This post is about an activity — so it begins with the ages the activity is intended for.
- Ages 3-6: just color the pictures, have fun!
- Ages 7-9: color the pictures but also try to answer the first question. A parent or teacher should help, and maybe look at this article on rotational symmetry.
- Ages 10-12: color the pictures and use them to answer all the questions if you can. Students may need help from a parent or teacher.
- Ages 13+: just color the pictures. Answer the questions if you’re interested.
This week’s Occupy Math is the first post in a new category: materials and activities. In response to a reader comment, Occupy Math is working up math-related activities. This one is a coloring book intended to introduce symmetry. A colored version of one of the images from the book appears above. The first page of the book asks five questions; all the other pages are intended to be colored. This post discusses and answers the five questions for a teacher or parent that might be using the activity.
How many different ways can you spin the shape around its center and get a shape that looks just the same?
This question has a more technical form, “what is the degree of rotational symmetry of the picture?”, but both these questions ask the number of ways that you can rotate the picture about its center and get an identical version of the picture. For the picture at the top of the post, the answer is “nine” (if you ignore the colors). For the picture below the answer is “five”. This question is supposed to help develop a person’s number sense, let them see concrete versions of numbers, and make the idea of rotational symmetry clear.
How many different types of shape are there that you could color in the picture?
The fact that all the pictures in the coloring book are rotationally symmetric means that the different shapes appear in rings around the center. The picture below numbers the shapes, of which there are four. Notice that all the shapes you can color in are the same as one of the four numbered shapes.
How many different areas are there to color in each picture?
This question is a bit of a trick question. Since the shapes you can color come in groups that go to one another when you rotate the picture, the answer is usually the number of shapes from Question 2 multiplied by the number of rotations that take the picture to itself from Question 1. Let’s color the example picture from the last question. It has six-fold rotation and four types of shapes to give use twenty-four total shapes: four colors, each with six shapes.
The exception that causes the word “usually” in the paragraph above is that some of the pictures later in the book have a single shape in the center. That central shape goes to itself when you rotate the picture — so you need to ignore that shape, use the counting trick above, and then add one to the total. This exception is teachable, but be careful. Here is one of these pictures, with 3×5+1=16 shapes to color.
Children, being more creative than adults, may think that coloring two adjacent areas the same color makes a new “area you can color”. Well, it does, but for counting the number of areas you can color, we ignore this possibility. That’s because, if you’re allowed to combine areas to make new ones, it makes way, way too many areas to count.
Can you use the rotations to make it easier to count the number of areas that could be colored?
Since the third question was fairly hard, at least for a beginning student, this question is a follow-up to question three. The example used for question three is simple enough that a student could solve the problem by just counting to 24 — but the picture at the top of the page has nine different shapes (some of them pretty small) and nine-fold rotation — so there are 81 different regions to be colored. The following shape has seven-fold rotation and, counting outward along a partial spiral, nine types of shapes, for a total of 7× 9=63 shapes to be colored.
If it’s hard to see that this shape has seven-fold rotational symmetry, look at the little flower right in the center. That’s the easy place to see the symmetry.
Can you make your own simple shape and answer the questions about it?
This question is an open-ended one. If you are working on symmetry, this is a chance to see how much of the idea of symmetry has been absorbed. If you want to help, you might provide a rotationally symmetric array of dots for the students to connect, but students will often surprise you. Another way to help is to give students shapes, like a drinking glass, that can be traced. If you are working with compasses already, those can be valuable tools as well.
This post is Occupy Math’s first activity post that comes with a big handout (the coloring book linked at the top of the post). Occupy Math would welcome scans of the way people color the images in the coloring book (curious!) If you have ideas or suggestions for other posts of this type, do not be shy, Comment or tweet!
I hope to see you here again,
University of Guelph,
Department of Mathematics and Statistics