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Number Pyramids

Age 11 to 14 Challenge Level:

Why do this problem?

This problem introduces the idea of number pyramids as a context to explore number relationships that can be generalised and justified using algebra. The interactivity in the problem provides a 'hook' to engage students' curiosity, and allows them to very quickly perceive how the pyramid changes as the input numbers change.

This can be followed by some pencil-and-paper work away from the computer to make predictions which can then be tested using the interactivity.

Possible approach

If computers or tablets are available, students could work in pairs to explore the first number pyramid. Alternatively, the concept of a number pyramid could be introduced by drawing a couple of examples on the board, and silently filling in each cell, pausing to let students reflect on what they think is happening at each stage. 

"What do you think the rule is, to generate each layer of the pyramid from the layer below?"
"What questions do you think a mathematician might want to explore?"
 
Give students time to discuss and suggest. If no suggestions are forthcoming, share the questions suggested in the problem:
  • If I tell you the numbers on the bottom layer, can you work out the top number without working out the middle layer?
  • If you change the order of the numbers on the bottom layer, will the top number change?
  • Given any three numbers for the bottom, how can you work out the largest possible number that could go at the top?
  • If I give you a target for the top number, can you quickly find three possible numbers for the bottom?
Give students time to work on the questions that have been raised. Encourage them to experiment before trying to draw more general conclusions. As students are working, make it clear that they are expected to be able to explain and justify any generalisations they make.

If students notice patterns but can't explain them, it may be helpful to introduce algebraic representation.

Similar questions can be asked about larger pyramids. Here is an interactive four-layer number pyramid.

To finish off the lesson, bring the whole class together to discuss their discoveries, and use the interactivities to test out their conjectures.
 

Key questions

How are the numbers in each layer generated?
If I increase one of the numbers on the bottom layer by one, how does the top number change?
 

Possible extension

More Number Pyramids follows on from this task.
 

Possible support

Encourage students to work systematically, changing just one number at a time, and looking at the effect on the top layer.