In *More Good Questions: Great Ways to Differentiate Secondary Mathematics Instruction*, Dr. Marian Small discusses the turn-around strategy to create open questions.

Instead of asking “The legs of a right triangle are 3 cm and 6 cm long. What is the hypotenuse?” the teacher can ask **“The answer is √45. What could the question be?”**

There are many possible questions. For example,

Determine the length of the hypotenuse.

Determine the length of *x*.

A square has an area of 45 cm². What is the side length?

What is an example of a square root that has a value between 6 and 7?

Which number is the greatest: √37, 6, 6½, √45?

Students will come up with a variety of questions. However, at first, I imagine the response to open questions such as “The answer is √45. What could the question be?” will be silence. Students are used to being asked questions where there is one correct answer. In math, you either get it or you don’t. It’s not just questions that need turning around. This black and white view of mathematics also needs turning around. With time and practice, class discussions about open questions can help change this attitude.

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I think if you ask students to provide something open-ended like that, you have to be prepared to to accept whatever questions the students may have,otherwise it becomes an exercise into trying to read your mind. In this case, there are an infinite amount of questions the students can come up with not related to triangles or even non-perfect squares. I’m of the school that being able to formulate a question is more important than solving one, but I don’t think training students to recognize solutions to textbook problems is the way to do that.

If it gets students communicating in math class. I’d accept all questions, even some non-math ones (see http://mythagon.wordpress.com/2011/07/24/whats-the-question/). I’d have some questions in mind, but I’d be willing to go in a different direction if that’s where the discussion headed. I agree there are many questions students can ask that have nothing to do with Pythagoras. I had my Math 8 class in mind when I wrote this. In my curriculum, this is when students are first introduced to square roots so the possible question types that could come up may not be infinite. There are some opportunites here to help students make connections, even if it involves students generating some questions that could be found in a textbook.

Thanks for replying. One concern I’d have is students using an excuse to use such an open-ended question as a cop-out. (What’s “root(45)-0?”) However, you know your students and I agree, if it gets them talking it’s a good thing. There’s certainly potential for interesting conversation starters.