One of these things is not like the others

When you read the title of this post, did you think Sesame Street? Foo Fighters? Or, like me, both?

Recently, Geoff shared seven (sneaky) activities to get students talking mathematically. One activity, ‘odd one out’, involves having students pick the one mathematical thing that doesn’t belong. This reminds me of one strategy used by Dr. Marian Small to create open questions – asking for similarities and differences.

Here’s my ‘odd one out’ question:

Which of the following quadratic functions doesn’t belong? (Dr. Small might ask “Which of these four functions are most alike?”)
y=2\left( x-1\right) ^{2}+3
y=\dfrac {1} {2}\left( x-3\right) ^{2}-5
y=3\left( x+2\right) ^{2}-4
y=-\dfrac {3} {2}\left( x-4\right) ^{2}+6

Students might say,
y=2\left( x-1\right) ^{2}+3 because it does not cross the x-axis
y=\dfrac {1} {2}\left( x-3\right) ^{2}-5 because it is a vertical compression of y = x²
y=3\left( x+2\right) ^{2}-4 because it is a horizontal translation to the left
y=-\dfrac {3} {2}\left( x-4\right) ^{2}+6 because it opens down

Do the graphs of these functions strengthen your choice or make you change your mind?

I carefully chose the values of ap, and q in y = a(x – p)² + q so that students could reasonably argue that any one of the functions could be picked as the odd one out. Because I am not looking for one particular answer, each student should be able to confidently answer the question and contribute to a mathematical discussion. Planning disagreement is key; it means students will have to justify their mathematical thinking.



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 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.