I wrote the blog on Proportions at home, and had my notes at school, so I thought I’d add a few pictures.
This is what I wrote for the kids to compare the Original Proportion, the Proportion with common denominators or common numerators, the Problem we had to solve to make equivalent ratios:
Here is the same sheet, now the kids told me you could cross multiply the original proportion (or simplified proportion) and get the same problem we solved from the common num/denom problem:
This is from the next day. The numbers 1, 2, and 3 are the order I introduced the problem. First we reviewed how to solve proportions (first look for a relationship vertically or horizontally, then if there isn’t a convenient one they can cross multiply), then looked at proportions with more than one term in a position, and finally written problems. For the written problems they needed to 1) write a word ratio to organize their thinking, 2) write their proportion, and 3) solve (any way they chose) and write a word answer.
Finally, here is some of their work (they told me what to write) solving the problems from section 2 in the picture above. Some kids simplified vertically, others simplified horizontally, but they still got the same answer!
There you go!