Thought for the day: It takes courage to let students create the observations and questions, because you will not always know the answers. Show them that you are willing to take risks, too, and that you want to learn math along with them! It is completely normal (and an excellent thing!) to have lots of questions left that you were unable to answer completely.

Concepts: visual patterns, attributes and attribute sorting, making polygons

Beginning

I notice three attributes: shape, color, and dot size.
I notice two kinds of shapes: squares and circles.
I notice two colors: blue and green.
I notice two kinds of dots: large and small.
I notice that the attributes are combined to make pictures that are arranged in a circle.
I wonder why there is a big circle connecting the drawings.
I notice that the dots are always right on the big circle.
I notice that it looks like a picture is missing on the left side of the circle.

Exploring

I wonder what the missing picture should look like.
I notice that there are: 2 greens and 1 blue; 2 squares and 1 circle; 2 small dots and 1 large dot.
I notice that when I go from the top to the bottom of the big circle, only the color stays the same.  (It stays green, but the shape and the size of the dot change.)
I notice that if I follow the shapes around the big circle, only one attribute stays the same each time. (Only the shape [square] stays the same from the Top to the Right. Only the dot-size [small] stays the same from the Right to the Bottom.)
I wonder if these observations can help me predict what the missing picture should be.
I wonder if there is more than one answer for what the missing picture can be.
I notice that these four pictures do not show every possible picture I can make from these attributes.
I wonder how many other pictures I can make using these attributes.

Creating

As they notice and wonder, students may create new ideas and drawings that flow out of things that they have wondered about. For example, they may:

Create predictions about how to continue patterns in the attributes that they see.
Create drawings to illustrate their predictions.
Create different drawings using the same three attributes.
Create their own patterns using their new drawings (and maybe some old ones, too).
Create new options within each attribute and explore the possible drawings they can make.
Create new attributes and new drawings using new patterns.

Reflecting and Extending

I notice that there are different ways to describe the same pattern.
I notice that I see different things when I look around the circle (in either direction) as compared to across the circle.
I notice that some patterns can go in a loop that repeats forever.
I notice that I can make a lot of drawings from a small number of attributes.
I wonder how many possible drawings I can make with these attributes.
I wonder if there are other ways to organize my drawings to show new patterns.
I wonder what happens if I allow more options for some attributes (for example: 3 kinds of shape, 3 colors, and 2 kinds of dot).

Notes

It seems to me that most ways of thinking about the patterns lead to filling the empty place on the left with a blue circle and a large dot:

However, you and your students may come up with good reasons for using a different picture! Let me know if you do!

In creating the picture, I used a rule that only one attribute can stay the same from one picture to the next as I go around the circle, including at the end when I go from the Left picture back to the original picture on the Top. For example, from the Top the the Right, the shape stays a square, but the color and the dot size change.

Using this rule, are there other possible ways to fill in the picture on the left?

There are eight drawings that you can make from these three attributes (with two options per attribute). They look like this:

I used half of them in my drawing. Can you make other drawings that use my rule with a different combination of four shapes? How about your own rules? How many drawings can you make? Is it possible to create a rule that uses all eight shapes in a loop? What other ways can you think of to arrange the eight shapes into patterns? (Tables are an interesting thing to explore!)

Older students may explore the idea of predicting the number of drawings they can make with n attributes and x options per attribute. (The answer is x to the power of n (or x • x • x …, with n of the xs multiplied together.)