Then I tried this this year: I told the kids that 3(2x-5) meant 3 sets of (2x-5) and made them expand the expression by writing out three separate sets of (2x-5). Then we combined like terms and figured out that it was 6x-15. I pounded in the idea that the ‘3 times’ meant ‘three sets’. The kids got the idea they needed three sets of 2x and three sets of -5. I hadn’t even mentioned the distributive property yet.

Then your talk came in… I wanted the ‘set’ idea to stick, so I gave them different headaches.

*Power move.*

I asked them to expand 7(6-8y)–minor grumbling… 1/2(4x+8)–they loved that… then 50(12x+7)–major whining. For a group of kids who were quite below grade level and also didn’t want to write (12x+7) fifty times, they came up with the aspirin awfully fast.

Then I taught them the convention and stuff. All in all, it was great and the kids felt super successful. (I also felt successful by using this strategy–I had a headache and you gave me an aspirin!)

]]>I really like the coin problem and how it showed why arrays come in handy for counting. I’ve been working with our elementary math coach (I’m middle school) on using arrays. I am definitely sharing this with her.

]]>It follows from two simple rules that students don’t usually have issues with — sum of a triangle and then adjacent angles on a line. As a result I have crossed this theorem off my list of ones to teach, since it can be quickly worked out from first principles — and I therefore I also don’t have to teach them when not to apply it (when the angle given is vertically opposite, not exterior) and have more time to teach the ones I do correctly.

I also don’t teach alternate exterior angles on parallel lines are equal. Again it follows from two simple rules we are already teaching, and is one less thing for them to confuse themselves about.

A plethora of rules they have to learn puts students off Geometry — so that’s where their headache tends to be. So I make it easier by cutting out surplus rules. If that means that some questions take an extra step, then so be it.

(I do actually teach both rules, because they are simple introductions to proofs. What I don’t do is expect them to add them to the list of ones they have to memorise.)

]]>Added to the document.

]]>Here’s a recent headache of mine. My regular geometry students have a hard time with the “exterior angle of a triangle is equal to the sum of the remote interior angles” theorem. I don’t know if 6 problems is enough of a headache for my students to want the cure, but it’s a start. You’re welcome to copy to your directory and share.

Stacie

https://drive.google.com/open?id=15V-73Rw3VPVPMgndfpUaiqNTB30XHNk9 ]]>

I think this is the one that Dan was speaking of: http://math.ucsd.edu/~jrabin/publications/ProblemFreeActivity.pdf

]]>For more:

– David Hawkins’s “I, Thou, It” – https://www.brandeis.edu/mandel/questcase/Documents/Readings/David%20Hawkins%201974%20I,%20Thou,%20and%20It.pdf

– Richard Elmore’s “Instructional Core” – http://www.fpsct.org/uploaded/Teacher_Resource_Center/Instructional_Practices/Resources/20091124152005.pdf ]]>