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    • This is both more fun, and more educational than “graph this, graph that, what do you notice?” (More about this.)

      Find equations whose solution is x = 2. Instead of the standard “simplify” process, students get to “complicate” the equation. This gives them an opportunity to practice “doing the same thing to both sides” as part of a creative challenge.

      All these examples show the power of reversing traditional questions. On the one hand, it makes it possible for stronger students to show off their fancy solutions, and to remain interested and engaged. At the same time, all students can find their own level of challenge, and you can support them as they take that next step. This is a sort of differentiation with no artificial ceiling: everyone gains.

      3. AIM HIGH

      When I first started teaching in a high school, I was given a terrible piece of advice: “Aim for the middle”. Our program was not tracked, and we had a wide range of talents and backgrounds in each class. Since that is difficult to manage, the idea of aiming for the middle seemed like common sense: if you aim too low, you are betraying your stronger students; if you aim to high, too many kids will be frustrated. As is often the case, common sense was not a good guide.

      Since then, I have developed some strategies to handle a wide range of students in the same class, which I share in Reaching the Full Range. One key component of these strategies is to aim high. In everyday class work and homework, by all means include some material that is too easy, and some material that is too difficult. But for the activities and challenges that anchor an important concept, it is best to be ambitious. If it turns out the question is too difficult, it is always possible to support students with hints, or some other kind of scaffolding. If it is too easy, it is not moving the class forward, and it is giving the wrong message. Here are some examples.

      Rich anchor problems. Start a new unit with a Big Question that encompasses the concepts you are about to teach. For example, to prepare students for sequences, explore what happens when you iterate the function y = mx + b. In other words, if you start with a certain value for x, and it yields a y, use that y as the next x. Repeat and see what happens. This is especially interesting when is between -1 and 1, but all cases are worth exploring. A full discussion provides an opportunity to introduce subscripts, to think about limits, and eventually to zero in on the usual arithmetic and geometric sequences.

      Add another representation. Do not limit yourself to a single approach to your topic. Look for other ways to think about it.  Can manipulatives provide a productive environment to explore these ideas? Can technology help? Is there a visual representation that could be helpful? For example, use well-chosen graph paper rectangles to illustrate the addition of fractions: