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### Number and algebra

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### Advanced mathematics

# Times Tables Shifts

## 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?

Levels 1 and 3 includes tables up to 10.

Level 2 and 4 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.

Level 1Level 2Level 3Level 4

Why do this problem?

#### Possible approach

#### Possible questions

#### Possible extension

#### Possible support

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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

Levels 1 and 3 includes tables up to 10.

Level 2 and 4 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.

Times Tables Shifts

Level 1Level 2Level 3Level 4

? | ? | ? | ? | ? |

Always enter the biggest times table it could be.

The shift is always less than the times table.

The shift is always less than the times table.

Table Shifted updown by

Why do this problem?

This problem will help learners become more familiar with multiplication (times tables) facts, by encouraging them to think about number properties and therefore gain a deeper understanding of our number system. It also makes the inverse relationship between multiplication and division explicit. The interactivity will capture learners' curiosity so that they are motivated to find solutions
and, in this way, they will be doing *meaningful* calculation practice.

In conjunction with the other tasks in this group, this task also offers the chance to focus on any of the five key ingredients that characterise successful mathematicians.

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? How do they know?

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 to the fact 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 or tablet 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, using level 2 or 3, 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.

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?

The Shifting Times Table problem is a Stage 3 version of this task and the interactivity includes an extra level, which includes numbers in the 21-50 times tables. The tasks in this group would also provide useful follow-up to this one.

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

There are three tables in a room with blocks of chocolate on each. Where would be the best place for each child in the class to sit if they came in one at a time?

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?