I started looking at the NRich website today and just loved their tasks for secondary students! Just on their front page alone, I found these excellent tasks.
http://nrich.maths.org/1019 looks very different from, but is actually very similar to the beehive traversal problem I had written about here. You can actually couple the two problems on the same day(s), to give one problem to one half of the class and the other problem to the other half of the class, and they can come together later to discuss/make connections.
http://nrich.maths.org/1785 is a really nice and sneaky lead-in to quadratics. Low-floor, high-ceiling indeed!
http://nrich.maths.org/7405 is also super fun to help us think about modular arithmetic. A nice extension task to the problem would be for the students to come up with values that could go into each bag, that would allow you to draw a set of 4 numbers and then have the result be divisible by 4, or to allow you to draw a set of 5 numbers and be divisible by 5, etc. Then, have the students go around and experiment this with other students or with their parents as a "Gee, Whiz!" homework assignment. As part of the homework, they should also explain how it works to their parent.
NRich also has a fun Problem of the Week, which also promotes inquiry and flexible thinking but is still very accessible to kids. This week's problem is reproduced as follows:
A snail is at one corner of the top face of a cube with side length 1m. The snail can crawl at a speed of 1m per hour. What proportion of the cube's surface is made up of points which the snail could reach within one hour? (Originally from UKMT Mathematical Challenges)
This snail problem is a simple introductory Geometry problem that all students can access in Grade 9 Geometry, that also reviews some basic proportional reasoning concepts. It is rich because it helps to raise questions such as, "What happens if the snail stops and turns direction? How do we find the maximum covered distance?"
If you are looking for a more challenging version of the surface traversal problem for use later in the year, I recommend this one from Exeter's Math 3 (thanks to Thomas Seidenberg for the generous sharing of their de-spiraled Exeter conical material). This one is not a low-floor problem, as you've probably figured out. But, I still like it because I don't think it's that intuitive without some hands-on manipulation of 2-D nets, that the shortest distance from the spider to the fly would not be along the conical rim.
A spider is on the rim of a conical cup when it spies a ﬂy one third of the way around
the rim. The cone is 36 cm in diameter and 24 cm deep. In a hurry for lunch, the
spider chooses the shortest path to the ﬂy. How long is this path?
I love NRich! (And Brits in general, but perhaps that is out of the scope of this post.)