### Paving Paths

How many different ways can I lay 10 paving slabs, each 2 foot by 1 foot, to make a path 2 foot wide and 10 foot long from my back door into my garden, without cutting any of the paving slabs?

### 1 Step 2 Step

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

### Farey Sequences

There are lots of ideas to explore in these sequences of ordered fractions.

# Pocket Money

### Why do this problem?

When students meet arithmetic and geometric sequences, they may not appreciate the different ways that they grow. This problem uses the context of pocket money to encourage students to take a fresh look at different types of sequence, compare the summation of sequences, and perhaps do some spreadsheet work to answer questions and pose questions of their own. The surprise that students experience when they see how quickly the exponential series grows intrigues them and piques their curiosity.

### Possible approach

Display the three Pocket Money options:
1. £10 every day
2. £3 on the first day, £3.50 on the second, £4 on the third, and so on, increasing 50p per day
3. 1p on the first day, 2p on the second, 4p on the third, and so on, doubling each day.
"If you were going to be receiving pocket money every day for a month, which of these options would you pick and why?"
Give students a while to think on their own, then invite them to discuss with a partner before bringing the class together. Possible suggestions might be:
I wouldn't pick option 3 because you only get 1p to start with!
I would pick option 1 because I know I will have £10 every day so £300 or £310 by the end of the month.
I would pick option 2 because even though it starts off as less than £10 it will eventually be more than £10 every day because it's growing.

Once students have shared their ideas and reasons, invite them to work out how much money they would have on each day of the month. This is a great opportunity to do some spreadsheet work, but if computers aren't available, students could work together in small groups and record their answers in a table.

You could invite students to come up with some questions of their own to answer, or share the ones from the problem:
• In which months would option 1 be better than option 2?
• If your family stopped your pocket money on day 8, which option would give you the most?
• On which day of the month does option 3 become the most fruitful?
• If you chose option 3, how many days would it be before you became a millionaire?
At the end of the lesson, set aside some time to reflect on their initial ideas. Were they surprised by how quickly the geometric sequence grew? Can they think of any real-life processes that grow exponentially like this?

This problem is inspired by the ancient legend described in Sissa's Reward. You may wish to end the lesson by showing this YouTube video which gives a sense of just how much rice you would need to fulfil the request!

### Key questions

How much money would I have on day 2? Day 3? Day n?
What would the graphs look like for the total money on each day?

### Possible extension

Students could come up with formulas to describe the total amount of money on the nth day.
Students could explore arithmetic sequences more formally in Slick Summing
Double Trouble provides a good introduction to geometric sequences.

The Legacy invites students to explore compound interest and investment models, and would provide a challenging follow-up activity.

### Possible support

Focus on the first 10 days. Students could set their work out in a table for each method, showing how much pocket money was given on each day and what the running total is.