
Year 8: Algebra  arithmagons
A lesson start (taken from a write up in
ACs' 2nd year of teaching!)
The students filed in as usual, in a noisyish
way, and slowly got their stuff out of their bags. I took the register
over murmurings of conversations  choosing to ignore them. I then
went to the board, with most eyes on me. I tapped with my pen and
waited for everyone's attention.
We were going to do some work on arithmagons
 I drew one in silence, put a number in two circles, paused and
filled the box inbetween, paused...put a number in the third circle...filled
in the other two boxes.
A few hands had gone up, there was silence. Another
example, still silence, a couple of students bursting to tell the
answers in the boxes, but still silence (this was surprising)...I
was making eye contact with many of the class and looking a lot
at a girl who I felt might be the last to pick up what was going
on  there was concentration, but still no understanding on her
face. A third example...I turned to look at the class and everyone's
eyes were burning into the board  I hadn't experienced this before.
Still silence...I now took two answers from the class for what should
go in the boxes, everyone's hand seemed to be up except the girl;
she was straining and seemed to have just twigged; she half whispered
an answer, not quite committing herself  but it was correct. One
more example...two boys had lost concentration, staring again brought
them back. The girl's hand was now up with the rest  the boys seemed
to be following, after all, so I nodded at her and a correct answer
came.
I then drew an arithmagon with only the
boxes filled in and invited the class to try to find what the numbers
in the circles could have been...no one needed a further explanation,
which, for me, is rare!
Questions:
· Can you find a method or strategy for solving any arithmagon?
· Are there any arithmagons that have no solution? (You might
want to write on the board any the class come up with which they
can't solve.)
· Is there ever more than one solution?
· Can you use algebra to find a solution? (or just to describe
how they work eg if a, b and c are in the circles, what is in the
boxes?)
· What happens with square arithmagons? pentagonal arithmagons?
