### 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.

# Shifting Times Tables

### Why do this problem?

This problem encourages students to think about the properties of numbers. The use of an interactivity provides an engaging "hook" to stimulate students' curiosity and draws them into the structure of linear sequences and straight line graphs. It also provides a natural language, that of the "times table" and "shift" for talking about remainders and modular arithmetic.

### Possible approach

This printable worksheet may be useful: Shifting Times Tables
The solutions are available here.

"I'm thinking of a times table. I wonder if you can work out which it is? $6, 12, 18, 24$" (writing the numbers on the board as you say them.)
"What about $33, 44, 55, 66?$"
"$48, 54, 60, 66?$"
"$135, 150, 165, 180?$"
Keep going until the class are confident and fluent at working out the times tables - to avoid shouting out, students could write their answers on mini whiteboards.

"What if I give you some random numbers from a times table instead? $55, 40, 105, 60$"
"What about $90, 60, 105, 45?$" These are all in the $3$, $5$ and $15$ times tables. Point out that we're only going to be interested in finding the largest possible times table, so we'll say these are numbers in the $15$ times table.

"What about $280, 160, 560, 720?$" We hope this will catch some students out, as they'll be tempted to suggest the $10$ or possibly the $20$ times table. In fact, these are all in the $40$ times table as well. This is another chance to highlight that we're interested in the largest possible times table.

Now show the interactivity from the problem, and alert the students that it does something slightly different (but don't tell them what!). Generate a set of numbers using Level 1 or 2, and give the class a short time to discuss with their partner what they think the computer has done.

Do the same a couple more times, without any whole-class sharing, but giving pairs a little time to refine their ideas. Then bring the class together and discuss what they think is going on. Link what they say to the terminology of "Table" and "Shift" used in the interactivity.

Emphasise that the table should always be the largest possible, and the shift should always be less than the table. This example could be used to bring these ideas out:

$82, 202, 122, 442$

Possible suggestions that might emerge:

Table: 10, Shift: 2, or 12, or 22...
Table: 5, Shift: 2, or 7, or 12...
Table: 20, Shift: 2, or 22, or 42...

But we are interested in

Table: 40, Shift: 2.

Group students in pairs at a computer or with a tablet and challenge them to develop a strategy to find the table and shift with ease for Levels 1 and 2. Once they can confidently answer Level 1 and 2 questions, they can move on to Levels 3 and 4 where they are given random terms from the shifted times table instead of the first five terms. While students are working, circulate and listen out for students who have developed useful strategies that they can share with the rest of the class.

If computers are not available for students, use the interactivity to generate a dozen or so examples at appropriate levels, and write them on the board for the class to work on. Students could also work in pairs and create examples for their partners to work out, or work on the examples on this worksheet.

Once students are confident at finding the times table and the shift, ask them to work on the following questions:

• What can you say if the numbers are all odd?
What about if they are all even?
Or a mixture of odd and even?
• What can you say if the units digits are all identical?
What if there are only two different units digits?
• What can you say if the difference between two numbers is prime?
What can you say if the difference between two numbers is composite (not prime)?

Finally, bring the class together to discuss these questions and then generate a Level 4 example. Invite students to explain how they would tackle it.

Here is an account of one teacher's approach to using this problem.

### Key questions

What is the same between numbers in a times table and numbers in the shifted times table?
What can you learn from the difference between any two numbers in a shifted times table?
How do you find the shift once you've worked out the table?