## Seven Circles … Again

24 Feb

We tried Seven Circles I from Illustrative Mathematics a few weeks ago.

At the end of class one day, I showed students the diagram and what question they might explore with it. I collected their responses using an Open Response Quick Poll and have shown the results below.

What does this figure have to do with geometry?            1

if we connected each top vertex of the triangles, will it make a hexagon?         1

whats the area of all the circles       1

Why are the circles in this shape?  1

what are the circles forming?          2

what is the area of all of the circles            1

why are we looking at circles?         1

are the spaces between the circles triangles?       1

what are the seven circles forming?           1

do all the circles have the same diameter?            1

do all the circles have the same diameter  1

Are the circles’ diameters the same?          1

Can the measures of the triangles that can be drawn through circles be calculated quickly  1

why are all the circles touching?     1

Why are the circles in that certain arrangement?            1

Can you find the area for that?       1

Is there a way to solve non 90° triangles? (With sin, cos, tan, or the other trig functions)      1

Can the circles be mapped onto each other with a rigid motion?           1

when you look at the image what do you see?      1

are the 6 figures that look like triangles in the gaps of the circles considered triangles since their sides arent straight    1

what are the circles for        1

are all of the circles congruent to each other?       1

What is the significance of the circular pattern?   1

How can you find the measurement of each circle           1

What are the triangular looking spaces in between the triangles called?          1

Some students were interested in the space between the circles. Other students wondered whether the circles were congruent. The task is given below.

My students felt like it was pretty obvious that this could work with 7 congruent circles. I gave them different sized coins so that they could play. What if the circle in the middle is not congruent to the others? Will this work for 6 congruent circles? Or 8 congruent circles?

After students played for a few minutes, I sent them a TNS document that a friend made to explore this task. I used Class Capture to watch while students used the technology to make sense of the necessary and sufficient conditions for 6 circles and 7 circles in the given arrangement. Who had something interesting to discuss with the whole class?

Many students saw the regular pentagon or regular hexagon with vertices at the centers of the outside circles and used that to make sense of the mathematics. While I was watching them, I was trying to figure out how we should proceed as a class. We started with Claire’s work. What do we know?

We saw a dilation. We saw central angles of a regular pentagon. We saw isosceles triangles, which we bisected to make right triangles. We saw an opportunity to use right triangle trigonometry. We looked for and made use of structure. We reasoned abstractly and quantitatively.

And before the bell rang, we looked back at the picture with 7 circles and recognized that the 30-60-90 triangles require that the radius of the center circle equal the radius of the outer circles.

We only touched the surface of what we can learn from this task. Last year, we didn’t even do that. Last year, I shared the task with students during their performance assessment lesson, but we spent all of our time on Hopewell Triangles. This year, we got to it, but I know that our exploration could have been better. We began to answer what are the necessary and sufficient conditions for 6 circles. And in the process, we came across an argument for why the 7 circles must be congruent. But we didn’t really solve the conditions for 6 circles.

I wanted to write about this as a reminder that we are all learning. In this journey, I am finding good tasks out there to try with my students. And I am more confident about some than about others. Even though I don’t know exactly how the tasks should play out in the classroom, I am going to keep trying them. And I’m not going to throw out the task just because we didn’t get as deep into the mathematics as I wish we had. I will try again next year as the journey continues …