the red pen in the hole and draw a circle. Indicate the internal part: this is a circle. It is the part of a plane enclosed by a very special closed curve. Identify the center of the circle (tack) and the radius (radius: Latin rod, spoke of a wheel) of the circle (stick). All of the points that make up this red line are the same distance away from the center. This is why this special closed curve line is called the circumference (circumference: Latin circumferre, to carry around, circum, around, ferre, bear).
Ask the child to take another stick identical to the first. Fix them together at the center and arrange them so that they are opposite rays. This is the diameter (diameter: Greek dia, through, metron, measure).
Take the longest stick from the box and place it so that it touches the circumference at its two ends. This is a cord (cord: Latin chorda, cord, string).
Each of the two parts of the circumference divided by this cord is an arc. This small one is an arc (arc: Latin arcus, bow, arch); the large one is an arc.
The diameter is a special cord because it is the longest cord possible, and it is the only cord which passes through the center of the circle. The two arcs which result from the division of the circumference by the diameter are special arcs. They are called semi-circumferences (semi: Latin half). Three period lesson.
Bring out the fraction insets and line them up left to right: whole, some fraction (3/3), halves, and the inscribed triangle. Take out the whole inset and place it on the table. This is a circle. Take out the 1/3. This is a sector (sector: Latin a cutter) of the circle. A sector is what we call each of the two parts into which the circle is divided by the two radii. Any of our fractions can serve as a sector. Perhaps the 1/2 could even be considered as a special sector. Remove the 1/2 inset. This is a semi-circle. It is exactly one-half of the circle.
Remove one of the “moon” pieces of the last inset. This is called a segment (segment: Latin secare, to cut) of the circle. A segment is the name given to each of the two parts resulting from the subdivision of a circle by a cord. Perhaps even the semi-circle can be considered as a segment.
Redefine sector and segment a little more precisely. This is a sector. It is formed by two radii, two radii which are not prolongations of each other. Thus the semi-circle cannot be called a sector. The segment is formed by a cord, but by a cord which does not pass through the center of the circle, that is a cord which is not the diameter. Thus the semi-circle cannot be called a segment either, but it is the limit of these two figures.
The ring of a circle is the last part of the plane enclosed between two circumferences having the same center. Three period lesson. Classified nomenclature and commands.
Age: Seven and a half years
B. Relationship between a circumference and a straight line
Presentation: Place a stick (straight line) and the large wooden circle (circumference) on opposite sides of the board. Move one toward the other, but do not let them touch. The teacher says, “external, external, external…” The straight line is external to the circumference and vice versa. repeat the experience this time sopping when the stick touches the circumference … external, tangent (tangent: Latin tangere, to touch). The line and circumference are touching each other at one point.
Repeat the experience, this time placing the stick on top of the circumference … external, tangent, secant (secant: Latin secans < seca, to cut). The line cuts the circumference at two points. Identify the two points of intersection. Three period lesson.
Note: Even if a short stick were used, the secant would intersect the circumference at two points, because the stick represents a straight line which goes on in both directions to infinity.
Exercises: Classified Nomenclature and commands
C. Relationship between two circumferences
Presentation: Place the two circumferences on the board and repeat the experiences of the previous presentation:
external – having on points in common
tangent – having one point in common
secant – having two points in common
Return the circumference to the tangent position. Ask the child: is the one outside or inside the other? Flip one circumference over to show that they would look like otherwise. So we can say that this circumference is external; we can also say that it is tangent. These two adjectives (external, tangent) refer to the same circumference. When we combine the qualities, one adjective becomes an adverb (externally, tangent).
Flip one circumference over. They are still tangent, but now one is internal. Repeat the transition of the adjectives. These circumferences are internally tangent.
Return to the first position and repeat the experience using these new names … external, externally tangent, internally tangent. Move the inner circumference so that it is neither tangent nor concentric. This circumference is internal. It is inside the other, but there are no points in common.
Move the inner circle so that the circumferences are concentric. Use two small sticks to check that one is equidistant from the other all the way around. This is a particular type of internal, called concentric (concentric: Latin con, with, together; center). They have the same center. Concentric circles are a subset of internal. Do a three-period lesson.
Exercises: Classified nomenclature and commands
Age: Eight years
D. Relationship between a circumference and a straight lineMaterials:
…Box of sticks, board, measuring angle
Presentation: Place the circumference and a long stick to serve as the straight line on the board. Allow the child to experiment to find the stick which may serve as a radius for this circle. The child chooses the one that looks right.
Move the straight line towards the circumference, external, external … We will use the radius to put this straight line in relation to the circumference. We’ll choose a radius which is perpendicular – use the measuring angle to check. Place one finger on the straight line where a perpendicular would intersect, and place another finger on the center. When the straight line is external, the distance between the center and the straight line is greater than the radius.
Move the straight line to the tangent position. Recall the first level definition: they touch at one point. When the straight line is tangent, the distance between the center and the straight line is equal to the radius.
Repeat the experience … external, tangent, … secant. Recall the first level definition. When the straight line is secant, the distance between the center and the straight line is less (shorter) than the radius.
Later we’ll learn how to write the symbols for these positions.
Age: Nine years
E. Relationship between two circumferences
…Box of sticks, supplies and board
…Two charts – Internal and Secant
Presentation: Place the two circumferences on the board and ask the child to find the two sticks which will serve as radii; place these in them appropriately. Use a long stick to align the radii in the same straight line. Remove the long stick.
(External) Place a finger on each center, to show the distance between the two centers. Is this distance equal to, greater than, or less than the sum of the two radii? Place two sticks like the radii, on the board end to end to allow the child to visualize the sum of the radii. The distance is greater; by how much? find an appropriate stick.
Two circumferences are external to each other when the distance between the two centers is greater than the sum of the radii.
This can be expressed in symbols. The symbol /d/ will represent the distance between the two centers (since d is used for diameter, we’ll use a small delta ). R will represent the longer radius and r will represent the shorter radius.
Thus > (R + r), or we can consider the sum of the radii. Is it equal to, greater than, or less than the distance between the two centers? Less, so (R + r) < . The two statements say the same thing, but different symbols reflect the different order of the terms.
(Internal) Refer to the chart. Placing a finger on each center, the distance between the centers is shown by the green line segment. The radii are shown side by side on the chart, but the sticks may be superimposed to leave a portion of the longer stick visible as the difference. Look at the chart. Is the distance between the centers equal to, greater than, or less than the difference between the radii? Formulate a statement about internal circumference and write the symbolized form. < (R – r) and (R – r) > .
(Externally tangent) Ask the child to place the two circumferences together again using the long stick to align the radii. Place the two extra sticks end to end to show the sum of the radii. Is the distance between the two centers equal to, greater than, or less than the sum of the radii? Formulate a statement and write the symbols. = (R + r) and (R +r) = .
(Internally tangent) Invite the child to position the circumference accordingly; the radii should be superimposed. Place the two extra sticks to demonstrate the difference between the radii. Ask the question as usual, formulate a statement and write the symbols. = (R – r) and (R – r) = .
(Secant) Refer to the chart. This time we need two series of extra sticks. Place out a pair to represent the sum and a second pair to represent the difference (Notice in all other cases we used one or the other). Begin with the sum. Ask the question, formulate a statement and write it in symbols in the two ways. < (R + r)
Discard the sticks for the sum. (R + r) >
Consider the difference the same way. > (R – r)
Using these four statements we can combine them (R – r) < to make one: (R – r) < < (R + r) or (R + r) > > (R – r) .
(Concentric) Fix the two radii to the board with an upholstery tack and position the circumferences concentrically. Invite the child to use two fingers to show the distance between the two radii. the two fingers are at the same place; there is no distance. Recall the meaning of concentric. = 0, which means there is no distance. Three period lesson. Classified nomenclature and command cards.
Age: Nine years