Thought for the day: Be equally interested in all ideas that your students produce, even when they aren’t related to the learning goals (which they often won't be—at least at first)!

Concepts: polygons; lengths and angles; parallel segments; parallelograms (possibly symmetry)



I notice that the bottom isn't flat.
I notice that the whole shape is tilted.
I notice that the shape has four sides.
I notice that it looks like a "squashed" rectangle.
I notice that the opposite sides look the same length.


I wonder why the shape looks so tall.
I wonder if (or how) I could change the shape into a rectangle.
I wonder if the shape has a name.
I wonder if the some of the sides are parallel.
I wonder if some of the angles are the same.



As they notice and wonder, students may create their own pictures that have certain things in common with the picture above. Many of their ideas will flow from things that they have wondered about. For example, they may

Create an accurate copy of the shape above. (First, think about what tools you could use.)
Create a copy of the shape in which on the sides is horizontal or vertical.
Create a lot of other parallelograms. Make each one as different as possible.
Create shapes that might look like parallelograms (to some people) but are not.
Create a game in which players must compare and contrast shapes that include some parallelograms.

Reflecting and Extending

I notice that opposite sides of all of my parallelograms are parallel (opposite). (This is what makes them parallelograms.)
I notice that the opposite sides of all of my parallelograms are also the same length (congruent).
I notice that the opposite angles of my parallelograms are also the same size (congruent).
I wonder if I can make a parallelogram where some of these things don’t happen.
I notice that all of these things also happen with rectangles.
I wonder if every rectangle is a parallelogram.
I notice that some shapes can have more than one hame.


Your students'  (and your own) ideas will be far more creative than the examples I showed above! Enjoy their inventiveness, and take the opportunity to learn about what they are seeing and thinking. These observations can inform your instruction.

Even the simplest images can lead to wonderful, rich conversations. If you look through your teaching manual and other materials, you will probably find countless images that you could use effectively as prompts. It's simply a matter of approaching them from a new perspective: begin by listening rather than telling.

You can use the image above with students of many different ages. Younger students will see more general features of the shape. You might eventually have them compare and contrast it with more familiar shapes such as rectangles. Older students may focus more on particulars such as side lengths, angles, parallelism, etc. For them, you might use the prompt to study the properties or develop possible definitions of a parallelogram.

It is very useful for students to see plenty of examples of geometric figures in "non-standard" positions or orientations. For example, none of the sides of this shape are vertical or horizontal. I once had a teacher in a professional development session exclaim that she wanted to "just squash it flat"! You can be sure that unusual orientations are bothersome to many students as well. They are not used to them, because textbooks often show such a limited range of possibilities. In fact, this can be a leading cause of misconceptions in identifying and analyzing geometric figures. 

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