Investigating Pascal's Triangle
In this investigation, we look at Pascal's Triangle in a slightly different way - rotated and with the top line of ones taken off.
Age
7 to 11
Challenge level
Problem
I think that it's time to look at Pascal's Triangle afresh. So, let's see what happens when we turn it around in a special way.
So we start with the layout as usual, turn it anticlockwise 45 degrees and then take off the top line of ones.
Image
Image
Image
So the first 9 lines would look like this ;
Image
BUT if we take the digital roots of the numbers and see the
first 30 lines we have a new look to Pascal's Triangle as shown
below.
Image
So here you have an awful lot of numbers in rows and columns to explore.
Those of you with some spreadsheet experience (or those of you that are quick to learn such things!) should go to the HINT at this point, before going further.
Some starting points for further explorations.
1. On my spreadsheet I decided to explore the rows that are multiples of 9 as I had noticed that there were a lot of 141's and 171's in those lines. Because that is what I wanted to explore I only wrote down the first number that comes after the patterns I was exploring [that's why there are lots of spaces in two out of every three rows.]
Image
So you could do a similar thing, looking at line numbers of a different multiple.
If you do a few you then could compare your discoveries.
2. You could take just one line [e.g.30] and use the spreadsheet to extend it very far.
Here I've taken line 30, and grouped the numbers in that row in 9's and then three 9's together and underneath each other. I did this because my eyes led me to some interesting repetitions. I then noted down things according to the columns I had placed them under.
Image
See what other things you can notice with this line or any other that you choose.
a) Look for additions that are
interesting;
b) Look at repeats of
numbers;
c) Look for reflections of
patterns.
AND see if you can explainWHY? these patterns are
occurring.
Getting Started
You can probably see all kinds of things to explore. Those of you who are familiar with spreadsheets can extend the pattern further. You would put 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, etc across the top and then you have to add the first 1 with nothing/zero that's beneath it to get the next line started. Let me show you clearly, here's the first part of the spreadsheet I used:
Image
So, row 2 is simply the numerals going 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, etc.
Row 3; C3 is made from B3 + B2; D3 = C2 + C3 etc.[C3 = 1 + 0, D3 = 2 + 1 etc.]
Row 4, is made up in the same way so, D4= C3 + C4, etc.
So those formulae can be put in easily and then dragged through the other cells.
BUT we are dealing with digital roots which the computer deals with as MOD 9 [a kind of clock going from 1 to 9].
Let's have a look at the formulae that we need for the 4 cells in row 3, they are as follows,
Image
Once you have these you can drag the formulae through more cells to the right and also down.
In this way you can extend the pattern as long as the spreadsheet will allow.
Enjoy!
Student Solutions
We haven't received any solutions for this investigation yet, but we'd love to hear from you.
Please don't worry that your solution is not "complete" - we'd like to hear about anything you have tried.
Teachers - you might like to send a summary of your children's work.
Teachers' Resources
Exploring numbers in sequences and patterns can prove to be a very useful way of engaging pupils in mathematics for a valuable length of time. It can teach pupils about perseverance and enable them to reinforce some mathematical ideas, skills and concepts at the same time.
Because there is so much in the first big table [showing lines 1 to 30 as digital roots], it may be necessary to present just a small part of it to younger pupils. Perhaps take 5 or 6 of them and go to 27 numbers in length. There are so many opportunities for spotting addition and subtraction, as well as different kinds of patterning. Older pupils will be able to look at the table as a whole and as well as what the younger pupils do they may be interested in the overall arrays of numbers, particularly the number 9!
The "starting points for further extensions" section can be a good tool to help to bring some newer skills into play when doing a maths investigation. Using a spreadsheet will obviously make some aspects clearer and easier as well as opening up even more possibilities. These explorations are also a good tool to use when helping the pupils discover more about the use of a spreadsheet.
I feel I MUST emphasise the question for more experienced pupils as to WHY certain patterns occur. Sometimes when exploring digital roots with the older pupils some answers to "WHY?" come from realising thet they are working in mod 9 and working in other mods will make some things easier to explore.