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More Number Pyramids

Age 11 to 14 Challenge Level:

Why do this problem?

The interactivity in the problem provides a 'hook' to engage students' curiosity, and allows them to experiment, 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. 

Possible approach

This problem follows on nicely from Number Pyramids
If computers or tablets are available, students could work in pairs using the interactivity. If not, they could construct their own pyramids using paper and pencil.
 
Start by showing the interactivity:
"I'm going to change the number in the bottom left cell, 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 linear 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." The follow-up interactivity can be used to test out conjectures when the bottom layer goes up by numbers other than one.

Finally, you could finish off the lesson by choosing a bottom left number and step size, and challenging students to predict what the top number would be, or choosing a target top number and a bottom left number and challenging them to work out the correct step size.
 

Key questions

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?
If I know the number in the bottom left hand corner, can you work out the top number without working out the middle layers?
How do we know that $8x+12$ is always a multiple of $4$ but never a multiple of $8$?
(for integer values of $x$)

Possible extension

Given the top number and either the starting number or the difference between the numbers on the bottom layer, can students work out the missing piece of information?

Possible support

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.