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# Example of Depth and Connectedness

### Proportional Reasoning

Through the process of field trials, we are able to develop content that results in student understanding of key ideas in depth. An example is illustrated in the way that CMP treats proportional reasoning, a fundamentally important topic for middle school mathematics and beyond. Conventional treatments of this central topic are often limited to a brief expository presentation of the ideas of ratio and proportion, followed by training in techniques for solving proportions. In contrast, the CMP curriculum materials develop core elements of proportional reasoning in a seventh- grade Unit, Comparing and Scaling, with the groundwork having been developed in four prior Units. Five succeeding Units build on and connect to students’ understanding of proportional reasoning. These Units and their connections are summarized as follows:

Comparison statements are used to introduce ratios and rates in Comparing Bits and Pieces. Students’ understanding of equivalent fractions from previous grades is extended and contrasted with equivalent ratios. In this Unit, since students’ understanding of ratios is just beginning ratios are not notated as fractions. The fraction notation is deferred until 7th grade. The extensive work with equivalent forms of fractions builds the skills needed to work with ratio and proportion problems. These ideas are developed further in Variables and Patterns as rate reemerges in rate tables and graphs.