There are many benefits to this. It may reveal strategies that make more sense to the students than the ones you come up with. It gives students the confidence they can recreate the formula if they don’t remember it. It makes “base” and “height” concrete and visible rather than numbers on a figure. A discussion of student solutions can lead to a discussion of how to generalize what was learned, leading up to a formula. And of course, starting this way does not prevent you from providing your own explanation later.

**2. REVERSE TRADITIONAL QUESTIONS**

Many traditional math class questions are, well, boring. What is 7 + 3? What is the greatest common factor of 12 and 15? Graph y = 6x + 4. Solve 6x + 4 = 3x + 10 for x. And so on. Reversing the traditional question often yields great engagement and powerful discussions. Here are some examples.

**Find pairs of numbers that add up to 10**. Even answering this with positive whole numbers gives more students a chance to contribute answers. But this can expand in many directions, depending on the class. You can ask for *sets* of numbers that add up to 10, such as 3, 3, and 4. If you are getting started with integer arithmetic, great patterns will emerge if you allow negative numbers in the pairs, and your students will be practicing their new skills as part of an interesting quest. Likewise, in Algebra 2, if you ask that same question about complex numbers.

**Find pairs of numbers whose greatest common factor is 3. **This is sure to reveal what understanding your students have about common factors. If some students don’t understand the underlying concepts well enough to do this, they can get help from their classmates, or from you. If even that is not sufficient, some review is in order!

**Find the equations that will yield these graphs. **