Why do this problem?
This problem offers an opportunity to explore functions of
functions. Although the context is powers and logs, the same
structure can be used to explore other functions. By considering
the questions in the task, students will gain a clearer
understanding of the functions in question.
If students all have access to computers, they could explore
the interactivity on their own or in pairs to make sense of how the
upper layers of the pyramid are generated. Alternatively,
display the interactivity to the whole class, and ask for
suggestions of numbers to enter in the bottom layer, and give them
time to discuss in pairs what they think is going on.
Once they have some ideas, discuss as a class what they have
If they have not yet met logarithms, they might express the
relationship as something along the lines of:
"It's the power of two that the product of the numbers on the
layer below is", and this is a good opportunity to introduce the
notation of logarithms, and perhaps to start deducing some of the
laws of logarithms.
Once the class have established how the pyramid works, set
them the four challenges from the problem, to work on away from the
computers, so they have to work out the answers rather than relying
on trial and error.
Finally, bring the group together and share the methods they
used for answering each challenge, before checking their answers
using the interactivity.
Can you find numbers for the bottom layer that give you
whole numbers on the second layer?
What is special about these numbers?
What happens if you change just one of the numbers on the
the same activity as the interativity in the problem, but could be
adapted to create other function pyramids to investigate.
Suggest students start by putting 4s in every cell on the bottom