Manu was singing a song. It went "one, two, three, one, two, …"

What are the next three numbers?

What is the tenth number he’ll sing?

This problem is about recognising number sequences and then being able to use them in some way. In this question the number sequence is 1, 2, 3, 1, 2, 3, …. The students first have to recognise this and then use it to find the tenth number in Manu's song.

### Problem

Manu was singing a song. It went "one, two, three, one, two, …" What are the next three numbers? What is the tenth number he’ll sing?

### Teaching sequence

- Start the following pattern: snap, clap, snap, clap....
*Join my pattern when you know how it goes.* - After the all the students can follow the pattern ask them if they can use counters (2 colours) to show the pattern.

Ask if they can use numbers to show this pattern: 1, 2, 1, 2... - Read the problem to the class.
- As the students work on the problem ask questions that focus on their description of patterns:
*What number comes next? How do you know?*

How will you find the tenth number? (ways of keeping track) - Share solutions.

#### Extension to the problem

Find other ways to show the pattern (counters, actions ...)

### Solution

The next three numbers are 3, 1, and 2 in that order.

To find the tenth number Manu will sing it may be easiest to write out all the numbers. (It can be done by drawing or by using counters numbered 1, 2, 3.) We have listed the first ten numbers with their ‘number’ underneath.

1, 2, 3, 1, 2, 3, 1, 2, 3, 1

1 2 3 4 5 6 7 8 9 10

So the tenth number that Manu sings is **one**!

#### Solution to the extension

Other ways to represent this repeating triplet include:

blue red yellow blue red yellow** ...**

click slap clap click slap clap