New ways to get attention of students
Ms. Rania Ali taught in a primary school located in a slum. Most students came there because they had to. Their parents got a certain amount of living support from the government if their children were in school. So the parents told the kids that they had to be in school. The kids had at best marginal interest in learning. Ms. Rania Ali was aware of her challenge but the great teacher found new ways to get the attention of her students. This is the story of the day when she had to teach about angles as one of the geometry lessons.
Ms. Ali asked one of the boys to come and face the class. Then she asked the boy to turn left, left again and again until he faced the class . Then she told the class that the boy had a rotated a full circle. Rotation was measured as an angle, and for a full circle the angle was 360°.
See-saw and angles
Some students were amused but Ms. Ali had to do more to get them interested. So she showed them the picture of a see-saw. The see-saw was a flat board with the sides which could go up or down. Measurement of the angle would show that the top formed an angle of 180°. The other half of the angle was formed below the board and that was also 180° (not shown here). Thus the angles below and above the board added to a full rotation of 360°.
Kids loved it so far but wondered what Ms. Ali was going to show next.
Ms. Ali: You saw that the angle on the flat surface of the board was 180°. Even if you were to cut this angle into two or three parts, their sum will be 180°. So remember that the sum of all angles made on a straight line is 180°.
Here is an example. Angles ABE, EBD and DBC are made on the straight line ABC. The angle EBD will be 180° – 45° -30° = 105°.
Do you remember a Ferris wheel?
Now, do you remember what a ferris wheel looks like? I am going to show you a ferris wheel with 8 gandolas for kids to sit in. The center of the wheel is marked with the letter C. It is connected with spokes to the gandolas. Alia, Arisha, Mehak, Fateh and Rehan are sitting in the gandolas as shown. First, tell me what is the angle between the spoke for Alia’s gandola bar, center and the spoke for Arisha’s gandola bar. I guess Alia and Arisha have to figure this out.
Alia: Ms. Ali, there are 8 spokes in the ferris wheel. That means the angle between two spokes next to each other is 360 divided by 8 which is 45°. So the Alia-Center-Arisha will be 45°.
Ms. Ali: Did everybody get that? Now Mehak has to tell me the angle Alia-Center-Mehak.
Mehak: This angle is three eighth of a full circle. So it will be 360 x 3/8 or 135°.
Ms. Ali: Rehan, tell me the angle between you and Alia.
Rehan: Ms., I can go to the bottom of the wheel to Alia. That will be three eighth of the whole circle. So it will be 135°. However, if Alia were to come all the way from top of the wheel, the angle will be 360 minus 135 which is 225°.
What about me?
Rajab: How come only five of them get to ride on the ferris wheel? What about me?
Ms. Ali: Everyone will get to ride. A big ferris wheel is coming next. Here it is. Rajab, how many gandolas does it have?
Rajab: I counted 36 gondolas. Everyone from our class can go on it with room left for others.
Ms. Ali: If you were sitting in one gandola and Imtiaz was in the next one, what would be the angle between the your spoke, center and the next spoke?
Rajab: The full circle is 360° and this would be 1/36th part of it or 10°.
Ms. Ali: Great job, Rajab. Hope everyone had fun with the ferris wheels. Go home and look for more of them on the Internet. See how many different ones you can find. Find the angles between one spoke and the one next to it. You can also look at wheels of cars and rims as they may also form angles. No homework, today. Just have fun with the wheels.
What is the smallest angle between the hour (short) and minute (long) hands at the different times shown in the clocks below.
Solution: A full circle is 360°. In the clock, the numbers 1 to 12 divide this circle into 12 equal parts each being 360/12 or 30°. At 1 O’clock the long hand is at 12 and the short hand points to 1 – only one number away. Therefore the angle between the two hands is 30°. At 3 the angle between the two hands will be 3 x 30 or 90° and at 4 it will be 4 x 30 or 120°. At 8 O’clock one could say that the hands are separated by 8 numbers if you count 1,2…8 but only 4 numbers if you count 8,9,10 and 11. So the angle between them could be 240° or 120° but we are asked the smallest angle. That will be 120°. Similarly, at 10, the smallest angle will be 60°.