These may be too hard, but the ACM's International Collegiate Programming Contest (ICPC) has a set of past programming problems that require algorithmic thinking. I took a class in college where we basically just worked on these for 3 hours a week. It was really good problem-solving experience.
Any puzzle game which requires students to plan the entire problem before executing it might help. Also, there are physical puzzles which can be solved algorithmically much more neatly than if they use 'trial and error'.
In English, one resource which I have not seen mentioned yet is Code.org which has themed coding puzzles for all ages. Other puzzles, ...
An "old school" answer (nearly 60 years old now!) which works for any age range is turtle graphics, which is (are?) implemented as a Python module.
We can only guess how much of your curriculum is officially labelled "geometry", but it will certainly teach algorithmic thinking, and also be fun.
There are two questions here.
The easier question is describing volume. There are two separate ways to measure volume. In metric terms, one standard measure would be liters and the other would be cubic meters. They are related in metric: one liter is 1000 cubic centimeters. Both of them (and their "American" counterparts) are extremely common and ...
Challenging question! Two ideas.
(1) Calculate the Greatest Common Divisor of two natural numbers, not
so easily accomplished by hand on moderately large numbers.
could be used to illustrate recursion/induction. Here is Python3 code:
trace = True # True turns on tracing prints.
def GCD( a, b ):
'''Returns the Greatest Common ...
One well-known source is Project Euler. The concept behind it is that each problem is mathematical and designed to be solved by an efficient algorithm on a "normal" computer in less than a minute. The early problems are all extremely accessible. As the problems go on, they become (in my mathematical opinion) far more esoteric from either a mathematical or ...
Another example is that if you have lets say $30$ guests coming to your home, you want to buy cups for the tea or coffee or whatever but these cups come in dozens, i.e. there is only bags of $12$ cups, so how many bags you need?
They start for example $12+12=24$ not enough cups, $12+12+12=36$ now we have enough cups, so we need $3$ bags.
(Posting my comment as @ChrisCunningham suggests.)
How about: The number of days to their next birthday? This could involve: number of days remaining in this month, plus the number of days in the next month, the next, ..., until the number of days in their birthday month. Or, similarly: The number of days from now until Christmas: $13$ in November
plus $25$ ...
I ... find myself troubled by lacking proper examples of why a sophisticated theory can actually turn out to be beneficial ...
I agree this is troubling. "I feel your pain," as the saying goes.
I would like to suggest that
computer graphics can serve as a source for a subset of the examples you seek: to motivate polynomials, and motivate finding roots of ...