### Modular Fractions

We only need 7 numbers for modulus (or clock) arithmetic mod 7 including working with fractions. Explore how to divide numbers and write fractions in modulus arithemtic.

Decipher a simple code based on the rule C=7P+17 (mod 26) where C is the code for the letter P from the alphabet. Rearrange the formula and use the inverse to decipher automatically.

### Double Time

Crack this code which depends on taking pairs of letters and using two simultaneous relations and modulus arithmetic to encode the message.

# Function Pyramids

### Why do this problem?

This problem offers an opportunity to explore functions of functions. Although the context is powers and logs, the same structure can be used to explore other functions. By considering the questions in the task, students will gain a clearer understanding of the functions in question.

### Possible approach

If students all have access to computers, they could explore the interactivity on their own or in pairs to make sense of how the upper layers of the pyramid are generated. Alternatively, display the interactivity to the whole class, and ask for suggestions of numbers to enter in the bottom layer, and give them time to discuss in pairs what they think is going on.
Once they have some ideas, discuss as a class what they have noticed.

If they have not yet met logarithms, they might express the relationship as something along the lines of:
"It's the power of two that the product of the numbers on the layer below is", and this is a good opportunity to introduce the notation of logarithms, and perhaps to start deducing some of the laws of logarithms.

Once the class have established how the pyramid works, set them the four challenges from the problem, to work on away from the computers, so they have to work out the answers rather than relying on trial and error.

Finally, bring the group together and share the methods they used for answering each challenge, before checking their answers using the interactivity.

### Key questions

Can you find numbers for the bottom layer that give you whole numbers on the second layer?
What is special about these numbers?
What happens if you change just one of the numbers on the bottom layer?

### Possible extension

This spreadsheet offers the same activity as the interativity in the problem, but could be adapted to create other function pyramids to investigate.

### Possible support

Suggest students start by putting 4s in every cell on the bottom layer.