PYR YIDS
and an'
~nitlmn cricuu
by Paul Andrews and Anne Sinkinson
One of the privileges of working in teacher education is
visiting large numbers of partnership schools as part of our
work with trainees. It frequently uncovers exciting ideas
which we can feed back to help in the development of
trainees' own resource banks of approaches and activities.
However, this is not always the case and not infrequently we
find trainees constrained by prescriptive schemes of work
Pyramids and Consecutive Integers
The manner in which the task is presented here - which,
incidentally, was how a local school presented it to its year
seven pupils as a coursework task which our PGCE trainees
were to assess - leads to an assumption that the base
numbers should be consecutive integers. This is not
which prevent their developing as autonomous professionals
necessarily the case but it does provide a structured
whose opinions and beliefs are valued. One of the most
disappointing aspects of many such departments is the
presentation to pupils of mathematics as a set of discrete
skills and techniques which are taught and practised in
isolation from other skills and techniques. For several
introduction which should facilitate pupils' entry into the
reasons we believe that such approaches, indicative of a
peculiarly English tradition, are unlikely to facilitate either
children's understanding of mathematics as a coherent and
interrelated set of ideas or their ability to engage with those
aspects of the subject - reasoning, logical thinking, problem-
solving and proof - which make it worth studying. Sadly,
there is sufficient evidence from around the world to suggest
that where mathematics is not presented as a coherent and
continuous experience overall attainment is likely to be
lower than in those countries where it is. This article is an
attempt to take a fairly well-known investigation and show
how it might form one element of a more coherent and
continuous curricular experience in secondary schools.
The investigation is frequently presented to children as
something like this:
task. Much of what follows we draw from the work of our
trainees who were invited to see where the task took them in
order to frame their understanding of the children's efforts.
For a base of three numbers the peak is 4f + 4 where f is
the first of the three consecutive integers. This fits with the
example above where the first number was 3 and the peak
was 16. The justification for this can be seen below.
4f+4
2f+l 2f+3
f f . f+2
As an aside, there is an interesting pedagogic perspective on
this element of the work. The image above is the general
form of the particular example represented by, say, n=7
which, it could be argued, looks like this:
7+7+7+7+4
Number Pyramids
The picture below shows a number pyramid where the
numbers in adjacent cells are added together to find the
contents of the cell above. Investigate further.
7+7+1 ]i...............iiiiiiiii.iil 7+7+3
7 """ . . 7+ 1 ................ ! i l 7+2
and which could be seen as being equivalent to
16
7
9
3F ~l~i!i 4 5 ii~ii
4.7+ 4
2.7+1
2.7+3
l 7 i "~iiii~iiiiiiii 7+ 1 lii "ii"iiiiiiiiii 7+2 1
Mathematics in School, November 2000 The Mathematical Association website www.m-a.org.uk 27
Thus, there is a fairly natural link between the particularity
of seven and the generality of n, or p or number or whatever.
As the number of rows changes the following results are
obtained for a first base number f:
Rows
1
2
3
4
5
Peak f 2f+ 1 4f+ 4 8f+12 16f+32
This is an interesting set of results because it offers both
comfort and threat. There is comfort in the apparently clear
The notion of starting at the middle and invoking
symmetry is a strategy worth pursuing with children. It
allows not only access to a generality but also a justification
for the result based on a principle of gain and loss. Also,
there is an obvious sense that children working on the task
in this way will be consolidating their skills of algebraic
manipulation within a context where the manipulation is
not just meaningful but necessary.
A consideration of the even pyramids necessitates a look at
the odds. In this case the peak is be equal to 2n-1x where x,
in this case, was the middle number of the bottom row. This,
pattern in the coefficients off, the first base number, and
threat in the less obvious constants. In other words, many
pupils might see that the rule is likely to be of the form p
trainees as it seemed at odds with the result obtained for the
(peak) = 2n-lf + ? where n was the number of rows in the
even pyramids. However, since both the middle number of
pyramid.
However, a relatively straightforward rearrangement of
the different functions gives the following:
Rows
1
2
3
4
5
Peak 0.5(2f+0) 1(2f+l) 2(2f+2) 4(2f+3) 8(2f+4)
which can be generalized as p = 2n-2(2f + n - 1).
There are other approaches to the same problem. Some
pupils may notice, for example, that the peak of a four-base
pyramid is always a multiple of the sum of the middle two
base numbers - that is, that 52 = 4(6+7).
52
24 iiiiiiiiii 28
55
tii~ii~iiil
~~iiiii~i~ii13
ii 157! 8
Hiiiiii~!
This could facilitate a structural argument based on
recognizing the general in the particular - in the case of the
four-row problem the peak is four times the sum of the
middle two base numbers because, if the sum of the middle
two base numbers is written as x, then the following would
be true:
4x
2x-2 iii!Siliii~i 2x+2
.... 6 8
Such an approach exploits nicely the symmetry of the
pyramid - that which is lost on one side of the pyramid is
gained on the other. This can be extended to other pyramids
with an even number of rows - any pyramid where the sum
of the middle two numbers could be found. Trainees who did
initially at least, caused consternation for some of the
an odd pyramid {f + (n-2)/2) and the mean of the middle
two of an even pyramid {(f + (n-1)/2 + f + n/2)/2} can be
expressed as functions off, the first term of the bottom row,
the two results can be reconciled.
It is one of the pleasures to be gained from mathematics
when results which, on the face of it, seem disparate are
reconciled as equivalent. This is often the case with results
which seem amenable to an odd or even approach. We also
believe, even though we are reporting the work of PGCE
students, that there is little here that could not have been
achieved by an able pupil at Key Stage four.
At some stage, many trainees realized that there was no
necessity for them to persist with consecutive numbers
although few deviated from exploring those based on
numbers in arithmetic progression. It is interesting,
however, that not one observed that pyramids based on
consecutive integers was a particular case of those based on
arithmetic progressions and that the general result could be
inferred from the structure of the former.
It is not inconceivable that trainees had never been
encouraged to recognize the generality in the particular case
- that the traditional reaction of, say, a university
mathematics department would be to eschew such a process
as lacking in rigour. This is a speculative comment but it
does seem disappointing that trainees, many of whom were
fresh from some of the finest mathematics departments in
the country, either did not recognize the context for what it
was or decided not to exploit it.
Pyramids and Pascal's Triangle
Many observed that the structure of the pyramids reflected
that of Pascal's triangle and showed, for example, that the
generalized peak for a three-row pyramid would take the
form of that shown below.
this in their work prior to assessing that done by the pupils
wrote something like this:
In a pyramid with an even number of say n, rows, the
second row will have n-1 cells and each of these will
a+2b+c
a+b BiiiiiiiiIiiii b+c
comprise x, the sum of the middle two base numbers, plus or
a liii!iiiiii iiiii b iiiiilii c
pyramid, will contain 2x plus or minus something. The next
The emergence of Pascal's triangle is one of the more
interesting elements of the Pyramids task and one which
minus something. On the row above will be n-2 cells and
each of these, because of the summative nature of the
row will contain n-3 cells containing 4x plus or minus
and so on. This process will continue up to the nth row
which will comprise a solitary cell containing the result
one might encourage children to explore. Indeed, there is no
reason to assume that a task like this should not be used with
2n-2x.
could be invited to investigate how each of the base numbers
any class of children to derive Pascal's triangle - pupils
28 Mathematics in School, November 2000 The Mathematical Association website www.m-a.org.uk
contributes to the peak. Indeed, there is no reason why many
secondary pupils, particularly those who are used to working
in a whole-class collaborative manner, should not access a
generalized result for, say, a three-row pyramid in the
three-row pyramids the peak is also a Fibonacci number.
What are the relationships here? What is the effect of
permuting the base numbers? What if we use Fibonacci
numbers other than consecutive?
manner of the one shown above.
34
Many trainees went on to note that for a pyramid of n
rows, the coefficient of the mth base number would be the
5 [ !iiiiiii!ii 8 l iiii i 13
mth element of the binomial expansion of (l+x)n-1 and
offered their idea in conventional series notation involving
sigmas and factorials. Interestingly, a small number noticed
that any peak is the sum of two peaks of a lower order. This
gave them the impetus to undertake a proof by induction the only difficulty lying in the algebraic manipulation
-which they succeeded in doing but which we choose not to
include here.
2 iiiiii]3 liiiiil5
One of the strengths of this particular strand of the
Pyramids investigation lies in the different levels of
accessibility. In most of the situations described above:
* novice investigators will be able to see Fibonacci
numbers emerging as the peaks;
Pyramids and the Fibonacci Sequence
Few trainees explored the effect of using numbers other than
consecutive integers. A few looked at numbers in arithmetic
progression but failed to notice that this was, in effect, the
same as working with consecutive integers. It is of interest to
us that good university graduates of mathematics appeared
unaware that consecutive integers were just particular
examples of an arithmetic sequence.
The pyramid below shows the result of using three
consecutive Fibonacci numbers.
21
* different levels and forms of generality can be recognized
by both novice and more experienced investigators;
* the context allows for a wide range of possibilities many
of which will prove productive and will allow for a
genuine sense of personal involvement;
* proof and justification can be encouraged at a range of
levels - from a structural articulation as to why a
particular peak should be a Fibonacci number to a
sophisticated proof by induction.
Thus, we would argue that if teachers propose doing some
work on the Fibonacci sequence then a revisitation of the
Pyramids investigation provides a context from which
many of this particular sequence's relationships might
8 [ii i~ i 13
emerge.
i~~ii~lS
This invites the question, given that the nth Fibonacci
number is in the first position, which Fibonacci will be the
Pyramids and Linear Equations
Look at the pyramids below. How might we encourage
children to find the missing values? What mathematical
peak?
Also, one can see that in the case above, the peak is exactly
ideas might we develop from such a situation?
seven times the first base number. Is this always the case
and, if not, what is the relationship?
The figure below shows the results of using alternative
Fibonacci numbers for the base of the pyramid. In this case
2
1
13
?
3 !~~~~if 1 ~~fifi l2iiiiiiiii 'iiiii
the peak looks as though it might bthe emiddle number
multiplied by five. Is this always the case?
65
18 i.iii47
5 1!!!!!!!!!13 !3iiii34
13. 16
-2
?
5
3
The next figure shows the results of using consecutive
Fibonacci numbers but with the largest forming the middle.
What result can be found here?
63
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rr***rr*aar
rr~*r***rr*r
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~**lt(ib*l
SS*SiS+C(~
8
r*rrrs****
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*a*r*a*r*r
B~
13
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21 aal~*r*lra
*II
~*I*b*~
I*X~II*CI*
SI(i*~l**r
Srci)r*erl
~C"'**l*tl
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r*rrra**r**S Ir*r rrraar*~
The situation we have here is one in which a range of
approaches could be adopted. We could try a trial and
improvement approach based on our understanding of
pyramids. Alternatively we could, as is shown below,
develop the linear equation 3x+10= 16 which gives us, as
teachers, a choice - to develop formal solution methods or to
continue the ad hoc approaches which pupils will have
already begun to develop. Either way, the problem will, at
Of course, the exploration of Fibonacci need not end at
three rows. A four-row pyramid can be seen below. As with
the very least, provide pupils with a reference point for when
they are introduced more formally to linear equations.
Mathematics in School, November 2000 The Mathematical Association website www.m-a.org.uk 29
3x+10
5+x iiiiiiiiifiiiit i 5+2x
Such a problem demands that pupils are systematic and
well organized. In the case of this particular peak the only
possible integral results are:
1141, 2,2 2,1i,3 2, 2, 1 3,1, 24,1
3 iiiiiiiiiitii~~ii
1 Iiiliiiil~i~iii~i!!H!iPii~ij
x Ii~~i~~I~
......
. 4 iii!iiiiii
iH +x .!iiiti!ii!
x +~~Ifiiiiii
4 4
There are other questions one might ask. For example, the
pyramid above has a unique solution. What would be the
The generation of data for a problem like this is greatly
simplified by the use of a spreadsheet on which has been
constructed an appropriate pyramid.
effect on that solution of increasing the peak number by 1?
How much would it have to increase in order to double the
Which leads to these results for three-rowed pyramids:
value of the missing number? What would be the effect on
the solution of increasing (or decreasing) the value of the
Peak 4 5 6 7 8 9 10 11 12
centre base number?
It is our conjecture that mathematical content which
emerges as a consequence of the need to solve a
mathematical problem is preferable to its being introduced
in arbitrary or, worse, contrived real-world contexts. Such
problems help in the development of pupils' understanding
the important message that mathematics has an integrity
and coherence.
Arrangements 1 2 4 6 9 12 16 20 25
These figures are simultaneously unexpected and delightful
- the odd numbered peaks seemingly achieved in twice a
triangular numbers of ways and the even numbered peaks
obtained in a square numbers of ways. That is, for an odd
peak number, n, the number of solutions is (n-3)(n-1)/4
whilst for even peak numbers the number of solutions is
(n-2)2/4. Of course, this is only part of the story. Why should
we be getting these numbers and is there something in the
However, work on pyramids does not necessarily restrict
us to the solution of equations in one variable. The pyramids
below show two missing values which, again, we can address
in a variety of ways.
13 3
. 3!!iii~iiiil 1 ?
Trial and improvement will lead to a range of solutions
which, when graphed, may point towards the algebra of
straight lines and (possibly) conversations about negative
values and the arithmetic of directed numbers.
structure of the problem that allows us to justify our claims?
The first thing we noticed was that it is only the even peak
numbers that can have symmetrical bottom rows - because
it is obviously impossible to get an odd peak from such a
start. Significantly, the number of such symmetrical
solutions is relatively simple to predict. For any even peak,
call it 2m, the smallest possible centre number is 1 which
is combined
m1[(2m-2)/2]
on each side.
largest possible
centre
numberwith
is m-1
whichThe
is combined
with 1
on each side. Both configurations can be seen below. The
centre number can also take every value between 1 and m-1.
2m
2m
lm , ll tiiiiii~iiiil ....[ii iiiiiiim l ........ . .
4
For example, suppose the centre number was p, where p was
3
an arbitrary integer between 1 and m-l, then the pyramid
would look like this:
2
2m
110
Significantly, a context such as this can serve to introduce
pupils to the ideas of continuity and the fact that any point
on the line will represent a valid solution to the problem and
mp ii ] -p
Thus, of the (2m-2)2/4 possible solutions for an even peak,
m-1 of them are symmetrical. Thus, with the total number of
that this is independent, for example, of the quadrant in
solutions simplified to (m-1)2, the number of asymmetric
which the point lies.
solutions is (m-1)2-(m-1) = (m-1)(m-2) which is, interestingly,
the number of solutions for the previous odd peak.
A Very Different Investigation
Look at the pyramid below. We know nothing of its base
although we do know its peak. How many possible solutions
are there if we assume that all the base numbers are positive
integers?
However, the symmetries of the even peaks are not
necessarily going to explain the general form for any peak
number. So let's look to see what we can glean from a
systematic analysis of the situation. We'll start with an even
peak and see where it leads.
Firstly, with three numbers on the base and a fixed peak,
two of the three numbers will define the pyramid
completely. So let's start by looking at what happens when
.... . ii ii i
we do a little specifying. If the three numbers are a, b and c
and the peak is even, 2m, say, then we can look at several
possible situations.
30 Mathematics in School, November 2000 The Mathematical Association website www.m-a.org.uk
Firstly, we know that a + 2b + c = 2m. Now, c can take all
values from 1, its smallest, to 2m-3, its largest. Our argument
will focus, initially, on the odd values of c.
When c = 1, we know that a+2b must be odd which tells
us that a must be odd (because 2b is necessarily even). Now,
the smallest such value a can take is 1 (when 2b is at its
maximum) and the largest occurs when 2m-1-2b is
minimized. This occurs when b=1. Thus, a can take all odd
values between 1 and 2m-3. This includes m-1 terms (to
calculate the number of terms in a sequence of odd numbers
beginning at 1 one should add one to the last term and then
halve the answer).
When c = 3, a + 2b must be odd so that a must be odd. The
smallest such value is obviously 1 (when 2b is at its largest)
and the largest occurs when 2m-3-2b is minimized. This, too,
offers an introduction to a variety of topics on the
mathematics curriculum. We hope we have shown that it
allows for the revisitation of topics or ideas covered earlier -
not as topics taught discretely every year but within an
integrated problem-solving framework. We hope we have
shown that Pyramids might form part of a coherent
sequence of activities and lessons from which mathematical
sense-making on the part of the learner might emerge. We
hope we have shown how an investigation like Pyramids
serves the interests of equality of opportunity - it facilitates
access to the curriculum for all pupils. We argue that it is one
tool in a kit of mathematical starting points which falsify the
belief, to which most teachers subscribe and politicians
choose to perpetuate, that mathematics cannot be taught to
mixed ability classes.
occurs when b= 1. Thus a can take every odd number from 1
to 2m-5 and includes n-2 terms.
However, social justice was but one of our objectives in
writing this article. We hope also that we have shown that
the mathematical experiences of many children could be
When c= 5, a+2b is odd so a must be odd. The smallest
made more coherent by the invocation of different aspects of
the Pyramids problem at different times in their schooling.
value a can take is one and the largest is when b= 1. Thus, a
can take every odd number from 1 to 2m-7 and includes
m-3 terms.
This process continues until c reaches its largest odd value
which occurs when both a and b are 1 and is clearly a unique
solution.
Thus we can conclude that for the odd cs the total number
of different solutions is the sum of all the integers 1, 2, 3, .....,
m-3, m-2, m-1. That is, the (m-1)th triangular number
m(m-1)/2.
A similar argument shows that the even cs gives a similar
We believe that mathematics which emerges as a
consequence of repeated exposure to such problems is likely
to acquire a meaning and purpose which might not be seen
in other contexts - the development of the need to work on
linear equations, for example, which work on Pyramids
might necessitate, becomes purposeful and offers a
coherence which cannot be achieved through the use of
pseudo-real-world contexts.
We close with some questions which have not been
addressed above but which might serve as the means by
which further mathematical ideas might be introduced.
sequence of integers from 1 to m-2. That is, the (m-2)th
If the base numbers of a three-row pyramid are consecutive
triangular number (m-1)(m-2)/2.
integers:
And, of course, the sum of two consecutive triangular
numbers is a square number. In this case we get:
m(m-1)/2 + (m-1)(m-2)/2 = (m + m-2)(m-1)/2 =
(2m-2)(m-1)/2 = (m-1)2.
Now, if the peak number were n, rather than the 2m we
used here, then (m-1)2 would be written (n-2)2/4, which is
exactly what we predicted earlier.
A similar process can be used to show that the number of
possible solutions for an odd peak number is (n-3)(n-1)/4 as
indicated above.
So, the results of this particular investigation are amenable
to some form of proof. It is not a particularly satisfying proof
as we feel that results invoking generalities pertaining to
* What conditions must apply to the base numbers of a
three-row pyramid to get an odd peak?
* When is the peak a multiple of three? four? five? ..
If the base numbers are in arithmetic progression:
* What conditions must apply to the base numbers of a
three-row pyramid to get an odd peak? When is it even?
* What conditions must apply to the base numbers of a
three-row pyramid in order for the peak to be double the
sum of the base numbers?
If the base numbers are consecutive triangular numbers
what can you say about the peaks? S1
either triangular or square numbers ought to fall to a
structural or geometric argument. However, we believe that
the sort of argument we offer is accessible to many learners
and further validates Pyramids as a productive means by which
mathematical activity is generated. Of course, we have considered
Keywords: Investigations; Proof; Equal Opportunities.
only pyramids with three rows - what if we looked at four,
five or more? What about pyramids with just two rows?
Authors
Conclusion
e-mail: [email protected]
We believe we have shown that Pyramids can lead to a wide
Paul Andrews and Anne Sinkinson, University of Cambridge Faculty of
Education, 17 Trumpington Street, Cambridge CB2 1QA.
range of learning outcomes for pupils of varying
Acknowledgements: We would like to thank Heather Massey and colleagues at
the St John Fisher Roman Catholic High School, Peterborough for their help
mathematical sophistication. We hope we have shown that it
in the setting-up and implementation of the work for our trainees.
Mathematics in School, November 2000 The Mathematical Association website www.m-a.org.uk 31