Why You Should Mix Easy and Hard Problems in Math Class

One of the more impactful paragraphs I’ve read about teaching mathematics came from Henri Picciotto in his post, Mapping Out a Course.

Throughout the unit, go back and forth between easy and hard the whole time. Avoid the trap of starting easy and gradually making it harder, for two reasons. One, starting too easy gives the wrong impression, and students don’t launch into the unit with the right attitude. Two, gradually getting more difficult gives the impression to some students that there’s a time where they might as well give up. Going back and forth is a way to keep everyone alert and challenged, and to have times where the pressure is lessened.

That idea made a significant difference in my classroom.

Instead of gradually increasing the difficulty of problems throughout a lesson, I began mixing easy and hard problems throughout the entire class period. The result was a classroom where challenging problems became a normal part of our routine rather than something students encountered only at the end.

Here’s what that can look like.

A Sample Algebra 1 Notes Handout

Here’s a sample Notes handout from our Algebra 1 Slope unit.

We started with something relatively straightforward: naming the four types of slope; positive, negative, zero, and undefined.

We started with an accessible task before introducing a more challenging problem.

But rather than gradually increasing the difficulty, the next section was a challenging problem.

The lesson moves to a challenging problem instead of gradually increasing difficulty.

When I give students a hard problem, I don't always leave them to work through it independently. Often, I use Think-Pair-Share to give students individual think time, partner discussion, and a whole-class debrief.

The goal is to provide the support students need while still asking them to take on meaningful challenges.

After that challenge, we moved into two sections of easier, skill-level problems involving calculating slope from different representations.

Easier skill practice gives students time to build fluency after tackling a challenging problem.

This gave students a chance to get into a rhythm and work toward the learning target for the day: improving their ability to calculate slope from different representations.

Then we went back to another challenge problem.

A second challenge appears before the lesson is over, rather than being saved for the end.

We continued this pattern for the rest of the notes: easy, hard, easy.

The completed notes alternate between easier practice and challenging problems throughout the lesson.

Why Mix Easy and Hard Problems?

I use this approach in notes, but I also incorporate challenging problems into practice assignments, warm-ups, and spiraling practice.

It’s important not to turn this into a rigid formula. If students need more focused time with a particular skill, give them that time. The goal isn't to force a hard problem into every few minutes.

Instead, mix challenging problems into the normal flow of instruction so students don't begin to expect that difficulty will continually increase.

1. Students are more likely to engage with challenging problems.

When difficulty gradually increases, students can start to recognize that the problems are getting harder. For some students, that can eventually lead to the feeling that they’ve reached the point where they can’t succeed.

Mixing easy and hard problems disrupts that pattern.

A challenging problem becomes something we simply encounter during class, not a signal that things are about to become increasingly difficult.

2. Challenging problems don't get pushed to the end.

This was one of the biggest practical benefits I noticed.

When I used a gradual difficulty ramp, the hardest problems often appeared toward the end of class. If we ran out of time, those were the problems we didn't get to.

Now, challenging problems appear throughout the lesson.

That means students get more opportunities to encounter difficult mathematics, discuss it with classmates, and see that struggling with a problem is a normal part of our classroom routine.

3. Easy problems still matter.

Mixing in challenging problems doesn't mean eliminating easier practice.

Students need opportunities to experience success, build fluency, and regain confidence after working through something difficult.

It’s okay to have several easier problems in a row when students need focused practice. It’s also okay to give students two challenging problems in a row.

The key is to pay attention to the flow of the lesson.

I often think about it as a balance: push students' thinking without pushing them so far that they disengage.

Make Challenging Problems Low-Pressure

Another way to help students become more comfortable with difficult problems is to give them opportunities to solve those problems without worrying about a grade.

In my classroom, I use minimal grading for daily notes and practice. Those are often the places where I mix in challenging problems.

Students can attempt a problem, make a mistake, talk through it with a partner, and try again without feeling that the attempt is going to hurt their grade.

You can learn more about this approach in our Standards-Based Grading system and philosophy.

The goal isn't to make every problem hard.

The goal is to make challenging mathematics a normal part of every lesson.

Give students easy wins. Give them time to build skills. Give them opportunities to struggle productively.

And, throughout all of it, mix the easy and the hard.

That simple change made challenging problems feel much more like a normal part of mathematics in my classroom and much less like something students needed to fear.