This session was a preview of a new (as yet unreleased) book by AIMS on solving equations. Four different discovery/investigation style projects were presented. Three of them are available in this pdf file.
The fourth lesson is not included in the above packet, and is a variation on the classic balancing scale.
The scale is made using:
The weighed objects are made using:
Make the strips into footballs. Once they are made, you can unfold them, and add pennies on the inside. Put the same number of pennies inside each football of the same color. Use a small bit of tape to seal the footballs.
You probably want 4-8 footballs of each color, with 0 to 4 pennies inside. This will actually give you weights of 1-5 pennies, since the paper accounts for one penny's worth of weight.
You can then create equations by putting, say, 2 footballs and 3 pennies on one side, and 4 footballs on the other side. Have the students add pennies to the second side until the scale balances. They may then add or remove items from either or both bags, with the condition that you hold the scale steady while they do so, and that the scale must balance when they are done adding or removing.