This
problem offers students the opportunity to notice
patterns, make conjectures, explain what they notice and prove
their conjectures. Generalisation provokes the need to use
algebraic techniques such as collecting like terms and representing
number sequences algebraically.

This problem follows on nicely from
Number Pyramids

What follows could be done in a classroom with students
working on paper, or in a computer room so that students can make
use of the interactivity and spreadsheet.

Start by showing the interactivity:

"I'm going to type in a number (2), and I'd like you to watch
what happens. Can you work out what is going on? Do you notice
anything interesting?"

Allow students a short time to discuss in pairs what they
saw.

"In a moment, I'm going to type in the number 7. Can you
predict what will happen?"

Give pairs a little time to discuss and decide, then show what
happens.

"In a while, I'm going to ask you to share anything
interesting you have noticed, and any questions that have arisen.
You might want to try some more examples to test out your ideas or
to give you more data before looking for patterns. Or you might
like to think about different ways of representing what's going on
in the pyramid."

After students have had plenty of time to explore, bring the
class together and share noticings and conjectures. If no-one has
considered using algebra, this would be a good time to suggest
representing the bottom left corner with $n$ for example, and
working out the other entries in terms of $n$.

Once the class have an algebraic expression for the top
number, this can be used in two ways:

- Can they explain why it's impossible for some numbers to appear at the top (when an integer is entered at the bottom)?
- Given a top number, can they use their expression to find what number should be entered at the bottom to generate it?

"In these number pyramids, the bottom layer is always a set of
consecutive numbers, but there's no reason why the bottom layer
couldn't be any other number sequence - starting at 13 and going up
in 4s for example. Is there a quick way to work out what the top
number will be? Explore some different sequences and use algebra to
help you predict and explain what happens."

Can you work out what is going on in this pyramid of
numbers?

What do you notice about the numbers on each row of the
pyramid?

How do we know that $8x+12$ is always a multiple of $4$ but
never a multiple of $8$?

(for integer values of $x$)Students could work on Number Pyramids
first in order to gain some familiarity with the structure
underlying the problem.

The group could be split so that some investigate sequences
that go up in 2s, some 3s, some 4s and so on. Then the class could
come together to share what they have found out before generalising
to any sequence.