## Times Tables Shifts

The numbers in the five times table are:
5, 10, 15, 20, 25 ...

I could shift these numbers up by 3 and they would become:
8, 13, 18, 23, 28 ...

In this activity, the computer chooses a times table and shifts it.
Can you work out the table and the shift each time?

Can you explain how you worked out the table and shift each time, and why your method will always work?

Level 1 includes tables up to 10.
Levels 2 and 3 include tables up to 20.
On levels 1 and 2, the numbers will always be the first five numbers in the times table.
On level 3, the numbers could be any five numbers from the shifted times table.

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#### Why do this problem?

This problem encourages children to think about the properties of numbers.  As well as helping learners to become more familiar with multiplication facts, this activity will also help give them a greater understanding of the structure of our number system.

#### Possible approach

Explain to the group that you're thinking of a times table and ask them if they can work out which it is.  Write these numbers on the board as you say them: 3, 6, 9, 12.  What about 4, 8, 12, 16?  45, 50, 55, 60?
Keep going until the class is confident and fluent at working out the times tables.  To avoid shouting out, learners could write their answers on mini whiteboards.

Explain that you will now give the class some random numbers from a times table rather than the first four numbers.  Write up, for example, 60, 20, 100, 50.  Discuss that these are all in the 2, 5 and 10 times tables, but we're only interested in finding the largest possible times table, so we'll say these are numbers in the 10 times table.

Now show the interactivity from the problem and alert the children that it does something slightly different (but don't tell them what!). Generate a set of numbers using Level 1 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.

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. Learners could also work in pairs and create examples for their partners to work out.

Once pairs 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.

#### Possible 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

The Shifting Times Table problem is a Stage 3 version of this task.

#### Possible support

Children may find a table square helpful as they work on this activity.