Introduction to the Material – Squares
Insets of the fractional parts of the square
Presentation: Bring out the insets arranging the frames in two rows with the whole inset centered at the left between the two rows.
Isolate the frame and inset of the whole. The child identifies it: square, whole. Continue bringing forth inset one at a time, identifying each: the number of equal parts, how the whole was divided each time. After identification of the halves, present charts 1 and 5. After the fourths have been identified, present charts 2 and 6. The eighths (rectangles) were formed by the division of each small square (1/4) by joining the midpoints of two opposite sides. The eight small triangles were formed by the two diagonals and the lines joining the midpoints of the opposite sides.
Note: If the two insets of the fourths were transparent, and were superimposed, the resulting lines would be those of the eight triangles.
Remove the 2/8 rectangles and 4/16 squares to show that the sixteenths were formed by joining the midpoints of the other opposite sides. This is a repetition of the passage from halves to to fourths. Likewise, use 2/8 triangles and 4/16 triangles to show that the sixteenths were formed by the other diagonal. Again this is a repetition of the passage from halves to fourths.
Note: For this presentation it is important that the pieces are arranged in the inset frame, just as they are shown on charts 1 – 4. Any arrangement showing an inscribed square is incorrect at this stage.
Identify the value of the pieces of each inset: whole, halves, halves, fourths, fourths, etc… Demonstrate that the pieces are equally divided by superimposing them back to back. Present charts 7 and 8.
Isolate the whole and the quadrilateral fractions. Identify the shape of the whole. Find the others that have the same shape. Identify the shape of the half. find the other having the same shape. Note that the shapes alternate: square (1), rectangle (1/2), square (1/4), rectangle (1/8), square (1/16).
Isolate the whole and the triangular fractions. Recall the shape of the whole. Identify the shape of the half and the others in order. Classify each triangle. All are right-angled isosceles triangles. Present charts 3 and 4. demonstrate the relationship between the lines of the figures using the arrangement pictured. As always, begin with the whole, and add one piece at a time.
· The side of the square is equal to one of the equal legs of the isosceles triangle (1/2).
· The equal side of the 1/2 triangle is equal to the hypotenuse of the 1/4 triangle.
· The equal side of the 1/4 triangle is equal to the hypotenuse of the 1/8 triangle.
· The equal side of the 1/8 triangle is equal to the hypotenuse of the 1/16 triangle.
Exercises: The child may show that the patterns demonstrated here go on infinitely, with regards to subdivisions, the shapes formed (alternate squares and rectangles), and the relationship between lines in the triangles. The child may make copies of the geometry charts.