## Constructive Triangles

### MATERIAL

The constructive triangles are used to demonstrate that all plane geometric figures can be constructed from triangles. There are five boxes: 2 rectangular, 1 triangular, and 1 large and 1 small hexagonal. Each box contains triangles of different sizes, shapes, and colors. With the exception of Rectangular Box 2, black guidelines are painted in different positions on the triangles to help the child to construct other figures. This should be encouraged as an exploratory work that will provide a foundation for later concepts of equivalency, similarity, and congruency.

### Rectangular Box 1

• three pairs of large right angled scalene triangles in three different colors
• a pair of red triangles that form an isosceles trapezoid bisected diagonally
• a pair of equilateral yellow triangles
• two different colored pairs of large right angled isosceles triangles

### Rectangular Box 2

• Two equilateral triangles
• Two right angled isosceles triangles
• Two right angles scalene triangles
• A trapezoid divided diagonally to form an obtuse angled scalene triangle and an acute angled scalene triangle
All of the figures are blue and there are no longer any guidelines.

### Triangular Box

• One gray equilateral triangle, the shape and size of the box
• One green equilateral triangle bisected from the midpoint of the base to the apex
• One yellow equilateral triangle divided into three equal pieces by lines drawn from each angle to the center of the triangle (along angle bisectors).
• ### Large Hexagonal Box

• One large yellow hexagon, the same size as the box, cut by joining the vertices of every other angle to form one large equilateral triangle and three obtuse angled isosceles triangles. There are black guidelines along the perimeter of the equilateral triangle and the bases of the smaller triangles.
• A second large equilateral triangle divided along its intersecting angle bisectors to form three obtuse angled isosceles triangles. There are black guidelines along the two equal sides of each triangle.
• Two equal red obtuse angled isosceles triangles the same size as the yellow ones, but with their guidelines along the base opposite the obtuse angle.
• Two equal gray obtuse angled isosceles triangles the same size as the others with black lines along one of the equal sides.

### Small Hexagonal Box

• 6 gray equilateral triangles with guidelines along two sides to form a hexagon, the same size as the box
• 3 green equilateral triangles (same size as above) which are put together to form an equilateral trapezoid. One triangle has black guidelines along two sides, the other two have a single guideline.
• A large yellow triangle which inscribes within the box, formed by joining every other vertex of the hexagon
• 2 additional red equilateral triangles (same size) each with a single black guideline
• 6 red obtuse angled isosceles triangles with guidelines along the base opposite the obtuse angle

### Rectangular Box 1

The teacher opens the box and says to the child, "We call these the constructive triangles. Why? Because we can construct other figures with them."

She asks the child to remove them from the box and group them by similar shapes. "Now can we group each set by color also?"

When the child has done so, beginning with the equilateral triangles, the teacher traces the black guide lines with her fingers and moves them together until they touch. "Now what do we call this?"

If the child does not know the name, the teacher should give it.

She might take the isosceles triangles next, and ask the child to do the same. There are two sets of isosceles triangles, one forms a square and the other forms a parallelogram.
"Let's try putting the scalene triangles together." The result is a rectangle, and a parallelogram."

"Now our last two red ones. Can you put those together on the guideline. What is the figure you have made? A trapezoid."

Review with the child the figures that have been made with the different kinds of triangles. With the younger children the attention is on the black line and it is a sensorial experience of shape, and vocabulary review of terms that have already been learned in the geometric cabinet.

The children can trace these new shapes and label them to put in their own geometry book.

### Rectangular Box 2

Here the child can see how many shapes can be made using one shape. With this material we have no guidelines to tell us what we must do. The child takes the equilateral triangles and discovers that there is only one shape to be made, no matter how he or she joins them. The child takes the other triangles in turn and discovers how many different shapes can be made with each pair. Here the teacher can check the child's work orally to be sure that the child knows the names of the figures and that the child can write and spell them correctly, since this is a sensitive period for reading and handwriting.

Use the same procedure with each of the successive constructive triangle boxes, allowing plenty of time for experimentation, practice and mastery before the child is invited to go on to the next box.