“Think About Your Favorite Problem From A Unit”

Bob Lochel, responding to commenter Jenni who wondered how, when, and where to integrate tasks into a unit:

In my years as math coach, the most efficient piece of advice I would give to teachers is this: think about your favorite problem from a unit, the problem you look forward to, or that problem which is number 158 in the last section which you know will generate all kinds of discussion. Without fail, this problem is often done last, as the summary of all ideas in the unit. Okay, why not do it first? Keep it simmering in the background, flesh it out as ideas are developed and pratice occurs. It often doesn’t take a sledgehammer to make a good unit great.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. Word.

    I like the word “motivate”. This can be hard to figure out, but find a question students want to answer. Pose it at the start of the unit. Then they will know what they are doing, why they are doing it, and have a touchstone to contextualize their work.

    That said, I think there’s nothing quite like seeing what students can do with a challenging task after they’ve been immersed in a new topic for a few days. My best solution is more tasks with a bit less time spent on them, to look at important concepts from as many angles as possible. But really rich tasks can do everything, and tie whole units together. Who wants to collate a ton of those?

  2. I’ve been thinking about doing this as my homework set for each unit . . . great for the whole -part – whole unit breakdown. As each piece of the puzzle is found, they can use it to describe the physical model (okay, I teach physics!) . . by the end of the unit, they should be ready for additional open ended scenarios . . . anyone have more experience with this on how to make it happen?

  3. This comment hits at the core of 3ACTS, right? Why does a student have to wait through 40 minutes of lecture to get to the mildly interesting part?

    Regarding complex tasks as homework, there should be some careful thought in the design. Often high school students aren’t totally ready for a completely open task. Many will freak out over the lack of structure, or wander off into conclusions driven by misconceptions. They aren’t going to good at fiddling with a problem if someone hasn’t shown them how to do it. A lot.

    If you want to design complex homework, you need to model it with complex/puzzling classroom content.

  4. And for teachers adopting technology, this works as well. For example, if your favorite problem is the Pulley Problem, then try to use your dynamic geometry software to model the problem with different radii and belt lengths. If you have a new L.M.S., then try it out for your favorite project.

  5. Totally agree. Perfect timing too; tomorrow I am doing Andrew Stadel’s “Stacking Cups” Three Act lesson as a way to INTRODUCE solving linear systems, rather than as an application at the end of the unit.

  6. I really like this idea — too often I leave the interesting problems for the end of the unit, and, more often or not, I don’t get to them, or give them the time they deserve. I could use this idea to introduce the Pythagorean Theorem to my 8th graders. I like to show students diagrams with details missing, requiring the Pythagorean Theorem to find the missing values, etc. It might make this more powerful to students by having them think about what values they need, and then asking the question of: what can I use to get these missing values?

  7. Jessica Ciampi

    April 29, 2014 - 4:43 am -

    I like this idea because most of the time by the time you get through all the BS and get to the most important part the period is almost over and by then you might have lost half of your class. I think we tend to leave the interesting parts till last because we feel like we need to build up the idea first. Sometimes this is good because it shows students where the idea comes from however sometimes we can lose the motivation and attention of our students.

  8. I have always loved related rates problems because they are great opportunities to apply the analytics of calculus to the very concrete. Even problems as simple as containers emptying over time require synthesis of concepts.
    Consider for example varying the parameters and the shapes of the large container; there are a good many talking points here, some being:
    does the shape of the container matter? (how/why?)
    does its hight/width matter? (how/why?)
    does the density of the fluid matter? (how/why?)
    does the radius of the hole matter? (how/why?)
    does the strength of gravity matter? (how/why?)

  9. I have never thought about this before. What a great idea! Most teachers tend to do the easy problems and work their way up to the challenging ones but often times never get to that point and end up stopping at a mediocre problem due to time or whatever reasons. I am really going to take this into action and build my lesson with my favorite problem!