Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
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|
As you read about how several students approached the problem, think about what connections the students made.
I used tiles and graph paper. I remembered when we made hexominos and tried to make all the ones with four in a row again. Some that I made looked like new 4-lines, but they really weren't new.
For example, these are actually all the same upper-case T hexomino:
I know they are congruent because you can flip H1 to match H2. You can rotate H2 to match H3. You can flip H4 to match H3. It's like when we worked on finding congruent figures before. I found seven hexominos with a line of 4 squares. I crossed off the ones that were the same:
I made lots of 1-by-4 rectangles on my grid paper. Then I added on two more squares above or below in as many ways as I could think of. I thought about using tiles in my mind. I tried to remember some of the hexominos that we made before, and thought about how we had folded them to see if they made cubes. I cut out all my shapes and threw away extras if they matched exactly when I turned them over. Then I drew my shapes on new grid paper. I found nine ways:
(After the initial assignment and class discussion, the teacher asked Jimmy to continue his investigation on his own. The following week he surprised her with this report.)
This project took a long time! I started with a line of four square-tiles and two extras. First I put the two extras on the end to make an upper-case L:
Then I moved the two extras over a space. After that, I moved the extras over one more space. But then I saw that I could flip that one over to have the same hexomino that I had just made. That's when I saw that many of my hexominos with 4-lines would just need to be flipped or rotated to match the shapes that I already found:
It was harder to use the two extras as separate squares. I made a line of four green squares and used a red and blue extra square. First I put the red above and on the end and tried to make all the possible ways by just moving the blue one. I recorded some ways on grid paper:
It seemed like the Valentines problem that we did, where we had to stay organized or it would be very confusing. So, I found all the ways with the red square in one place, then all the ways with the red square starting in a new place. I started to record them on grid paper but saw that some of the new shapes matched old shapes because they could be flipped or rotated, so I had to cross a lot of them off. It got confusing, so I used letters to name the places where the red and blue squares could be attached:
I made a table to keep track of the new ones. The red and blue colors didn't really matter because AC and CA make the same hexomino, so I crossed off all the ones like CA, etc.:
Here's a list of my hexominos:
AA, AB, AC, AD, AE, AF, AG, AH, BB, BC . . . BF, BG.
The others had to be crossed off because BD is like AC, BE is like AF, BH is like AG, CD is like AB, CE is like BH, DE is like AH, etc.
My friend showed me one more way:
So, I found 13 ways to make a 4-line hexomino.
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