Follow the Numbers
1 out of 3
What happens when you add the digits of a number then multiply the result by 2 and you keep doing this? You could try for different numbers and different rules.
Problem
Follow the Numbers printable sheet
This is your chance to follow some numbers and see where they go!
A simple rule is all you need.
My first suggestion is to add the digits together then multiply (times) by 2.
The first number that I chose happened to be 56.
So, let's start:
We add the 5 and 6, $5+6=11$
We multiply the 11 by 2, $2 \times 11 = 22$, and that's the first part of the journey.
So, where now?
We carry on with this rule:
We add the 2 and 2, $2+2=4$
We multiply the 4 by 2, $4\times 2 = 8$, and that's the second part the journey.
Now, $8 + 0 = 8$ and $8\times 2 = 16$ and that was the third part.
And, $1 + 6= 7$, and $7\times 2 = 14$
Do the same to 14 and we get 10
10 leads to 2, 2 leads to 4, 4 leads to 8 and we are back to where we got to in the second part of the journey.
If we went on and on and wrote down where we got to after each part we would see something like:
56, 22, 8, 16, 14, 10, 2, 4, 8, 16, 14, 10, 2, 4, 8, ...
After exploring that journey it's time to start somewhere new, for example 11 which goes along like this:
11, 4, 8, 16, 14, 10, 2, 4, 8, ...
Oh! So we are on the same bit as before, a circular bit that goes 2, 4, 8, 16, 14, 10 and then back to the 2 again.
Now a new starting place, 96.
This goes like this:
96, 30, 6, 12, 6, 12, 6, ...
Oh! So we now have a smaller circular bit of the journey that goes 6, 12 then back to the 6.
I explored further trying to start with each number from 1 to 99.
Then I tried similar, but different rules.
I found I needed a big piece of plain paper and used arrows to show the journeys. Here are just some bits of them to tempt you to go further:
and
There are 99 starting points to try and I've only show you 8 on each of the two above so there are lots more to explore!
Decide on the rules you will use and investigate what happens with different starting points.
You might invent your own way of recording your findings.
We'd love to hear how you got on.
Getting Started
What number will you start with?
What could you try next?
What is the same and what is different about the 'journeys' you have tried?
Student Solutions
From Year $4$ at Queen Edith School, Cambridge we had the following rather good idea sent in.
After we had got the idea of following a number on its journey, we split up the work of checking out lots of numbers. Some of us began with numbers in the $30$s, some with numbers in the $40$s, and so on.
We found that all the numbers we tried ended up on one of three journeys:
$2, 4, 8, 16, 14, 10, 2, 4, 8,$ ... which we called the "red" journey
$6, 12, 6, 12,$ ... which we called the "green" journey
$18, 18, 18,$ ... which we called the "blue" journey
Next, we used a $100$ square on the Smartboard, and coloured the numbers to match their journeys. After we had coloured a few of the numbers, some of us spotted patterns beginning to show, like the blue diagonal from $81$ up to $9$. We predicted that other numbers on the diagonal would also be blue and checked them out. We also saw green squares along diagonals and made more predictions.
Finally, we made a display using the 100 square and some of our work to challenge other children to predict the journeys for some of the squares we had not coloured.
Can you predict a journey and then check if you were right?
From Krystof in Prague and Matthew from Hamworthy Middle School we had had similar results. From Karin in West Acton in London we had a clever further idea sent in.
My rule for "Follow the Numbers" is to work out the difference between the $2$ digits and add $5$ to the difference.
Here is some of my "Follow the Numbers"
Starting number:$24 24,07,12,06,11,05,10,06...$
Starting number:$39 39,11,05,10,06,11...$
Starting number:$83 83,10,06,11,05,10...$
Starting number:$63 63,08,13,07,12,06,11,05,10,06... $
On my "Follow the Numbers", most of my numbers had a pattern of $08,13,07,12,06,11,05,10,06.$
Well done Karin, I like this very much, others of you could try your own rules.
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
This activity is one which is accessible to a very wide range of children. It also opens out the possibility of pupils asking, "I wonder what would happen if ...?" Pupils' curiosity may be easily aroused while finding exploring further. It enables them to enjoy 'playing with' numbers, and discover peculiar yet interesting things that can happen! It could be used to raise some pupils' self esteem in mathematics, whilst still giving scope for others to explore further and further showing their resilience and perseverance.
Possible approach
Ask pupils to choose a number then tell them the rule: add the digits and multiply by two. Follow the rule a few times so that the children are happy with it. For some pupils that act of suggesting the starting number helps them feel able to participate. Once the rule has been applied then of course you could try a new starting number.
Give children the chance to work in pairs or small groups. Introducing the need to record what you get could be approached after a short time so that "numbers are not lost"!
This activity may be suitable for 'simmering' over a period of several days or weeks. You could dedicate some wall space to it so children could post up their 'journeys' for particular starting numbers. It could then become a whole class investigation.
Key questions
Whilst they are working on their own or in small groups you can ask them to tell you anything that they notice.
Can you find attractive ways of displaying their numbers for others to see?
What would you like to call this group of numbers that you've discovered?
Have you got to an end? If so how do you know?
What could you try now that you seem to have got to the end of that journey of numbers?
Possible extension
Make sure that every number from 1 to 99 has been used to start the journeys, sticking to the same rule. Pupils should be able to ask themselves "I wonder what would happen if I ...?", so they can change one part of the rule.
When two or more have been completed these can then be compared and pupils can be encouraged to make predictions about future results.
When quite a few have been completed, encourage pupils to compare them carefully. What is similar, different and the same? Some might like to show the sequences using their choice of appropriate software. They could then derive hypotheses relating to the sequences and test them.
Possible support
Plenty of digit cards will help children who find recording more difficult.