Materials: Box of sticks, supplies, board Red cord Paper and scissors, red pen Labels for writing
Presentation: The teacher places the red cord on the board so that a simple open curve is formed. What is this? It is a curve that has a beginning and an end. Which is the beginning and which is the end? I can’t tell, but what is important is that my hand can go all around, inside and out without ever leaving the plane. This is an open curve; it is like an open gate.
The teacher ties the two ends of the cord and places it on the board again so that it forms a simple closed curve. Again the teacher moves her hand around the figure on the plane: outside the figure. The teacher lifts her hand and touches the plane inside the figure and identifies it – inside the region of the figure. Identify the internal and external parts of the plane. This is a closed curve region. We can also add the quality “simple” since it is not overlapping (demonstrate this) Thus, it is a simple closed curve. Set it aside.
The teacher places a single stick on the board, moves her hand all around it on the plane. A second stick is joined to this one and they are placed on the board forming any angle. The teacher’s hand still can move all around the angle. A third stick is united to these two and placed on the board; the teacher moves her hand all around it. This is a broken line. It is open; it does not form a region. Since it is straight in places, it is not a curve. It is like a line that has been broken.
The three sticks are united to form a triangle. This time my hand must go around the figure, but I must lift it to go inside. Identify the internal and external regions. We can call this figure a polygon, which is just a general name.
Place the simple closed curve region (cord) and the polygon (stick) side by side on the board. The polygon is the opposite of the closed curve region. These are the two big families with which we will be working.
To reinforce the concepts just given, the teacher draws figures in red on the paper and invites the child to cut them out. (remove the cord and sticks)
We can classify all the figures into two groups.(place out a label which says figures (or regions)) Beneath this heading place the two sub-headings – closed curve regions; and polygons. Sort the four cut-outs into these two groups.
All of these are figures (regions). They can be closed curve regions, limited by a curved line. They can be polygons limited by broken lines.
Materials: Geometry cabinet insets
Additional insets for the geometry cabinet
Two concave figures constructed previously with the child now in red on laminated cardboard
Seven classification cards
A stick from the box of sticks
A. Classification of closed curve regions and polygons
The teacher places out the label “Figures or Regions” and below it: “Closed Curve Regions” and “Polygons”. The child is invited to classify all of the figures into these two columns, emptying the cabinet, drawer by drawer, placing the figures into the two groups. On a large sheet of paper the child draws the result of this classification, drawing each figure.
B. A sensorial classification of convex and concave (re-entrant)
The teacher takes any convex polygon and the stick and passes the stick over the surface of the inset saying – internal… internal… internal. The internal region is not interrupted by any part of the external region. This is a convex figure. the teacher repeats the experience with any convex closed curve figure and arrives at the same conclusions. Taking a concave polygon, the experience is repeated – internal…internal….internal… but, look… the stick touches an internal part, then an external part and then an internal part again. This is a concave (re-entrant) figure. The experience is repeated with a concave closed curve region, and the same conclusions are made. The teacher places out the label “Figures (Regions)” and below it “convex” and “concave (Re-Entrant)”. The child is invited to classify these four figures. Notice that in each column there is one closed curve figure and one polygon. the child continues classifying all of the other figures. All of the figures will end up in the convex group except the three flowers. These three figures are a special case. It seems that we should classify these flowers as concave, but concave flowers have a very different shape (this will be shown later). these flowers are convex. The child copies the situation on a large piece of paper.
Note: At a later age, these figures will be classified according to the presence of convex or reflex angles.
C. Closed curve regions/polygons – convex/concave
The teacher places out the heading card – “Figures (Region)” and below it – “Closed Curve Regions” and “Polygons”. Below each of these subheadings is placed “Convex” and “Concave (Re-Entrant)”. the child classifies each figure, making four columns. The child copies the situation on a large piece of paper.
At the end the teacher isolates the concave figures and their classification cards. We will only be interested in convex figures. When we talk about closed curve regions and polygons, we can assume from now on that they will be convex. One exception will be considered later.