Nets

Materials: Handouts, grid paper, card stock, Scotch tape

Goals:

¨ Make and use manipulatives in problem solving.

¨ Move from experimentation (i.e., trial and error) to a more systematic approach to problem solving.

¨ Carefully record the results of each step of a solution.

¨ Visualize 3-dimensional objects from 2-dimensional representations.

¨ Visualize reflections (flipping) and rotations (spinning).

Step 1:

¨ There is only one way to join 2 squares along an edge and the resulting shape is the usual domino. How many different ways can 3 squares be joined together along their edges? Do not count shapes that are simply flips or spins of another shape. These shapes are called triominoes. List all triominoes.

Suggestion: To help you find all the triominoes, quadominoes, etc., I recommend that you make or find some squares and physically move them around. You can easily record your figures on grid paper.

¨ How many different ways can 4 squares be joined together along their edges? These shapes are called quadominoes. List all the quadominoes.

Suggestion: To find all the quadominoes you might start with a specific triomino and find all the different ways to add one square to it. Be on the lookout for shapes that are made by flipping or spinning another shape.

¨ There are 12 different ways 5 squares be joined together along their edges. These shapes are called pentominoes. List all the pentominoes.

¨ There are 35 different ways can 6 squares be joined together along their edges? These shapes are called hexominoes. List as many hexominoes as you can.

¨ Actively read "The Third Dimension" from Math for Fun.

¨ Decide which of the hexominoes you found are nets and which are not nets. Can you draw some conclusions about how to look at a hexomino to decide whether it is a net or not? What characteristic can you look for?

Suggestion: The idea here is to try to build your ability to visualize in three-dimensions. However, to strengthen our ability to visualize sometimes we need to work with something physical first. So don't hesitate to do what they suggest in Math for Fun and actually make the hexominoes out of cardboard and tape to see if they can be folded into cubes.

¨ Complete Activity 1.10: Making Dice from Mathematics for Elementary Teachers via Problem Solving: Student Activity Manual.