A Simple Dissection Puzzle

One cut

Make only one cut, and then assemble the pieces into a square.

This is my new introduction to completing the square. This time around I’m trying to give a strong geometric intuition of what’s going on. I’m hoping to build off of this by excluding the little square, and asking the students how one would figure out the area of the missing square. Then I’ll work up to using variables for the sides, and then finally from that derive a procedure.

Easier said than done, I know. I also have some history I was going to toss in, and I’m hoping to slip in a decent application at the end. Any suggestions of where to look would be appreciated.

8 Responses

  1. looks like an interesting start, though this looks like an almost gimme for algebra tiles.

  2. Indeed, this seems to have “algebra tiles” written all over it, although it doesn’t make as good of a hook , plus doesn’t illustrate “divide by 2” as well as physical chopping with a pair of scissors.

    If they’ve been used in class before though they’d make a great segue.

    • I was thinking about giving a few of these as a light homework assignment.

      I like the idea, but agree with you and Mr. K that it’s simple and doesn’t “hook” or get the “divide by 2”.

      I thought that after doing maybe one of these “gimmes” they could do another, similar one that asks the needed area of the missing “corner piece”.

      The third would be an x^2 and 3x. They have to make 1 cut and then find the missing area. At that point they have to think a little “outside the box” to realize that they have to cut the 3x’s in half – not on a dotted line.

      The last part is to do a quick write up explaining what they did. I’ll be trying this this week (Tuesday probably), I’ll get back with how it went.

  3. I use pictures in place of tiles
    With (x+10)^2 we spend some time considering the two long rectangles.

    Months later we solve equations by taking the square root of each side.
    We solve (x+10)^2 = 0
    We solve x^2 + 14x + 49 = 0
    And we draw the picture. We look at the rectangles.
    I tell a horrible story about my sister nibbling the corners off of triscuits and sticking them back in the box when we were little kids.
    And then I put up We solve x^2 + 10x + 23 = 0, and I with some discussion I put up a “triscuit” with the corner nibbled off…

    Not the only way. Not necessarily the best way. But it’s been effective for me.


  4. although it doesn’t make as good of a hook

    True, though it seems like the hook might want a little bit *more*, I’m just not sure what. I there with you on the “make one cute & solve” part, it just seem to translate. I suspect that may because I’m not familiar with how you’ve taught the lead up to this.

    It seems that the algebra tile approach would be to start off providing a big square and an even number of rectangles, and asking them how many small squares would be needed to make a square out of the pieces. Eventually you give them a couple, and ask how many more (or less) they need.

    BTW – I’ve found at least two different solutions to your problem above, one of which is the intended solution, the other which breaks the “completing the square” idea.

  5. re: breaking the solution. That, Mr. K, makes it so much better. I’m really hoping the students do that. It’s another reason why I want to do it as a puzzle rather than with algebra tiles.

    I think to make it more interesting as a hook I want to start with multiple puzzles, only one of them being completing the square.

    Anyway, thanks both of you for the food for thought. I will keep mulling.

  6. I want to start with multiple puzzles, only one of them being completing the square.

    Good plan, I think. Please do a write up of the results.

  7. I have tried this now and it worked well. The writeup may take a little time because I’m busy with being observed this week and our school pass-to-graduate test next week.

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