Wonderful Number Patterns

Well, let's form some patterns of our own. But this time we will use two maths operations [ 'operations' mean adding, multiplying, dividing, subtracting etc.] to produce each new number in the pattern.

You will use multiplication with either adding or subtraction. BUT you can have the multiplication last if you like.

So let's get started, you will need to do $5$ things for this work:

$1$) Choose a STARTING NUMBER (In my example suppose it's $7$)
$2$) Decide which OPERATION to do first [addition, subtraction or multiplication]
$3$) Decide what NUMBER to use with the operation you chose in ($2$)
$4$) Decide on your next OPERATION
$5$) Decide what NUMBER to use with the operation you chose in d.

My Choices are $7  x  4  -  5 $

Starting number $7 [ x 4 - 5 ]$ gives $23$

Now we use that $23$.....$ [ x 4 - 5 ]$ gives $87$

Now we use that $87$.....$ [ x 4 - 5 ]$ gives $343$

Remember we do have to do both operations and then write the result down.

See this happening with Alex in this "Number Plumber" . You'll need to drag the $7$ where it says "number equals". Then click "drop" under the name "Alex"

I now write down ALL THE THINGS THAT I NOTICE ABOUT THIS PATTERN.

Things like:-

The answers go odd, even, odd, even . . . . .

The units figures go $7,  8,  1,  0,  7,  8,  1,  0 ,$ . . . etc.

You might be able to see lots more, I've just written down a few quick ones.

When I've had a good long, hard look and talked with others, perhaps, I do the next stage; which is:-

Take a look at a, b, c, d, e, that I used and make a SMALL CHANGE

SUPPOSE I decide to change e, to subtracting $4$ instead of $3$ BUT everything else stays the same so:-

Starting Number $7 [ x 3 - 4 ]$ gives ......

TO BE REPLACED BY MOVIE? OR http://nrich.maths.org/DataFlow/DataFlow.html?config=/content/id/6928/np6928.xml    OR PPT?

Like before I write down all the things that I notice :-

LIKE :-

The answers all end in a $7$

The tens figures go $4,  3,  0,  1,  4,  3,  0,  1,$ . . .

AND maybe many more things.

Then I COMPARE what I noticed this time with last time, similar things and rather different things.

SO I might write something like:-

In the first pattern the units went in a pattern of $4$ repeating different figures, in the second pattern it's the tens that do that. Both patterns end with a $7$ every $4th$ one. I also notice that in the units of the first you could say that the $8 - 1$ gives $7$!

WELL after that L O N G introduction it's time for you to have a go!

$1$) Choose you five things to start off with a, b, c, d, e.
$2$) Produce at least $8$ answers underneath each other in good columns [it helps to see tens patterns etc.].
$3$) Write about the things you notice.
$4$) Make one small change in the $5$ starting things a, b, c, d, e.
$5$) Do $2$) and $3$) again.
$6$) Compare what's happened this time with the first time.

Then CHOOSE :-

i) It might be good to make a similar change again and see what happens [like subtracting $5$ in our example]
ii) Make a very different change [so it might be the starting number this time changed to $10$ in our example]
iii) Make a totally fresh new start [for example $3 ( x 11 + 1)]$
iv) Follow something through that it interesting you when you compare what you notice.

If you notice something very interesting happen then you may be able to do some predicting and then asking yourself WHY does this particular thing happen.