### 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 uses the 'hook' of a mystery function to engage students' curiosity about inverse functions of exponentials. Requiring no prior knowledge of logarithms, the interactivity in this task could be used to start off a conversation about the law $\log(a) + \log(b) = log(ab)$. The function used in the task uses powers of 2 as a starting point.

### 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.

Students who have not yet met logarithms might express the relationship in terms of the powers of 2 in the row below:
"If the bottom two numbers are $2^x$ and $2^y$ then the row above is $x+y$"
"Multiply the two bottom numbers together. If you get a power of 2, it's the power that goes in the next row."

These two different ways of thinking about the function could be used to introduce the law $\log(a)+\log(b)=\log(ab)$.

One interesting discussion point for students who are already familiar with logarithms could be whether these two functions are in fact equivalent, and which one the computer is using - if $a$ and $b$ are both negative, the computer does in fact produce an answer, which indicates that it is performing the product first before taking logs!

Once everyone is clear about how the function pyramid works, invite students to work on the following challenges:
• Can you choose numbers for the bottom layer so that the number 1 appears in the top cell? Can you do it in more than one way?
• Can you choose numbers for the bottom layer so that the number 5 appears in the top cell? Can you do it in more than one way?
• Can you choose numbers for the bottom layer so that the numbers in the middle layer are negative?
• Can you choose numbers for the bottom layer so that the number in the top cell is negative?
To finish off the lesson, share students' answers to these challenges and discuss how they approached them, as well as any other observations they have about the function pyramid.

### Key questions

Can you find numbers for the bottom layer that give you whole numbers on the second layer?
What is special about the numbers on the bottom layer that give whole number answers for the second layer?

### Possible extension

Students could create function pyramids of their own, perhaps using a spreadsheet, for other functions.

### Possible support

Suggest that students pick one number on the bottom row to change at a time. How does the value in the second layer change? When is the output a whole number?