Why do this problem?
requires students to draw firm mathematical conclusions following a period of experimentation. It provides an interesting context for students to begin to appreciate the value of using algebraic notation.
The problem encourages students to ask more general questions beyond the initial context.
Demonstrate how number pyramids work. This could be done with the teacher remaining silent throughout, and expecting the students to make sense of what is happening.
"What do you notice?"
"What questions do you think a mathematician might ask next?"
Give students time to discuss and suggest. If no suggestions are forthcoming, share the questions suggested in the problem:
- Given the numbers on the bottom layer in order, can you find a quick way to work out the number at the top?
- If you change the order of the numbers on the bottom layer, will the top number change?
- If you can rearrange the numbers on the bottom layer, can you find a quick way to work out the largest possible number that could go at the top?
- Given the number at the top, how can you come up with possible numbers to go at 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 the lesson draws on, make it clear that they are expected to be able to explain
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. This spreadsheet
may be useful.
How are these numbers generated?
How does the position of a number on the bottom row affect the total at the top?
With the starting numbers $10$, $1$, $6$, $4$, why is it impossible to make a top total which is a multiple of $3$?
If a 100-layer pyramid had $1$s in every cell on the bottom layer, how could you work out the number at the top?
"Increase the first number on the bottom layer by $1$. What happens to the total at the top?
Now try with the other numbers. What do you notice?"