I have a rule about the first day of school: always do some math. No, that doesn't mean you have to start Lesson 1.1 as soon as students walk through the door, but it does mean that you should give your students a preview of the kind of thinking, reasoning, puzzling, and sense-making that they'll be doing in your class this year. Ideally, students will be so highly engaged that they barely even recognize they're doing math -- and certainly not the kind of math they're used to in school.
I also have an inordinate appreciation for what I call "interesting problems". These are tasks that use mathematical thinking and strategy, but don't require specific content knowledge like the formula for the equation of a circle or knowing what a composite function does. They are highly accessible, highly engaging, and have multiple solution strategies. The task itself can be explained in a few sentences and students can work on them for 20 minutes or 2 hours, depending on how far they want to take it. I scour the internet for tasks like these and have been collecting them for YEARS on my computer. I decided this was the year to bring them to the light and share them with the Math Medic community.
These tasks don't require formal content knowledge, but they do help students engage in the mathematical practices and develop mathematical habits of mind, such as:
Looking for and making use of structure
Representing one's thinking
Working systematically
Visualizing
Developing a convincing argument
Conjecturing and generalizing
While I've curated this list with high school students in mind, many of these tasks could be done with middle schoolers or even with adults. The inspiration for these questions came from all over this great big internet, but have been adapted and reformatted for classroom use. So, without further ado, here are my (current) top 10 "interesting problems" to do on the first day of school.
10 Interesting Problems
A linear context in a LOT of disguise. Many solution strategies and great opportunities for representing one's thinking with a model or visual.
This one is set up with multiple parts providing lots of natural extensions. Thinking about a number's properties is key to this task! Make sure to print the 100s chart that is on page two on a separate sheet of paper. You can offer it to everyone or as an optional support.
Loads of solution strategies on this one as well. Your teacher brain might scream system of equations with 4 variables, but you'll be surprised at the intuitive solutions your students find to solve this problem.
Perfect after a summer of olympics. Students deal with rates in this problem, which is an important concept for any age group and relevant for any math course.
This one's been famous for a long time but I'm sharing it anyway because students do great with it!
This one and the next two all encourage students to think systematically. There's a brute force solution but making use of structure will illuminate an easier way.
This is a good intro to thinking systematically and has a nice extension. I would use this in an Algebra 1 or Geometry course.
This one is very difficult, so we recommend saving it for your Precalc or above courses.
This one is the most recent in my collection and I'm still thinking about the extension part!
I've often done this one with Geometry students because of the shapes and visual reasoning components.
Editable versions of these tasks can be found in this Google Drive folder.
How to use these in your classroom:
Pick ONE task for students to work on. We don't recommend giving multiple tasks back-to-back because it can start to feel like a worksheet, rather than a puzzle.
Solve the problem yourself first! We are purposefully not giving solutions here, so make sure you've wrestled with the problem yourself before handing it out to students.
Have students work on these in groups of 2, 3, or 4. Make sure they have enough materials available to hash out their ideas and represent their strategies. These are great to do on vertical non-permanent surfaces or poster paper.
Decide how long you will let students work. If doing this on the first day of school, we recommend about 20 minutes. If students don't have a solution by then, that is totally fine. A surprising number of them will keep thinking about the problem throughout the day or even at home.
Be ready with some extensions for groups who finish early, but make sure they understand what "done" means. Have they clearly communicated their strategy? Have they convinced themselves and others that their strategy will hold up? If giving an extension, make sure it's related to the given task, not just a different task. It's important that students are challenged in the depth of their reasoning, not in the quantity of problems.
If you're looking for more tasks like these, I highly recommend the NRICH site from the University of Cambridge.