Geometry
Squares and Divided Figures
Material
- 9 square metal insets 10cm by 10cm, each set on a frame (the same size) with a white background for the inset to rest on.
- An entire square
- A square divided into 2 equal rectangles
- A square divided into 4 equal squares
- A square divided into 8 equal rectangles
- A square divided into 16 equal squares
- A square divided into 2 equal triangles
- A square divided into 4 equal triangles
- A square divided into 8 equal triangles
- A square divided into 16 equal triangles
Direct Aim
- To interchange and build the square with the different figures.
- To gain an understanding of;
a) equivalent
b) identical
c) similar figures
Indirect Aim
To gain an intuitive idea of fractions.
Control of Error
An exact fit of the interchanged figures.
Presentation
Interchange the square divided into 2 rectangles and the one divided into 2 triangles. Superimposed by placing them in contact at the underside – showing their equivalence.
Exercise 1
He builds the first square with triangles and the second with rectangles and goes on experimenting.
Exercise 2
The child finds the 2 triangles are identical, the 2 rectangles are identical; the triangles and rectangles are equivalent.
The child is encouraged to make comparisons by placing the triangle on the rectangle. The small triangle remaining from the rectangle equals the small triangle which remains uncovered on the larger triangle and therefore although they are not the same shape, they still must have the same area.
Exercise 3
Repeat with all the insets, discovering the difference between equivalent and identical figures.
Exercise 4
Let the child find similar figures, i.e. those which have the same form but differ in dimension. Let him compare and find the rectangle is half of the original square. The one which is half of the original square and the one that is half of the smaller square (quarter of the original square) are similar figures. The children compare and contrast to see that the same may be said of the large square and of the smaller ones that are 1/4, 1/16, etc.
Measuring Angles
Materials
- Fraction Insets (Circles) – 10 metal frames, each of which has a circular opening 10 cms in diameter. One inset is a complete circle, the others are divided into 2, 3, 4, 5, 6, 7, 8, 9, and 10 equal parts.
- Instrument for measuring angles, (illustrated below)
.
Presentation
Place the vertex of the angle touching the center of the diameter, with the side resting on the radius marked 0.° Together you look at the other end of the arc and read the degrees of the angle.
Exercise 1
Then go on to measure the angle of the circle, the semi-circle and right angle.
Exercise 2
Repeat with all the different sectors to find:
1/3 of a circle =120°
1/4 of a circle=90°
1/s of a circle=72°
1/6 of a circle=60°
1/7 of a circle=51°
1/8 of a circle =45°
1/9 of a circle=40°
1/10 of a circle=36°
Decimal Fractions
Material
Instrument for reduction of common fractions into decimal fractions, (illustrated below).
Direct Aim
The children work with the fraction insets and learn to measure the angles of each sector and to count the degrees.
Presentation
To show the common fraction as a decimal fraction, place the sector carefully against the raised radius with the arc touching the inset circumference. At the end of the arc the number represents the hundredths corresponding to the sector. Use the 1/4 sector. It’s arc ends at 25, so 1/4 = 0.25°
Equivalent Figures
Material
A large rectangular metal frame with two large white voids, the equivalent triangle and rectangle. The pieces of the rectangle fit the opening of the rectangle or the triangle.
Presentation
Demonstrate the equivalence of the rectangle and triangle by filling in the triangle space with the 3 pieces from the rectangle.
Exercise 1
A triangle is seen to be able to be reduced to a rectangle.
Exercise 2
Find the area of a triangle by multiplying the base by half the height.
Similar material
- A rhombus divided into 2 triangles and a rectangle with pieces which can be put into the rhombus when the triangles are removed.
- Material to show the equivalence of a trapezoid and a rectangle.
- Material showing the equivalence of a regular polygon, a decagon, and a rectangle.