4X Project: 'T numbers' Investigation
by Ashley Pape (PGS)
I adapted a previously used investigation to encourage pupils to explain what they had found and not
just get an answer. I hope that this approach will have improved pupils' understanding of the activity
and the algebra involved.
Many teachers will be familiar with the investigation so the first page of this document is a summary
of the answers to the written questions that I posed near the end of the investigation. If you want to
read on further the second page is a more detailed explanation of the lesson.
Class: Year 8
Lesson length: 75mins (investigation could be completed in a shorter period)
Topic: Algebra – writing nth term formula...generalising
1
2
3
4
5
6
7
8
9
10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
n-1
n
n+1
n+10
n+20
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
The sum of the numbers in T2 (above) = 40
The general formula Tn = (n-1) + n + (n+1) + (n+10) + (n+20) = 5n + 30
When the pupils had found a number of formulae (as above) for different letters placed on the grid I
wrote these questions on the board...
1) How is the number of 'n's (coefficient of n) related to the letter on the number grid?
Pupils' responses However many squares you use is how many 'n's you need.....the number of squares it
covers... the number of numbers you use
2) Why is there a common formula for the same letter placed anywhere on the 100 square?
Pupils' responses because the shape never changes the formula will always be the same.... this is because it is a
10 x 10 grid and rows go up 10 and columns go up in 1s, there is a pattern that is always
consistent......the numbers are related in the same way in each shape... relationship of numbers
the same eg 33=n, 32 =n-1, 34=n+1 and 52 = n, 53 = n+1, 51=n-1 etc....
because they are all connected to n and also related to each other... this is a 10x10grid the
numbers go up as you go across so relationships from number to number stays the same...all
numbers based around n eg n+10, n+20 because it is on a 100 square... they follow a set
pattern.....
3) Why is it better to place the n in the middle of top line of the letter T, rather than at the top
left?
Pupils' responses you have to add 20 and 10 not 21 and 11 [to make the other numbers in the T]......so either
side can cancel....if you put n at 1 the 12 would be adding 11 not 10 [to make the number on
the next row in the T], if we use 2, then it will be adding 10.
It was clear from these responses that most of the group could not explain themselves coherently.
The investigation 'T numbers' involves finding the sums of numbers in particular arrangements on a
100 square. Essentially pupils will move from actually adding up the numbers to generalising using
algebra.
I actually started the pupils on calculating the sums of numbers in L-shapes.
For example:
1
2
3
4
5
6
7
8
9
10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
The L shape shown has a
sum, L2= 2+12+22+23 = 59
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
Next the class had to calculate the sum of other Ls on the 100 square. I then asked them how they
could spot a pattern connecting the top number of the L (ie 2 in the shape shown) to sum, There
wasn't much response from the class so I re-worded the question...”Is there a systematic way, to give
a number sequence/progression that would allow us to find a formula for the sum?”
Responses included; finding the sum of L2 , L12, L22 ......
and finding the sum of L2 L4 L6 ...
eventually we got to the idea of calculating L1, L2, L3 the answers were then calculated ( 55, 59, 63)
Pupils quickly said “the numbers go up in fours”, “the formula is n+4”. I encouraged the pupils to
think back (and even look back in their notes) to the nth term of a linear sequence that they had
covered just before the Christmas holiday.
The general formula was soon arrived at Ln= 4n + 51 I asked the the pupils to check that the
formula worked.
I then encouraged the pupils to look at starting from the algebra, rather than the numerical answer:
N
N+10
N+20 N+21
The next stage was to find the formula for other letters placed on the grid and to check numerically if
their formula worked (letters suggested were T, then I ). The class were then asked to choose their
own letter or letters to work on.
I then posed the questions listed on page 1 asking then to write complete sentences (with about ten
minutes of the lesson left), I encouraged the class to share their answers to see if other pupils
understood what they were saying or to give them ideas.
Extension: I drew a different number grid (5 x 20, with number 1 to 5 down the first column) on the
board and asked for the formula for a T again.
On the whole I was very pleased with the outcome of the lesson; the children were excited that they
could find a formula for their own letters and shapes, there was also a good deal of competition at the
beginning to get the formulae first. However, it was clear to me that many pupils in the class found it
difficult to express themselves (coherently) on paper, something I will try to develop in future
lessons.