Number Pyramids
Why do this problem?
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?"
- 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.
Key questions
Possible support
Possible extension
You may also like
Summing Consecutive Numbers
15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?
Always the Same
Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?
Fibs
The well known Fibonacci sequence is 1 ,1, 2, 3, 5, 8, 13, 21.... How many Fibonacci sequences can you find containing the number 196 as one of the terms?