A Triangle of Circles

Students are likely to begin by noticing interesting patterns in the colors. The colors may also guide them to recognize other features of the design.

After you have talked about the colors for a while, give students a copy of this image in which the circles are not colored in. Ask them to color them in ways that highlight or create other objects or patterns that they see (or can imagine!) in the picture. As a class, you can eventually discuss the uncolored image and focus on observations and questions that are unrelated to color.


Some responses from 2nd grade students

 A second grade teacher (whose class saw only the uncolored image) sent me a few sample "noticings and wonderings" from her students:

I wonder if it is a bond.
I wonder if it is a dot-to-dot.
I wonder if it has something to do with math.
I wonder why the lines don't go through the middle.
I wonder if it's like some sort of vehicle that travels round and round on its wheels, crushing everything in its path!
I notice it is made up of lines and circles to form a triangle.

It appears that the teacher must have connected adjacent circles with small line segments. By a "bond," perhaps the student was thinking of a number bond, which is often shown as numbers inside of circles connected by segments. At any rate, notice the creativity of the replies. Some observations and questions may not lead to a mathematical follow-up, but the students are engaged and the teacher is learning what is going through their minds. She may be able to refer to many of the responses during class discussion.  And some of the students' comments will lead to interesting investigations. For example, would it make sense to have a number bond that shows more than three numbers? What would this look like? What happens if you do allow the lines to go through the middle of the triangle? What would that look like? How many lines would there be? Will there still be symmetry? Listening carefully to students' ideas is one the main ways that I come up with new ideas for mathematical problems!


Some responses from older students and adults

Here are a few other questions and observations that I have heard when showing this to older students and adults:

I wonder why there is a circle missing in the middle.
I wonder how many circles would be missing if the triangle had 4, 5, or more circles on each side.
I notice a hexagon surrounded by three circles at the corners of the triangle.
I notice that the big triangle is made up of three small triangles, one at the top, the left, and the right.
I wonder how many small triangles I can find in the picture.
I wonder how many circles you would need to make larger triangles.
I notice a trapezoid with a circle on top.
I notice three overlapping rectangles in the picture.
I wonder how many parallelograms I could find in the picture.
I wonder how the answers to some of the questions would change if the triangle did not have the same number of circles on each side.
I wonder what would happen if I filled the circles with numbers.


Noticing and Wondering

Can you come up with other "noticings" and "wonderings"? Do any of them lead to interesting problems to explore?

Encouraging students to notice and wonder promotes curiosity and flexible thinking in math. It helps you differentiate instruction, because students naturally create questions that fit their current level of understanding. Developing a habit of awareness and reflection before, while, and after solving problems is a great way to foster mathematical talent. And isn't it astonishing how much there is to notice and wonder about such a simple picture when you stop and pay attention? 

The final "wondering" in the list above (about filling the circles with numbers) is what I had in mind when I created the image. But others' observations and questions led to many more ideas for things to investigate. To see more about both the original problem and the new ones inspired by others, visit the page for Numbers and Operations under the ACCME Books menu, and look at the activity titled "Triangle Sums."