Dissecting Polygons

I appreciate any math lesson where I give the students minimal instruction, yet the lesson is rich in content and it gets kids to do math. (I’m learning to do more and more of not doing much.)

Each student got two pieces of grid paper.

Part 1

In portrait orientation, fold paper in half vertically and approximately into thirds horizontally. Unfold paper to see the six sections created by the folds. Repeat with the other paper. Number the two papers from 1 to 6, like this.


Then I gave each student a ruler and these instructions:

In section 1 of your paper, draw a parallelogram. Find a way to cut your parallelogram into pieces that you can rearrange to form a rectangle. (Here I clarified what I meant: all the cut pieces from the original figure must be put together again to form the requested figure without gaps or overlaps — like a jigsaw puzzle.)

I gave students more graph paper and encouraged them to redraw the parallelogram so they could actually make the cut(s) using scissors. But for this first question, the kids were able to just make a “visual” cut — showing it as a dashed segment — and the requested figure was drawn in the section to the right of the original figure, something like this. (Two students’ original parallelograms were rectangles, and that was perfectly okay.)


When everyone understood the directions, I gave out the next five challenges — all as oral instructions.

In section 2: Start with a right triangle. Cut it into pieces that will form a rectangle.

In section 3: Start with a scalene, non-right triangle. Cut it into pieces that will form a parallelogram.

In section 4: Start with a scalene, non-right triangle. Cut it into pieces that will form a rectangle.

In section 5: Start with a trapezoid. Cut it into pieces that will form a rectangle.

And in section 6: Start with a trapezoid. Cut it into pieces that will form a triangle. (See if you can do this with a single cut.)

The kids were really engrossed in this activity! They were doing a lot of drawing and re-drawing; they were cutting each polygon into pieces and manipulating them.

What they said during the activity:

What the… Oh, I gotcha.

No cheating, J. (to which J. responded with “I am ahead of you, dude!”)

I have number six done!

Yeah, but did you do it with one cut?

Z. is on number six because he skipped 4 and 5.

I’m still working on number three…

Why can’t I draw a scalene triangle?! UGH!!

Can I flip a piece? (I didn’t think of this, so it certainly wasn’t part of my instruction. We discussed and agreed that you may NOT flip a piece over. Kids started labeling their pieces or coloring one side so they wouldn’t accidentally flip them.)

I later asked them to add color to the different edges of the polygons/cut pieces with different colors to make it easier for someone to follow how the pieces had translated after the cut.

No one had finished all 6 tasks; most were working on task 4, so I asked them to spend about 30 minutes more on it for homework.

Tomorrow I’ll check their progress on Part 1. I’ll ask kids to come up and share their ways of dissecting the polygons. Then we’ll move on to Part 2. But I’ll be sure to collect their work on Part 1 first because I don’t want them to look back at Part 1 to do Part 2 — I’m just mean that way.

Part 2

Sort of like the reverse of Part 1. In Part 1, students were asked to dissect various shapes and rearrange their pieces to form mostly rectangles. While in Part 2, students will begin with rectangles and dissect each one to form these shapes: 1) an isosceles triangle, 2) a right triangle, 3) a scalene triangle, 4) a non-rectangular parallelogram and 5) a trapezoid.

I’m so excited for tomorrow because this went so well today! I love math and I love my students!!

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