Cogs
After learning about moving bridges, the children at a nursery became interested in learning about cogs. They spent some time playing with a set of cogs in the nursery, and some of their comments are documented below.
A: “They look like wheels! They spin!”
B: “Like loop the loop!”
C: “Cogs are like teeth; they’re like wheels , they spin.”
Child: “A baby one, a big one and a middle-sized one.”
Child: “They go round and round; big ones go slow and the little ones go fast.”
Child: “These are teeth, they fit together.”
The children spent time thinking about the different numbers of teeth each cog had and comparing the ones that had an even and an odd number. They also noticed that the smaller cogs spun faster than the larger cogs.
Cogs are an excellent way to learn about ratios, as the number of rotations two cogs make depends on the ratio of the number of teeth between the cogs. In the images below, we look at some different cogs and how they rotate.
In the image above, we can see that the 4-toothed cog rotates three times in the time it takes for the 12-toothed cog to rotate once. This is because the 12-toothed cog has three times as many teeth as the 4-toothed cog. We see that when the 4-toothed cog starts in the twelfth indent, it will then be in the fourth indent after one rotation, and the eighth indent after two rotations, and finally back in the twelfth indent after three rotations. We can then notice that the 4-toothed cog will be facing upright on every fourth indent. As there are exactly 12 indents on the 12-toothed cog, the 12-toothed cog will be facing upright for every third rotation of the 4-toothed cog.
In the image above, we can see that a cog with 6 teeth rotates twice in the time it takes for a cog with 12 teeth to rotate once. This is because the cog with 12 teeth has twice as many teeth as the cog with 6 teeth.
When the teeth of two cogs do not have a common factor, the number of rotations it will take for the cogs to be upright is dependent on their lowest common multiple. The lowest common multiple of two numbers is the smallest number that is divisible by both numbers. For example, the lowest common multiple of 5 and 12 is 60, as no number lower than 60 in the 12 times table (12,24,36,48) is also divisible by 5.
If we had a 5-toothed cog and a 12-toothed cog, their lowest common multiple would be 60. This means that the 5-toothed cog would rotate \(\frac{60}{5} = 12\) times, and the 12-toothed cog would rotate \(\frac{60}{12} = 5\) times before the two cogs were again facing upright at the same time.
In the table below, we can see the number of rotations it would take for the system of 4 cogs shown above to be facing upright at the same time. We can see that the smaller the cog, the more turns it will complete. It could be fun for the children to learn about these ratios by assigning a group of children to each cog. Each time their cog is facing upright, the children could shout the colour of their cog. This way, the children could hear the comparative speeds of the cogs and when their positions align.
| Number of Teeth | Number of Rotations Until Whole System is Upright Again |
|---|---|
| 4 | 15 |
| 5 | 12 |
| 6 | 10 |
| 12 | 5 |
After experimenting with the wooden cogs in the classroom, the children noticed that there were cogs in the objects around them. They noticed that a hand drill had several cogs, meaning that when they turned the handle, the end spun very quickly. The end spins quicker as the cog at the end of the drill has fewer teeth, meaning that for each turn of the handle, the little cog spins several times. Using a sticker, the children were able to determine that one turn of the large cog meant four turns of the smaller cog.
