Observing Student Connections
 Introduction | Hexominos | Sorting Hexominos | Student Work | Problem Reflection #1 | Hexominos Into Cubes | Problem Reflection #2 | Classroom Practice | Observe a Classroom | Your Journal

While sorting hexominos, a group of fifth-graders noticed that three of their hexominos could be folded to make a cube and the others could not. Their teacher followed up with a task for all the student groups to consider:

 Last week the class found three hexominos that fold into a cube: T, t, Z. Are there any others? What makes a hexomino fold or not fold into a cube? When might people care about a problem like this? Ask several adults for suggestions of real-world connections.

Here's how the class responded:

Teacher: Let's start with the last question: What connections did you find between our work with hexominos and things outside of math class?

Liana: Gift boxes and packaging boxes are sometimes cut out at a factory and shipped to stores in a flat stack to save space. The box designer needs to make sure that the design will work and that it isn't a really strange shape for shipping.

Teacher: What does Liana mean by "strange shape"? Who has an idea?

Whitney: Well, if it was like our Z-shaped hexomino, they might be harder to cut out and to keep neatly on a shelf before you fold them.

Teacher: What other connections did you think of or find out about through your interviews?

A.J.: My grandpa said that these problems are important in the sheet-metal shop where he works. Sometimes they cut out really big pieces of metal and fold and weld them for air-conditioning in big buildings. If they make a mistake, it can cost lots of money and waste time.

Carlos: My sister said that it's kind of like some geometry problems that she has solved.

Whitney: My neighbor said that this is really important when she cuts out lots of pieces for quilts. She doesn't fold them to make boxes, but she has to be very careful about the angles and measurements and how they fit together. She fits lots of pieces together to make a design and doesn't want any empty spaces.

Teacher: What about the question, "Why do so many of our hexominos not fold up to make a cube?"

Liana: That's hard to explain! But most of the hexominos have faces that overlap when you try to make a box, so they don't make a six-sided cube.

Teacher: Who can tell us more about that idea?

Whitney: Well, Carlos and I noticed that the upper- and lower-case T-shaped hexominos work easily because they have four squares in a row to make a ring to start. Then, the two side flaps can be folded.

Teacher: What makes a hexomino not fold into a cube?

A.J.: Well, if you have overlapping, it won't work. So you have to check for that. You can't have the two side flaps on the same side, for example.

Teacher: How might we check for overlapping? Is there a rule?

Liana: It's not so hard to see that some of them, like the rectangle shape, will overlap, because not enough squares stick out to make six different faces for a cube. It seems really hard to make a rule.

Teacher: Instead of trying to make a rule, why don't you do more investigating? You might see what is alike about the three that do make a cube. And, since the T-shapes always work, why don't you start by checking to see how some pieces that don't work are different from a T-shape? A.J. already made a conjecture that you can't have the two side flaps on the same side. You could also work on making sure that we've found all the possibilities.

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