Here is a "themes, tools, concepts" map for proportional relationships. The themes, in red italics, are possible "real world" or visual contexts where the concept can arise and be approached by students. The tools, in blue, are mathematical environments where students can deepen their understanding and learn how to work with it. The concepts, in bold green, are related ideas in other parts of the curriculum. Some obvious connections are indicated by line segments.

Of course, the diagram is incomplete, and I'm sure you could find additional items in all three categories, and moreover establish additional connections. I know I could. (For example, I didn't include any electronics among the tools!) Still, even this incomplete diagram is illuminating. It shows that there is no answer to "what is the best way to teach proportional relationships?" There is no one way! If we want students to develop a robust understanding, we need to explore all these connections.

Don't panic: this is quite possible.

Most of the items in the diagram can be found in chunks of curriculum I link to on the Middle School page of my Web site. Click on Proportional Relationships, on Scaling, and on Dilations. Moreover, those files are the tips of an educational iceberg. The connected view of the subject means that each of the worksheets in these packets are related to much more downloadable material in other parts of my site.

The beauty of the multidimensional approach is that it allows needed repetition without boredom: each time the subject is approached, it feels different because of a different context, a different tool, or a different mathematical connection. This gives weaker students many opportunities to get on board, and it keeps things interesting for stronger students. Much better than doing the same thing again if it didn't work for everyone the first time.

--Henri

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