Here are two math problems, and I only care about one of them:
Problem 1: A person has 3,000 bananas and a camel. The person wants to transport the maximum number of bananas to a destination 1,000 km away, using only the camel as transportation. The camel cannot carry more than 1,000 bananas at a time and eats one banana every km it travels. What is the maximum number of bananas that can be transferred to the destination? (geeksforgeeks.org)
Problem 2: A recipe requires ½ cup of flour for every batch of cookies. How many full batches of cookies can be made with 5½ cups of flour? (SBAC, Grade 6, 2019)
I like Problem #1 a lot because I don’t care that it’s not real-world. The scenario is ridiculous, but the problem-solving required isn’t.
The problem with Problem #2 is that it’s supposedly real-world, and yet it’s not. Find me a recipe for a “batch” of cookies that requires only half a cup of flour. This token attempt at making a problem ‘real-world’ is worse than just giving me the bananas and the camel.
So I changed Problem #2 to this:
A recipe requires 1½ cups of sugar to make a batch of 4 dozen sugar cookies. Amy has a 4-pound bag of sugar, which contains about 9 cups. She can sell a dozen sugar cookies at the local farmer’s market for $20.
What I did was turn the problem into a scenario—the kind Annie Fetter and her Math Forum colleagues championed to encourage notice-and-wonder. There’s a nice bank of these at Goal Free Problems. Without a question attached, students can focus on making sense of the scenario first instead of jumping straight to ‘how do I solve this?’
Then, borrowing Peter Liljedahl’s terminology, I would ask students to create three questions from the scenario—mild, medium, and spicy.
Why this matters:
When students create the questions instead of just solving them, something shifts. They have to think about what makes a question easy versus hard, which forces them to actually understand the mathematical structure. That’s metacognition at work.
When a student labels something “mild” that’s actually spicy (or vice versa), you immediately see where their confidence doesn’t match their understanding. That’s actionable. Students who aren’t sure what to do can start with their own “mild” and build up—they’re essentially creating their own scaffolded question set instead of you doing all the differentiating.
Question-posing is higher-order thinking. They have to understand the scenario well enough to extract different mathematical relationships from it. And you’ll see solution diversity—one student’s “spicy” might use systems of equations while another’s uses proportional reasoning for the same scenario. Plus, they made these. There’s inherent investment when it’s their work.
Have students share their questions (gallery walk, digital board, whatever) and solve each other’s work. The calibration conversation that emerges is where the learning compounds.
Final note:
The main reason I wanted to share this is how often I hear teachers asking for extra practice problems. I’m wary of asking students to do four problems when two or three would be sufficient. And we’ve all seen those worksheets with 30+ problems—that’s where math goes to die.
I’m hoping this gives you another strategy when you run out of practice problems. Take existing questions, turn them into scenarios, and have students create the tiered questions themselves. It’s also a good use for AI—give it a prompt like:
Turn this problem into a scenario for 6th graders:
Remove the question entirely
Use realistic numbers (look them up if needed)
Keep it to 3-4 sentences
Make sure there’s enough information for students to create mild, medium, and spicy questions
Check that the context is age-appropriate
I made a few examples if you want to see what this looks like. The spacing is intentional—you can cut them into strips for thin slicing.
When you say, "I’m wary of asking students to do four problems when two or three would be sufficient," I wonder what you're wary of?
As a teacher who's tried plenty of activities like what you describe, what I'm wary of is that many of my students will engage and do a lot of great thinking about the structure of the concept, but a significant minority will say idk, or opt out, or take the easy way out and write something trivial. And I spend my time trying to get them to engage but it's a bit like whack-a-mole and I know that some students aren't getting much out of the activity.
I can often get those same students engaged with the same ideas by giving them a well-sequenced set of practice problems, starting easy and getting gradually harder. And in that case, I often give 10 or 15 problems (the exact number will vary depend on the topic) and err on the side of more, not less. What would you say I should watch out for when giving more practice?