## Pythagorean Exploration

I don’t love this textbook task.

Too many substeps before students return toÂ theÂ question: what’s the relationship between the length of the sides of a right triangle?

“For each right triangle, write anÂ additionÂ statementâ€¦”? C’mon!

But I’m hesitant to join the down with textbooks revolution; I don’t want to associate myself with the back to basics movement. SoÂ in conversations where the suggested alternative is more worked examples,Â I soften my criticism.

Besides, it gives me something to modify.Â Instead of completing the table, I could challenge students to find right triangles and ask “What do you notice?”

One problem: this requires “attend to precision” to do some heavy lifting.

This leads to some truly awkward feedback: “Are you sure it’s a right triangle? You might want to measure again.”

GeoGebra may provide a solution.

Click to view on GeoGebraTube

## Pythagorean Mistakes

Consider the math mistakesÂ below. Not real samples of student work (for that, go here), but real mistakes. I’ve seen each one. I think you’ll recognize them.

Homework

1. What math mistake did each student make?

2. What are some implications for our work?

Good.Â Now answer questions 3 and 4.

3. What role did memorization of the times table play?

4. What are some implications for the conversations we could be having?

## Turn it Around

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 isÂ âˆš45. 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 cmÂ². 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:Â âˆš37, 6, 6Â½,Â âˆš45?

Students will come up with a variety of questions. However, at first, I imagine the response to open questions such asÂ “The answer isÂ âˆš45. 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.