Investigational Work

Following through and seeing the steps and paths that MAY be taken

Choose any four consecutive even numbers. (For example: 6, 8, 10, 12). Multiply the two middle numbers together. (e.g. 8 x 10 = 80) Multiply the first and last numbers. (e.g. 6 x 12 = 72) Now subtract your second answer from the first. (e.g. 80 - 72 = 8) Try it with your own numbers. Why is the answer always 8?

The Problem as seen on Web

Remember to

• Choose four consecutive EVEN numbers;• Multiply the two outer number• Multiply the two middle numbers• Find the difference.

Allow time for the thoughts and ideas to develop to try and answer WHY?

So we have a go!

You might get …

“I tried a few and then did a kind of algebra, so I wrote 2n, 2n+2, 2n+4, 2n+6Multiplying : 2n x (2n+6) and (2n+2)x(2n+4) . . . . [various calculations to end up with]This left me with 8.”

2

46

8

24 16

Or they may show it spatially …

Click to show the 2,4,6,8, which has to be two wide, when countingin twos.

Click to show the 4 x 6 inyellow

Click toshow the 2 x 8 inblue

Click [three times slowly] to showthe (4 x 6)minus the (2 x 8, bentover)

and so wesee the 8that remain !

There may be a lot more going on to explore these two very different ways of working (thinking/learning), but this may be

enough.

BUT asking “I wonder what would happen if … “will most likely lead to other opportunities.

SO, “I wonder what would happen if instead of using 2as the ‘step’ size we tried other sizes?”

OR , “I wonder what would happen if instead of using fourlots of numbers we had three, five, six, seven etc.?”

Either or both of these questions can be explored, the next slide shows the results that follow and allow further investigation into what can be

seen.

Siz

e of

Ste

pclick

We could also go up to, say, … twenty one numbers in the lineclick

and what about taking bigger steps in the line?

click

8

So if we have not just four numbers in a line, stepping up in 1s and 2’s see what we get …

Number in a Line

It’s also a good idea, when you have a lot of numbers

in a sequence or a table, to have a look at the Digital Roots of those numbers.

And now …

The next slide gives an opportunity to do this,and it’s worthwhile to allow lots of time forfurther exploration of these new numbers.

Numbers in a Line

Size

of

Ste

p

to see their Digital Roots. Look and explore … thenClickClick again to see the top line extended

Numbers in a LineDifference

Digital Root

So let’s go for the Digital Roots

Exploration may focus on the reflective pattern seen in the Digital Roots.

click

This may be exciting in itself and so stop here.

BUT you could ask“What is so special about the numbers and their reflective partner?”

A further investigation can start just now!

Let’s have a look at the numbers either side of the middle 9.We’ll have to have a look at where they came from.

click

The next slide looks at pairing off those numbers and then looks at the difference between the differences that were found earlier on.

click

Number of numbersDifference

Digital Root

31

42

54

66

79

812

916

1020

1125

1230

1336

1442

1549

1656

1764

1872

1981

2090

21100

22110

23121

24132

25144

26156

27169 28

182 29

196 30210 31

225 32240

33256

34272

35289

31

42

54

66

79

812

916

1020

1125

1230

1336

1442

1549

1656

1764

1872

1981

2090

21100

22110

23121

24132

25144

26156

27169

28182

29196

30210

31225

32240

33256

34272

35289

Remember the top number is how many numbers in the line and the lower number is the difference between thetwo multiplications.

18

---36---

------54------

---------72---------

------------90------------

---------------108---------------

------------------126------------------

---------------------144---------------------

------------------------162------------------------

---------------------------180---------------------------

------------------------------198------------------------------

---------------------------------216---------------------------------

------------------------------------234------------------------------------

---------------------------------------252---------------------------------------

------------------------------------------270------------------------------------------

---------------------------------------------288---------------------------------------------

Click to see the differences between the pairs

So here are the same ones.Now click to see them

paired

Investigating these relationships only came about because of looking at the Digital Roots

Click to see the difference of the differences spatially

Here is the yellow showing 18 numbers in line with a difference of 90

Here is the brown showing 20 numbers in line with a difference of 72

Click to show 90 - 72

Click to show thedifference of 18separated

Click again

and Click again

Where have we got to?1/ We completed the question, allowing for different kinds of answers.

3/ We’ve changed the numbers into digital roots, which gave new ideas that could be investigated further.

2/ We extended the rules to look at new arrays of numbers, that could be investigated.

Notice how the numerical can lead to the spatial and visa versa.Each investigation may give rise to number patterns that the pupils have come across before.Going back to the original challenge and then changing something allows more work to develop and comparisons can be made between the sets of results

So one more suggestionbefore I sign off.

Working effectively with pupils doing an investigation I believe thatthe teacher has to resist taking the pupils down a route that they have seen.

I hope that this presentation will just open a door for some to see that sovery much can be explored from a simple starting point.

Some pupils may be prepared to go further with asking the question,“I wonder what would happen if …”

So, let’s take as an example, considerably changing the initial rule to …

“Multiply the first and middle numbers [lower middle in the case of an even number]

Multiply the last and middle numbers [upper middle in the case of an even number]”

(e.g. 1, 2, 3, 4 so multiply 1 x 2 and 3 x 4 we get 12 - 2 = 10 )

(e.g. 1, 2, 3, 4, 5 so multiply 1 x 3 and 3 x 5

we get 15 - 3 = 12 )

EVENnumber of numbers

ODDnumber of numbers

Working with this new rule …

… we can get a new table for stepping up in 1s and 2s

Again, an array of numbers to explore, as was done with the original rule. The next slide looks at just one of these investigations.

1 x 2 click 3 x 4 click12 - 2 click and click again for the next step

move those extra ones-click

Looking at this spatiallyUsing 1,2,3,4 as an example

So we suddenly find that we have a triangular number formed . There are of course many questions that can now be pursued further. What about …?