### In Proportion

This week Mrs. Chew at A Schoolyard Blog posed a proportional reasoning problem. I've been watching to see what kind of comments she would get in response to this problem, as I've been reading about proportional reasoning in the middle grades chapter of The Mathematical Education of Teachers. According to MET, proportional reasoning is one of the most important abilities taught in the middle grades.

I really like her problem because it is phrased in such an open ended way. While most adults would recognize that this problem really refers to

With practice just about everyone can be taught how to recognize these scaling problems and how to use proportional thinking to solve them. Someone with a real understanding of this idea should be able to solve problems which involve a proportional relationship between three or more things:

If 100 workers can make 100 widgets in 100 hours, how many widgets will 10 workers make in 10 hours?

I really like her problem because it is phrased in such an open ended way. While most adults would recognize that this problem really refers to

*us*and that we need to get a ruler and measure, many students wouldn't know how best to approach the lack of information. Most proportional reasoning problems are of the form, "In the original photograph of a man and a tree, the man is 2 inches tall and the tree is 5 inches tall. In an enlargement, the man is 5 inches tall. How tall is the tree in the enlargement?" What's so interesting about this type of question is that so many students will respond that the tree is 8 inches tall in the enlarged photo. With one question, we can get a good idea whether a student can use proportional reasoning or whether he or she is still using linear thinking. According to MET, it is the role of the middle school teachers to help students make this transition. I'd like to give this question to a class of pre-service teachers. How many of them have made this shift in their thinking?With practice just about everyone can be taught how to recognize these scaling problems and how to use proportional thinking to solve them. Someone with a real understanding of this idea should be able to solve problems which involve a proportional relationship between three or more things:

If 100 workers can make 100 widgets in 100 hours, how many widgets will 10 workers make in 10 hours?