Safety: This activity uses a craft knife. Younger mathematicians should ask an adult to help. More information can be found at the CSIRO Education Safety page.

#### You will need

- 2 x A4 sheets of foam core
- Printout of this template
- Glue
- Scalpel or sharp craft knife
- Cutting mat
- Nails
- Markers

#### What to do

- Glue the template printout onto a sheet of foam core.
- Cut out the 4 gears and 3 smaller circles with the craft knife.
- Poke a hole in the middle of the 3 circles and 4 gears with a nail.
- To assemble the first axle, stick a 1-toothed gear on a nail, and then put 2 circles on the same nail.
- Stick this nail into a sheet of foam core. The gear should be on top of the circles.
- To assemble the second axle, put a 10-toothed gear on a second nail. Put some glue on the remaining circle and then put it on the same nail, so it sticks to the gear. Finally, put some glue on the remaining 1-toothed gear and put it on the nail so it sticks to the small circle piece.
- Stick the second axle into the sheet of foam core so it interlocks with the first axle. Rotate the first axle – every time the first axle completes a revolution, the second axle should move 1 tooth.
- To make the third axle, simply put the remaining 10-toothed gear on a nail.
- Stick the third axle into the foam core so it interlocks with the 1-tooth gear on the second axle.
- Turn the first axle. Every 10 turns, the second axle will turn completely, and the third axle will move 1 tooth.
- Turn the first axle until the third axle moves 1 tooth.
- Draw an arrow to 1 of the teeth of the gear on top of the second axle. Label that tooth 0 and the remaining teeth from 1–9, increasing clockwise.
- Draw an arrow to a tooth on the final gear. Label that tooth 0, and the remaining teeth from 1–9, increasing anticlockwise.
- As you turn the 1-toothed gear clockwise, the arrows will point to the number of rotations!

#### What’s happening?

We have 10 symbols we use to write numbers, from 0–9. When you count higher than 9, the units digit goes back to 0 and starts counting again. To keep a track of how many times this has happened, we increase the value of the digit in the tens place.

These gears do the same thing. The middle axle counts how many times the 1-toothed gear has rotated, and the final gear counts the number of times the middle gear has rotated.

#### Applications

Our number system has 10 symbols to write numbers with, but you can make number systems with more or fewer symbols than 10. The Maya had a number system with 20 symbols. If you wanted to make counting gears for their number system, you would need 20 teeth on each gear, instead of 10.

Computers also use a different counting system, but they have only 2 symbols. This system of writing numbers is called binary. It takes a lot more digits to write a number in binary – nine is written 1001 and ninety is written 1011010.