Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
As an introduction, Nan Sepeda reminds her students that they have studied pentominoes in the past, and asks them, in their groups, to classify the twelve pentominoes according to a scheme they devise.
Now it's time to create hexominoes, figures made up of six squares. As with pentominoes, the squares may not overlap and they must join along whole sides.
The groups use grid paper and color tiles to create as many different hexominoes as they can. Then Ms. Sepeda asks them to sort the hexominoes according to their own scheme. Finally, students organize the hexominoes on a poster to make a display.
These schemes are extremely varied. They include sorting by symmetry, by perimeter, by whether the hexomino folds into a cube, and by whether it looks like a letter.
An Exploration for Teacher Workshops
Materials: Pattern Blocks (3 blue rhombuses per pair); optionally, pattern block templates or isometric dot paper
Working with a partner, use Pattern Blocks to make all possible triominoes out of three blue rhombi. (A triomino is composed of three shapes.)
In your triominoes, the three rhombi may not overlap, and they must join along whole sides. Two triominoes are the same if you can rotate or flip the one shape to match the other exactly.
Record your triominoes on paper.
- How can you tell whether you have a duplicate?
- How can you tell you have them all?
- There are only two triominoes made from squares. Why are there so many more made from rhombi?
- What strategies did you use to record your triominoes? Did you have any difficulties?
- How many tetrominoes (4 shapes) or pentominoes (5) do you think you can make with rhombi?
Additional Discussion Topics
Here are some additional ideas for discussion that arise in the video:
About the Mathematical Content
- Ms. Sepeda says, "the most effective way to teach children is to put manipulatives in their hands." She was talking about the cut-out hexominoes they made themselves. But what role did the color tiles play? How would the activity have been different if the tiles had not been available?
- This is a class of students used to working together. What aspects of the lesson would not work with students who seldom work in groups? What aspects would?
- In many pentomino activities, much of the lesson goes into defining what makes one pattern different from another (it can't be rotated or flipped to match). Yet Ms. Sepeda didn't mention it at all. How would that work with your students? How would you cope with the "what's different" issue?
- Discuss the different classification schemes students developed. What mathematical ideas do students use in each? What's the mathematical benefit of inventing a classification scheme?
- The next day, students tried to determine the complete set of hexominoes. What issues or difficulties do you foresee arising in that lesson?
These questions appear at the end of the video. Here are some follow-up ideas and prompts to help get a discussion going.
What is your role when a lesson goes in an unexpected direction?
Ms. Sepeda originally intended this lesson to focus more strongly on the fact that these shapes have the same areas but different perimeters. While that came up, it was not central to the lesson. What could she do in later lessons to make sure the idea of "same area but different perimeter" gets addressed? Ms. Sepeda argues that open-ended, student-centered investigations give students ownership of the mathematics. On the other hand, they can make it hard to cover the content in your syllabus. What do you think about open-ended lessons?
Discuss Ms. Sepeda's reaction to the group who sorted by letters.
One group sorted shapes into ones that looked like upper- and lowercase letters. Ms. Sepeda says that, even though it was not mathematical, it was important to let them do it their way.
Why do you think Ms. Sepeda said the letter scheme was not mathematical? Do you agree or disagree?
Looking at the lesson as a whole, what is the mathematical benefit for students in that group?
What is the benefit to the class of having that group sort the shapes as they did?
How else might you focus this lesson?
Suppose you didn't want the lesson to be so open-ended. What could you do to focus students on what you want while leaving them as much autonomy as possible?
What different mathematical goals can you make from this activity?
What extensions can you design to focus the work?