# 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:

#### Grade 6

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.

#### Grade 7

Proportional reasoning is a dominant thread that runs throughout the 7th grade units. *Stretching and Shrinking* introduces proportionality concepts through the context of geometric Problems involving similarity. Students connect visual ideas about enlarging (stretching) and reducing (shrinking) figures, numerical ideas about scale factors and ratios, and applications of similarity through work with problems focused around the question: "What would it mean to say two figures are similar?"

Next in the seventh grade is a core proportional reasoning Unit, *Comparing and Scaling*, which connects fractions, percents, and ratios through investigation of various situations in which the central question is: “What strategies make sense in describing how much greater one quantity is than another?” Through a series of problem-based Investigations, students explore the meaning of ratio comparison and through a progression of Problems that builds on intuition and moves to developing and articulating procedures, students develop a variety of techniques for dealing with questions that include unit rates and constants of proportionality.

*Moving Straight Ahead *follows* Comparing and Scaling* and is about linear relationships and equations. Proportional thinking is connected and extended to the core ideas of linearity (constant rate of change and slope), as well as recognizing the coefficients of x in y = mx as the constant of proportionality. *In What Do You Expect?*, students use ratios to make comparisons of probabilities. In *Filling and Wrapping*, students investigate the effects on volume and surface area from scaling up the dimensions of prisms. Lastly, Samples and Populations uses proportional reasoning to compare data situations and to choose samples from populations.

#### Grade 8

*Thinking With Mathematical Models; Looking For Pythagoras; Growing, Growing, Growing; and Frogs, Fleas, and Painted Cubes* extend the understanding of proportional relationships in the eighth grade by investigating and contrasting the relationships between linear functions, exponential functions, quadratic functions, and inverse relationships. This description of how CMP treats proportional reasoning illustrates two things about CMP: the in-depth development of fundamental ideas and the connected use of these important ideas throughout the program.