Why do this problem?
This problem is simple to explain yet involves quite a
complicated solution process. This problem will hone skills of
addition of two digit numbers whilst challenging the organised
mathematical thinking of students. The problem may be done by
trial and error or with some appeal to algebra.
Possible approach
It is possible for this problem to be done entirely
individually but a group discussion may lead to more insights
about the strucure of the number pyramid. There are a great
number of possible combinations of base numbers; ideally
students should be encouraged to understand some of the
structure of the pyramid in order to reduce the number of
possibilities that they have to try out.
You might initially discuss the problem as a group. Can
anyone see any structure or offer a solution strategy?
Students could then experiment individually with various
combinations of numbers. Encourage students to devise a clear
recording system. Encourage them to decide sensibly on the
next combination of numbers to try rather than randomly. For
example, if a top number is too small then how can the
numbers at the bottom be altered to increase this?
You could use a
simple
spreadsheet to model the pyramid. Could students
construct one of these themselves? This is an interesting
structural challenge which allows the creation and
investigation of larger pyramids.
Key questions
Key questions should lead to understanding the strucutre of
the pyramids
- What is the total for the order $1, 2, 4, 8, 9, 12$?
Would we get the same total with a different order? Why?
- What is the largest possible top number for the
pyramid?
- What is the smallest possible top number total for the
pyramid?
- Which pairs of numbers can be switched without changing
the value at the top of the pyramid?
Possible extension
Once an student has found a solution they could be asked
these questions:
- Are there any other possible combinations of $1, 3, 4, 8,
9, 12$ which lead to the answer?
- What other top numbers are possible? Can you find top
numbers which are not possible?
- Are there any other combinations of 6 base numbers which
lead to the top number being $200$?
Possible support
Students who struggle with the level of addition might be
provided with a simple
spreadsheet to do the calculations. They could also be
asked simply to work out $5$ pyramids with different numbers to
see who can get the largest number or the answer closest to
$200$.