### Pebbles

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

### Happy Numbers

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

### Intersecting Circles

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

# Shifting Times Tables

## Shifting Times Tables

The interactivity displays five numbers from a shifted times table.
On Levels 1 and 2 it will always be the first five numbers.
On Levels 3 and 4 it could be any five numbers from the shifted times table.

Use the interactivity to generate some sets of five numbers.
Can you work out the times table and by how much it has been shifted?

Shifting Times Tables

? ? ? ? ?

Always enter the biggest times table it could be.
The shift is always less than the times table.

Table
Shifted by

### Why do this problem?

This problem encourages students to think about the properties of numbers. It could be used as an introduction to work on linear sequences and straight line graphs.

### 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 $90, 60, 105, 45?$" Discuss that these are all in the $3$, $5$ and $15$ times tables, but we're only 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! 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" that the computer uses.

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.

Ideally, each pair would now work at a computer to develop a method of finding the table and shift with ease. If that isn't possible, 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.

Once students are finding the table and shift easily, bring the class together. Generate a new example and ask a pair to talk through their thinking as they work towards the solution, but ask them to stop short of actually giving the answer. The rest of the class could write the answer on mini whiteboards once they've heard enough to work it out. Repeat, giving other pairs the opportunity to share their thinking.

Finally, allow the class some time to work in pairs on the questions at the bottom of the problem, and then discuss their ideas, emphasising the need to justify any conclusions they reach.

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?

### Possible extension

Here are some follow-up resources that may build on students' thinking about this problem:
The Remainders Game
Expenses
Modular Arithmetic (article)

### Possible support

Perhaps start with the Factors and Multiples Game to practise working with multiples and factors. This could then be followed up by looking at the problem Remainders.