Question 7
After each of the following cuts, predict the shape of the hole when the paper is opened. After you have made your prediction, make the cutouts and verify your answer.
When we cut folded paper, the cut is mirrored across the fold lines, creating symmetrical shapes.
Step 1 — Predicting for (a)
Let us take a square piece of paper. We fold this paper exactly in half. Now we have two layers of paper. We cut a wavy shape along the folded edge. This cut goes through both layers. When we open the paper, the cut shape is mirrored. It creates a hole that is symmetrical. The hole will look like a butterfly or a decorative shape.
Step 2 — Predicting for (b)
Let us take a square piece of paper. We fold it in half. Then we fold it in half again. Now we have a smaller square. We fold this square diagonally to make a triangle. We fold the two outer corners of the triangle inwards. This makes a shape with eight layers of paper. We make a straight cut across the tip of this folded paper. When we open the paper, the cut is repeated eight times. This creates a hole with eight sides. The hole will be a symmetric octagonal shape.
Step 3 — Predicting for (c)
Let us take a square piece of paper. We fold it in half vertically. Then we fold it in half horizontally. Now we have a quarter of the original square. We fold this quarter square in half vertically again. This makes a small rectangle with eight layers of paper. We make two small square cuts. One cut is from the corner where all folds meet. This corner is the center of the original paper. The other cut is from a corner with open edges. This corner will be an edge of the original paper. When we open the paper, the cuts are repeated. The cut from the center creates a square hole in the middle. The cut from the edge creates a square cutout at the bottom.
Step 4 — Predicting for (d)
Let us take a square piece of paper. We fold this paper exactly in half. Now we have two layers of paper. We cut a T-shape from the folded edge. The top of the T is a horizontal line. The stem of the T goes downwards. This cut goes through both layers. When we open the paper, the T-shape is mirrored. This creates two rectangular cutouts. These cutouts are on opposite sides of the paper.
Answer
(a) A symmetric butterfly-like or decorative shape. (b) A symmetric hexagonal/octagonal shape. (c) A square hole at the center and a square cutout at the bottom. (d) Two rectangular cutouts on opposite sides.
More questions in FIO
Do you see any line of symmetry in the figures at the start of the chapter? What about in the picture of the cloud?
For each of the following figures, identify the line(s) of symmetry if it exists.
Punching Game
The fold is a line of symmetry. Punch holes at different locations of a folded square sheet of paper using a punching machine and create different symmetric patterns.
In each of the following figures, a hole was punched in a folded square sheet of paper and then the paper was unfolded. Identify the line along which the paper was folded. Figure (d) was created by punching a single hole. How was the paper folded?
Given the line(s) of symmetry, find the other hole(s):
Here are some questions on paper cutting. Consider a vertical fold. We represent it this way:
Vertical Fold
Similarly, a horizontal fold is represented as follows:
Horizontal Fold
After each of the following cuts, predict the shape of the hole when the paper is opened. After you have made your prediction, make the cutouts and verify your answer.
Trace each figure and draw the lines of symmetry, if any:
Copy the following on a dot grid. For each figure draw two more lines to make a shape that has a line of symmetry.
Find the angles of symmetry for the given figures about the point marked •.
(a) (b) (c)
Which of the following figures have more than one angle of symmetry?
Give the order of rotational symmetry for each figure:
Colour the sectors of the circle below so that the figure has:
(i) 3 angles of symmetry
(ii) 4 angles of symmetry
(iii) What are the possible numbers of angles of symmetry you can obtain by colouring the sectors in different ways?
Draw two figures other than a circle and a square that have both reflection symmetry and rotational symmetry.
Draw, wherever possible, a rough sketch of:
a. A triangle with at least two lines of symmetry and at least two angles of symmetry.
b. A triangle with only one line of symmetry but not having rotational symmetry.
c. A quadrilateral with rotational symmetry but no reflection symmetry.
d. A quadrilateral with reflection symmetry but not having rotational symmetry.
In a figure, 60° is the smallest angle of symmetry. What are the other angles of symmetry of this figure?
In a figure, 60° is an angle of symmetry. The figure has two angles of symmetry less than 60°. What is its smallest angle of symmetry?
Can we have a figure with rotational symmetry whose smallest angle of symmetry is:
a. 45°?
b. 17°?