Thought for the day: Some prompts are so open-ended that almost anything can happen! In order to meet specific learning goals around content, you often want to guide students gently in certain directions. But sometimes you may find a golden opportunity to give their ideas and questions free rein!

Concepts: visual patterns; counting patterns; even-odd patterns; equivalent names for numbers; rules; composing and decomposing geometric figures; making polygons; informal line symmetry

Beginning

I notice five circles making an upside-down V.
I notice 1 + 2 + 2 (top row + middle row + bottom row).
I notice 3 + 3 – 1 (left + right – top (overlap)).
I notice 3 + 2 (top triangle + 2 on the bottom).
I notice that the left and right sides look the same.
I wonder how I could color in the circles to show patterns.
I wonder how I can keep the pattern going.
I notice a missing circle on the bottom.

Exploring

I notice that I can add two circles at a time to make the V longer and longer.
I notice that when I do this, there is always an odd number of circles.
I wonder why the number of circles is always odd.
I notice that when I make the V longer, there are more missing circles (1, then 2, then 3, etc.).
I notice that I can put more circles on top to make X shapes.
I notice that when I do this, the shape looks the same on the top and bottom (as well as the left and right).
I notice that there is still always an odd number of circles when I do this.
I notice that I could also extend the pattern to the left or right to make W or M kinds of shapes.
I notice that I can flip the top three circles down to make a diamond shape, and this shape has
an even number of circles!

Creating

As they notice and wonder, students may create and explore their own designs by extending patterns in the prompt. Many of their creations may flow out of things that they have wondered about. For example, they may

Create longer V shapes by joining circles below.
Create X shapes by joining circles above.
Create extended M or W shapes by joining circles on the left or right.
Create “diamond” shapes by flipping (reflecting) the design over a horizontal line at the bottom.

Create different addition expressions to count the circles in their designs (and look for patterns in their expressions).
Color their designs in different ways in order to highlight counting or symmetry patterns.

Reflecting and Extending

I notice the bigger Vs always have an odd number of circles, because each row has a pair of circles, and there is an extra circle on top.

I wonder what it would look like if I extended the pattern up and down, left and right at the same time.

Note: Patterns like this may be too challenging for young students to create, but you may be able to work together to discover ways to do it! Alternately, you could simply show them the design and ask then to notice and wonder. How many copies of the Vs, Xs, Ms, Ws, and diamonds can they see? What other patterns and shapes do they see? Can they find quick ways to count the circles in the design?

Notes

The ideas and pictures above give just a small taste of the endless possibilities! Encourage kids to make drawings, look for patterns, find creative ways to count the circles in their drawings, write addition and subtraction expressions to show their patterns and counting strategies, etc.

To help students create complex designs, It may be helpful to make some templates of circles that they can trace around. Even better, make a template of the original image with the five circles cut out from a piece of poster board so that they can trace inside of them.